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: $x[n]=\sum _{k=-\infty }^{k=\infty } x[k\left]\delta \right[n–k]$. 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.

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 $x[n]=\sum _{k=-\infty }^{k=\infty } x[k\left]\delta \right[n–k]$, i.e. a linear combination of weighted shifted impulses.

If the input of the linear system is $x\left[n\right]$, then the output $y[n]:\mathrm{y[}n]=\sum _{k=-\infty }^{\infty } x[k\left]h_k\right[n]$. 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 $h_k[n]=h_0[n–k]$. Let’s assume that $h_0[n]=h[n]$. $h\left[n\right]$ is the output for the input $\delta \left[n\right]$ of the LTI system. So we have $y[n]=\sum _{k=-\infty }^{\infty } x[k\left]h\right[n–k]$. 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 $y\left[n\right]=x\left[n\right]*h\left[n\right]$.

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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