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 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 the chapters 

• System Description in Frequency Domain  and 
• System Description in Time Domain 

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:

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

$\text{Definitions:}$ 

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

• 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 (in German language).

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 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  (in German language)  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  Exercise 2.1Z  for example, the time variant  2D–Transfer function  is considered:

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

Exercises to the chapter

Exercise 2.1: Two-Dimensional Impulse Response

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