Example 1

Let's consider the simple system having p=1 p1 and q=0 q0.

To compute the output at some index, this difference equation says we need to know what the previous output yn−1 y n1 and what the input signal is at that moment of time. In more detail, let's compute this system's output to a unit-sample input: xn=δn xn δn . Because the input is zero for negative indices, we start by trying to compute the output at n=0 n0.

What is the value of y-1 y-1? Because we have used an input that is zero for all negative indices, it is reasonable to assume that the output is also zero. Certainly, the difference equation would not describe a linear system if the input that is zero for all time did not produce a zero output. With this assumption, y-1=0 y-1 0 , leaving y0=b y0 b . For n>0 n0, the input unit-sample is zero, which leaves us with the difference equation ∀n,n>0:yn=ayn−1 n n0 yn a y n1 . We can envision how the filter responds to this input by making a table.

Coefficient values determine how the output behaves. The parameter bb can be any value, and serves as a gain. The effect of the parameter aa is more complicated (Table 1). If it equals zero, the output simply equals the input times the gain bb. For all non-zero values of aa, the output lasts forever; such systems are said to be IIR ( Infinite Impulse Response). The reason for this terminology is that the unit sample also known as the impulse (especially in analog situations), and the system's response to the "impulse" lasts forever. If aa is positive and less than one, the output is a decaying exponential. When a=1a1, the output is a unit step. If aa is negative and greater than -11, the output oscillates while decaying exponentially. When a=1 a1, the output changes sign forever, alternating between bb and -bb. More dramatic effects when |a|>1 a1; whether positive or negative, the output signal becomes larger and larger, growing exponentially.

Positive values of aa are used in population models to describe how population size increases over time. Here, nn might correspond to generation. The difference equation says that the number in the next generation is some multiple of the previous one. If this multiple is less than one, the population becomes extinct; if greater than one, the population flourishes. The same difference equation also describes the effect of compound interest on deposits. Here, nn indexes the times at which compounding occurs (daily, monthly, etc.), aa equals the compound interest rate plus one, and b=1 b1 (the bank provides no gain). In signal processing applications, we typically require that the output remain bounded for any input. For our example, that means that we restrict |a|=1 a1 and chose values for it and the gain according to the application.

Example 2

A somewhat different system has no "a" coefficients. Consider the difference equation

Because this system's output depends only on current and previous input values, we need not be concerned with initial conditions. When the input is a unit-sample, the output equals 1q 1 q for n=0…q−1 n 0 … q 1 , then equals zero thereafter. Such systems are said to be FIR (Finite Impulse Response) because their unit sample responses have finite duration. Plotting this response (Figure 2) shows that the unit-sample response is a pulse of width qq and height 1q 1 q . This waveform is also known as a boxcar, hence the name boxcar filter given to this system. (We'll derive its frequency response and develop its filtering interpretation in the next section.) For now, note that the difference equation says that each output value equals the average of the input's current and previous values. Thus, the output equals the running average of input's previous qq values. Such a system could be used to produce the average weekly temperature ( q=7 q 7 ) that could be updated daily.