Digital Signal Processing Reference
In-Depth Information
12
FREQUENCY DOMAIN
REPRESENTATION
CHAPTER OUTLINE
12.1 DFT and IDFT Equations 85
12.1.1 First DFT Example 86
12.1.2 Second DFT Example
88
12.1.3 Third DFT Example 89
12.1.4 Fourth DFT Example 90
12.2 Fast Fourier Transform 93
12.3 Discrete Cosine Transform 97
In this chapter we are going to examine the DFT (Discrete
Fourier Transform) and its more popular cousin, the FFT (Fast
Fourier Transform). The FFT is used in many applications, and
we will see its uses in broadcast modulation and distribution
systems. It also lays the basis for the DCT (Discrete Cosine
Transform), commonly used in image compression, which is also
covered.
This chapter has more mathematics than most of the others in
this topic. It requires some familiarity with basic trigonometry,
complex numbers and the complex exponential. A refresher on
these topics is provided as an appendix for those who haven't
used these in a long time.
The DFT is simply a transform. It takes a sequence of sampled
data (a signal), and computes the frequency content of that
sampled data sequence. This will give the representation of the
signal in the frequency domain, as opposed to the familiar time
domain representation. This can be done in both the vertical and
horizontal dimensions, although for now we will assume a one-
dimensional signal. The result is a frequency-domain representa-
tion of the signal, providing the spectral content of a given signal.
All of the signal information is preserved, but in a different form.
The IDFT (Inverse Discrete Fourier Transform) computes the
time-domain representation of the signal from the frequency-
domain information. Using these transforms, it is possible to
