Discrete-time Fourier series and transform - Continuous and Discrete Time Signals and Systems - page 464

CHAPTER

11

Discrete-time Fourier series
and transform

In Chapter 10, we developed analysis techniques for LTID systems based on the
convolution sum by representing the input sequence x [ k ] as a linear combination
of time-shifted unit impulse functions. In this chapter, we introduce frequency-
domain representations for DT sequences and LTID systems based on weighted
superpositions of complex exponential functions. For periodic sequences, the
resulting representation is referred to as the discrete-time Fourier series (DTFS),
while for aperiodic sequences the representation is called the discrete-time
Fourier transform (DTFT). We exploit the properties of the discrete-time Fourier
series and Fourier transform to develop alternative techniques for analyzing DT
sequences. The derivations of these results closely parallel the development of
the CT Fourier series (CTFS) and CT Fourier transform (CTFT) as presented
in Chapters 4 and 5.
The organization of this chapter is as follows. In Section 11.1, we intro-
duce the exponential form of the DTFS and illustrate the procedure used to
calculate the DTFS coefficients through a series of examples. The DTFT pro-
vides frequency representations for aperiodic sequences and is presented in
Section 11.2. Section 11.3 defines the condition for the existence of the DTFT,
and Section 11.4 extends the scope of the DTFT to represent periodic sequences.
Section 11.5 lists the properties of the DTFT and DTFS, including the time-
convolution property, which states that the convolution of two DT sequences
in the time domain is equivalent to the multiplication of the DTFTs of the
two sequences in the frequency domain. The convolution property provides
us with an alternative technique to compute the output response of the LTID
system. The DTFT of the impulse response is referred to as the transfer func-
tion, which is covered in Section 11.6. Section 11.7 defines the magnitude and
phase spectra for LTID systems, and Section 11.8 relates the CTFT and DTFT
of periodic and aperiodic waveforms to each other. Finally, the chapter is con-
cluded in Section 11.9 with a summary of important concepts covered in the
chapter.

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