As with analog linear systems, we need to find the frequency response of discrete-time systems. We used impedances to derive directly from the circuit's structure the frequency response. The only structure we have so far for a discrete-time system is the difference equation. We proceed as when we used impedances: let the input be a complex exponential signal. When we have a linear, shift-invariant system, the output should also be a complex exponential of the same frequency, changed in amplitude and phase. These amplitude and phase changes comprise the frequency response we seek. The complex exponential input signal is x (n)= Xej2πfn. 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)= Yej2πfn. Plugging these signals into the fundamental difference equation (5.42), we have
The assumed output does indeed satisfy the diference equation if the output complex amplitude is related to the input amplitude by
This relationship corresponds to the system's frequency response or, by another name, its transfer function. We find that any discrete-time system defined by a diference equation has a transfer function given by
Furthermore, because any discrete-time signal can be expressed as a superposition of complex exponential signals and because linear discrete-time systems obey the Superposition Principle, the transfer function relates the discrete-time Fourier transform of the system's output to the input's Fourier transform.
The frequency response of the simple IIR system (difference equation given in a previous example (Discrete-Time Systems in the Time-Domain) is given by
This Fourier transform occurred in a previous example; the exponential signal spectrum (Figure 5.10: Spectra of exponential signals) portrays the magnitude and phase of this transfer function. When the filter coefficient a is positive, we have a lowpass filter; negative a results in a highpass filter. The larger the coefficient in magnitude, the more pronounced the lowpass or highpass filtering.
The length-q boxcar filter (difference equation found in a previous example (Discrete-Time Systems in the Time-Domain) has the frequency response
This expression amounts to the Fourier transform of the boxcar signal (5.13). There we found that this frequency response has a magnitude equal to the absolute value of dsinc(πf); see the length-10 filter's frequency response (Figure 5.11: Spectrum of length-ten pulse). We see that boxcar filters length-q signal averages have a lowpass behavior, having a cutoff frequency of .
Suppose we multiply the boxcar filter's coefficients by a sinusoid: Use Fourier transform properties to determine the transfer function. How would you characterize this system: Does it act like a filter? If so, what kind of filter and how do you control its characteristics with the filter's coefficients?
These examples illustrate the point that systems described (and implemented) by difference equations serve as filters for discrete-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 response desired can be designed. This design flexibility can't be found in analog systems. In the next section, we detail how analog signals can be filtered by computers, offering a much greater range of filtering possibilities than is possible with circuits.