Digital Signal Processing Reference
In-Depth Information
6
APPLICATIONS OF THE FOURIER
TRANSFORM
In this chapter, several engineering applications of the Fourier transform are con-
sidered. The mathematical basis and several properties of the Fourier transform are
presented in Chapter 5. We present additional examples of how the Fourier trans-
form, and the frequency domain in general, can be used to facilitate the analysis and
design of signals and systems.
6.1
IDEAL FILTERS
The concept of the transfer function, which is one of the ways the Fourier transform
is applied to the analysis of systems, is discussed in Chapter 5:
V
2
(v)
V
1
(v)
.
[eq(5.46)]
H(v) =
V
1
(v)
V
2
(v)
Here, is the Fourier transform of the input signal to a system and is the
Fourier transform of the output signal. Consideration of this concept leads us to the
idea of developing transfer functions for special purposes. Filtering is one of those spe-
cial purposes that is often applied in electronic signal processing. Figure 6.1 shows the
frequency-response characteristics of the four basic types of filters:
the ideal low-pass
filter,
the
ideal high-pass filter,
the
ideal bandpass filter,
and the
ideal bandstop filter
.
These
ideal
filters have transfer functions such that the frequency components
of the input signal that fall within the
passband
are passed to the output without
modification, whereas the frequency components of the input signal that fall into
the
stopband
are completely eliminated from the output signal.
Consider the frequency response shown in Figure 6.1(a). This is the magni-
tude frequency spectrum of an
ideal low-pass filter
. As can be seen, this filter has a
unity magnitude frequency response for frequency components such that
and zero frequency response for
ƒvƒ F v
c
ƒvƒ 7 v
c
.
ƒvƒ F v
c
The range of frequencies
is
ƒvƒ 7 v
c
called the
passband
of the filter, and the range of frequencies
is called the
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