数字信号处理双语版ppt第四章

DT FT [ x1 (n) x2 (n)] X1 () X 2 ()
2. Time Shifting
DTFT[ x(n n0 )] e
3. Frequency Shifting
j n0
X ()
DTFT[ e j0n x(n)] X ( 0 )
θ () arg[ X ()] called thephasefunctionor spectrum
Understanding DSP, Second Edition
(3) Relations between the Rectangular and Polar Forms
X r ( ) | X () | cos (ω)
1.5 DFT Leakage
1.6 Windows 1.7 DFT Resolution, Zero Padding, and FrequencyDomain Sampling
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4.1 DTFT
In the literature of DSP you’ll encounter the topics of continuous Fourier transform, Fourier series, discrete-time Fourier transform, discrete Fourier transform, and periodic spectra.
Figure 4-1-2
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Example : The DTFT of a sequence is given by
X ( e j )
2πδ (ω ω0 2πk ) , π ω0 π
k

The function δ(ω ω0 2πk ) train. Determine its IDTFT. Solution:
(5) Principal-Value Phase Function
π θ(ω) π
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It is a geometric series and can be evaluated as:
Understanding DSP, Second Edition
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Properties of the Discrete-Time Fourier Transform
4. Time Reversal
DT FT [ x(n)] X ()
Furthermore, if sequence is real, then
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四种付里叶变换形式的归纳
时间函数 连续和周期（T） 连续和非周期 离散（T）和非周期 频谱函数 非周期和离散 非周期和连续
2 ）和连续 周期（ T
周期（ 2 T ）和离散
离散（T）和周期
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DT FT [ x(n)] X * ()
5. Differentiation in Frequency
dX () DT FT [ nx(n)] j dω
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Properties of the Discrete-Time Fourier Transform
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3
Figure 4-1-1 (a) Infinite-length continuous time signal with transforms
X () x(t )e jt dt


1 x(t ) 2
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n
e j0n e jn
2πδ(ω ω0 2πk )
k

ω0 0, then j n DTFT[1] e 2πδ (ω 2πk )
n k
Example: Find DTFTs for the sequences sin(ω0n) , respectively. Solution:
DTFT: IDTFT:
x(n)
X ( )e 2
1

j n
d
(1) Rectangular Form
(2) PHale Waihona Puke Baidular Form
X () X r () jX i ()
X () | X () | e j ( )
| X () | called themagnitudefunctionor spectrum
X i () | X () | sin (ω)
| X () | [ X r2 () X i2 ()]1/2 X i (ω) (ω) arctan[ ] X r (ω) (4) Periodicity
X ( 2k )
n j ( ω 2 πk ) n x ( n ) e n jn x ( n ) e X ( )
x(n) 1 N
m 0
m n X ( m ) W , 0 n N 1 N
N 1
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Eq 4-2
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Although it looks more complicated than Eq. (4–1), Eq. (4–2) turns out to be easier to understand. The N separate DFT analysis frequencies are：
Eq 4-3
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Quite often we’re interested in both the magnitude and the power (magnitude squared) contained in each X(m) term, and the standard definitions for right triangles apply here as depicted in the following Figure.
With the advent of the digital computer, the efforts of early digital processing pioneers led to the development of the DFT defined as the discrete frequency-domain sequence X(m), where: DFT:
Figure 4-2-1
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If we represent an arbitrary DFT output value, X(m), by its real and imaginary parts:

and
DTFT[sin( 0 n)] j [ ( 0 2k ) ( 0 2k )]
k

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Properties of the Discrete-Time Fourier Transform 1. Linearity
Properties of the Discrete-Time Fourier Transform
8. Time-Domain Convolution Theorem and if
y ( n) x ( n) h( n) x ( m) h( n m)
m

then
Y ( ) X ( ) H ( )
10
Xo(ω) is continuous and periodic with a period of 2π, whose magnitude shown in the following Figure. This is an example of a sampled (or discrete) timedomain sequence having a periodic spectrum.
where we have used the sampling property of
.) (ω
Understanding DSP, Second Edition
Thus we get following results:
X (e j ) DTFT[ e j0n ]
And if
cos(ω0n)
and
DTFT[cos( 0 n)] 1 {DTFT[ e j0n ] DTFT[ e j0n ]} 2 [ ( 0 2k ) ( 0 2k )]
k
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is called the periodic impulse
j )e jn d x(n) 1 - X ( e 2 1 jn dω [ 2 ( ω ω 2 πk ) ] e 0 2 - k j n dω e j0 n , n - ( ω ω ) e 0



X ()e jt d
4
Figure 4-1-1 (b) Infinite-length signal of periodic pulses with transforms
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Figure 4-1-1 (c)(d) Infinite-length discrete time sequence and discrete periodic samples with transforms
6. Complex Conjugation
DT FT[ x* (n)] X * ()
DT FT[ x* (n)] X * ()
7. Parseval’s Theorem
1 | x ( n ) | 2 n
2




| X ( ) |2 d
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Chapter 4 The Discrete Fourier Transform
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Outline
1.1 DTFT 1.2 Understanding the DFT Equation 1.3 DFT Properties 1.4 Inverse DFT
9. Frequency-Domain Convolution Theorem
y(n) x(n)w(n)
Y () 1 2
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X ()W ( )d


4.2 Understanding the DFT Equation
Eq 4-1
IDFT:
1 j 2nm / N x(n) X (m)e N m1
19
N 1
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Let
WN e j (2π / N ) ,
N 1 n 0
Then
mn X (m) x(n)WN , 0 m N 1