Difference Equation

The difference equation is a formula for computing an output sample at time $n$ based on past and present input samples and past output samples in the time domain.^6.1We may write the general, causal, LTI difference equation as follows:

$$y(n) = b_0 \,x(n) + b_1 \,x(n - 1) + \cdots + b_M \,x(n - M)$$

$$\qquad\quad\; - a_1 \,y(n - 1) - \cdots - a_N \,y(n - N)$$

$$= \sum_{i=0}^M b_i \,x(n-i) - \sum_{j=1}^N a_j \,y(n-j) \protect$$ (6.1)

where $x$ is the input signal, $y$ is the output signal, and the constants $b_i, i = 0, 1, 2, \ldots, M$, $a_i, i = 1, 2, \ldots, N$ are called the coefficients

As a specific example, the difference equation

$$y(n) = 0.01\, x(n) + 0.002\, x(n - 1) + 0.99\, y(n - 1)$$

specifies a digital filtering operation, and the coefficient sets $(0.01, 0.002)$ and $(0.99)$ fully characterize the filter. In this example, we have $M = N = 1$.

When the coefficients are real numbers, as in the above example, the filter is said to be real. Otherwise, it may be complex.

Notice that a filter of the form of Eq. (5.1) can use ``past'' output samples (such as $y(n-1)$) in the calculation of the ``present'' output $y(n)$. This use of past output samples is called feedback. Any filter having one or more feedback paths ($N>0$) is called recursive. (By the way, the minus signs for the feedback in Eq. (5.1) will be explained when we get to transfer functions in §6.1.)

More specifically, the $b_i$ coefficients are called the feedforward coefficients and the $a_i$ coefficients are called the feedback coefficients.

A filter is said to be recursive if and only if $a_i\neq 0$ for some $i>0$. Recursive filters are also called infinite-impulse-response (IIR) filters. When there is no feedback ( $a_i=0, \forall i>0$), the filter is said to be a nonrecursive or finite-impulse-response (FIR) digital filter.

When used for discrete-time physical modeling, the difference equation may be referred to as an explicit finite difference scheme.^6.2

Showing that a recursive filter is LTI (Chapter 4) is easy by considering its impulse-response representation (discussed in §5.6). For example, the recursive filter

$$\begin{eqnarray*} y(n) &=& x(n) + \frac{1}{2}y(n-1) \\ &=& x(n) + \frac{1}{2}x(n-1) + \frac{1}{4}x(n-2) + \frac{1}{8}x(n-3) + \cdots, \end{eqnarray*}$$

has impulse response $h(m) = 2^{-m}$, $m=0,1,2,\ldots\,$. Since the impulse response is the same no matter when the impulse occurs (time invariant), and since the impulse response values do not depend on the input amplitude (linear), the filter is LTI.