Mobile Communications/General Description of Time Variant Systems - Revision history

Guenter at 13:41, 29 January 2023

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Hwang at 18:14, 24 January 2023

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Hwang at 18:10, 24 January 2023

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Hwang at 13:21, 24 January 2023

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Hwang at 23:46, 12 November 2022

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Guenter at 11:08, 5 November 2021

2021-11-05T11:08:30Z

Javier: Text replacement - "time-discrete" to "discrete-time"

2021-10-11T09:01:00Z

Text replacement - "time-discrete" to "discrete-time"

Javier: Text replacement - "„" to """

2021-05-28T14:24:29Z

Text replacement - "„" to """

@@ Line 6: / Line 6: @@
 == # OVERVIEW OF THE SECOND MAIN CHAPTER # ==
 <br>
-After the time variance, the term&nbsp; &raquo;'''Frequency Selectivity'''&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.
+After the time variance, the term&nbsp; &raquo;'''frequency selectivity'''&laquo;&nbsp; is now introduced and illustrated with examples,&nbsp; a channel property which is also of great importance for mobile communications.&nbsp; As in the entire book,&nbsp; the description is given in the equivalent low-pass range.
 It is covered in detail:
-*the difference between time invariant '''KORREKTUR: time-invariant''' and time variant '''KORREKTUR: time-variant''' systems,
+#The&nbsp; &raquo;difference between time-invariant and time-variant systems&laquo;,
-*the time variant '''KORREKTUR: time-variant''' impulse response as an important descriptive function of time variant '''KORREKTUR: time-variant''' systems,
+#the&nbsp; &raquo;time-variant impulse response&laquo;&nbsp; as an important descriptive function of time-variant systems,
-*multi-way reception as the cause of frequency-selective behaviour,
+#&raquo;multi-way reception&laquo;&nbsp; as the cause of frequency-selective behaviour,
-*a detailed derivation and interpretation of the GWSSUS channel model,
+#a detailed derivation and interpretation of the&nbsp; &raquo;GWSSUS channel model&laquo;,
-*the characteristics of the GWSSUS model: &nbsp; coherence bandwidth, correlation duration, etc.
+#the characteristics of the GWSSUS model: &nbsp; &raquo;coherence bandwidth,&nbsp; correlation duration&laquo;,&nbsp; etc.
@@ Line 27: / Line 27: @@
-The most important results are briefly explained again here.&nbsp; We assume a&nbsp; linear and time invariant '''KORREKTUR: 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:
+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:
 *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;:$$H(f) = R(f)/S(f).$$
@@ Line 107: / Line 107: @@
 *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.
-*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 '''KORREKTUR: time-variant''' two-dimensional&nbsp; &raquo;'''2D transfer function'''&laquo;&nbsp; is considered:
+*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  &nbsp; &raquo;'''two-dimensional transfer function'''&laquo;&nbsp; is considered:
 ::<math>H(f,\hspace{0.05cm} t)

@@ Line 107: / Line 107: @@
 *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.
-*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:
+*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 '''KORREKTUR: time-variant''' two-dimensional&nbsp; &raquo;'''2D transfer function'''&laquo;&nbsp; is considered:
 ::<math>H(f,\hspace{0.05cm} t)

@@ Line 10: / Line 10: @@
 It is covered in detail:
-*the difference between time invariant and time variant systems,
+*the difference between time invariant '''KORREKTUR: time-invariant''' and time variant '''KORREKTUR: time-variant''' systems,
-*the time variant impulse response as an important descriptive function of time variant systems,
+*the time variant '''KORREKTUR: time-variant''' impulse response as an important descriptive function of time variant '''KORREKTUR: time-variant''' systems,
 *multi-way reception as the cause of frequency-selective behaviour,
 *a detailed derivation and interpretation of the GWSSUS channel model,
@@ Line 27: / Line 27: @@
-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:
+The most important results are briefly explained again here.&nbsp; We assume a&nbsp; linear and time invariant '''KORREKTUR: 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:
 *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;:$$H(f) = R(f)/S(f).$$
@@ Line 49: / Line 49: @@
 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>
-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 '''KORREKTUR: time invariant''' and time-variant '''KORREKTUR: time variant''' systems can be shown clearly.<br>
+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>
 <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;

@@ Line 49: / Line 49: @@
 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>
-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>
+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 '''KORREKTUR: time invariant''' and time-variant '''KORREKTUR: time variant''' systems can be shown clearly.<br>
 <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;

@@ Line 28: / Line 28: @@
 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:
-*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|$\text{transfer function}$]]&nbsp; $H(f)$&nbsp; which is also referred to as the&nbsp; "frequency response".&nbsp; By definition&nbsp;:$$H(f) = R(f)/S(f).$$
+*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;:$$H(f) = R(f)/S(f).$$
-*Similarly, the system is defined by the&nbsp; [[Linear_and_Time_Invariant_Systems/Systembeschreibung_im_Zeitbereich#Impulsantwort|$\text{impulse response}$]]&nbsp; $h(t)$&nbsp;, which is the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#Das_zweite_Fourierintegral|$\text{inverse Fourier transform}$]]&nbsp; of&nbsp; $H(f)$.&nbsp; &nbsp; The output signal results from the convolution:
+*Similarly, the system is defined by the&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Impulse_response|$\text{impulse response}$]]&nbsp; $h(t)$&nbsp;, which is the&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_second_Fourier_integral|$\text{inverse Fourier transform}$]]&nbsp; of&nbsp; $H(f)$.&nbsp; &nbsp; The output signal results from the convolution:
 ::<math>r(t) = s(t) \star h(t) \hspace{0.4cm} {\rm with} \hspace{0.4cm} h(t)
@@ Line 38: / Line 38: @@
 {{BlaueBox|TEXT=
 $\text{Definitions:}$&nbsp; &nbsp; The following input signals are suitable for detecting the linear distortions caused by&nbsp; $H(f)$&nbsp; or &nbsp; $h(t)$:&nbsp;
-*a&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Dirac_Delta_Impulse|$\text{Dirac delta}$]]&nbsp; or&nbsp; "impulse":
+*a&nbsp; [[Signal_Representation/Special_Cases_of_Pulses#Dirac_delta_or_impulse|$\text{Dirac delta}$]]&nbsp; or&nbsp; "impulse":
 :$$s(t) = \delta(t) \hspace{0.3cm}\Rightarrow \hspace{0.3cm}  &nbsp; r(t) = \delta(t) \star h(t)= h(t)\hspace{0.3cm}\Rightarrow \hspace{0.3cm} \text{impulse response,}$$
 *a&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Step_response|$\text{step function}$]]&nbsp; or&nbsp; "Heaviside step function":
 :$$s(t) = \gamma(t) \hspace{0.3cm}\Rightarrow \hspace{0.35cm}  &nbsp; r(t) = \gamma(t) \star h(t)\hspace{1.5cm}\Rightarrow \hspace{0.3cm} \text{step response,}$$
-*a&nbsp; [[Signal_Representation/Time_Discrete_Signal_Representation#Dirac_Comb_in_Time_and_Frequency_Domain|$\text{Dirac comb}$]]&nbsp; or&nbsp; "Dirac delta train":
+*a&nbsp; [[Signal_Representation/Discrete-Time_Signal_Representation#Dirac_comb_in_time_and_frequency_domain|$\text{Dirac comb}$]]&nbsp; or&nbsp; "Dirac delta train":
 :$$s(t) = p_\delta(t) \hspace{0.25cm}\Rightarrow \hspace{0.3cm}  &nbsp; r(t) = p_\delta(t) \star h(t)\hspace{1.3cm}\Rightarrow \hspace{0.3cm} \text{impulse response train.}$$}}

@@ Line 49: / Line 49: @@
 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>
-On the next page 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>
+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>
 <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;
@@ Line 99: / Line 99: @@
 <br clear=all>
 Regarding the last equation and the above graph, it should be noted
-*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; On the last pages we spoke about&nbsp; $h(t)$&nbsp;.<br>
+*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>
 *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>

@@ Line 4: / Line 4: @@
 |Nächste Seite=Multipath Reception in Mobile Communications}}
-== # SYNOPSIS OF THE SECOND MAIN CHAPTER # ==
+== # OVERVIEW OF THE SECOND MAIN CHAPTER # ==
 <br>
 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.

@@ Line 103: / Line 103: @@
 *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>
-*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 $($"echoes"$)$.
+*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"$)$.
 *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.