Discrete LTI systems

This post answers the question “What is discrete LTI system?”. It is useful to consider discrete-time signals as a sequence of impulses. For example, a discrete-time signal is on show in Figure 1. Figure 2 shows its mathematical representation, where the signal is divided into the single impulses. S,  the sum of these individual impulses, form the initial discrete-time signal.

The sum of the impulses is:

$x[n]=\sum _{k=-\infty }^{k=\infty } x[k\left]\delta \right[n–k]$

. In the other words the discrete-time signal is the linear combination of shifted impulses

$\delta \left[n–k\right]$

with the weight

$x\left[k\right]$

. This equation is called the shifting property of the discrete-time unit impulse.

If the $x\left[n\right]$

is a linear time-invariant function, then the convolution sum

$y\left[n\right]$

is a linear time-invariant function too.

Let’s consider the response of a linear discrete-time function x[n], that can be represented by the sum of impulses

$x[n]=\sum _{k=-\infty }^{k=\infty } x[k\left]\delta \right[n–k]$

, i.e. a linear combination of weighted shifted impulses.

If the input of the linear system is

$x\left[n\right]$

, then the output

$y[n]:\mathrm{y[}n]=\sum _{k=-\infty }^{\infty } x[k\left]h_k\right[n]$

. Here the

$h_k\left[n\right]$

are the responses to the signals .

Generally speaking, the functions

$h_k\left[n\right]$

are not related to each other for each  particular

$k$

. In our case  is a response of impulse function, then

$h_k\left[n\right]$

is a linear shifted version of itself.

So

$h_k[n]=h_0[n–k]$

. Let’s assume that

$h_0[n]=h[n]$

.

$h\left[n\right]$

is the output for the input

$\delta \left[n\right]$

of the LTI system. So we have

$y[n]=\sum _{k=-\infty }^{\infty } x[k\left]h\right[n–k]$

. This equation is called superposition (convolution) sum of the sequences

$x\left[k\right]$

and

$h\left[n\right]$

. Symbolically superposition(convolution) function is represented by

$y\left[n\right]=x\left[n\right]*h\left[n\right]$

.

In order to find the superposition sum $y\left[n\right]=x\left[n\right]*h\left[n\right]$

, where

$h\left[n\right]$

is a response of the impulse

$\delta \left[k–n\right]$

, we have to make the following actions:
1. Determine the function

$x\left[n\right]\delta \left[k–n\right]$

;
2. Determine the function

$y\left[n\right]=x\left[n\right]\delta \left[k–n\right]h\left[k\right]$

.

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