Digital Signal Processing Reference
In-Depth Information
3
Time and Frequency Analysis
of Discrete-T
ime Signals
3.1
Brief Theory of Discrete-Time Fourier Transform,
Discrete Fourier Transform, and Fast Fourier Transform
In the previous chapter, the
Z
-transform was shown to be an effective tool
in linking the time and frequency domains of a discrete-time signal
x
).
However, in order to specify practical properties of discrete-time systems,
such as low-pass filtering or high-pass filtering, it is necessary to transform
the complex
z
(
n
-plane to the real-frequency,
ω
, axis. Specifically, the region of
the complex
z
-plane that is used in this transformation is the unit circle,
specified by the region
z = e
j
. The resulting transform is the
Discrete-Time
ω
FourierTransform
), which will be discussed first in this chapter.
Due to the need for a more applicable and easily computable transform,
the
Discrete Fourier Transform
(
DTFT
was introduced, which is very homoge-
neous in both forward (time to frequency) and inverse (frequency to time)
formulations. The crowning moment in the evolution of DSP came when the
Fast Fourier Transform
(
DFT
)
(
FFT
) was discovered by Cooley and Tukey in 1965.
1,2
The FFT, which is essentially a very fast algorithm to compute the DFT,
makes it possible to achieve real-time audio and video processing.
3.1.1
Discrete-Time Fourier Transform
The frequency response of the system is very important in defining the
practical property of the system, such as low-pass or high-pass filtering. It
can be obtained by considering the system function
H
(
z
) on the unit circle,
as discussed in
Chapter 2
. Similarly, for a discrete-time sequence
x
(
n
), we
can define the
Z
-transform
X
(
z
) on the unit circle as follows:
∞
∑
j
ω
−
jn
ω
Xe
() ()
=
Xz
=
xne
()
(3.1)
j
ω
ze
=
n
=−∞
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