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.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 . Let’s assume that . 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].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].