5.12 Discrete-time systems in the frequency domain

As with analog linear systems, we need to find the frequency response of discrete-time systems. We used impedances to derivedirectly from the circuit's structure the frequency response. The only structure we have so far for a discrete-timesystem is the difference equation. We proceed as when we used impedances: let the input be a complex exponential signal. Whenwe have a linear, shift-invariant system, the output should also be a complex exponential of the same frequency, changed inamplitude and phase. These amplitude and phase changes comprise the frequency response we seek. The complex exponential inputsignal is $x(n)=Xe^{i\times 2\pi fn}$ . Note that this input occurs for all values of $n$ . No need to worry about initial conditions here. Assume the output has a similar form: $y(n)=Ye^{i\times 2\pi fn}$ . Plugging these signals into the fundamental difference equation , we have

These examples illustrate the point that systems described (and implemented) by difference equations serve as filters fordiscrete-time signals. The filter's order is given by the number $p$ of denominator coefficients in the transfer function (if the system is IIR) or by the number $q$ of numerator coefficients if the filter is FIR. When a system's transfer function has both terms,the system is usually IIR, and its order equals $p$ regardless of $q$ . By selecting the coefficients and filter type, filters having virtually any frequency responsedesired can be designed. This design flexibility can't be found in analog systems. In the next section, we detail how analogsignals can be filtered by computers, offering a much greater range of filtering possibilities than is possible with circuits.