presentation-9 fast fourier transformehm.kocaeli.edu.tr/upload/duyurular/111219055246ca490.pdf · Title: Microsoft PowerPoint - presentation-9 fast fourier transform Author: aysun

MEH329DIGITAL SIGNAL PROCESSING

-9-Fast Fourier Transform

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


Fast Fourier Transform Introduction


• 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

Fast Fourier Transform Introduction


• 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


Fast Fourier TransformIntroduction


• 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)

Fast Fourier TransformDecimation in Time


• 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



/ 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



/ 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







• 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





2-pointDFT

2-pointDFT

2-pointDFT

Fast Fourier TransformDecimation in Time (Final Graph)


Nlog2N complex multiplications and additions



• Butterfly Structure:

WN(r+N/2) = -WN

r

(just one multiplication!)Reduces complex multiplications and additions to (N/2)log2N



• Indexing rule:

111x111X7x7X

011x110X3x6X

101x101X5x5X

001x100X1x4X

110x011X6x3X

010x010X2x2X

100x001X4x1X

000x000X0x0X

00

00

00

00

00

00

00

00

Bit reversed order!



• 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)

Fast Fourier TransformDecimation in Frequency


• 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







• N=8 point decimation in frequency: -- Twiddle factors



• 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)





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



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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presentation-9 fast fourier transformehm.kocaeli.edu.tr/upload/duyurular/111219055246ca490.pdf · Title: Microsoft PowerPoint - presentation-9 fast fourier transform Author: aysun