What is convolution; convolution sum

Convolution

Convolution is the operation that gives the output of an LTI system from its input and its impulse response. For discrete time, the output is a weighted sum of the present and past (and possibly future) inputs, with the impulse response values as weights:

$$y[n] = x[n] * h[n]$$

Here $h[n]$ is the impulse response, the output when the input is the unit impulse $\delta[n]$.

Expression for the convolution sum

Step 1: Represent the input with impulses. Using the sifting property, any sequence is a sum of scaled, shifted impulses:

$$x[n] = \sum_{k=-\infty}^{\infty} x[k]\,\delta[n-k]$$

e.g. $\{\underset{\uparrow}{3}, 1\} = 3\delta[n] + \delta[n-1]$.

Step 2: Response to one impulse. By definition $\delta[n] \rightarrow h[n]$.

Step 3: Time invariance. $\delta[n-k] \rightarrow h[n-k]$.

Step 4: Homogeneity. $x[k]\,\delta[n-k] \rightarrow x[k]\,h[n-k]$ ($x[k]$ is a constant for each $k$).

Step 5: Additivity. The output for the sum of all these inputs is the sum of their outputs:

$$y[n] = \sum_{k=-\infty}^{\infty} x[k]\,h[n-k]$$

This is the convolution sum. With $m = n - k$ it can also be written $y[n] = \sum_{k} h[k]\,x[n-k]$ (commutative property).

For a causal system with a causal input: $y[n] = \sum_{k=0}^{n} x[k]\,h[n-k]$.

Steps to evaluate: fold $h[k] \rightarrow h[-k]$; shift to $h[n-k]$; multiply with $x[k]$; add all products; repeat for each $n$. If the lengths are $N_1$ and $N_2$, the output length is $N_1 + N_2 - 1$, starting at (start of $x$) + (start of $h$).

Example: $x[n] = \{\underset{\uparrow}{1}, 1\}$, $h[n] = \{\underset{\uparrow}{2}, 3\}$:

• $y[0] = 1\cdot2 = 2$
• $y[1] = 1\cdot3 + 1\cdot2 = 5$
• $y[2] = 1\cdot3 = 3$

So $y[n] = \{\underset{\uparrow}{2}, 5, 3\}$.