View source for Mobile Communications/General Description of Time Variant Systems

 
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== # SYNOPSIS 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. 

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: &nbsp; coherence bandwidth, correlation duration, etc.



== Transfer function and impulse response ==
<br>
The description parameters of a communication system have already been described in the chapters&nbsp;
* [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain|System Description in Frequency Domain]]&nbsp; and&nbsp;
* [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain|System Description in Time Domain]]&nbsp; 


[[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; '''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/Systembeschreibung_im_Frequenzbereich#.C3.9Cbertragungsfunktion_-_Frequenzgang|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|impulse response]]&nbsp; $h(t)$&nbsp;, which is the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#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 with} \hspace{0.4cm} h(t)
 \hspace{0.2cm}  \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.2cm} H(f)  
 \hspace{0.05cm}.</math>

{{BlaueBox|TEXT=
$\text{Definitions:}$&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 impulse]]: &nbsp;&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; [[Linear_and_Time_Invariant_Systems/Systembeschreibung_im_Zeitbereich#Sprungantwort|Heaviside step function]]: 
:$$s(t) = \gamma(t) \hspace{0.3cm}\Rightarrow \hspace{0.3cm}  &nbsp; r(t) = \gamma(t) \star h(t)\hspace{1.5cm}\Rightarrow \hspace{0.3cm} \text{step response,}$$
*a&nbsp; [[Signal_Representation/Time_Discrete_Signal_Representation#Diracpuls_im_Zeit-_und_im_Frequenzbereich|Dirac impulse train]]: 
:$$s(t) = p_\delta(t) \hspace{0.25cm}\Rightarrow \hspace{0.3cm}  &nbsp; r(t) = p_\delta(t) \star h(t)\hspace{1.3cm}\Rightarrow \hspace{0.3cm} \text{impulse response train.}$$}}
 


: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; 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;  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;(in German language).<br>



== Time invariant vs. time variant channels ==
<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>

[[File:EN_Mob_T_2_1_S2.png|right|frame|Time invariant and time variant channel|class=fit]]

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):

::<math>s(t) = p_{\rm \delta} (t) = \sum_{n = -\infty}^{+\infty} {\rm \delta} (t - n \cdot T)
 \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 entire received signal after the LTI channel, according to the middle graph, can then be written as:

::<math>r(t) = p_{\rm \delta} (t) \star h(t) = \sum_{n = -\infty}^{+\infty} h (t - n \cdot T)
 \hspace{0.05cm}.</math>

*For a time-variant channel (lower graph) this equation is not applicable.&nbsp; In each time interval, a (slightly) different signal shape is obtained. 


{{BlaueBox|TEXT=  
$\text{Conclusion:}$&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>

== Two-dimensional impulse response==
<br>
[[File:EN_Mob_T_2_1_S3.png|right|frame|Two-dimensional impulse response|class=fit]]

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 following relations apply:

::<math>\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},</math>
::<math>\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}.</math>
<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&ndash;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>

*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;$)$.

*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.

*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; '''2D&ndash;Transfer function'''&nbsp; is considered:

::<math>H(f,\hspace{0.05cm} t)
 \hspace{0.2cm}  \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ \hspace{0.2cm} h(\tau,\hspace{0.05cm}t)  
 \hspace{0.05cm}.</math>


==Exercises to the chapter==
[[Aufgaben:Exercise 2.1: Two-Dimensional Impulse Response]]

[[Aufgaben:Exercise 2.1Z: 2D-Frequency and 2D-Time Representations]]



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