General Description of Time Variant Systems

From LNTwww


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:

  1. The  »difference between time-invariant and time-variant systems«,
  2. the  »time-variant impulse response«  as an important descriptive function of time-variant systems,
  3. »multi-way reception«  as the cause of frequency-selective behaviour,
  4. a detailed derivation and interpretation of the  »GWSSUS channel model«,
  5. 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  $\text{transfer function}$  $H(f)$  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  $H(f)$:      $r(t) = A \cdot H(f= 0)$.

In the next section 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.  In the last sections 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 discrete-timely  $($at multiples of  $T)$  while the  $\tau$–axis can be continuous in time as in the example shown.   However, in mobile communications, a discrete-time   $h(\tau, \hspace{0.05cm}t_0)$  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 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 for the chapter

Exercise 2.1: Two-Dimensional Impulse Response

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