Impulse Responses and Difference Equations

Convolution and Difference Equations

We previously demonstrated that the output of any linear, time-invariant system can be expressed by a convolution of an input and system impulse response. Here, we show that convolutions can be expressed with differences equations. Hence, any linear, time-invariant system can be expressed by difference equations.

The Difference Equation Form of Convolution

If we use the commutativity property ( $x[n] * h[n] = h[n] * x[n]$ ), we see that the convolution can be written as

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

Hence, we can write this as:

$y[n] = \cdots + h[-2] x[n+2] + h[-1] x[n+1] + h[0] x[n-0] + h[1] x[n-1] + h[2] x[n-2] + \cdots$

Since $\ldots,h[-2],h[-1],h[0],\ldots$ are all constant values, the expression above is a difference equation.

Impulse Response Properties

Using our new definition, we can we can derive relationship between our impulse response $h[n]$ and system properties.

Memoryless

An LTI system with impulse response $h[n]$ is memoryless if

$h[n] = a \, \delta[n]$

for any scalar constant $a$ .

Causal / Anti-Causal

An LTI system with impulse response $h[n]$ is causal if

$h[n] = 0 \qquad \textrm{for } n < 0$

An LTI system with impulse response $h[n]$ is anti-causal if

$h[n] = 0 \qquad \textrm{for } n \geq 0$

A causal system must be dependent on only past and present values. As a difference equation, this can be written as

$y[n] = h[0] x[n-0] + h[1] x[n-1] + h[2] x[n-2] + \cdots$

Hence, this is equivalent to stating $h[n] = 0$ for all $n < 0$ :

$y[n] = \cdots + (0) x[n+2] + (0) x[n+1] + h[0] x[n-0] + h[1] x[n-1] + h[2] x[n-2] + \cdots$

BIBO Stable

An LTI system with impulse response $h[n]$ is BIBO stable if

$\sum_{n=-\infty}^{\infty} |h[n]| < \infty$

Finite / Infinite Impulse Responses

FIR Filters

If the impulse response $h[n]$ is finite in length (i.e., the number of non-zero values is finite), then the number of terms in the difference equation is finite. We refer to these types of linear, time-invariant systems as finite impulse response filters.

IIR Filters

If the impulse response $h[n]$ is infinite in length, the linear, time-invariant systems is a infinite impulse response filter.

Relative to the difference equation, the impulse response $h[n]$ may be infinite to two reasons.

The number of difference equation terms is infinite.

The difference equation has recursion (i.e., $y[n]$ is a function of $y[n-N]$ for any $N$ )

When we have recursive terms, the difference equation can be written as

$\cdots + g[-2] y[n+2] + g[-1] y[n+1] + g[0] y[n-0] + g[1] y[n-1] + g[2] y[n-2] + \cdots \ = \ \cdots + h[-2] x[n+2] + h[-1] x[n+1] + h[0] x[n-0] + h[1] x[n-1] + h[2] x[n-2] + \cdots$