Signals and SystemsYear 2

Signals and Systems: Discrete LTI systems

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: x[n]=k=-∞k=∞ x[k]δ[nk]. 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 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.
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 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 x[n], then the output y[n]:y[n]=k=-∞ x[k]hk[n]. 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 hk[n]=h0[nk]. Let’s assume that h0[n]=h[n]. h[n] is the output for the input δ[n] of the LTI system. So we have y[n]=k=-∞ x[k]h[nk]. 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.
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].

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