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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:

$\displaystyle y(n)$ $\displaystyle =$ $\displaystyle b_0 \,x(n) + b_1 \,x(n - 1) + \cdots + b_M \,x(n - M)$  
    $\displaystyle \qquad\quad\;
- a_1 \,y(n - 1) - \cdots - a_N \,y(n - N)$  
  $\displaystyle =$ $\displaystyle \sum_{i=0}^M b_i \,x(n-i) - \sum_{j=1}^N a_j \,y(n-j)
\protect$ (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

$\displaystyle 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

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,

has impulse response $ h(m) = 2^{-m}$, $ m=0,1,2,\ldots\,$. It is now straightforward to apply the analysis of the previous chapter to find that time-invariance, superposition, and the scaling property hold.

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