DFT和FFT处理
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2 For exam N = 64, 64log2(64) = 384 ple, 642 = 4096 N2 → N log2 ( N) or log2 ( N)
• Using redundancies in the DFT calculation.
Spectrum analysis:频谱分析
• • • • • • Radix-2:基2 基 Decimation:抽取 抽取 Zero padding:补零 补零 Decimation-in-time:时间抽取 时间抽取 Twiddle factor:旋转因子 旋转因子 Butterfly:蝶形运算 蝶形运算
FIGURE 11-5 Signal for Example 11.2.
Find the magnitude spectrum using both the DTFT and the DFT for the signal shown in Figure11.5.
FIGURE 11-6
DTFT magnitude spectrum for Example 11.2.
11.1 Definition of DFT
Discrete Fourier transform (DFT) X [k ] = ∑ x[n]e
n=0 N −1 − j 2π k n N
for k = 0,1,⋯ , N − 1
Inverse DFT 1 x[n] = N
∑ X [k ]e
k =0
N −1
FIGURE 11-39
One Stage of FFT.
Joyce Van de Vegte Fundamentals of Digital Signal Processing
Copyright ©2002 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.
j 2π
k n N
for n = 0,1,⋯ , N − 1
0 ≤ n ≤ N − 1 DFT window 0 ≤ k ≤ N − 1 X [k ] = X [k ] e
jθ [k ]
X [k ] θ [k ]
DFT magnitude spectrum DFT phase spectrum
X (Ω ) =
n = −∞
x[n]e − jnΩ = 2 − e − jΩ + 3e − j 2Ω + 3e − j 3 Ω ∑
∞
FIGURE 11-7
DFT magnitude spectrum for Example 11.2.
X [k ] = ∑ x[n]e
n=0
N −1
−j
2π nk N
FIGURE 11-41
Regrouping into even and odd sequences.
FIGURE 11-42 Comparing DFT and FFT efficiency.
in detaiages of 8-point FFT.
Joyce Van de Vegte Fundamentals of Digital Signal Processing
Copyright ©2002 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.
• periodicity
X (Ω + 2π ) = X (Ω )
FIGURE 7-2 Signal for Example 7.1.
X (Ω ) = ∑ x[n]e − jnΩ
−∞
∞
= 2 − e − jΩ + 3e − j 2 Ω + e − j 4 Ω
Joyce Van de Vegte Fundamentals of Digital Signal Processing
j 2π N −1 k n N N −1 − j 2π k n N
n is the sample number, index k runs from 0 to N - 1 ck are Fourier coefficients
FIGURE 8-13 Digital signal for Example 8.4.
Copyright ©2002 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.
FIGURE 8-11 Comparing DTFTs and DFSs for digital signals.
DFT和FFT处理
• DTFT:Discrete Time Fourier Transform
X (Ω ) =
n=∞
X (Ω ) = F {x[n]}
n = −∞
x[n]e − jnΩ ∑
Properties of DTFT
• Time delay
F {x[n − n0 ]} = e − jn0Ω X (Ω )
Copyright ©2002 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.
• DFS:Discrete Fourier Series(离散傅里叶 离散傅里叶 级数) 级数
1 x[n] = ∑ ck e c k = ∑ x[n]e N k =0 n= 0 N is the period of x[n];
= 2−e
− j 2π
k 4
+ 3e
− j 2π
k 2 4
+ 3e
− j 2π
k 3 4
FIGURE 11-8
DTFT and DFT results superimposed for Example 11.2.
Fast Fourier Transform and why?
• The Fast Fourier Transform, developed originally by Cooley and Tukey in 1965, in a significantly less computationally method of evaluating the DFT, and thus particularly attractive for ‘realtime’ spectral analysis DSP technology. • The FFT reduces the number of complex multiplication. N
the period N = 8
ck = ∑ x[n]e
n=0
N −1
− j 2π
k n N
= ∑ x[n]e
7 n=0
− j 2π
k n 8
= 1+ e
−j
πk
4
+e
−j
πk
2
+e
−j
3πk 4
(k from 0 to 7)
Joyce Van de Vegte Fundamentals of Digital Signal Processing
• Using redundancies in the DFT calculation.
Spectrum analysis:频谱分析
• • • • • • Radix-2:基2 基 Decimation:抽取 抽取 Zero padding:补零 补零 Decimation-in-time:时间抽取 时间抽取 Twiddle factor:旋转因子 旋转因子 Butterfly:蝶形运算 蝶形运算
FIGURE 11-5 Signal for Example 11.2.
Find the magnitude spectrum using both the DTFT and the DFT for the signal shown in Figure11.5.
FIGURE 11-6
DTFT magnitude spectrum for Example 11.2.
11.1 Definition of DFT
Discrete Fourier transform (DFT) X [k ] = ∑ x[n]e
n=0 N −1 − j 2π k n N
for k = 0,1,⋯ , N − 1
Inverse DFT 1 x[n] = N
∑ X [k ]e
k =0
N −1
FIGURE 11-39
One Stage of FFT.
Joyce Van de Vegte Fundamentals of Digital Signal Processing
Copyright ©2002 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.
j 2π
k n N
for n = 0,1,⋯ , N − 1
0 ≤ n ≤ N − 1 DFT window 0 ≤ k ≤ N − 1 X [k ] = X [k ] e
jθ [k ]
X [k ] θ [k ]
DFT magnitude spectrum DFT phase spectrum
X (Ω ) =
n = −∞
x[n]e − jnΩ = 2 − e − jΩ + 3e − j 2Ω + 3e − j 3 Ω ∑
∞
FIGURE 11-7
DFT magnitude spectrum for Example 11.2.
X [k ] = ∑ x[n]e
n=0
N −1
−j
2π nk N
FIGURE 11-41
Regrouping into even and odd sequences.
FIGURE 11-42 Comparing DFT and FFT efficiency.
in detaiages of 8-point FFT.
Joyce Van de Vegte Fundamentals of Digital Signal Processing
Copyright ©2002 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.
• periodicity
X (Ω + 2π ) = X (Ω )
FIGURE 7-2 Signal for Example 7.1.
X (Ω ) = ∑ x[n]e − jnΩ
−∞
∞
= 2 − e − jΩ + 3e − j 2 Ω + e − j 4 Ω
Joyce Van de Vegte Fundamentals of Digital Signal Processing
j 2π N −1 k n N N −1 − j 2π k n N
n is the sample number, index k runs from 0 to N - 1 ck are Fourier coefficients
FIGURE 8-13 Digital signal for Example 8.4.
Copyright ©2002 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.
FIGURE 8-11 Comparing DTFTs and DFSs for digital signals.
DFT和FFT处理
• DTFT:Discrete Time Fourier Transform
X (Ω ) =
n=∞
X (Ω ) = F {x[n]}
n = −∞
x[n]e − jnΩ ∑
Properties of DTFT
• Time delay
F {x[n − n0 ]} = e − jn0Ω X (Ω )
Copyright ©2002 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.
• DFS:Discrete Fourier Series(离散傅里叶 离散傅里叶 级数) 级数
1 x[n] = ∑ ck e c k = ∑ x[n]e N k =0 n= 0 N is the period of x[n];
= 2−e
− j 2π
k 4
+ 3e
− j 2π
k 2 4
+ 3e
− j 2π
k 3 4
FIGURE 11-8
DTFT and DFT results superimposed for Example 11.2.
Fast Fourier Transform and why?
• The Fast Fourier Transform, developed originally by Cooley and Tukey in 1965, in a significantly less computationally method of evaluating the DFT, and thus particularly attractive for ‘realtime’ spectral analysis DSP technology. • The FFT reduces the number of complex multiplication. N
the period N = 8
ck = ∑ x[n]e
n=0
N −1
− j 2π
k n N
= ∑ x[n]e
7 n=0
− j 2π
k n 8
= 1+ e
−j
πk
4
+e
−j
πk
2
+e
−j
3πk 4
(k from 0 to 7)
Joyce Van de Vegte Fundamentals of Digital Signal Processing