This chapter treats frequency-domain analysis by developing the discrete-time Fourier transform (DTFT) and the spectral representation of discrete-time signals. The development, properties, and applications of the DTFT closely parallel those for the continuous-time Fourier transform. In fact, the DTFT and the CTFT are closely related, a fact that we can use to our advantage in many applications. Discrete-time Fourier analysis provides an important tool for the study of discrete-time signals and systems. A frequency-domain perspective is particularly useful to better understand the digital processing of analog signals and digital resampling techniques. In preparation for the next chapter, this chapter concludes by generalizing the DTFT to the z-transform.
The Discrete-Time Fourier Transform
The continuous-time Fourier transform (CTFT), sometimes simply called the Fourier transform (FT), is a tool to represent an aperiodic continuous-time signal in terms of its frequency components, thereby providing a spectral representation of the signal. We now develop a similar tool, the discrete-time Fourier transform (DTFT), to represent an aperiodic discrete-time signal in terms of its frequency components, which also leads to a spectral representation of the signal. The principal difference between the two types of transforms is that the former represents signals with continuous-time sinusoids or exponentials, while the latter uses discrete-time sinusoids or exponentials.
One possible way to develop the continuous-time Fourier transform is to start with the Fourier series representation of periodic signals and then, letting the period go to infinity, extend the results to aperiodic signals.
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