5.11 Discrete-time systems in the time-domain  (Page 2/2)

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

To compute the output at some index, this difference equationsays we need to know what the previous output $y(n-1)$ and what the input signal is at that moment of time. In moredetail, let's compute this system's output to a unit-sample input: $x(n)=\delta (n)$ . Because the input is zero for negative indices, we start by trying to compute the output at $n=0$ .

Coefficient values determine how the output behaves. The parameter $b$ can be any value, and serves as a gain. The effect of the parameter $a$ is more complicated ( [link] ). If it equals zero, the output simply equals the input times the gain $b$ . For all non-zero values of $a$ , the output lasts forever; such systems are said to be IIR ( I nfinite I mpulse R esponse). 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 $a$ is positive and less than one, the output is a decaying exponential. When $a=1$ , the output is a unit step. If $a$ is negative and greater than $-1$ , the output oscillates while decaying exponentially. When $a=-1$ , the output changes sign forever, alternating between $b$ and $-b$ . More dramatic effects when $\left|a\right|> 1$ ; whether positive or negative, the output signal becomes largerand larger, growing exponentially.

Positive values of $a$ are used in population models to describe how population size increasesover time. Here, $n$ might correspond to generation. The difference equation says thatthe number in the next generation is some multiple of the previous one. If this multiple is less than one, thepopulation becomes extinct; if greater than one, the population flourishes. The same difference equation alsodescribes the effect of compound interest on deposits. Here, $n$ indexes the times at which compounding occurs (daily, monthly, etc.), $a$ equals the compound interest rate plus one, and $b=1$ (the bank provides no gain). In signal processingapplications, we typically require that the output remain bounded for any input. For our example, that means that we restrict $\left|a\right|< 1$ and choose values for it and the gain according to the application.