文档介绍：INTRODUCTION
Discrete-time Fourier transform and inverse Fourier transform.
Similarities and differences between continuous-time and discrete-time Fourier transforms.
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1. Representation of Aperiodic Signals: The Discrete-Time Fourier Transform
As ,
…
…
–N –N1 0 N2 N
n
–N1 0 N2
x[n]
n
1) Development of the Discrete-Time Fourier Transform
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Defining a function:
Thus,
Consequently,
As ,
discrete-time Fourier transform
discrete-time inverse Fourier transform
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Differences between the continuous-time and discrete-time Fourier transform are: periodicity of the discrete-time transform and the finite interval of integration in the synthesis equation.
In discrete time,
Low frequencies are the values of ω near even multiple of π;
high frequencies are those values of ω near odd multiples of π.
–2π–π 0 π 2π
ω
–2π–π 0 π 2π
ω
0
n
x1[n]
0
n
x2[n]
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Example Consider the signal
1/(1–α)
1/(1+α)
–2π–π 0 π 2πω
–2π–π 0 π 2πω
1 >α> 0
0 >α> -1
1/(1+α)
1/(1–α)
–2π–π 0 π 2πω
–2π–π 0 π 2πω
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Example Consider the signal
0
x[n]
n
(1+α)/(1–α)
(1–α)/(1+α)
–2π 0 2π
ω
for 0 < α< 1,
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Example Consider the rectangular pulse
–N10 N1
x[n]
1
n
5
–2π–π 0 π 2π
ω
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2) Convergence Issues of the Discrete-Time Fourier Transform
If x[n] is an infinite duration signal, we must consider the question of convergence of the infinite summation in the analysis equation.
The analysis equation will converge if x[n] is absolutely summable; that is
In contrast to the situation for the analysis equation, there are generally no convergence issues associated with the synthesis equation.
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Approximation to the unit sample obtained as in eq.() plex exponentials with frequencies |ω| ≤ W: (a) W = π/4; (b) W = 3π/8; (c) W = π/2; (d) W = 3π/4; (e) W = 7π/8; (f) W = π. Note that for W = π,
.
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2. The Fourier Transform For Periodic Signals
First consider the Fourier transform of the sequence
To check the validity of this expression,
2π