Difference between revisions of "Mobile Communications/General Description of Time Variant Systems"

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{{Header
 
{{Header
|Untermenü=Frequency-selective transmission channels |Vorherige Seite=Non-frequency selective fading with direct component
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|Untermenü=Frequency-Selective Transmission Channels |Vorherige Seite=Non-Frequency Selective Fading With Direct Component
|Nächste Seite=Multipath reception in mobile communications}}
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|Nächste Seite=Multipath Reception in Mobile Communications}}
  
== # SYNOPSIS TO SECOND MAIN CHAPTER # ==
+
== # SYNOPSIS OF THE SECOND MAIN CHAPTER # ==
 
<br>
 
<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 communication.&nbsp; As in the entire book, the description is given in the equivalent low-pass range.  
+
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.  
  
 
It is covered in detail:
 
It is covered in detail:
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== Transfer function and impulse response ==
 
== Transfer function and impulse response ==
 
<br>
 
<br>
The description parameters of a message transmission system have already been described in the chapters&nbsp;
+
The description parameters of a communication system have already been described in two chapters of the book "Linear Time Variant Systems":
* [[Linear_and_Time_Invariant_Systems/Systembeschreibung_im_Frequenzbereich|Systembeschreibung im Frequenzbereich]]&nbsp; and&nbsp;
+
* [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain|System Description in Frequency Domain]],
* [[Linear_and_Time_Invariant_Systems/Systembeschreibung_im_Zeitbereich|Systembeschreibung im Zeitbereich]]&nbsp;  
+
* [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain|System Description in Time Domain]].&nbsp;  
  
 +
[[File:EN_Mob_T_2_1_S1_neu.png|right|frame|Considered LTI system|class=fit]]
  
[[File:EN_Mob_T_2_1_S1.png|right|frame|Considered LTI system|class=fit]]
 
  
of the book "Linear Time Variant Systems".  
+
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|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 most important results are briefly explained again here.<br>
+
*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:
  
We assume a&nbsp; ''linear and time invariant system'' &nbsp; &#8658; &nbsp; '''LTI&ndash;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:
+
::<math>r(t) = s(t) \star h(t) \hspace{0.4cm} {\rm with} \hspace{0.4cm} h(t)
*The system can be completely characterized by the&nbsp; [[Linear_and_Time_Invariant_Systems/Systembeschreibung_im_Frequenzbereich#.C3.9Cbertragungsfunktion_-_Frequenzgang|transfer function]]&nbsp; $H(f)$&nbsp; Which is also referred to as the&nbsp; <i> Frequency response</i>.&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|Impulse response]]&nbsp; $h(t)$&nbsp;, which is the&nbsp; [[Signal_Representation/Fouriertransformation_und_-rücktransformation#Das_zweite_Fourierintegral|inverse Fourier transformation]]&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 mit} \hspace{0.4cm} h(t)
 
 
  \hspace{0.2cm}  \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.2cm} H(f)   
 
  \hspace{0.2cm}  \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.2cm} H(f)   
 
  \hspace{0.05cm}.</math>
 
  \hspace{0.05cm}.</math>
  
 
{{BlaueBox|TEXT=
 
{{BlaueBox|TEXT=
$\text{Definitions:}$&nbsp;
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$\text{Definitions:}$&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|Dirac delta]]&nbsp; or&nbsp; "impulse":
The following input signals are suitable for detecting the linear distortions caused by&nbsp; $H(f)$&nbsp; or &nbsp; $h(t)$&nbsp;
+
:$$s(t) = \delta(t) \hspace{0.3cm}\Rightarrow \hspace{0.3cm}  &nbsp; r(t) = \delta(t) \star h(t)= h(t)\hspace{0.3cm}\Rightarrow \hspace{0.3cm} \text{impulse response,}$$   
*a&nbsp; [[Signal_Representation/Einige_Sonderf%C3%A4lle_impulsartiger_Signale#Diracimpuls|Dirac impulse]]: &nbsp;&nbsp;  
+
*a&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Step_response|step function]]&nbsp; or&nbsp; "Heaviside step function":  
:$$s(t) = \delta(t) \hspace{0.3cm}\Rightarrow \hspace{0.3cm}  &nbsp; r(t) = \delta(t) \star h(t)= h(t)\hspace{0.3cm}\Rightarrow \hspace{0.3cm} \text{Impulse response,}$$   
+
:$$s(t) = \gamma(t) \hspace{0.3cm}\Rightarrow \hspace{0.35cm}  &nbsp; r(t) = \gamma(t) \star h(t)\hspace{1.5cm}\Rightarrow \hspace{0.3cm} \text{step response,}$$
*a&nbsp; [[Linear_and_Time_Invariant_Systems/Systembeschreibung_im_Zeitbereich#Sprungantwort|Heaviside step function]]:  
+
*a&nbsp; [[Signal_Representation/Time_Discrete_Signal_Representation#Dirac_Comb_in_Time_and_Frequency_Domain|Dirac comb]]&nbsp; or&nbsp; "Dirac delta train":  
:$$s(t) = \gamma(t) \hspace{0.3cm}\Rightarrow \hspace{0.3cm}  &nbsp; r(t) = \gamma(t) \star h(t)\hspace{1.5cm}\Rightarrow \hspace{0.3cm} \text{step response,}$$
 
*a&nbsp; [[Signal_Representation/Zeitdiskrete_Signaldarstellung#Diracpuls_im_Zeit-_und_im_Frequenzbereich|Dirac impulse train]]:  
 
 
:$$s(t) = p_\delta(t) \hspace{0.25cm}\Rightarrow \hspace{0.3cm}  &nbsp; r(t) = p_\delta(t) \star h(t)\hspace{1.3cm}\Rightarrow \hspace{0.3cm} \text{impulse response train.}$$}}
 
:$$s(t) = p_\delta(t) \hspace{0.25cm}\Rightarrow \hspace{0.3cm}  &nbsp; r(t) = p_\delta(t) \star h(t)\hspace{1.3cm}\Rightarrow \hspace{0.3cm} \text{impulse response train.}$$}}
 
   
 
   
  
  
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; &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 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 impulse train&nbsp; $p_\delta(t)$&nbsp; as an input signal&nbsp; $s(t)$&nbsp;: &nbsp; <br> &nbsp; Hereby the similarities and differences between time-invariant and time-variant systems can be shown clearly.<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>
  
<i>Note:</i> &nbsp; The properties of&nbsp; $H(f)$&nbsp; and&nbsp; $h(t)$&nbsp; are covered in detail in the learning video&nbsp; [[Einige_Anmerkungen_zur_Übertragungsfunktion_(Lernvideo)|Some remarks on the transfer function]]&nbsp;.<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;
 +
[https://www.lntwww.de/Eigenschaften_des_%C3%9Cbertragungskanals_(Lernvideo) Eigenschaften des Übertragungskanals] &nbsp; &rArr; &nbsp; "Some remarks on the transfer function".<br>
  
  
  
== Time invariant vs. time variant channels ==
+
== Time&ndash;invariant vs. time&ndash;variant channels ==
 
<br>
 
<br>
The graphic is intended to illustrate the difference between a time invariant channel&nbsp; $\rm (LTI)$&nbsp; and a time variant channel &nbsp; $\rm (LTV)$&nbsp;.<br>
+
The graphic is intended to illustrate the difference between a linear time&ndash;invariant channel&nbsp; $\rm (LTI)$&nbsp; and a linear  time&ndash;variant channel &nbsp; $\rm (LTV)$&nbsp;.<br>
  
[[File:EN_Mob_T_2_1_S2.png|right|frame|Time invariant and time variant channel|class=fit]]
+
[[File:EN_Mob_T_2_1_S2.png|right|frame|Time&ndash;invariant and time&ndash;variant channel|class=fit]]
  
 
One can see from this illustration:
 
One can see from this illustration:
*The transmitted signal&nbsp; $s(t)$&nbsp; is a Dirac impulse train&nbsp; $p_\delta(t)$, i.e. an infinite sequence of Dirac impulses in equidistant intervals&nbsp; $T$,&nbsp; all with the weight&nbsp; $1$&nbsp; (see upper graph):
+
*The transmitted signal&nbsp; $s(t)$&nbsp; is a Dirac delta train&nbsp; $p_\delta(t)$, i.e. an infinite sequence of Dirac deltas in equidistant intervals&nbsp; $T$,&nbsp; all with the weight&nbsp; $1$&nbsp; (see upper graph):
  
 
::<math>s(t) = p_{\rm \delta} (t) = \sum_{n = -\infty}^{+\infty} {\rm \delta} (t - n \cdot T)
 
::<math>s(t) = p_{\rm \delta} (t) = \sum_{n = -\infty}^{+\infty} {\rm \delta} (t - n \cdot T)
 
  \hspace{0.05cm}.</math>
 
  \hspace{0.05cm}.</math>
  
*The Dirac impulse at&nbsp; $t = 0$&nbsp; is marked in green. The signal at the channel output is equal to&nbsp; $r(t) = h(t)$&nbsp;, with&nbsp; $s(t) = {\rm \delta}(t)$&nbsp;, also indicated in green. &nbsp; As a condition, it is assumed that the extension of the impulse response&nbsp; $h(t)$&nbsp; is smaller than $T$.<br>.
+
*The Dirac delta at&nbsp; $t = 0$&nbsp; is marked in green. The signal at the channel output is equal to&nbsp; $r(t) = h(t)$&nbsp;, with&nbsp; $s(t) = {\rm \delta}(t)$&nbsp;, also indicated in green. &nbsp; As a condition, it is assumed that the extension of the impulse response&nbsp; $h(t)$&nbsp; is smaller than $T$.<br>.
  
 
*The entire received signal after the LTI channel, according to the middle graph, can then be written as:
 
*The entire received signal after the LTI channel, according to the middle graph, can then be written as:
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$\text{Conclusion:}$&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>
 
$\text{Conclusion:}$&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>
  
<i>Note:</i> &nbsp; The learning video&nbsp; [[Eigenschaften_des_Übertragungskanals_(Lernvideo)|Properties of the transmission channel]]&nbsp; describes the differences between LTI and LTV systems.<br>
+
<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;
 +
[https://www.lntwww.de/Eigenschaften_des_%C3%9Cbertragungskanals_(Lernvideo) Eigenschaften des Übertragungskanals] &nbsp; &rArr; &nbsp; "Some remarks on the transfer function".<br>
 +
 
  
 
== Two-dimensional impulse response==
 
== Two-dimensional impulse response==
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To identify a time-variant impulse response, a second parameter is used and the impulse response is preferably mapped in a three-dimensional coordinate system.<br>
 
To identify a time-variant impulse response, a second parameter is used and the impulse response is preferably mapped in a three-dimensional coordinate system.<br>
  
The condition for this is that the channel is still linear; one speaks then of a&nbsp; $\rm LTV System$&nbsp;&nbsp; ("linear time-variant").  
+
The condition for this is that the channel is still linear.&nbsp; One speaks then of a&nbsp; $\text{LTV system}$&nbsp;&nbsp; ("linear time-variant").  
  
 
The following relations apply:
 
The following relations apply:
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<br clear=all>
 
<br clear=all>
 
Regarding the last equation and the above graph, it should be noted
 
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&ndash;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; On the last pages 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>
+
*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 $($&bdquo;echoes&rdquo;$)$.
+
*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"$)$.
  
*The LTV&ndash;equation is only applicable if the time change of the channel is slow in comparison to the maximum delay &nbsp; $\tau_{\rm max}$&nbsp; &nbsp; $($marked in the figure by the parameter&nbsp; $T$&nbsp; $)$. In mobile communications this condition &nbsp; &#8658; &nbsp; $\tau_{\rm max} < T$ &nbsp; is almost always fulfilled.
+
*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|Excercise 2.1Z]]&nbsp; for example, the time variant&nbsp; '''2D&ndash;Transfer function'''&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 two-dimensional&nbsp; '''2D transfer function'''&nbsp; is considered:
  
 
::<math>H(f,\hspace{0.05cm} t)
 
::<math>H(f,\hspace{0.05cm} t)
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==Excercises to the chapter==
+
==Exercises about the chapter==
 
[[Aufgaben:Exercise 2.1: Two-Dimensional Impulse Response]]
 
[[Aufgaben:Exercise 2.1: Two-Dimensional Impulse Response]]
  

Revision as of 15:24, 28 May 2021

# SYNOPSIS OF THE SECOND MAIN CHAPTER #


After the time variance, the term  Frequency Selectivity   is now introduced and illustrated with examples, a channel property which is also of great importance for mobile communications.  As in the entire book, the description is given in the equivalent low-pass range.

It is covered in detail:

  • the difference between time invariant and time variant systems,
  • the time variant impulse response as an important descriptive function of time variant systems,
  • multi-way reception as the cause of frequency-selective behaviour,
  • a detailed derivation and interpretation of the GWSSUS channel model,
  • the characteristics of the GWSSUS model:   coherence bandwidth, correlation duration, etc.


Transfer function and impulse response


The description parameters of a communication system have already been described in two chapters of the book "Linear Time Variant Systems":

Considered LTI system


The most important results are briefly explained again here.  We assume a  linear and time invariant system   ⇒   $\text{LTI system}$  with the signal  $s(t)$  at the input and the output signal  $r(t)$.   For the sake of simplicity, let  $s(t)$  and  $r(t)$  be real.  Then the following applies:

  • The system can be completely characterized by the  transfer function  $H(f)$  which is also referred to as the  "frequency response".  By definition :$$H(f) = R(f)/S(f).$$
\[r(t) = s(t) \star h(t) \hspace{0.4cm} {\rm with} \hspace{0.4cm} h(t) \hspace{0.2cm} \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.2cm} H(f) \hspace{0.05cm}.\]

$\text{Definitions:}$    The following input signals are suitable for detecting the linear distortions caused by  $H(f)$  or   $h(t)$: 

$$s(t) = \delta(t) \hspace{0.3cm}\Rightarrow \hspace{0.3cm}   r(t) = \delta(t) \star h(t)= h(t)\hspace{0.3cm}\Rightarrow \hspace{0.3cm} \text{impulse response,}$$
$$s(t) = \gamma(t) \hspace{0.3cm}\Rightarrow \hspace{0.35cm}   r(t) = \gamma(t) \star h(t)\hspace{1.5cm}\Rightarrow \hspace{0.3cm} \text{step response,}$$
$$s(t) = p_\delta(t) \hspace{0.25cm}\Rightarrow \hspace{0.3cm}   r(t) = p_\delta(t) \star h(t)\hspace{1.3cm}\Rightarrow \hspace{0.3cm} \text{impulse response train.}$$


On the other hand, a DC signal  $s(t) = A$  is not suitable to make the frequency dependence of the LTI system visible:  
  ⇒   With a low-pass system the output signal would then be always constant, independent of  $H(f)$:      $r(t) = A \cdot H(f= 0)$.

On the next page we consider a Dirac delta train  $p_\delta(t)$  as an input signal  $s(t)$:  
  ⇒   Hereby the similarities and differences between time-invariant and time-variant systems can be shown clearly.

Note:  The properties of  $H(f)$  and  $h(t)$  are covered in detail in the  $\text{LNTwww learning video}$  (in German language):
    Eigenschaften des Übertragungskanals   ⇒   "Some remarks on the transfer function".


Time–invariant vs. time–variant channels


The graphic is intended to illustrate the difference between a linear time–invariant channel  $\rm (LTI)$  and a linear time–variant channel   $\rm (LTV)$ .

Time–invariant and time–variant channel

One can see from this illustration:

  • The transmitted signal  $s(t)$  is a Dirac delta train  $p_\delta(t)$, i.e. an infinite sequence of Dirac deltas in equidistant intervals  $T$,  all with the weight  $1$  (see upper graph):
\[s(t) = p_{\rm \delta} (t) = \sum_{n = -\infty}^{+\infty} {\rm \delta} (t - n \cdot T) \hspace{0.05cm}.\]
  • The Dirac delta at  $t = 0$  is marked in green. The signal at the channel output is equal to  $r(t) = h(t)$ , with  $s(t) = {\rm \delta}(t)$ , also indicated in green.   As a condition, it is assumed that the extension of the impulse response  $h(t)$  is smaller than $T$.
    .
  • The entire received signal after the LTI channel, according to the middle graph, can then be written as:
\[r(t) = p_{\rm \delta} (t) \star h(t) = \sum_{n = -\infty}^{+\infty} h (t - n \cdot T) \hspace{0.05cm}.\]
  • For a time-variant channel (lower graph) this equation is not applicable.  In each time interval, a (slightly) different signal shape is obtained.


$\text{Conclusion:}$  With a   time-variant channel   you cannot specify neither a one-parameter impulse response  $h(t)$  nor a transfer function  $H(f)$ .


Note:  The differences between LTI and LTV systems are clarified with the  $\text{LNTwww learning video}$  (in German language):
    Eigenschaften des Übertragungskanals   ⇒   "Some remarks on the transfer function".


Two-dimensional impulse response


Two-dimensional impulse response

To identify a time-variant impulse response, a second parameter is used and the impulse response is preferably mapped in a three-dimensional coordinate system.

The condition for this is that the channel is still linear.  One speaks then of a  $\text{LTV system}$   ("linear time-variant").

The following relations apply:

\[\text{LTI:}\hspace{0.5cm} r(t) = \int_{-\infty}^{+\infty} h(\tau) \cdot s(t-\tau) \hspace{0.15cm}{\rm d}\tau \hspace{0.05cm},\]
\[\text{LTV:}\hspace{0.5cm} r(t) \hspace{-0.1cm} = \hspace{-0.1cm} \int_{-\infty}^{+\infty} h(\tau, \hspace{0.1cm}t) \cdot s(t-\tau) \hspace{0.15cm}{\rm d}\tau \hspace{0.05cm}.\]


Regarding the last equation and the above graph, it should be noted

  • The parameter  $\tau$  specifies the   delay time   to denote the time dispersion.  By writing out the convolution operation, it was possible to make  $\tau$  also the parameter of the LTI impulse response.  On the last pages we spoke about  $h(t)$ .
  • The second parameter of the impulse response or the second axis marks the   absolute time  $t$, which is used, among other things, to describe the time variance.  At different times  $t$  the impulse response  $h(\tau, \hspace{0.05cm}t)$  has a different form.
  • A peculiarity of the 2D representation is that the  $t$–axis is always plotted time-discretely  $($at multiples of  $T)$  while the  $\tau$–axis can be continuous in time as in the example shown.   However, in mobile communications, a time-discrete   $h(\tau, \hspace{0.05cm}t_0)$  with respect to  $\tau$  is assumed $($"echoes"$)$.
  • The LTV equation is only applicable if the change of the channel  $($marked in the figure by the parameter  $T)$  proceeds slowly in comparison to the maximum delay   $\tau_{\rm max}$.  In mobile communications this condition   ⇒   $\tau_{\rm max} < T$   is almost always fulfilled.
  • Selecting whether to apply the first Fourier integral to the parameter  $\tau$  or  $t$  leads to different spectral functions.  In the  Exercise 2.1Z  for example, the time variant two-dimensional  2D transfer function  is considered:
\[H(f,\hspace{0.05cm} t) \hspace{0.2cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ \hspace{0.2cm} h(\tau,\hspace{0.05cm}t) \hspace{0.05cm}.\]


Exercises about the chapter

Exercise 2.1: Two-Dimensional Impulse Response

Exercise 2.1Z: 2D-Frequency and 2D-Time Representations