How does convolution work?

Published by Arun Isaac on

Tags: math, signals

The process of convolution works on the twin concepts of linearity and shift invariance. Here is an animated explanation of the concept.

The process of convolution works on the twin concepts of linearity and shift invariance.

To calculate the output of a system with impulse response \( h[n] = \{1,2,3\} \) when the input is \( x[n] = \{4,1,2\} \), we convolve x[n] and h[n] to get the output.

Mathematically,

\[ y[n] = x[n] \ast h[n] \]

Instead of considering the input signal as a whole, we break it up into a combination of impulses.

Therefore

\[ x[n] = 4\delta[n]+\delta[n-1]+2\delta[n-2] \]

So, we calculate the response for each sample (which is proportional to the impulse response), and add up all the responses.

Partial output for \( 4\delta[n] \) is \[ 4h[n] = \{4, 8, 12\} \] Partial output for \( \delta[n-1] \) is \[ h[n-1] = \{0, 1, 2, 3\} \] Partial output for \( 2\delta[n-2] \) is \[ 2h[n-2] = \{0, 0, 2, 4, 6\} \]

Summing up all the partial outputs, we get the total output,

\[ y[n] = 4h[n]+h[n-1]+2h[n-2] = \{4,9,16,7,6\} \]

Convolution - Animated

Figure 1: Convolution - Animated