presentation-9 fast fourier transformehm.kocaeli.edu.tr/upload/duyurular/111219055246ca490.pdf ·...
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MEH329DIGITAL SIGNAL PROCESSING
-9-Fast Fourier Transform
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Fast Fourier TransformIntroduction
MEH329 Digital Signal Processing 2
• Computational Complexity:• For each DFT coefficient:
• N complex multiplication (4N real multiplication+2N real addition)• N-1 complex addition (2(N-1) real addition)
• For all DFT coefficients:• NxN complex multiplication• Nx(N-1) complex addition
As N gets larger, the number of computations required for DFT becomesvery large
1
2 /
0
1[ ]
Nj N kn
k
x n X k eN
1
2 /
0
[ ]N
j N kn
n
X k x n e
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MEH329 Digital Signal Processing 3
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Fast Fourier Transform Introduction
MEH329 Digital Signal Processing 4
• The approaches that aim to provide a fast computation of DFTare termed fast Fourier transform (FFT). Most suchapproaches are motivated by the following two properties ofDFT:1. Conjugate symmetry
2 /2 /2
Nj N k
j N ke e
2
Nk kN NW W
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Fast Fourier Transform Introduction
MEH329 Digital Signal Processing 5
• The approaches that aim to provide a fast computation of DFTare termed fast Fourier transform (FFT). Most suchapproaches are motivated by the following two properties ofDFT:
2. Periodicity with N
2 / 2 /j N k N j N ke e k N kN NW W
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MEH329 Digital Signal Processing 6
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Fast Fourier TransformIntroduction
MEH329 Digital Signal Processing 7
• Fast Fourier Transform (FFT).• 2 Decimation Algorithms:
• Decimation in Time (x[n] is decomposedinto successively smaller subsequences)
• Decimation in Frequency (X[k] is intosuccessively smaller subsequences)
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Fast Fourier TransformDecimation in Time
MEH329 Digital Signal Processing 8
• FFT based on decimation in time depends on decomposing x[n] into successively smaller subsequences
• Separate x[n] into two sequence of length N/2
– Even indexed samples in the first sequence– Odd indexed samples in the other sequence
12 /
0
1 12 / 2 /
n even n odd
[ ]
[ ] [ ]
Nj N kn
n
N Nj N kn j N kn
X k x n e
x n e x n e
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Fast Fourier TransformDecimation in Time
MEH329 Digital Signal Processing 9
/ 2 1 /2 1
2 12
0 r 0
[2 ] [2 1]
N Nr krk
N Nr
X k x r W x r W
• Using = 2 for even , and = 2 + 1 for odd
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Fast Fourier TransformDecimation in Time
MEH329 Digital Signal Processing 10
/ 2 1 /2 12 12
0 r 0
/2 1 /2 1
/2 /2r 0 r 0
[2 ] [2 1]
[2 ] [2 1]
N Nr krk
N Nr
N Nrk k rkN N N
kN
X k x r W x r W
x r W W x r W
G k W H k
N/2 point DFT of x[2r] N/2 point DFT of x[2r]
G[k] and H[k] are the N/2-point DFT’s of even and odd x[n] samples, and are periodic with N/2
• Using = 2 for even , and = 2 + 1 for odd
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MEH329 Digital Signal Processing 11
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Fast Fourier TransformDecimation in Time
MEH329 Digital Signal Processing 12
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MEH329 Digital Signal Processing 13
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Fast Fourier TransformDecimation in Time
MEH329 Digital Signal Processing 14
• 8-point DFT example using decimation-in-time
• Two N/2-point DFTs– 2(N/2)2 complex multiplications– 2(N/2)2 complex additions
• Combining the DFT outputs– N complex multiplications– N complex additions
• Total complexity– N2/2+N complex multiplications– N2/2+N complex additions– More efficient than direct DFT
• Repeat same process – Divide N/2-point DFTs into – Two N/4-point DFTs– Combine outputs
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Fast Fourier TransformDecimation in Time
MEH329 Digital Signal Processing 15
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Fast Fourier TransformDecimation in Time
MEH329 Digital Signal Processing 16
2-pointDFT
2-pointDFT
2-pointDFT
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Fast Fourier TransformDecimation in Time (Final Graph)
MEH329 Digital Signal Processing 17
Nlog2N complex multiplications and additions
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Fast Fourier TransformDecimation in Time
MEH329 Digital Signal Processing 18
• Butterfly Structure:
WN(r+N/2) = -WN
r
(just one multiplication!)Reduces complex multiplications and additions to (N/2)log2N
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Fast Fourier TransformDecimation in Time
MEH329 Digital Signal Processing 19
• Indexing rule:
111x111X7x7X
011x110X3x6X
101x101X5x5X
001x100X1x4X
110x011X6x3X
010x010X2x2X
100x001X4x1X
000x000X0x0X
00
00
00
00
00
00
00
00
Bit reversed order!
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Fast Fourier TransformDecimation in Time
MEH329 Digital Signal Processing 20
• Example: Find the DFT coefficients of x[n] = [1 1 2 2 -1 -1 2 2] using N=8 point FFT.
x[0]=1
x[4]=-1
x[2]=2
x[6]=2
x[1]=1
X[5]=-1
X[3]=2
X[7]=2
K[0]=1-1=0
K[1]=1+1=2
L[0]=2+2=4
L[1]=2-2=0
M[0]=1-1=0
M[1]=1+1=2
P[0]=2+2=4
P[1]=2-2=0
G[0]=0+4=4
G[1]=2+0j=2
G[2]=0-4=-4
G[3]=2-j0=2
H[0]=0+4=4
H[1]=2-j0=2
H[2]=0-4=-4
H[3]=2-j0=2
X[0]=4+4=8
X[1]=2+2(.707-j.707)
X[2]=-4-4(-j)
X[3]=2+2(-.707-j.707)
X[4]=4-4=0
X[5]=2+2(-.707+j.707)
X[6]=-4-4j
X[7]=2+2(.707+j.707)
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Fast Fourier TransformDecimation in Frequency
MEH329 Digital Signal Processing 21
• Separate X[k] into two coefficient sequence of length N/2
1
0
[ ]N
nkN
n
X k x n W
1 /2 1 1
2 2 2
0 0 /2
2 [ ] [ ] [ ]N N N
n r n r n rN N N
n n n N
X r x n W x n W x n W
/ 2 1 / 2 1 /2 1
/2 22/2
0 0 0
2 [ ] [ / 2] [ ] [ / 2]N N N
n N rn r nrN N N
n n n
X r x n W x n N W x n x n N W
/ 2 1
/20
2 1 [ ] [ / 2]N
n rnN N
n
X r x n x n N W W
Even-indexed frequencies
Odd-indexed frequencies
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Fast Fourier TransformDecimation in Frequency
MEH329 Digital Signal Processing 22
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Fast Fourier TransformDecimation in Frequency
MEH329 Digital Signal Processing 23
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Fast Fourier TransformDecimation in Frequency
MEH329 Digital Signal Processing 24
• N=8 point decimation in frequency: -- Twiddle factors
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Fast Fourier TransformDecimation in Frequency
MEH329 Digital Signal Processing 25
• N=8 point decimation in frequency:
x[0]=1
X[1]=1
x[2]=2
x[3]=2
x[4]=-1
X[5]=-1
X[6]=2
X[7]=2
0
0
4
4
2
2(.707-j.707)
0
0
4
4
-4
j4
2
2(.707-j.707)
2
-1.414-j1.414
X[0]=8
X[4]=0
X[2]=-4+j4
X[6]=-4-j4
X[1]=2+2(.707-j.707)
X[5]=2-2(.707-j.707)
X[3]=2-2(.707+j.707)
X[7]=2+2(.707+j.707)
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MEH329 Digital Signal Processing 26
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MEH329 Digital Signal Processing 27
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MEH329 Digital Signal Processing 28
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MEH329 Digital Signal Processing 29
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Windowing Prior to DFT• • In this case, frequency components that are not
actually in the signal occur
• • This is caused by the potential discontinuity that is caused by the periodicity in DFT
• • This can be considered as a leak of the energy to other frequencies, and therefore is termed spectral leakage
• • This problem can be mitigated by windowing
MEH329 Digital Signal Processing 30
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MEH329 Digital Signal Processing 31
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Discrete Fourier TransformWindowing
32
• Windowing is utilized to overcome this effect.• Well known window functions:
• Triangular• Trapezoid• Hamming• Hanning• Blackman• Parzen• Welch• Nuttall• Kaiser
0 5 10 15 20 250
0.2
0.4
0.6
0.8
1
Window Function (Hamming)
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MEH329 Digital Signal Processing 33