209
CHAPTER
11
Fourier Transform Pairs
For every
time domain
waveform there is a corresponding
frequency domain
waveform, and vice
versa.
For example, a rectangular pulse in the time domain coincides with a sinc function [i.e.,
sin(x)/x] in the frequency domain.
Duality provides that the reverse is also true; a rectangular
pulse in the frequency domain matches a sinc function in the time domain.
Waveforms that
correspond to each other in this manner are called
Fourier transform pairs
.
Several common
pairs are presented in this chapter.
Delta Function Pairs
For discrete signals, the delta function is a simple waveform, and has an
equally simple Fourier transform pair.
Figure 111a shows a delta function in
the time domain, with its frequency spectrum in (b) and (c).
The magnitude
is a constant value, while the phase is entirely zero.
As discussed in the last
chapter, this can be understood by using the expansion/compression property.
When the time domain is compressed until it becomes an impulse, the frequency
domain is expanded until it becomes a constant value.
In (d) and (g), the time domain waveform is shifted four and eight samples to
the right, respectively.
As expected from the properties in the last chapter,
shifting the time domain waveform does not affect the magnitude, but adds a
linear component to the phase.
The phase signals in this figure have not been
unwrapped
, and thus extend only from 
B
to
B
.
Also notice that the horizontal
axes in the frequency domain run from 0.5 to 0.5.
That is, they show the
negative
frequencies in the spectrum, as well as the
positive
ones.
The
negative frequencies are redundant information, but they are often included in
DSP graphs and you should become accustomed to seeing them.
Figure 112 presents the same information as Fig. 111, but with the
frequency domain in
rectangular form
.
There are two lessons to be learned
here.
First, compare the polar and rectangular representations of the
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d.
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a.
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g.
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Frequency Domain
Time Domain
Amplitude
Phase (radians)
FIGURE 111
Delta function pairs in
polar form
.
An impulse in the time domain corresponds to a
constant magnitude and a linear phase in the frequency domain.
frequency domains.
As is usually the case, the polar form is much easier to
understand; the magnitude is nothing more than a constant, while the phase is
a straight line.
In comparison, the real and imaginary parts are sinusoidal
oscillations that are difficult to attach a meaning to.
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 Spring '10
 SAGHRI
 Digital Signal Processing, Frequency, Signal Processing, DFT

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