Signals and Systems

# 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: . 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. The mathematical representation of the discrete-time function x[n]. 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 $x\left[n\right]$ 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 , i.e. a linear combination of weighted shifted impulses.

If the input of the linear system is $x\left[n\right]$, then the output . 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 ${h}_{k}\left[n\right]={h}_{0}\left[n–k\right]$. Let’s assume that ${h}_{0}\left[n\right]=h\left[n\right]$$h\left[n\right]$ is the output for the input $\delta \left[n\right]$ of the LTI system. So we have . 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 $y\left[n\right]=x\left[n\right]*h\left[n\right]$. 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 $y\left[n\right]=x\left[n\right]*h\left[n\right]$, 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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