[9.8 SUMMARY](#page-14-0)

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9.8 SUMMARY

This chapter deals with the analysis and processing of discrete-time signals. For analysis, our approach is parallel to that used in continuous-time signals. We first represent a periodic x[n] as a Fourier series formed by a discrete-time exponential and its harmonics. Later we extend this representation to an aperiodic signal x[n] by considering x[n] to be a limiting case of a periodic signal with period approaching infinity.

Periodic signals are represented by discrete-time Fourier series (DTFS); aperiodic signals are represented by the discrete-time Fourier integral. The development, although similar to that of continuous-time signals, also reveals some significant differences. The basic difference in the two cases arises because a continuous-time exponential ejω*^t* has a unique waveform for every value of ω in the range −∞ to ∞. In contrast, a discrete-time exponential ejⁿ has a unique waveform only for values of in a continuous interval of 2π. Therefore, if ⁰ is the fundamental frequency, then at most 2π/⁰ exponentials in the Fourier series are independent. Consequently, the discrete-time exponential Fourier series has only N⁰ = 2π/⁰ terms.

The discrete-time Fourier transform (DTFT) of an aperiodic signal is a continuous function of and is periodic with period 2π. We can synthesize x[n] from spectral components of X() in any band of width 2π. In a basic sense, the DTFT has a finite spectral width of 2π, which makes it bandlimited to π radians.

Linear, time-invariant, discrete-time (LTID) systems can be analyzed by means of the DTFT if the input signals are DTF-transformable and if the system is stable. Analysis of unstable (or marginally stable) systems and/or exponentially growing inputs can be handled by the z-transform, which is a generalized DTFT. The relationship of the DTFT to the z-transform is similar to that of the Fourier transform to the Laplace transform. Whereas the z-transform is superior to the DTFT for analysis of LTID systems, the DTFT is preferable in signal analysis.

If H() is the DTFT of the system’s impulse response h[n], then |H()| is the amplitude response, and H() is the phase response of the system. Moreover, if X() and Y() are the DTFTs of the input x[n] and the corresponding output y[n], then Y() = H()X(). Therefore, the output spectrum is the product of the input spectrum and the system’s frequency response.

Because of the similarity between the DFT and DTFT relationships, numerical computations of the DTFT of finite-length signals can be handled by using the DFT and the FFT, introduced in Secs. 8.5 and 8.6. For signals of infinite length, we use a window of suitable length to truncate the signal so that the final results are within a given error tolerance.