# Signals and Systems: Discrete LTI systems 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: . In the other words the discrete-time signal is the linear combination of shifted impulses δ[n-k] with the weight x[k]. This equation is called the shifting property of the discrete-time unit impulse. Figure 1. The mathematical representation of the discrete-time function x[n]. Figure 2. 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 x[n] is a linear time-invariant function, then the convolution sum y[n] 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 , i.e. a linear combination of weighted shifted impulses.

If the input of the linear system is x[n], then the output . Here the hk[n] are the responses to the signals . Representation of this formula is depicted in Figure 3.

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

So ${h}_{k}\left[n\right]={h}_{0}\left[n–k\right]$. Let’s assume that ${h}_{0}\left[n\right]=h\left[n\right]$. h[n] is the output for the input δ[n] of the LTI system. So we have . This equation is called superposition (convolution) sum of the sequences x[k] and h[n. Symbolically superposition(convolution) function is represented by y[n]=x[n]*h[n]. Figure 3. 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 y[n]=x[n]*h[n], where h[n] is a response of the impulse δ[k-n], we have to make the following action:
1. Determine the function x[n]δ[k-n];
2. Determine the function y[n]=x[n]δ[k-n]h[k].