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]=k=-∞k=∞ x[k]δ[nk]. In the other words the discrete-time signal is the linear combination of shifted impulses δnk with the weight xk. This equation is called the shifting property of the discrete-time unit impulse.

The mathematical representation of the discrete-time function x[n].
The mathematical representation of the discrete-time function x[n].
What is discrete LTI system?
The mathematical representation of the function x[k]δ[n-k]. (a) for k=0, (b) for k=1, (c) for k=2, (d) for k=3.
If the xn is a linear time-invariant function, then the convolution sum yn 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]=k=-∞k=∞ x[k]δ[nk], i.e. a linear combination of weighted shifted impulses.

If the input of the linear system is xn, then the output y[n]:y[n]=k=-∞ x[k]hk[n]. Here the hkn are the responses to the signals .

Generally speaking, the functions hkn are not related to each other for each  particular k. In our case  is a response of impulse function, then hkn is a linear shifted version of itself.

So hk[n]=h0[nk]. Let’s assume that h0[n]=h[n]hn is the output for the input δn of the LTI system. So we have y[n]=k=-∞ x[k]h[nk]. This equation is called superposition (convolution) sum of the sequences xk and hn. Symbolically superposition(convolution) function is represented by yn=xn*hn.

What is discrete LTI system?
Mathematical representation of the h[k] response of δ[n-k] impulse function (a), the mathematical representation of y[n] response of x[n], (b) for n=1, (c) for n=2, (d) for n=-1.
In order to find the superposition sum yn=xn*hn, where hn is a response of the impulse δkn, we have to make the following actions:
1. Determine the function xnδkn;
2. Determine the function  yn=xnδknhk.

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