Convolution Sum in LTI systems

Mental Model

You’ve got a linear system (that given an input signal, spits out an output one), but u don’t have an explicit formula to compute the output from the input, the system is like a black box to u. How do u then get an explicit input/output formula?

The output is the only thing that can help u observe how the system works. For every point in time ($k$), feed a unit impulse into the system and see how it reacts to it.

Note

The reaction at time $k$ is a signal that shows what output the unit impulse triggered across the whole time-domain.

It’s known as the system’s unit impulse response at time $k$, and is denoted by $h_k[n]$.

This info. is useful because an actual arbitrary input signal will also be a collection of unit impulses scaled by the input at each time ($k$).

For a general linear system, you’ll need to do this for every single point in time ($k$), to rightfully claim that you know how the system reacts across its whole lifespan. However, if the system is also time-invariant, you’ll only need to do this ONCE, since all the impulse responses are basically identical.

Once u acquire the impulse response knowledge, u can compute the system’s output to any arbitrary input signal.

To get the overall system output of an input signal $x[n]$,

How it works

1. Iterate through every sample of the input $x$ at time $k$.
   1. grab the system’s unit impulse response at that time $h_k[n]$. (The reaction signal triggered by the input impulse at time $k$) = $x[k]\cdot h_k[n]$
   2. Add this reaction to the overall output $y[n]$.

Here’s a GIF illustrating how this works as an input signal gets processed. (Note: In this example, I used the same exponential decay response for every time $k$, implying an LTI system).

Formal Definition

• (the System is linear)

  $y[n]={\sum }_{k=-\infty }^{+\infty }x[k]\cdot h_k[n]$

  • (the System is also time-invariant)

    $h_k[n]=h[n-k]$

TL;DR

An LTI system’s total output is its reaction to each individual input sample, added together.

This output is obtained by computing the convolution sum between its impulse response and input signal.

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