https://en.lntwww.de/index.php?title=Mobile_Communications/General_Description_of_Time_Variant_Systems&feed=atom&action=history
Mobile Communications/General Description of Time Variant Systems - Revision history
2024-03-29T13:02:16Z
Revision history for this page on the wiki
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https://en.lntwww.de/index.php?title=Mobile_Communications/General_Description_of_Time_Variant_Systems&diff=52312&oldid=prev
Guenter at 13:41, 29 January 2023
2023-01-29T13:41:03Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 13:41, 29 January 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l6" >Line 6:</td>
<td colspan="2" class="diff-lineno">Line 6:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== # OVERVIEW OF THE SECOND MAIN CHAPTER # ==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== # OVERVIEW OF THE SECOND MAIN CHAPTER # ==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>After the time variance, the term&nbsp; &raquo;'''<del class="diffchange diffchange-inline">Frequency Selectivity</del>'''&laquo; &nbsp; is now introduced and illustrated with examples, a channel property which is also of great importance for mobile communications.&nbsp; As in the entire book, the description is given in the equivalent low-pass range. </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>After the time variance, the term&nbsp; &raquo;'''<ins class="diffchange diffchange-inline">frequency selectivity</ins>'''&laquo;&nbsp; is now introduced and illustrated with examples,<ins class="diffchange diffchange-inline">&nbsp; </ins>a channel property which is also of great importance for mobile communications.&nbsp; As in the entire book,<ins class="diffchange diffchange-inline">&nbsp; </ins>the description is given in the equivalent low-pass range. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It is covered in detail:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It is covered in detail:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">*the </del>difference between <del class="diffchange diffchange-inline">time invariant '''KORREKTUR: </del>time-invariant<del class="diffchange diffchange-inline">''' </del>and <del class="diffchange diffchange-inline">time variant '''KORREKTUR: </del>time-variant<del class="diffchange diffchange-inline">''' </del>systems,</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">#The&nbsp; &raquo;</ins>difference between time-invariant and time-variant systems<ins class="diffchange diffchange-inline">&laquo;</ins>,</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">*</del>the <del class="diffchange diffchange-inline">time variant '''KORREKTUR: </del>time-variant<del class="diffchange diffchange-inline">''' </del>impulse response as an important descriptive function of <del class="diffchange diffchange-inline">time variant '''KORREKTUR: </del>time-variant<del class="diffchange diffchange-inline">''' </del>systems,</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">#</ins>the<ins class="diffchange diffchange-inline">&nbsp; &raquo;</ins>time-variant impulse response<ins class="diffchange diffchange-inline">&laquo;&nbsp; </ins>as an important descriptive function of time-variant systems,</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">*</del>multi-way reception as the cause of frequency-selective behaviour,</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">#&raquo;</ins>multi-way reception<ins class="diffchange diffchange-inline">&laquo;&nbsp; </ins>as the cause of frequency-selective behaviour,</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">*</del>a detailed derivation and interpretation of the GWSSUS channel model,</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">#</ins>a detailed derivation and interpretation of the<ins class="diffchange diffchange-inline">&nbsp; &raquo;</ins>GWSSUS channel model<ins class="diffchange diffchange-inline">&laquo;</ins>,</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">*</del>the characteristics of the GWSSUS model: &nbsp; coherence bandwidth, correlation duration, etc.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">#</ins>the characteristics of the GWSSUS model: &nbsp<ins class="diffchange diffchange-inline">; &raquo</ins>;coherence bandwidth,<ins class="diffchange diffchange-inline">&nbsp; </ins>correlation duration<ins class="diffchange diffchange-inline">&laquo;</ins>,<ins class="diffchange diffchange-inline">&nbsp; </ins>etc.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l27" >Line 27:</td>
<td colspan="2" class="diff-lineno">Line 27:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The most important results are briefly explained again here.&nbsp; We assume a&nbsp; linear and <del class="diffchange diffchange-inline">time invariant '''KORREKTUR: </del>time-invariant<del class="diffchange diffchange-inline">''' </del>system &nbsp; &#8658; &nbsp; $\text{LTI system}$&nbsp; with the signal&nbsp; $s(t)$&nbsp; at the input and the output signal&nbsp; $r(t)$. &nbsp; For the sake of simplicity, let&nbsp; $s(t)$&nbsp; and&nbsp; $r(t)$&nbsp; be real.&nbsp; Then the following applies:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The most important results are briefly explained again here.&nbsp; We assume a&nbsp; linear and time-invariant system &nbsp; &#8658; &nbsp; $\text{LTI system}$&nbsp; with the signal&nbsp; $s(t)$&nbsp; at the input and the output signal&nbsp; $r(t)$. &nbsp; For the sake of simplicity, let&nbsp; $s(t)$&nbsp; and&nbsp; $r(t)$&nbsp; be real.&nbsp; Then the following applies:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The system can be completely characterized by the&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_.E2.80.93_Transfer_function|$\text{transfer function}$]]&nbsp; $H(f)$&nbsp; which is also referred to as the&nbsp; "frequency response".&nbsp; By definition&nbsp;:$$H(f) = R(f)/S(f).$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The system can be completely characterized by the&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_.E2.80.93_Transfer_function|$\text{transfer function}$]]&nbsp; $H(f)$&nbsp; which is also referred to as the&nbsp; "frequency response".&nbsp; By definition&nbsp;:$$H(f) = R(f)/S(f).$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l107" >Line 107:</td>
<td colspan="2" class="diff-lineno">Line 107:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The LTV equation is only applicable if the change of the channel&nbsp; $($marked in the figure by the parameter&nbsp; $T)$&nbsp; proceeds slowly in comparison to the maximum delay &nbsp; $\tau_{\rm max}$.&nbsp; In mobile communications this condition &nbsp; &#8658; &nbsp; $\tau_{\rm max} < T$ &nbsp; is almost always fulfilled.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The LTV equation is only applicable if the change of the channel&nbsp; $($marked in the figure by the parameter&nbsp; $T)$&nbsp; proceeds slowly in comparison to the maximum delay &nbsp; $\tau_{\rm max}$.&nbsp; In mobile communications this condition &nbsp; &#8658; &nbsp; $\tau_{\rm max} < T$ &nbsp; is almost always fulfilled.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Selecting whether to apply the first Fourier integral to the parameter&nbsp; $\tau$&nbsp; or&nbsp; $t$&nbsp; leads to different spectral functions.&nbsp; In the&nbsp; [[Aufgaben:Exercise 2.1Z: 2D-Frequency and 2D-Time Representations|"Exercise 2.1Z"]]&nbsp; for example, the <del class="diffchange diffchange-inline">time variant '''KORREKTUR: </del>time-variant<del class="diffchange diffchange-inline">''' two-dimensional</del>&nbsp; &raquo;'''<del class="diffchange diffchange-inline">2D </del>transfer function'''&laquo;&nbsp; is considered:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Selecting whether to apply the first Fourier integral to the parameter&nbsp; $\tau$&nbsp; or&nbsp; $t$&nbsp; leads to different spectral functions.&nbsp; In the&nbsp; [[Aufgaben:Exercise 2.1Z: 2D-Frequency and 2D-Time Representations|"Exercise 2.1Z"]]&nbsp; for example, the time-variant <ins class="diffchange diffchange-inline"> </ins>&nbsp; &raquo;'''<ins class="diffchange diffchange-inline">two-dimensional </ins>transfer function'''&laquo;&nbsp; is considered:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>H(f,\hspace{0.05cm} t)</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>H(f,\hspace{0.05cm} t)</div></td></tr>
</table>
Guenter
https://en.lntwww.de/index.php?title=Mobile_Communications/General_Description_of_Time_Variant_Systems&diff=52198&oldid=prev
Hwang at 18:14, 24 January 2023
2023-01-24T18:14:56Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 18:14, 24 January 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l107" >Line 107:</td>
<td colspan="2" class="diff-lineno">Line 107:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The LTV equation is only applicable if the change of the channel&nbsp; $($marked in the figure by the parameter&nbsp; $T)$&nbsp; proceeds slowly in comparison to the maximum delay &nbsp; $\tau_{\rm max}$.&nbsp; In mobile communications this condition &nbsp; &#8658; &nbsp; $\tau_{\rm max} < T$ &nbsp; is almost always fulfilled.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The LTV equation is only applicable if the change of the channel&nbsp; $($marked in the figure by the parameter&nbsp; $T)$&nbsp; proceeds slowly in comparison to the maximum delay &nbsp; $\tau_{\rm max}$.&nbsp; In mobile communications this condition &nbsp; &#8658; &nbsp; $\tau_{\rm max} < T$ &nbsp; is almost always fulfilled.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Selecting whether to apply the first Fourier integral to the parameter&nbsp; $\tau$&nbsp; or&nbsp; $t$&nbsp; leads to different spectral functions.&nbsp; In the&nbsp; [[Aufgaben:Exercise 2.1Z: 2D-Frequency and 2D-Time Representations|"Exercise 2.1Z"]]&nbsp; for example, the time variant two-dimensional&nbsp; &raquo;'''2D transfer function'''&laquo;&nbsp; is considered:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Selecting whether to apply the first Fourier integral to the parameter&nbsp; $\tau$&nbsp; or&nbsp; $t$&nbsp; leads to different spectral functions.&nbsp; In the&nbsp; [[Aufgaben:Exercise 2.1Z: 2D-Frequency and 2D-Time Representations|"Exercise 2.1Z"]]&nbsp; for example, the time variant <ins class="diffchange diffchange-inline">'''KORREKTUR: time-variant''' </ins>two-dimensional&nbsp; &raquo;'''2D transfer function'''&laquo;&nbsp; is considered:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>H(f,\hspace{0.05cm} t)</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>H(f,\hspace{0.05cm} t)</div></td></tr>
</table>
Hwang
https://en.lntwww.de/index.php?title=Mobile_Communications/General_Description_of_Time_Variant_Systems&diff=52197&oldid=prev
Hwang at 18:10, 24 January 2023
2023-01-24T18:10:37Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 18:10, 24 January 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l10" >Line 10:</td>
<td colspan="2" class="diff-lineno">Line 10:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It is covered in detail:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It is covered in detail:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*the difference between time invariant and time variant systems,</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*the difference between time invariant <ins class="diffchange diffchange-inline">'''KORREKTUR: time-invariant''' </ins>and time variant <ins class="diffchange diffchange-inline">'''KORREKTUR: time-variant''' </ins>systems,</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*the time variant impulse response as an important descriptive function of time variant systems,</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*the time variant <ins class="diffchange diffchange-inline">'''KORREKTUR: time-variant''' </ins>impulse response as an important descriptive function of time variant <ins class="diffchange diffchange-inline">'''KORREKTUR: time-variant''' </ins>systems,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*multi-way reception as the cause of frequency-selective behaviour,</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*multi-way reception as the cause of frequency-selective behaviour,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*a detailed derivation and interpretation of the GWSSUS channel model,</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*a detailed derivation and interpretation of the GWSSUS channel model,</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l27" >Line 27:</td>
<td colspan="2" class="diff-lineno">Line 27:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The most important results are briefly explained again here.&nbsp; We assume a&nbsp; linear and time invariant system &nbsp; &#8658; &nbsp; $\text{LTI system}$&nbsp; with the signal&nbsp; $s(t)$&nbsp; at the input and the output signal&nbsp; $r(t)$. &nbsp; For the sake of simplicity, let&nbsp; $s(t)$&nbsp; and&nbsp; $r(t)$&nbsp; be real.&nbsp; Then the following applies:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The most important results are briefly explained again here.&nbsp; We assume a&nbsp; linear and time invariant <ins class="diffchange diffchange-inline">'''KORREKTUR: time-invariant''' </ins>system &nbsp; &#8658; &nbsp; $\text{LTI system}$&nbsp; with the signal&nbsp; $s(t)$&nbsp; at the input and the output signal&nbsp; $r(t)$. &nbsp; For the sake of simplicity, let&nbsp; $s(t)$&nbsp; and&nbsp; $r(t)$&nbsp; be real.&nbsp; Then the following applies:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The system can be completely characterized by the&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_.E2.80.93_Transfer_function|$\text{transfer function}$]]&nbsp; $H(f)$&nbsp; which is also referred to as the&nbsp; "frequency response".&nbsp; By definition&nbsp;:$$H(f) = R(f)/S(f).$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The system can be completely characterized by the&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_.E2.80.93_Transfer_function|$\text{transfer function}$]]&nbsp; $H(f)$&nbsp; which is also referred to as the&nbsp; "frequency response".&nbsp; By definition&nbsp;:$$H(f) = R(f)/S(f).$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l49" >Line 49:</td>
<td colspan="2" class="diff-lineno">Line 49:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>On the other hand, a DC signal&nbsp; $s(t) = A$&nbsp; is not suitable to make the frequency dependence of the LTI system visible: &nbsp; <br>&nbsp; &rArr; &nbsp; With a low-pass system the output signal would then be always constant, independent of&nbsp; $H(f)$:&nbsp; &nbsp; &nbsp; $r(t) = A \cdot H(f= 0)$.<br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>On the other hand, a DC signal&nbsp; $s(t) = A$&nbsp; is not suitable to make the frequency dependence of the LTI system visible: &nbsp; <br>&nbsp; &rArr; &nbsp; With a low-pass system the output signal would then be always constant, independent of&nbsp; $H(f)$:&nbsp; &nbsp; &nbsp; $r(t) = A \cdot H(f= 0)$.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In the next section we consider a Dirac delta train&nbsp; $p_\delta(t)$&nbsp; as an input signal&nbsp; $s(t)$: &nbsp; <br>&nbsp; &rArr; &nbsp; Hereby the similarities and differences between time-invariant <del class="diffchange diffchange-inline">'''KORREKTUR: time invariant''' </del>and time-variant <del class="diffchange diffchange-inline">'''KORREKTUR: time variant''' </del>systems can be shown clearly.<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In the next section we consider a Dirac delta train&nbsp; $p_\delta(t)$&nbsp; as an input signal&nbsp; $s(t)$: &nbsp; <br>&nbsp; &rArr; &nbsp; Hereby the similarities and differences between time-invariant and time-variant systems can be shown clearly.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><i>Note:</i>&nbsp; The properties of&nbsp; $H(f)$&nbsp; and&nbsp; $h(t)$&nbsp; are covered in detail in the&nbsp; $\text{LNTwww learning video}$&nbsp; (in German language):<br> &nbsp; &nbsp;</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><i>Note:</i>&nbsp; The properties of&nbsp; $H(f)$&nbsp; and&nbsp; $h(t)$&nbsp; are covered in detail in the&nbsp; $\text{LNTwww learning video}$&nbsp; (in German language):<br> &nbsp; &nbsp;</div></td></tr>
</table>
Hwang
https://en.lntwww.de/index.php?title=Mobile_Communications/General_Description_of_Time_Variant_Systems&diff=52161&oldid=prev
Hwang at 13:40, 24 January 2023
2023-01-24T13:40:25Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 13:40, 24 January 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l49" >Line 49:</td>
<td colspan="2" class="diff-lineno">Line 49:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>On the other hand, a DC signal&nbsp; $s(t) = A$&nbsp; is not suitable to make the frequency dependence of the LTI system visible: &nbsp; <br>&nbsp; &rArr; &nbsp; With a low-pass system the output signal would then be always constant, independent of&nbsp; $H(f)$:&nbsp; &nbsp; &nbsp; $r(t) = A \cdot H(f= 0)$.<br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>On the other hand, a DC signal&nbsp; $s(t) = A$&nbsp; is not suitable to make the frequency dependence of the LTI system visible: &nbsp; <br>&nbsp; &rArr; &nbsp; With a low-pass system the output signal would then be always constant, independent of&nbsp; $H(f)$:&nbsp; &nbsp; &nbsp; $r(t) = A \cdot H(f= 0)$.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In the next section we consider a Dirac delta train&nbsp; $p_\delta(t)$&nbsp; as an input signal&nbsp; $s(t)$: &nbsp; <br>&nbsp; &rArr; &nbsp; Hereby the similarities and differences between time-invariant and time-variant systems can be shown clearly.<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In the next section we consider a Dirac delta train&nbsp; $p_\delta(t)$&nbsp; as an input signal&nbsp; $s(t)$: &nbsp; <br>&nbsp; &rArr; &nbsp; Hereby the similarities and differences between time-invariant <ins class="diffchange diffchange-inline">'''KORREKTUR: time invariant''' </ins>and time-variant <ins class="diffchange diffchange-inline">'''KORREKTUR: time variant''' </ins>systems can be shown clearly.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><i>Note:</i>&nbsp; The properties of&nbsp; $H(f)$&nbsp; and&nbsp; $h(t)$&nbsp; are covered in detail in the&nbsp; $\text{LNTwww learning video}$&nbsp; (in German language):<br> &nbsp; &nbsp;</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><i>Note:</i>&nbsp; The properties of&nbsp; $H(f)$&nbsp; and&nbsp; $h(t)$&nbsp; are covered in detail in the&nbsp; $\text{LNTwww learning video}$&nbsp; (in German language):<br> &nbsp; &nbsp;</div></td></tr>
</table>
Hwang
https://en.lntwww.de/index.php?title=Mobile_Communications/General_Description_of_Time_Variant_Systems&diff=52160&oldid=prev
Hwang at 13:38, 24 January 2023
2023-01-24T13:38:51Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 13:38, 24 January 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l28" >Line 28:</td>
<td colspan="2" class="diff-lineno">Line 28:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The most important results are briefly explained again here.&nbsp; We assume a&nbsp; linear and time invariant system &nbsp; &#8658; &nbsp; $\text{LTI system}$&nbsp; with the signal&nbsp; $s(t)$&nbsp; at the input and the output signal&nbsp; $r(t)$. &nbsp; For the sake of simplicity, let&nbsp; $s(t)$&nbsp; and&nbsp; $r(t)$&nbsp; be real.&nbsp; Then the following applies:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The most important results are briefly explained again here.&nbsp; We assume a&nbsp; linear and time invariant system &nbsp; &#8658; &nbsp; $\text{LTI system}$&nbsp; with the signal&nbsp; $s(t)$&nbsp; at the input and the output signal&nbsp; $r(t)$. &nbsp; For the sake of simplicity, let&nbsp; $s(t)$&nbsp; and&nbsp; $r(t)$&nbsp; be real.&nbsp; Then the following applies:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*The system can be completely characterized by the&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#<del class="diffchange diffchange-inline">Transfer_function_</del>.E2.80.<del class="diffchange diffchange-inline">93_Frequency_response</del>|$\text{transfer function}$]]&nbsp; $H(f)$&nbsp; which is also referred to as the&nbsp; "frequency response".&nbsp; By definition&nbsp;:$$H(f) = R(f)/S(f).$$</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*The system can be completely characterized by the&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#<ins class="diffchange diffchange-inline">Frequency_response_</ins>.E2.80.<ins class="diffchange diffchange-inline">93_Transfer_function</ins>|$\text{transfer function}$]]&nbsp; $H(f)$&nbsp; which is also referred to as the&nbsp; "frequency response".&nbsp; By definition&nbsp;:$$H(f) = R(f)/S(f).$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Similarly, the system is defined by the&nbsp; [[Linear_and_Time_Invariant_Systems/<del class="diffchange diffchange-inline">Systembeschreibung_im_Zeitbereich</del>#<del class="diffchange diffchange-inline">Impulsantwort</del>|$\text{impulse response}$]]&nbsp; $h(t)$&nbsp;, which is the&nbsp; [[Signal_Representation/<del class="diffchange diffchange-inline">Fourier_Transform_and_Its_Inverse</del>#<del class="diffchange diffchange-inline">Das_zweite_Fourierintegral</del>|$\text{inverse Fourier transform}$]]&nbsp; of&nbsp; $H(f)$.&nbsp; &nbsp; The output signal results from the convolution:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Similarly, the system is defined by the&nbsp; [[Linear_and_Time_Invariant_Systems/<ins class="diffchange diffchange-inline">System_Description_in_Time_Domain</ins>#<ins class="diffchange diffchange-inline">Impulse_response</ins>|$\text{impulse response}$]]&nbsp; $h(t)$&nbsp;, which is the&nbsp; [[Signal_Representation/<ins class="diffchange diffchange-inline">Fourier_Transform_and_its_Inverse</ins>#<ins class="diffchange diffchange-inline">The_second_Fourier_integral</ins>|$\text{inverse Fourier transform}$]]&nbsp; of&nbsp; $H(f)$.&nbsp; &nbsp; The output signal results from the convolution:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>r(t) = s(t) \star h(t) \hspace{0.4cm} {\rm with} \hspace{0.4cm} h(t)</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>r(t) = s(t) \star h(t) \hspace{0.4cm} {\rm with} \hspace{0.4cm} h(t)</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l38" >Line 38:</td>
<td colspan="2" class="diff-lineno">Line 38:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT=</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT=</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$\text{Definitions:}$&nbsp; &nbsp; The following input signals are suitable for detecting the linear distortions caused by&nbsp; $H(f)$&nbsp; or &nbsp; $h(t)$:&nbsp;</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$\text{Definitions:}$&nbsp; &nbsp; The following input signals are suitable for detecting the linear distortions caused by&nbsp; $H(f)$&nbsp; or &nbsp; $h(t)$:&nbsp;</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*a&nbsp; [[Signal_Representation/<del class="diffchange diffchange-inline">Special_Cases_of_Impulse_Signals</del>#<del class="diffchange diffchange-inline">Dirac_Delta_Impulse</del>|$\text{Dirac delta}$]]&nbsp; or&nbsp; "impulse": </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*a&nbsp; [[Signal_Representation/<ins class="diffchange diffchange-inline">Special_Cases_of_Pulses</ins>#<ins class="diffchange diffchange-inline">Dirac_delta_or_impulse</ins>|$\text{Dirac delta}$]]&nbsp; or&nbsp; "impulse": </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$s(t) = \delta(t) \hspace{0.3cm}\Rightarrow \hspace{0.3cm} &nbsp; r(t) = \delta(t) \star h(t)= h(t)\hspace{0.3cm}\Rightarrow \hspace{0.3cm} \text{impulse response,}$$ </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$s(t) = \delta(t) \hspace{0.3cm}\Rightarrow \hspace{0.3cm} &nbsp; r(t) = \delta(t) \star h(t)= h(t)\hspace{0.3cm}\Rightarrow \hspace{0.3cm} \text{impulse response,}$$ </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*a&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Step_response|$\text{step function}$]]&nbsp; or&nbsp; "Heaviside step function": </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*a&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Step_response|$\text{step function}$]]&nbsp; or&nbsp; "Heaviside step function": </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$s(t) = \gamma(t) \hspace{0.3cm}\Rightarrow \hspace{0.35cm} &nbsp; r(t) = \gamma(t) \star h(t)\hspace{1.5cm}\Rightarrow \hspace{0.3cm} \text{step response,}$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$s(t) = \gamma(t) \hspace{0.3cm}\Rightarrow \hspace{0.35cm} &nbsp; r(t) = \gamma(t) \star h(t)\hspace{1.5cm}\Rightarrow \hspace{0.3cm} \text{step response,}$$</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*a&nbsp; [[Signal_Representation/<del class="diffchange diffchange-inline">Time_Discrete_Signal_Representation</del>#<del class="diffchange diffchange-inline">Dirac_Comb_in_Time_and_Frequency_Domain</del>|$\text{Dirac comb}$]]&nbsp; or&nbsp; "Dirac delta train": </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*a&nbsp; [[Signal_Representation/<ins class="diffchange diffchange-inline">Discrete-Time_Signal_Representation</ins>#<ins class="diffchange diffchange-inline">Dirac_comb_in_time_and_frequency_domain</ins>|$\text{Dirac comb}$]]&nbsp; or&nbsp; "Dirac delta train": </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$s(t) = p_\delta(t) \hspace{0.25cm}\Rightarrow \hspace{0.3cm} &nbsp; r(t) = p_\delta(t) \star h(t)\hspace{1.3cm}\Rightarrow \hspace{0.3cm} \text{impulse response train.}$$}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$s(t) = p_\delta(t) \hspace{0.25cm}\Rightarrow \hspace{0.3cm} &nbsp; r(t) = p_\delta(t) \star h(t)\hspace{1.3cm}\Rightarrow \hspace{0.3cm} \text{impulse response train.}$$}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
</table>
Hwang
https://en.lntwww.de/index.php?title=Mobile_Communications/General_Description_of_Time_Variant_Systems&diff=52158&oldid=prev
Hwang at 13:21, 24 January 2023
2023-01-24T13:21:53Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 13:21, 24 January 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l6" >Line 6:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== # OVERVIEW OF THE SECOND MAIN CHAPTER # ==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== # OVERVIEW OF THE SECOND MAIN CHAPTER # ==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>After the time variance, the term&nbsp; '''Frequency Selectivity''' &nbsp; is now introduced and illustrated with examples, a channel property which is also of great importance for mobile communications.&nbsp; As in the entire book, the description is given in the equivalent low-pass range. </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>After the time variance, the term&nbsp<ins class="diffchange diffchange-inline">; &raquo</ins>;'''Frequency Selectivity'''<ins class="diffchange diffchange-inline">&laquo; </ins>&nbsp; is now introduced and illustrated with examples, a channel property which is also of great importance for mobile communications.&nbsp; As in the entire book, the description is given in the equivalent low-pass range. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It is covered in detail:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It is covered in detail:</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l21" >Line 21:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The description parameters of a communication system have already been described in two chapters of the book "Linear Time Variant Systems": </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The description parameters of a communication system have already been described in two chapters of the book "Linear Time Variant Systems": </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain|System Description in Frequency Domain]],</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain|<ins class="diffchange diffchange-inline">"</ins>System Description in Frequency Domain<ins class="diffchange diffchange-inline">"</ins>]],</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain|System Description in Time Domain]].&nbsp; </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain|<ins class="diffchange diffchange-inline">"</ins>System Description in Time Domain<ins class="diffchange diffchange-inline">"</ins>]].&nbsp; </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:EN_Mob_T_2_1_S1_neu.png|right|frame|Considered LTI system|class=fit]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:EN_Mob_T_2_1_S1_neu.png|right|frame|Considered LTI system|class=fit]]</div></td></tr>
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<td colspan="2" class="diff-lineno">Line 28:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The most important results are briefly explained again here.&nbsp; We assume a&nbsp; linear and time invariant system &nbsp; &#8658; &nbsp; $\text{LTI system}$&nbsp; with the signal&nbsp; $s(t)$&nbsp; at the input and the output signal&nbsp; $r(t)$. &nbsp; For the sake of simplicity, let&nbsp; $s(t)$&nbsp; and&nbsp; $r(t)$&nbsp; be real.&nbsp; Then the following applies:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The most important results are briefly explained again here.&nbsp; We assume a&nbsp; linear and time invariant system &nbsp; &#8658; &nbsp; $\text{LTI system}$&nbsp; with the signal&nbsp; $s(t)$&nbsp; at the input and the output signal&nbsp; $r(t)$. &nbsp; For the sake of simplicity, let&nbsp; $s(t)$&nbsp; and&nbsp; $r(t)$&nbsp; be real.&nbsp; Then the following applies:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*The system can be completely characterized by the&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Transfer_function_.E2.80.93_Frequency_response|transfer function]]&nbsp; $H(f)$&nbsp; which is also referred to as the&nbsp; "frequency response".&nbsp; By definition&nbsp;:$$H(f) = R(f)/S(f).$$</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*The system can be completely characterized by the&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Transfer_function_.E2.80.93_Frequency_response|<ins class="diffchange diffchange-inline">$\text{</ins>transfer function<ins class="diffchange diffchange-inline">}$</ins>]]&nbsp; $H(f)$&nbsp; which is also referred to as the&nbsp; "frequency response".&nbsp; By definition&nbsp;:$$H(f) = R(f)/S(f).$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Similarly, the system is defined by the&nbsp; [[Linear_and_Time_Invariant_Systems/Systembeschreibung_im_Zeitbereich#Impulsantwort|impulse response]]&nbsp; $h(t)$&nbsp;, which is the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#Das_zweite_Fourierintegral|inverse Fourier transform]]&nbsp; of&nbsp; $H(f)$.&nbsp; &nbsp; The output signal results from the convolution:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Similarly, the system is defined by the&nbsp; [[Linear_and_Time_Invariant_Systems/Systembeschreibung_im_Zeitbereich#Impulsantwort|<ins class="diffchange diffchange-inline">$\text{</ins>impulse response<ins class="diffchange diffchange-inline">}$</ins>]]&nbsp; $h(t)$&nbsp;, which is the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#Das_zweite_Fourierintegral|<ins class="diffchange diffchange-inline">$\text{</ins>inverse Fourier transform<ins class="diffchange diffchange-inline">}$</ins>]]&nbsp; of&nbsp; $H(f)$.&nbsp; &nbsp; The output signal results from the convolution:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>r(t) = s(t) \star h(t) \hspace{0.4cm} {\rm with} \hspace{0.4cm} h(t)</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>r(t) = s(t) \star h(t) \hspace{0.4cm} {\rm with} \hspace{0.4cm} h(t)</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l38" >Line 38:</td>
<td colspan="2" class="diff-lineno">Line 38:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT=</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT=</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$\text{Definitions:}$&nbsp; &nbsp; The following input signals are suitable for detecting the linear distortions caused by&nbsp; $H(f)$&nbsp; or &nbsp; $h(t)$:&nbsp;</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>$\text{Definitions:}$&nbsp; &nbsp; The following input signals are suitable for detecting the linear distortions caused by&nbsp; $H(f)$&nbsp; or &nbsp; $h(t)$:&nbsp;</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*a&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Dirac_Delta_Impulse|Dirac delta]]&nbsp; or&nbsp; "impulse": </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*a&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Dirac_Delta_Impulse|<ins class="diffchange diffchange-inline">$\text{</ins>Dirac delta<ins class="diffchange diffchange-inline">}$</ins>]]&nbsp; or&nbsp; "impulse": </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$s(t) = \delta(t) \hspace{0.3cm}\Rightarrow \hspace{0.3cm} &nbsp; r(t) = \delta(t) \star h(t)= h(t)\hspace{0.3cm}\Rightarrow \hspace{0.3cm} \text{impulse response,}$$ </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$s(t) = \delta(t) \hspace{0.3cm}\Rightarrow \hspace{0.3cm} &nbsp; r(t) = \delta(t) \star h(t)= h(t)\hspace{0.3cm}\Rightarrow \hspace{0.3cm} \text{impulse response,}$$ </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*a&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Step_response|step function]]&nbsp; or&nbsp; "Heaviside step function": </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*a&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Step_response|<ins class="diffchange diffchange-inline">$\text{</ins>step function<ins class="diffchange diffchange-inline">}$</ins>]]&nbsp; or&nbsp; "Heaviside step function": </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$s(t) = \gamma(t) \hspace{0.3cm}\Rightarrow \hspace{0.35cm} &nbsp; r(t) = \gamma(t) \star h(t)\hspace{1.5cm}\Rightarrow \hspace{0.3cm} \text{step response,}$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$s(t) = \gamma(t) \hspace{0.3cm}\Rightarrow \hspace{0.35cm} &nbsp; r(t) = \gamma(t) \star h(t)\hspace{1.5cm}\Rightarrow \hspace{0.3cm} \text{step response,}$$</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*a&nbsp; [[Signal_Representation/Time_Discrete_Signal_Representation#Dirac_Comb_in_Time_and_Frequency_Domain|Dirac comb]]&nbsp; or&nbsp; "Dirac delta train": </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*a&nbsp; [[Signal_Representation/Time_Discrete_Signal_Representation#Dirac_Comb_in_Time_and_Frequency_Domain|<ins class="diffchange diffchange-inline">$\text{</ins>Dirac comb<ins class="diffchange diffchange-inline">}$</ins>]]&nbsp; or&nbsp; "Dirac delta train": </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$s(t) = p_\delta(t) \hspace{0.25cm}\Rightarrow \hspace{0.3cm} &nbsp; r(t) = p_\delta(t) \star h(t)\hspace{1.3cm}\Rightarrow \hspace{0.3cm} \text{impulse response train.}$$}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$s(t) = p_\delta(t) \hspace{0.25cm}\Rightarrow \hspace{0.3cm} &nbsp; r(t) = p_\delta(t) \star h(t)\hspace{1.3cm}\Rightarrow \hspace{0.3cm} \text{impulse response train.}$$}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l52" >Line 52:</td>
<td colspan="2" class="diff-lineno">Line 52:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><i>Note:</i>&nbsp; The properties of&nbsp; $H(f)$&nbsp; and&nbsp; $h(t)$&nbsp; are covered in detail in the&nbsp; $\text{LNTwww learning video}$&nbsp; (in German language):<br> &nbsp; &nbsp;</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><i>Note:</i>&nbsp; The properties of&nbsp; $H(f)$&nbsp; and&nbsp; $h(t)$&nbsp; are covered in detail in the&nbsp; $\text{LNTwww learning video}$&nbsp; (in German language):<br> &nbsp; &nbsp;</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[https://www.lntwww.de/Eigenschaften_des_%C3%9Cbertragungskanals_(Lernvideo) Eigenschaften des Übertragungskanals] &nbsp; &rArr; &nbsp; "Some remarks on the transfer function".<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[https://www.lntwww.de/Eigenschaften_des_%C3%9Cbertragungskanals_(Lernvideo) <ins class="diffchange diffchange-inline">"</ins>Eigenschaften des Übertragungskanals<ins class="diffchange diffchange-inline">"</ins>] &nbsp; &rArr; &nbsp; "Some remarks on the transfer function".<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l79" >Line 79:</td>
<td colspan="2" class="diff-lineno">Line 79:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT= </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT= </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>$\text{Conclusion:}$&nbsp; With a &nbsp; '''time-variant channel''' &nbsp; you cannot specify neither a one-parameter impulse response&nbsp; $h(t)$&nbsp; nor a transfer function&nbsp; $H(f)$&nbsp;.}}<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>$\text{Conclusion:}$&nbsp; With a &nbsp<ins class="diffchange diffchange-inline">; &raquo</ins>;'''time-variant channel'''<ins class="diffchange diffchange-inline">&laquo; </ins>&nbsp; you cannot specify neither a one-parameter impulse response&nbsp; $h(t)$&nbsp; nor a transfer function&nbsp; $H(f)$&nbsp;.}}<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><i>Note:</i>&nbsp; The differences between LTI and LTV systems are clarified with the&nbsp; $\text{LNTwww learning video}$&nbsp; (in German language):<br> &nbsp; &nbsp;</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><i>Note:</i>&nbsp; The differences between LTI and LTV systems are clarified with the&nbsp; $\text{LNTwww learning video}$&nbsp; (in German language):<br> &nbsp; &nbsp;</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[https://www.lntwww.de/Eigenschaften_des_%C3%9Cbertragungskanals_(Lernvideo) Eigenschaften des Übertragungskanals] &nbsp; &rArr; &nbsp; "Some remarks on the transfer function".<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[https://www.lntwww.de/Eigenschaften_des_%C3%9Cbertragungskanals_(Lernvideo) <ins class="diffchange diffchange-inline">"</ins>Eigenschaften des Übertragungskanals<ins class="diffchange diffchange-inline">"</ins>] &nbsp; &rArr; &nbsp; "Some remarks on the transfer function".<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l99" >Line 99:</td>
<td colspan="2" class="diff-lineno">Line 99:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br clear=all></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br clear=all></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Regarding the last equation and the above graph, it should be noted</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Regarding the last equation and the above graph, it should be noted</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*The parameter&nbsp; $\tau$&nbsp; specifies the &nbsp; '''delay time''' &nbsp; to denote the time dispersion.&nbsp; By writing out the convolution operation, it was possible to make&nbsp; $\tau$&nbsp; also the parameter of the LTI impulse response.&nbsp; In the last sections we spoke about&nbsp; $h(t)$&nbsp;.<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*The parameter&nbsp; $\tau$&nbsp; specifies the &nbsp<ins class="diffchange diffchange-inline">; &raquo</ins>;'''delay time'''<ins class="diffchange diffchange-inline">&laquo; </ins>&nbsp; to denote the time dispersion.&nbsp; By writing out the convolution operation, it was possible to make&nbsp; $\tau$&nbsp; also the parameter of the LTI impulse response.&nbsp; In the last sections we spoke about&nbsp; $h(t)$&nbsp;.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*The second parameter of the impulse response or the second axis marks the &nbsp; '''absolute time'''&nbsp; $t$, which is used, among other things, to describe the time variance.&nbsp; At different times&nbsp; $t$&nbsp; the impulse response&nbsp; $h(\tau, \hspace{0.05cm}t)$&nbsp; has a different form.<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*The second parameter of the impulse response or the second axis marks the &nbsp<ins class="diffchange diffchange-inline">; &raquo</ins>;'''absolute time'''<ins class="diffchange diffchange-inline">&laquo;</ins>&nbsp; $t$, which is used, among other things, to describe the time variance.&nbsp; At different times&nbsp; $t$&nbsp; the impulse response&nbsp; $h(\tau, \hspace{0.05cm}t)$&nbsp; has a different form.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*A peculiarity of the 2D representation is that the&nbsp; $t$&ndash;axis is always plotted discrete-timely&nbsp; $($at multiples of&nbsp; $T)$&nbsp; while the&nbsp; $\tau$&ndash;axis can be continuous in time as in the example shown. &nbsp; However, in mobile communications, a discrete-time &nbsp; $h(\tau, \hspace{0.05cm}t_0)$&nbsp; with respect to&nbsp; $\tau$&nbsp; is assumed $($"echoes"$)$.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*A peculiarity of the 2D representation is that the&nbsp; $t$&ndash;axis is always plotted discrete-timely&nbsp; $($at multiples of&nbsp; $T)$&nbsp; while the&nbsp; $\tau$&ndash;axis can be continuous in time as in the example shown. &nbsp; However, in mobile communications, a discrete-time &nbsp; $h(\tau, \hspace{0.05cm}t_0)$&nbsp; with respect to&nbsp; $\tau$&nbsp; is assumed $($"echoes"$)$.</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l107" >Line 107:</td>
<td colspan="2" class="diff-lineno">Line 107:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The LTV equation is only applicable if the change of the channel&nbsp; $($marked in the figure by the parameter&nbsp; $T)$&nbsp; proceeds slowly in comparison to the maximum delay &nbsp; $\tau_{\rm max}$.&nbsp; In mobile communications this condition &nbsp; &#8658; &nbsp; $\tau_{\rm max} < T$ &nbsp; is almost always fulfilled.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The LTV equation is only applicable if the change of the channel&nbsp; $($marked in the figure by the parameter&nbsp; $T)$&nbsp; proceeds slowly in comparison to the maximum delay &nbsp; $\tau_{\rm max}$.&nbsp; In mobile communications this condition &nbsp; &#8658; &nbsp; $\tau_{\rm max} < T$ &nbsp; is almost always fulfilled.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*Selecting whether to apply the first Fourier integral to the parameter&nbsp; $\tau$&nbsp; or&nbsp; $t$&nbsp; leads to different spectral functions.&nbsp; In the&nbsp; [[Aufgaben:Exercise 2.1Z: 2D-Frequency and 2D-Time Representations|Exercise 2.1Z]]&nbsp; for example, the time variant two-dimensional&nbsp; '''2D transfer function'''&nbsp; is considered:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*Selecting whether to apply the first Fourier integral to the parameter&nbsp; $\tau$&nbsp; or&nbsp; $t$&nbsp; leads to different spectral functions.&nbsp; In the&nbsp; [[Aufgaben:Exercise 2.1Z: 2D-Frequency and 2D-Time Representations|<ins class="diffchange diffchange-inline">"</ins>Exercise 2.1Z<ins class="diffchange diffchange-inline">"</ins>]]&nbsp; for example, the time variant two-dimensional&nbsp<ins class="diffchange diffchange-inline">; &raquo</ins>;'''2D transfer function'''<ins class="diffchange diffchange-inline">&laquo;</ins>&nbsp; is considered:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>H(f,\hspace{0.05cm} t)</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<math>H(f,\hspace{0.05cm} t)</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l114" >Line 114:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>==Exercises <del class="diffchange diffchange-inline">about </del>the chapter==</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>==Exercises <ins class="diffchange diffchange-inline">for </ins>the chapter==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Aufgaben:Exercise 2.1: Two-Dimensional Impulse Response]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Aufgaben:Exercise 2.1: Two-Dimensional Impulse Response]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>
Hwang
https://en.lntwww.de/index.php?title=Mobile_Communications/General_Description_of_Time_Variant_Systems&diff=50076&oldid=prev
Hwang at 23:46, 12 November 2022
2022-11-12T23:46:48Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 23:46, 12 November 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l49" >Line 49:</td>
<td colspan="2" class="diff-lineno">Line 49:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>On the other hand, a DC signal&nbsp; $s(t) = A$&nbsp; is not suitable to make the frequency dependence of the LTI system visible: &nbsp; <br>&nbsp; &rArr; &nbsp; With a low-pass system the output signal would then be always constant, independent of&nbsp; $H(f)$:&nbsp; &nbsp; &nbsp; $r(t) = A \cdot H(f= 0)$.<br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>On the other hand, a DC signal&nbsp; $s(t) = A$&nbsp; is not suitable to make the frequency dependence of the LTI system visible: &nbsp; <br>&nbsp; &rArr; &nbsp; With a low-pass system the output signal would then be always constant, independent of&nbsp; $H(f)$:&nbsp; &nbsp; &nbsp; $r(t) = A \cdot H(f= 0)$.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">On </del>the next <del class="diffchange diffchange-inline">page </del>we consider a Dirac delta train&nbsp; $p_\delta(t)$&nbsp; as an input signal&nbsp; $s(t)$: &nbsp; <br>&nbsp; &rArr; &nbsp; Hereby the similarities and differences between time-invariant and time-variant systems can be shown clearly.<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">In </ins>the next <ins class="diffchange diffchange-inline">section </ins>we consider a Dirac delta train&nbsp; $p_\delta(t)$&nbsp; as an input signal&nbsp; $s(t)$: &nbsp; <br>&nbsp; &rArr; &nbsp; Hereby the similarities and differences between time-invariant and time-variant systems can be shown clearly.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><i>Note:</i>&nbsp; The properties of&nbsp; $H(f)$&nbsp; and&nbsp; $h(t)$&nbsp; are covered in detail in the&nbsp; $\text{LNTwww learning video}$&nbsp; (in German language):<br> &nbsp; &nbsp;</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><i>Note:</i>&nbsp; The properties of&nbsp; $H(f)$&nbsp; and&nbsp; $h(t)$&nbsp; are covered in detail in the&nbsp; $\text{LNTwww learning video}$&nbsp; (in German language):<br> &nbsp; &nbsp;</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l99" >Line 99:</td>
<td colspan="2" class="diff-lineno">Line 99:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br clear=all></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br clear=all></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Regarding the last equation and the above graph, it should be noted</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Regarding the last equation and the above graph, it should be noted</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*The parameter&nbsp; $\tau$&nbsp; specifies the &nbsp; '''delay time''' &nbsp; to denote the time dispersion.&nbsp; By writing out the convolution operation, it was possible to make&nbsp; $\tau$&nbsp; also the parameter of the LTI impulse response.&nbsp; <del class="diffchange diffchange-inline">On </del>the last <del class="diffchange diffchange-inline">pages </del>we spoke about&nbsp; $h(t)$&nbsp;.<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*The parameter&nbsp; $\tau$&nbsp; specifies the &nbsp; '''delay time''' &nbsp; to denote the time dispersion.&nbsp; By writing out the convolution operation, it was possible to make&nbsp; $\tau$&nbsp; also the parameter of the LTI impulse response.&nbsp; <ins class="diffchange diffchange-inline">In </ins>the last <ins class="diffchange diffchange-inline">sections </ins>we spoke about&nbsp; $h(t)$&nbsp;.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The second parameter of the impulse response or the second axis marks the &nbsp; '''absolute time'''&nbsp; $t$, which is used, among other things, to describe the time variance.&nbsp; At different times&nbsp; $t$&nbsp; the impulse response&nbsp; $h(\tau, \hspace{0.05cm}t)$&nbsp; has a different form.<br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The second parameter of the impulse response or the second axis marks the &nbsp; '''absolute time'''&nbsp; $t$, which is used, among other things, to describe the time variance.&nbsp; At different times&nbsp; $t$&nbsp; the impulse response&nbsp; $h(\tau, \hspace{0.05cm}t)$&nbsp; has a different form.<br></div></td></tr>
</table>
Hwang
https://en.lntwww.de/index.php?title=Mobile_Communications/General_Description_of_Time_Variant_Systems&diff=42740&oldid=prev
Guenter at 11:08, 5 November 2021
2021-11-05T11:08:30Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 11:08, 5 November 2021</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4" >Line 4:</td>
<td colspan="2" class="diff-lineno">Line 4:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|Nächste Seite=Multipath Reception in Mobile Communications}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|Nächste Seite=Multipath Reception in Mobile Communications}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>== # <del class="diffchange diffchange-inline">SYNOPSIS </del>OF THE SECOND MAIN CHAPTER # ==</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>== # <ins class="diffchange diffchange-inline">OVERVIEW </ins>OF THE SECOND MAIN CHAPTER # ==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>After the time variance, the term&nbsp; '''Frequency Selectivity''' &nbsp; is now introduced and illustrated with examples, a channel property which is also of great importance for mobile communications.&nbsp; As in the entire book, the description is given in the equivalent low-pass range. </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>After the time variance, the term&nbsp; '''Frequency Selectivity''' &nbsp; is now introduced and illustrated with examples, a channel property which is also of great importance for mobile communications.&nbsp; As in the entire book, the description is given in the equivalent low-pass range. </div></td></tr>
</table>
Guenter
https://en.lntwww.de/index.php?title=Mobile_Communications/General_Description_of_Time_Variant_Systems&diff=41887&oldid=prev
Javier: Text replacement - "time-discrete" to "discrete-time"
2021-10-11T09:01:00Z
<p>Text replacement - "time-discrete" to "discrete-time"</p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 09:01, 11 October 2021</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l103" >Line 103:</td>
<td colspan="2" class="diff-lineno">Line 103:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The second parameter of the impulse response or the second axis marks the &nbsp; '''absolute time'''&nbsp; $t$, which is used, among other things, to describe the time variance.&nbsp; At different times&nbsp; $t$&nbsp; the impulse response&nbsp; $h(\tau, \hspace{0.05cm}t)$&nbsp; has a different form.<br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The second parameter of the impulse response or the second axis marks the &nbsp; '''absolute time'''&nbsp; $t$, which is used, among other things, to describe the time variance.&nbsp; At different times&nbsp; $t$&nbsp; the impulse response&nbsp; $h(\tau, \hspace{0.05cm}t)$&nbsp; has a different form.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*A peculiarity of the 2D representation is that the&nbsp; $t$&ndash;axis is always plotted <del class="diffchange diffchange-inline">time</del>-<del class="diffchange diffchange-inline">discretely</del>&nbsp; $($at multiples of&nbsp; $T)$&nbsp; while the&nbsp; $\tau$&ndash;axis can be continuous in time as in the example shown. &nbsp; However, in mobile communications, a time<del class="diffchange diffchange-inline">-discrete </del>&nbsp; $h(\tau, \hspace{0.05cm}t_0)$&nbsp; with respect to&nbsp; $\tau$&nbsp; is assumed $($"echoes"$)$.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*A peculiarity of the 2D representation is that the&nbsp; $t$&ndash;axis is always plotted <ins class="diffchange diffchange-inline">discrete</ins>-<ins class="diffchange diffchange-inline">timely</ins>&nbsp; $($at multiples of&nbsp; $T)$&nbsp; while the&nbsp; $\tau$&ndash;axis can be continuous in time as in the example shown. &nbsp; However, in mobile communications, a <ins class="diffchange diffchange-inline">discrete-</ins>time &nbsp; $h(\tau, \hspace{0.05cm}t_0)$&nbsp; with respect to&nbsp; $\tau$&nbsp; is assumed $($"echoes"$)$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The LTV equation is only applicable if the change of the channel&nbsp; $($marked in the figure by the parameter&nbsp; $T)$&nbsp; proceeds slowly in comparison to the maximum delay &nbsp; $\tau_{\rm max}$.&nbsp; In mobile communications this condition &nbsp; &#8658; &nbsp; $\tau_{\rm max} < T$ &nbsp; is almost always fulfilled.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The LTV equation is only applicable if the change of the channel&nbsp; $($marked in the figure by the parameter&nbsp; $T)$&nbsp; proceeds slowly in comparison to the maximum delay &nbsp; $\tau_{\rm max}$.&nbsp; In mobile communications this condition &nbsp; &#8658; &nbsp; $\tau_{\rm max} < T$ &nbsp; is almost always fulfilled.</div></td></tr>
</table>
Javier
https://en.lntwww.de/index.php?title=Mobile_Communications/General_Description_of_Time_Variant_Systems&diff=38816&oldid=prev
Javier: Text replacement - "„" to """
2021-05-28T14:24:29Z
<p>Text replacement - "„" to """</p>
<table class="diff diff-contentalign-left" data-mw="interface">
<col class="diff-marker" />
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<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 14:24, 28 May 2021</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l103" >Line 103:</td>
<td colspan="2" class="diff-lineno">Line 103:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The second parameter of the impulse response or the second axis marks the &nbsp; '''absolute time'''&nbsp; $t$, which is used, among other things, to describe the time variance.&nbsp; At different times&nbsp; $t$&nbsp; the impulse response&nbsp; $h(\tau, \hspace{0.05cm}t)$&nbsp; has a different form.<br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The second parameter of the impulse response or the second axis marks the &nbsp; '''absolute time'''&nbsp; $t$, which is used, among other things, to describe the time variance.&nbsp; At different times&nbsp; $t$&nbsp; the impulse response&nbsp; $h(\tau, \hspace{0.05cm}t)$&nbsp; has a different form.<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*A peculiarity of the 2D representation is that the&nbsp; $t$&ndash;axis is always plotted time-discretely&nbsp; $($at multiples of&nbsp; $T)$&nbsp; while the&nbsp; $\tau$&ndash;axis can be continuous in time as in the example shown. &nbsp; However, in mobile communications, a time-discrete &nbsp; $h(\tau, \hspace{0.05cm}t_0)$&nbsp; with respect to&nbsp; $\tau$&nbsp; is assumed $($<del class="diffchange diffchange-inline">&bdquo;</del>echoes"$)$.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*A peculiarity of the 2D representation is that the&nbsp; $t$&ndash;axis is always plotted time-discretely&nbsp; $($at multiples of&nbsp; $T)$&nbsp; while the&nbsp; $\tau$&ndash;axis can be continuous in time as in the example shown. &nbsp; However, in mobile communications, a time-discrete &nbsp; $h(\tau, \hspace{0.05cm}t_0)$&nbsp; with respect to&nbsp; $\tau$&nbsp; is assumed $($<ins class="diffchange diffchange-inline">"</ins>echoes"$)$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The LTV equation is only applicable if the change of the channel&nbsp; $($marked in the figure by the parameter&nbsp; $T)$&nbsp; proceeds slowly in comparison to the maximum delay &nbsp; $\tau_{\rm max}$.&nbsp; In mobile communications this condition &nbsp; &#8658; &nbsp; $\tau_{\rm max} < T$ &nbsp; is almost always fulfilled.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*The LTV equation is only applicable if the change of the channel&nbsp; $($marked in the figure by the parameter&nbsp; $T)$&nbsp; proceeds slowly in comparison to the maximum delay &nbsp; $\tau_{\rm max}$.&nbsp; In mobile communications this condition &nbsp; &#8658; &nbsp; $\tau_{\rm max} < T$ &nbsp; is almost always fulfilled.</div></td></tr>
</table>
Javier