Mobile Communications/General Description of Time Variant Systems

1. Mobile Communications
2. Frequency-selective transmission channels
3. General Description of Time Variant Systems

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# SYNOPSIS TO 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 communication.  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 message transmission system have already been described in the chapters 

• Systembeschreibung im Frequenzbereich  and 
• Systembeschreibung im Zeitbereich 

Considered LTI system

of the book "Linear Time Variant Systems".

The most important results are briefly explained again here.

We assume a  linear and time invariant system   ⇒   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 :$$H(f) = R(f)/S(f).$$

• Similarly, the system is defined by the  Impulse response  $h(t)$ , which is the  inverse Fourier transformation  of  $H(f)$    The output signal results from the convolution:

\[r(t) = s(t) \star h(t) \hspace{0.4cm} {\rm mit} \hspace{0.4cm} h(t) \hspace{0.2cm} \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.2cm} H(f) \hspace{0.05cm}.\]

$\text{Definitions:}$ 

The following input signals are suitable for detecting the linear distortions caused by  $H(f)$  or   $h(t)$ 

• a  Dirac impulse:   

$$s(t) = \delta(t) \hspace{0.3cm}\Rightarrow \hspace{0.3cm}   r(t) = \delta(t) \star h(t)= h(t)\hspace{0.3cm}\Rightarrow \hspace{0.3cm} \text{Impulse response,}$$

• a  Heaviside step function:

$$s(t) = \gamma(t) \hspace{0.3cm}\Rightarrow \hspace{0.3cm}   r(t) = \gamma(t) \star h(t)\hspace{1.5cm}\Rightarrow \hspace{0.3cm} \text{step response,}$$

• a  Dirac impulse train:

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

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 impulse 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 learning video  Some remarks on the transfer function .

Time invariant vs. time variant channels

The graphic is intended to illustrate the difference between a time invariant channel  $\rm (LTI)$  and a 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 impulse train  $p_\delta(t)$, i.e. an infinite sequence of Dirac impulses 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 impulse 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 learning video  Properties of the transmission channel  describes the differences between LTI and LTV systems.

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  $\rm LTV System$   ("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 time change of the channel is slow in comparison to the maximum delay   $\tau_{\rm max}$    $($marked in the figure by the parameter  $T$  $)$. 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  Excercise 2.1Z  for example, the time variant  2D–Transfer function  is considered:

\[H(f,\hspace{0.05cm} t) \hspace{0.2cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ \hspace{0.2cm} h(\tau,\hspace{0.05cm}t) \hspace{0.05cm}.\]

Excercises to the chapter

Exercise 2.1: Two-Dimensional Impulse Response

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