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10/5/2012 1 Fourier transform of discrete time signals Discrete Fourier Transform Inverse Discrete Fourier Transform Properties of Discrete Fourier Transform Computation of discrete Fourier transform FFT algorithms Decimation in Time (DIT) Decimation in Frequency (DIF) 10/5/2012 1 S. THAI SUBHA CHAPTER-III Time domain signal Frequency domain signal 10/5/2012 2 S. THAI SUBHA CHAPTER-III
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Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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Page 1: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

1

• Fourier transform of discrete time signals

• Discrete Fourier Transform

• Inverse Discrete Fourier Transform

• Properties of Discrete Fourier Transform

• Computation of discrete Fourier transform

– FFT algorithms

• Decimation in Time (DIT)

• Decimation in Frequency (DIF)

10/5/2012 1S. THAI SUBHA CHAPTER-III

Time domain signal

Frequency domain signal

10/5/2012 2S. THAI SUBHA CHAPTER-III

Page 2: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

2

• Fourier series (Periodic signals) & Fourier Transform

(Finite energy signals)

– Useful in the analysis & design of LTI system.

• Decomposes the signals in terms of sinusoidal or

complex exponential components

– To represent in frequency domain

• Converts Time domain signal to a frequency domain

signal

)eor t(sin jωω

10/5/2012 3S. THAI SUBHA CHAPTER-III

• Fourier Transform (FT) is a way of transforming a continuous

signal into the frequency domain.

• Discrete Time Fourier Transform (DTFT) is a Fourier Transform of

a sampled signal.

• Discrete Fourier Transform (DFT) is a discrete numerical

equivalent using sums instead of integrals that can be computed on a

digital computer.

• Finally, FFT is a highly elegant and efficient algorithm, which is

still one of the most used algorithms in speech processing,

communications, frequency estimation, etc – one of the most highly

developed area of DSP.

10/5/2012 4S. THAI SUBHA CHAPTER-III

Page 3: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

3

Fourier series Fourier Transform

Gives the harmonic content of a

periodic time function

Gives the frequency information

for an aperiodic signals (Finite

energy signals)

Discrete frequency spectrum Continuous frequency spectrum

DTFT DFT

Sampling is performed only in

time domain

Obtained by performing

sampling operation in both the

time and frequency domains

Continuous function ofω Discrete frequency spectrum

10/5/2012 5S. THAI SUBHA CHAPTER-III

The Fourier transform of a continuous time aperiodic signal is given

by

The conditions for the existence of the Fourier transform is follows

the Dirichlet conditions which are:

1 .The signal x(t) has a finite no. of discontinuities.

2.The signal x(t) has a finite no. of maxima and minima.

3.The signal x(t) is absolutely integrable that is

∫∞

∞−

ω−ω=ω allfor dte)t(x)j(X tj

∫∞

∞−

∞< dt)t(x

10/5/2012 6S. THAI SUBHA CHAPTER-III

Page 4: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

4

The discrete time Fourier transform of finite energy aperiodic signal

x(n) is representation of signal in terms of complex exponential

sequence e-jωn where ω is a real frequency variable.

The discrete time Fourier transforms of x(n) is defined as

where X(ω) represents the frequency content if signal is x(n).

• Physically X(ω) is a decomposition of x(n) into its frequency

components.

∑∞

−∞=

ω−ω=ω=

n

njje)n(x)(X)e(X

10/5/2012 7S. THAI SUBHA CHAPTER-III

ITDFT

The Inverse Discrete Time Fourier Transform (ITDFT)

is given by,

∫π

π−

ωωω

π= de)(X

2

1)n(x nj

10/5/2012 8S. THAI SUBHA CHAPTER-III

Page 5: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

5

Relationship between Z-Transform &

Fourier Transform

• Let us consider a sequence x(n) having z-

transform with ROC that includes the unit

circle.

(1)

• The Fourier transform of x(n) is given by

(2)

9

ω=

jrez where

∑∞

−∞=

ω−−ω=

n

njnjer)n(x)re(X

∑∞

−∞=

ω−ω=

n

njje)n(x)e(X

10/5/2012 S. THAI SUBHA CHAPTER-III

• In the z-plane this corresponds to the locus of

points on the unit circle . Hence X(ejω) is

equal to X(z) evaluated along the unit circle, or

• For X(ejω) to exists, the ROC of X(z) must

include the unit circle.

10

1z =

ω=

ω= jez

j )z(X)e(X

10/5/2012 S. THAI SUBHA CHAPTER-III

Page 6: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

6

Discrete Fourier Transform

• DFT of a discrete time signal x(n) is a finite durationdiscrete frequency sequence.

• DFT sequence is denoted by X(k).

• DFT is obtained by sampling one period of the FourierTransform X(ω) of the signal x(n) at a finite no. offrequency points.

• Sampling is performed at N equally spaced points in theperiod 0 ≤ ω ≤ 2π at ωk = 2πk /N; 0 ≤ k ≤ N-1

1)-3...(N 2, 1, 0,kfor ;)(X)k(XN

k2 =ω= π=ω

10/5/2012 11S. THAI SUBHA CHAPTER-III

• DFT converts the continuous function of ω to adiscrete function of ω.

• Thus, DFT allows us to perform frequencyanalysis on a digital computer.

DFT is important for two reasons:

1. To perform spectral analysis.

2. To perform filtering operations in frequency domain.

10/5/2012 12S. THAI SUBHA CHAPTER-III

Page 7: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

7

Compute the N-point DFT of the following finite length

sequence given as,

(a).

(b).

≤≤

≤≤=

1-Nn0 odd,-n 0,

1-Nn0 even,-n ,1)n(x

1-Nn0 ,ax(n) n≤≤=

10/5/2012 13S. THAI SUBHA CHAPTER-III

2

N

2

N.... 2, 0,kfor e

1)-....(N 2, 1, 0,kfor e)n(x)k(X)}n(x{DFT

1

0n

nk 2j

1N

0n

N

nk 2j

2N

2N

=

==

===

∑−

=

π−

=

π−

≤≤

≤≤=

1-Nn0 odd,-n 0,

1-Nn0 even,-n ,1)n(x ).a(

==

otherwise 0,

2

N.... 2, 0,k ,

2

N

)k(X

10/5/2012 14S. THAI SUBHA CHAPTER-III

Page 8: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

8

1-Nn0 ,a x(n)(b). n≤≤=

∑−

=

π−

===1N

0n

N

nk 2j

1)-....(N 2, 1, 0,kfor e)n(x)k(X)}n(x{DFT

N

k 2j

N

N

k 2j

xNN

k 2j

N

1N

0n

n

N

k 2j

1N

0n

N

nk 2j

n

ae1

a1

ae1

ea1

ae

1)-....(N 2, 1, 0,kfor ea)k(X

π−

π−

π−

=

π−

=

π−

−=

−=

=

==

10/5/2012 15S. THAI SUBHA CHAPTER-III

1. Linearity :

The DFT obeys the law of linearity.

If

10/5/2012 16S. THAI SUBHA CHAPTER-III

Page 9: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

9

2. Periodicity

10/5/2012 17S. THAI SUBHA CHAPTER-III

3. Time reversal of a sequence:

10/5/2012 18S. THAI SUBHA CHAPTER-III

Page 10: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

10

4. Circular shift of a sequence:

10/5/2012 19S. THAI SUBHA CHAPTER-III

5. Circular frequency shift:

10/5/2012 20S. THAI SUBHA CHAPTER-III

Page 11: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

11

6. Multiplication of two sequences:

10/5/2012 21S. THAI SUBHA CHAPTER-III

7. DF of even and odd sequences : • The DFT of an even sequence is purely real.

• The DFT of an odd sequence is purely imaginary.

• Therefore DFT can be evaluated using cosine and sine transforms for even and

odd sequences respectively.

π=

π=

=

=

N

nk2sinx(n)X(k)

sequence, odd For

N

nk2cosx(n)X(k)

sequence,even For

1-N

0n

1-N

0n

10/5/2012 22S. THAI SUBHA CHAPTER-III

Page 12: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

12

N-point DFT of a finite duration sequence x(n)

of length L, where N ≥ L, is defined as,

The Inverse Discrete Fourier Transform (IDFT)

is given by,

( )∑−

=

==π

1N

0k

knj1)-...(N 1, 0, n ,e)k(X

N

1)n(x N

2

∑−

=

π−

===1N

0n

N

nk 2j

1)-....(N 2, 1, 0,kfor e)n(x)k(X)}n(x{DFT

10/5/2012 23S. THAI SUBHA CHAPTER-III

Compute 4-point DFT of causal three sample

sequence given by,

≤≤=

else 0,

2n0 ,3

1

)n(x

∑−

=

π−

==1N

0n

N

nk 2j

1)-....(N 2, 1, 0,kfor e)n(x)k(X

e)n(x)k(X3

0n

4

nk 2j

∑=

π−

=

10/5/2012 24S. THAI SUBHA CHAPTER-III

Page 13: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

13

+++=

+++=

π−

π−

π−

π−

π−

0ee13

1

e)3(xe)2(xe)1(x)0(x)k(X

k j2

k j

2

k 3j

k j2

k j

The values of X(k) can be evaluated for k = 0, 1, 2, 3

π−π+

π−

π+= k sinjk cos

2

k sinj

2

k cos1

3

1)k(X

[ ] 01 33

1 X(0) 0;k When ∠===

[ ]23

1

3

j 1j1

3

1 X(1) 1;k When

π−∠=−=−−==

10/5/2012 25S. THAI SUBHA CHAPTER-III

[ ] 03

1 111

3

1X(2) 2;k When ∠=+−==

[ ]23

1

3

j 1j1

3

1X(3) ;3k When

π∠==−+==

ππ

−=∠

=

π

∠∠π

−∠∠=

2 ,0 ,

2 0,X(k) ,Phase

3

1 ,

3

1 ,

3

1 1,X(k) Magnitude,

23

1 ,0

3

1 ,

23

1 0,1X(k)

by,given is x(n)sequence DFTpoint -4 The

10/5/2012 26S. THAI SUBHA CHAPTER-III

Page 14: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

14

ππ

−=∠

=

π

∠∠π

−∠∠=

2 ,0 ,

2 0,X(k) ,Phase

3

1 ,

3

1 ,

3

1 1,X(k) Magnitude,

23

1 ,0

3

1 ,

23

1 0,1X(k)

by,given is x(n)sequence DFTpoint -4 The

0 1 2 3 k

2

π

2

π−

Magnitude response Phase response

0 1 2 3 4 k

X(k) .

. ..

.

10/5/2012 27S. THAI SUBHA CHAPTER-III

• Ex. Pr:

Find 4 - point DFT of x(n) = {1, 2, 3, 4} and draw magnitude spectrum.

• Ans: X(k) = {10, 2.828, 2, 6.32}

2

k 3j

k j2

k j

2

k 3j

k j2

k j

e4e3e21

e)3(xe)2(xe)1(x)0(x)k(X

π−

π−

π−

π−

π−

π−

+++=

+++=

10/5/2012 28S. THAI SUBHA CHAPTER-III

Page 15: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

15

• Find IDFT for X(k) = {3, -1, 3, -1}

( )∑−

=

==π

1N

0k

knj1)-...(N 1, 0, n ,e)k(X

N

1)n(x N

2

( )∑=

==π

3

0k

knj3 2, 1, 0, n ,e)k(X

4

1)n(x N

2

( ) ( )[ ]njnjnj2

32 ee3e3

4

1)n(x

ππ

−+−=π

0} 2, 0, ,1{)n(x =

10/5/2012 29S. THAI SUBHA CHAPTER-III

• Find the 8-point DFT and IDFT for the given

sequence x(n) = {1, 2, 3, 4}

Solution: N-point DFT is given by,

∑−

=

π−

==1N

0n

N

nk 2j

1)-....(N 2, 1, 0,kfor e)n(x)k(X

∑=

π−

==7

0n

4

nk j

....7 2, 1, 0,kfor e)n(x)k(X

10/5/2012 30S. THAI SUBHA CHAPTER-III

Page 16: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

16

• For k = 0,

• For k = 1,

4

k 3j

2

k j

4

k j

2

k 7j

2

k 3j

4

k 5j

k j4

k 3j

2

k j

4

k j

e4e3e21 )k(X

e)7(xe)6(xe)5(x

e)4(xe)3(xe)2(xe)1(x)0(x)k(X

π−

π−

π−

π−

π−

π−

π−

π−

π−

π−

+++=

+++

++++=

104321 )0(X =+++=

7.2426 j-0.4142 -

e4e3e21 )1(X 4

3j

2

j

4

j

=

+++=

π−

π−

π−

10/5/2012 31S. THAI SUBHA CHAPTER-III

• For k = 2,

• For k = 3,

• For k = 4,

j2 2-

e4e3e21 )2(X 2

3j

j2

j

+=

+++=

π−

π−

π−

j1.2426 - 2.4142

e4e3e21 )3(X 4

9j

2

3j

4

3j

=

+++=

π−

π−

π−

2- 4-32-1

e4e3e21 )4(X 3j2jj

=+=

+++=π−π−π−

4

k 3j

2

k j

4

k j

e4e3e21 )k(X π

−π

−π

+++=

10/5/2012 32S. THAI SUBHA CHAPTER-III

Page 17: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

17

• For k = 5,

• For k = 6,

• For k = 7,

• The 8 - point DFT of the given x(n) is,

4

k 3j

2

k j

4

k j

e4e3e21 )k(X π

−π

−π

+++=

j1.2426 2.4142

e4e3e21 )5(X 4

15j

2

5j

4

5j

+=

+++=

π−

π−

π−

j2- 2-

e4e3e21 )6(X 2

9j

3j2

3j

=

+++=

π−

π−

π−

7.2426 j0.4142 -

e4e3e21 )7(X 4

21j

2

7j

4

7j

+=

+++=

π−

π−

π−

( ) ( ) ( ){

( ) ( ) ( ) ( )}7.2426 j0.4142 - ,j2- 2- ,j1.2426 2.4142 ,2-

,j1.2426 - 2.4142 ,j2 2- ,7.2426 j-0.4142 - 10,X(k)

++

+=

10/5/2012 33S. THAI SUBHA CHAPTER-III

• Pr.: Find the circular convolution of the given

data sequences,

x1(n) = {1, 3, 5, 7} & x2(n) = {2, 4, 6,8}

x1(0)=1

x1(1)=3

x1(3)=7

x1(2)=5

.

.

.

.x2(1)=4

x2(0)

=2

x2(3)=8

x2(2)

=6

+

+

+

+

y(0) = x1(0)x 2(0) + x1(1) x2(3)

+ x1(2) x2(2) + x1(3) x2(1)

= 2 + 28 + 30 +24

= 84

∑−

=

−=1N

0k

N21 )]kn([x)k(x)n(y

10/5/2012 34S. THAI SUBHA CHAPTER-III

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18

x1(n) = {1, 3, 5, 7} & x2(n) = {2, 4, 6,8}

x1(0)=1

x1(1)=3

x1(3)=7

x1(2)=5

.

.

.

.x2(1)

=4

x2(0)=2

x2(3)

=8

x2(2)=6

+

+

+

+

y(1) = x1(0)x 2(1) + x1(1) x2(0)

+ x1(2) x2(3) + x1(3) x2(2)

= 4 + 42 + 40 +6

= 92

10/5/2012 35S. THAI SUBHA CHAPTER-III

x1(n) = {1, 3, 5, 7} & x2(n) = {2, 4, 6,8}

x1(0)=1

x1(1)=3

x1(3)=7

x1(2)=5

.

.

.

.+

+

+

+

y(2) = x1(0)x 2(2) + x1(1) x2(1)

+ x1(2) x2(0) + x1(3) x2(3)

= 6 + 56 + 10 +12

= 84

x2(1)=4

x2(0)

=2

x2(3)=8

x2(2)

=6

10/5/2012 36S. THAI SUBHA CHAPTER-III

Page 19: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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19

x1(n) = {1, 3, 5, 7} & x2(n) = {2, 4, 6,8}

y(n) = {84, 92, 84, 60}

x1(0)=1

x1(1)=3

x1(3)=7

x1(2)=5

.

.

.

.+

+

+

+

y(3) = x1(0)x 2(3) + x1(1) x2(2)

+ x1(2) x2(1) + x1(3) x2(0)

= 8 + 14 + 20 +18

= 60

x2(1)

=4

x2(0)=2

x2(3)

=8

x2(2)=6

10/5/2012 37S. THAI SUBHA CHAPTER-III

• Method for computing DFT with reduced number of calculations.

• Decomposition of N-point DFT into successively smaller DFTs.

• In an N-point sequence, if N = rm, then sequence can be decimated into r-point sequences.

• For each r-point sequence, r-point DFTs are computed.

• From the results of r-point DFTs, r2-point DFTs are computed.

• From the results of r2-point DFTs, r3-point DFTs are computed and so on until we get rm-point DFT.

• Hence, m - number stages of computation and r - radix of the FFT algorithm.

10/5/2012 38S. THAI SUBHA CHAPTER-III

Page 20: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

20

RadixRadix--2 FFT2 FFT

• Efficient algorithm for computing N-point DFT.

• N-point sequence is decimated into 2-point sequences and the 2-point DFT for each decimated sequence is computed.

• From the results of 2-point DFTs, the 4-point DFTs are computed.

• From the results of 4-point DFTs, the 8-point DFTs are computed and so on until we get N-point DFT.

10/5/2012 39S. THAI SUBHA CHAPTER-III

• For performing radix-2 FFT, the value of N

should be such that, N - even.

– Total number of complex additions = Nlog2N

– Total number of complex multiplications

= (N/2) log2N.

�For DFT

�Total number of complex additions = N(N-1)

�Total number of complex multiplications = N2

10/5/2012 40S. THAI SUBHA CHAPTER-III

Page 21: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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21

N DFT FFT

4 16 8

8 64 24

16 256 64

32 1024 160

64 4096 384

The number of complex multiplications to perform an

N-point DFT using the conventional (DFT) algorithm

and the FFT algorithm.

10/5/2012 41S. THAI SUBHA CHAPTER-III

• Time domain N-point sequence is decimated

into 2 - point sequences - DIT.DIT.

DIF:DIF:� N-point sequences---N/2-point sequences---

N/4-point sequences are obtained by

decimation of frequency domain sequences.

10/5/2012 42S. THAI SUBHA CHAPTER-III

Page 22: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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22

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 43

DITDIT--FFT FFT ImplementationImplementation

�� To efficiently implement the FFT To efficiently implement the FFT algorithm a few observations are made:algorithm a few observations are made:

�� Each stage has the same number of Each stage has the same number of butterflies (number of butterflies = N/2, N is butterflies (number of butterflies = N/2, N is number of points).number of points).

�� The number of DFT groups per stage is equal The number of DFT groups per stage is equal to (N/2to (N/2stagestage).).

�� The The number of butterflies in the group is number of butterflies in the group is equal to 2equal to 2stagestage--11..

FFT

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 44

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

Example: 8 point FFTExample: 8 point FFT

FFT

Page 23: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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23

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 45

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

FFT

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 46

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 1= 1

Stage 1Stage 1

FFT

Page 24: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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24

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 47

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 2= 2

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

Stage 2Stage 2Stage 1Stage 1

FFT

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 48

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

FFT

Page 25: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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25

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 49

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

(2)(2) Blocks/stage:Blocks/stage:

�� Stage 1:Stage 1:

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

FFT

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 50

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

(2)(2) Blocks/stage:Blocks/stage:

�� Stage 1: NStage 1: Nblocksblocks = 1= 1

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

Block 1Block 1

FFT

Page 26: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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26

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 51

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

(2)(2) Blocks/stage:Blocks/stage:

�� Stage 1: NStage 1: Nblocksblocks = 2= 2

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

Block 1Block 1

Block 2Block 2

FFT

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 52

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

(2)(2) Blocks/stage:Blocks/stage:

�� Stage 1: NStage 1: Nblocksblocks = 3= 3

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

Block 1Block 1

Block 2Block 2

Block 3Block 3

FFT

Page 27: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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27

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 53

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

(2)(2) Blocks/stage:Blocks/stage:

�� Stage 1: NStage 1: Nblocksblocks = 4= 4

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

Block 1Block 1

Block 2Block 2

Block 3Block 3

Block 4Block 4

FFT

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 54

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

(2)(2) Blocks/stage:Blocks/stage:

�� Stage 1: NStage 1: Nblocksblocks = 4= 4

�� Stage 2: NStage 2: Nblocksblocks = 1= 1

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

Block 1Block 1

FFT

Page 28: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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28

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 55

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

(2)(2) Blocks/stage:Blocks/stage:

�� Stage 1: NStage 1: Nblocksblocks = 4= 4

�� Stage 2: NStage 2: Nblocksblocks = 2= 2

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

Block 1Block 1

Block 2Block 2

FFT

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 56

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

(2)(2) Blocks/stage:Blocks/stage:

�� Stage 1: NStage 1: Nblocksblocks = 4= 4

�� Stage 2: NStage 2: Nblocksblocks = 2= 2

�� Stage 3: NStage 3: Nblocksblocks = 1= 1

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

Block 1Block 1

FFT

Page 29: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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29

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 57

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

(2)(2) Blocks/stage:Blocks/stage:

�� Stage 1: NStage 1: Nblocksblocks = 4= 4

�� Stage 2: NStage 2: Nblocksblocks = 2= 2

�� Stage 3: NStage 3: Nblocksblocks = 1= 1

(3)(3) B’flies/block:B’flies/block:

�� Stage 1:Stage 1:

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

FFT

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 58

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

(2)(2) Blocks/stage:Blocks/stage:

�� Stage 1: NStage 1: Nblocksblocks = 4= 4

�� Stage 2: NStage 2: Nblocksblocks = 2= 2

�� Stage 3: NStage 3: Nblocksblocks = 1= 1

(3)(3) B’flies/block:B’flies/block:

�� Stage 1: NStage 1: Nbtfbtf = 1= 1

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

FFT

Page 30: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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30

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 59

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

(2)(2) Blocks/stage:Blocks/stage:

�� Stage 1: NStage 1: Nblocksblocks = 4= 4

�� Stage 2: NStage 2: Nblocksblocks = 2= 2

�� Stage 3: NStage 3: Nblocksblocks = 1= 1

(3)(3) B’flies/block:B’flies/block:

�� Stage 1: NStage 1: Nbtfbtf = 1= 1

�� Stage 2: NStage 2: Nbtfbtf = 1= 1

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

FFT

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 60

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

(2)(2) Blocks/stage:Blocks/stage:

�� Stage 1: NStage 1: Nblocksblocks = 4= 4

�� Stage 2: NStage 2: Nblocksblocks = 2= 2

�� Stage 3: NStage 3: Nblocksblocks = 1= 1

(3)(3) B’flies/block:B’flies/block:

�� Stage 1: NStage 1: Nbtfbtf = 1= 1

�� Stage 2: NStage 2: Nbtfbtf = 2= 2

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

FFT

Page 31: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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31

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 61

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

(2)(2) Blocks/stage:Blocks/stage:

�� Stage 1: NStage 1: Nblocksblocks = 4= 4

�� Stage 2: NStage 2: Nblocksblocks = 2= 2

�� Stage 3: NStage 3: Nblocksblocks = 1= 1

(3)(3) B’flies/block:B’flies/block:

�� Stage 1: NStage 1: Nbtfbtf = 1= 1

�� Stage 2: NStage 2: Nbtfbtf = 2= 2

�� Stage 3: NStage 3: Nbtfbtf = 1= 1

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

FFT

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 62

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

(2)(2) Blocks/stage:Blocks/stage:

�� Stage 1: NStage 1: Nblocksblocks = 4= 4

�� Stage 2: NStage 2: Nblocksblocks = 2= 2

�� Stage 3: NStage 3: Nblocksblocks = 1= 1

(3)(3) B’flies/block:B’flies/block:

�� Stage 1: NStage 1: Nbtfbtf = 1= 1

�� Stage 2: NStage 2: Nbtfbtf = 2= 2

�� Stage 3: NStage 3: Nbtfbtf = 2= 2

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

FFT

Page 32: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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32

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 63

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

(2)(2) Blocks/stage:Blocks/stage:

�� Stage 1: NStage 1: Nblocksblocks = 4= 4

�� Stage 2: NStage 2: Nblocksblocks = 2= 2

�� Stage 3: NStage 3: Nblocksblocks = 1= 1

(3)(3) B’flies/block:B’flies/block:

�� Stage 1: NStage 1: Nbtfbtf = 1= 1

�� Stage 2: NStage 2: Nbtfbtf = 2= 2

�� Stage 3: NStage 3: Nbtfbtf = 3= 3

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

FFT

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 19, Slide 64

Example: 8 point FFTExample: 8 point FFT

(1)(1) Number of stages:Number of stages:

�� NNstagesstages = 3= 3

(2)(2) Blocks/stage:Blocks/stage:

�� Stage 1: NStage 1: Nblocksblocks = 4= 4

�� Stage 2: NStage 2: Nblocksblocks = 2= 2

�� Stage 3: NStage 3: Nblocksblocks = 1= 1

(3)(3) B’flies/block:B’flies/block:

�� Stage 1: NStage 1: Nbtfbtf = 1= 1

�� Stage 2: NStage 2: Nbtfbtf = 2= 2

�� Stage 3: NStage 3: Nbtfbtf = 4= 4

FFT ImplementationFFT Implementation

WW00 --11

WW00 --11

WW00 --11

WW00 --11

WW22 --11

WW00

--11WW00

WW22 --11

--11WW00

WW11 --11

WW00

WW33 --11

--11WW22

--11

Stage 2Stage 2 Stage 3Stage 3Stage 1Stage 1

�� Decimation in time FFT:Decimation in time FFT:

�� Number of stages = logNumber of stages = log22NN

�� Number of blocks/stage = N/2Number of blocks/stage = N/2stagestage

�� Number of butterflies/block = 2Number of butterflies/block = 2stagestage--11

FFT

Page 33: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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33

Compute 2-

point DFT

Compute 2-

point DFT

Compute 2-

point DFT

Compute 2-

point DFT

Compute 4-

point DFT

Compute 4-

point DFT

Compute 8-

point DFT

x(0)

x(5)

x(2)

x(3)

x(7)

x(6)

x(1)

x(4)

X(0)

X(1)

X(2)

X(3)

X(4)

X(5)

X(6)

X(7)

10/5/2012 65S. THAI SUBHA CHAPTER-III

• Initially, do bit reversal for grouping even &

odd sequences.

• Find Twiddle factor,

– First stage, (N = 2), k = 0, 1

– Second stage, (N = 4), k = 0, 1, 2, 3

N

kj2 -

ke W

N

π

=

1e W 2

)0(j2 -

0

2==

π

j - e W

1 e W

4

j2 -

1

4

4

)0(j2 -

0

4

==

==

π

π

10/5/2012 66S. THAI SUBHA CHAPTER-III

Page 34: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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34

– Third stage, (N = 8), k = 0, 1, ..7

• First stage output:

707.0j707.0e W j; - e W

707.0j707.0e W 1; e W

8

j6 -

3

88

j4 -

2

8

8

j2 -

1

88

)0(j2 -

0

8

−−====

−====

ππ

ππ

)4(xW)0(x)1(V

)4(xW)0(x)0(V

0

211

0

211

−=

+=

)6(xW)2(x)1(V

)6(xW)2(x)0(V

0

212

0

212

−=

+=

)5(xW)1(x)1(V

)5(xW)1(x)0(V

0

221

0

221

−=

+=

)7(xW)3(x)1(V

)7(xW)3(x)0(V

0

222

0

222

−=

+=

V11(0)

V11(1)0

2W

x(0)

X(4)

Butterfly diagram

-1

10/5/2012 67S. THAI SUBHA CHAPTER-III

• Second stage output:

)1(VW)1(V)3(F

)0(VW)0(V)2(F

)1(VW)1(V)1(F

)0(VW)0(V)0(F

12

1

4111

12

0

4111

12

1

4111

12

0

4111

−=

−=

+=

+=

)1(VW)1(V)3(F

)0(VW)0(V)2(F

)1(VW)1(V)1(F

)0(VW)0(V)0(F

22

1

4212

22

0

4212

22

1

4212

22

0

4212

−=

−=

+=

+=

10/5/2012 68S. THAI SUBHA CHAPTER-III

Page 35: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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35

• Third stage output:

)3(FW)3(F)7(X

)2(FW)2(F)6(X

)1(FW)1(F)5(X

)0(FW)0(F)4(X

)3(FW)3(F)3(X

)2(FW)2(F)2(X

)1(FW)1(F)1(X

)0(FW)0(F)0(X

2

3

81

2

2

81

2

1

81

2

0

81

2

3

81

2

2

81

2

1

81

2

0

81

−=

−=

−=

−=

+=

+=

+=

+=

10/5/2012 69S. THAI SUBHA CHAPTER-III

Find 8-point DFT of the sequence

x(n) = {1, 2, 2,1, 1, 2, 2, 1} using DIT-FFT algorithm.

Solution:

Bit reversal: Group odd and even sequences:

Normal

Form

Bit reversed

Form

x(0) (000) 1 (000) x(0) 1

x(1) (001) 2 (100) x(4) 1

x(2) (010) 2 (010) x(2) 2

x(3) (011) 1 (110) x(6) 2

x(4) (100) 1 (001) x(1) 2

x(5) (101) 2 (101) x(5) 2

x(6) (110) 2 (011) x(3) 1

x(7) (111) 1 (111) x(7) 1

10/5/2012 70S. THAI SUBHA CHAPTER-III

Page 36: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

36

• Find Twiddle factor:

• First stage output:

1e W 2

)0(j2 -

0

2==

π

j - e W 1; e W 4

j2 -

1

44

)0(j2 -

0

4 ====

ππ

707.0j707.0e W j; - e W

707.0j707.0e W 1; e W

8

j6 -

3

88

j4 -

2

8

8

j2 -

1

88

)0(j2 -

0

8

−−====

−====

ππ

ππ

0)4(xW)0(x)1(V

2)4(xW)0(x)0(V

0

211

0

211

=−=

=+=

0)6(xW)2(x)1(V

4)6(xW)2(x)0(V

0

212

0

212

=−=

=+=

0)5(xW)1(x)1(V

4)5(xW)1(x)0(V

0

221

0

221

=−=

=+=

0)7(xW)3(x)1(V

2)7(xW)3(x)0(V

0

222

0

222

=−=

=+=

Bit reversed

Form

(000) x(0) 1

(100) x(4) 1

(010) x(2) 2

(110) x(6) 2

(001) x(1) 2

(101) x(5) 2

(011) x(3) 1

(111) x(7) 1

10/5/2012 71S. THAI SUBHA CHAPTER-III

1

x(0)

x(4)

x(2)

x(6) 1

1

x(1)

x(5)

x(3)

x(7)1

-1

-1

-1

-1

2

0

4

0

4

0

2

0

Bit reversed

Form

(000) x(0) 1

(100) x(4) 1

(010) x(2) 2

(110) x(6) 2

(001) x(1) 2

(101) x(5) 2

(011) x(3) 1

(111) x(7) 1

0

2W

0

2W

0

2W

0

2W

10/5/2012 72S. THAI SUBHA CHAPTER-III

Page 37: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

37

• Second stage output:

0)1(VW)1(V)3(F

2)0(VW)0(V)2(F

0)1(VW)1(V)1(F

6)0(VW)0(V)0(F

12

1

4111

12

0

4111

12

1

4111

12

0

4111

=−=

−=−=

=+=

=+=

0)1(VW)1(V)3(F

2)0(VW)0(V)2(F

0)1(VW)1(V)1(F

6)0(VW)0(V)0(F

22

1

4212

22

0

4212

22

1

4212

22

0

4212

=−=

=−=

=+=

=+=

0)4(xW)0(x)1(V

2)4(xW)0(x)0(V

0

211

0

211

=−=

=+=

0)6(xW)2(x)1(V

4)6(xW)2(x)0(V

0

212

0

212

=−=

=+=

0)5(xW)1(x)1(V

4)5(xW)1(x)0(V

0

221

0

221

=−=

=+=

0)7(xW)3(x)1(V

2)7(xW)3(x)0(V

0

222

0

222

=−=

=+=

10/5/2012 73S. THAI SUBHA CHAPTER-III

1

1x(0)

x(4)

1x(2)

x(6) 1

1

1x(1)

x(5)

1x(3)

x(7)1

-1

-1

-1

-1

2

0

4

0

4

0

2

0

1

1

-j

-j

0

4W

1

4W

0

4W

1

4W

-1

-1

-1

-1

6

0

-2

0

6

0

0

2

0

2W

0

2W

0

2W

0

2W

10/5/2012 74S. THAI SUBHA CHAPTER-III

Page 38: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

38

• Third stage output:

0)3(FW)3(F)7(X

j22)2(FW)2(F)6(X

0)1(FW)1(F)5(X

0)0(FW)0(F)4(X

0)3(FW)3(F)3(X

j22)2(FW)2(F)2(X

0)1(FW)1(F)1(X

12)0(FW)0(F)0(X

2

3

81

2

2

81

2

1

81

2

0

81

2

3

81

2

2

81

2

1

81

2

0

81

=−=

+−=−=

=−=

=−=

=+=

−−=+=

=+=

=+=

0)1(VW)1(V)3(F

2)0(VW)0(V)2(F

0)1(VW)1(V)1(F

6)0(VW)0(V)0(F

12

1

4111

12

0

4111

12

1

4111

12

0

4111

=−=

−=−=

=+=

=+=

0)1(VW)1(V)3(F

2)0(VW)0(V)2(F

0)1(VW)1(V)1(F

6)0(VW)0(V)0(F

22

1

4212

22

0

4212

22

1

4212

22

0

4212

=−=

=−=

=+=

=+=

10/5/2012 75S. THAI SUBHA CHAPTER-III

1

1x(0)

x(4)

1x(2)

x(6) 1

1

1x(1)

x(5)

1x(3)

x(7)1

-1

-1

-1

-1

2

0

4

0

4

0

2

0

1

1

-j

-j

0

4W

1

4W

0

4W

1

4W

-1

-1

-1

-1

6

0

-2

0

6

0

0

2

0

8W

1

8W

2

8W

3

8W

1

-j

0.707 - j 0.707

- 0.707 - j 0.707

-1

-1

-1

-1

X(0)=12

X(1)= 0

X(3)= 0

X(5)= 0

X(7)= 0

X(4)=0

X(2)

= - 2–j2

X(6)

= - 2+j2

X(k) = {12, 0, (-2 - j2), 0, 0, 0, (-2 + j2), 0}

0

2W

0

2W

0

2W

0

2W

10/5/2012 76S. THAI SUBHA CHAPTER-III

Page 39: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

39

Find 8-point DFT of the sequence x(n) = {2, 1, 4, 6, 5, 8, 3, 9}

using DIT-FFT algorithm.

Solution:

Bit reversal: Group odd and even sequences:

Normal

Form

Bit reversed

Form

x(0) (000) 2 (000) x(0) 2

x(1) (001) 1 (100) x(4) 5

x(2) (010) 4 (010) x(2) 4

x(3) (011) 6 (110) x(6) 3

x(4) (100) 5 (001) x(1) 1

x(5) (101) 8 (101) x(5) 8

x(6) (110) 3 (011) x(3) 6

x(7) (111) 9 (111) x(7) 9

10/5/2012 77S. THAI SUBHA CHAPTER-III

• Find Twiddle factor:

• First stage output:

1e W 2

)0(j2 -

0

2==

π

j - e W 1; e W 4

j2 -

1

44

)0(j2 -

0

4 ====

ππ

707.0j707.0e W j; - e W

707.0j707.0e W 1; e W

8

j6 -

3

88

j4 -

2

8

8

j2 -

1

88

)0(j2 -

0

8

−−====

−====

ππ

ππ

3)4(xW)0(x)1(V

7)4(xW)0(x)0(V

0

211

0

211

−=−=

=+=

1)6(xW)2(x)1(V

7)6(xW)2(x)0(V

0

212

0

212

=−=

=+=

7)5(xW)1(x)1(V

9)5(xW)1(x)0(V

0

221

0

221

−=−=

=+=

3)7(xW)3(x)1(V

15)7(xW)3(x)0(V

0

222

0

222

−=−=

=+=

Bit

reversed

Form

(000) x(0) 2

(100) x(4) 5

(010) x(2) 4

(110) x(6) 3

(001) x(1) 1

(101) x(5) 8

(011) x(3) 6

(111) x(7) 9

10/5/2012 78S. THAI SUBHA CHAPTER-III

Page 40: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

40

1

1x(0)

x(4)

1x(2)

x(6) 1

1

1x(1)

x(5)

1x(3)

x(7)1

-1

-1

-1

-1

7

-3

7

1

9

-7

15

-3

Bit

reversed

Form

(000) x(0) 2

(100) x(4) 5

(010) x(2) 4

(110) x(6) 3

(001) x(1) 1

(101) x(5) 8

(011) x(3) 6

(111) x(7) 9

0

2W

0

2W

0

2W

0

2W

10/5/2012 79S. THAI SUBHA CHAPTER-III

• Second stage output:

j3)1(VW)1(V)3(F

0)0(VW)0(V)2(F

j3)1(VW)1(V)1(F

14)0(VW)0(V)0(F

12

1

4111

12

0

4111

12

1

4111

12

0

4111

+−=−=

=−=

−−=+=

=+=

3j7)1(VW)1(V)3(F

6)0(VW)0(V)2(F

3j7)1(VW)1(V)1(F

24)0(VW)0(V)0(F

22

1

4212

22

0

4212

22

1

4212

22

0

4212

−−=−=

−=−=

+−=+=

=+=

3)4(xW)0(x)1(V

7)4(xW)0(x)0(V

0

211

0

211

−=−=

=+=

1)6(xW)2(x)1(V

7)6(xW)2(x)0(V

0

212

0

212

=−=

=+=

7)5(xW)1(x)1(V

9)5(xW)1(x)0(V

0

221

0

221

−=−=

=+=

3)7(xW)3(x)1(V

15)7(xW)3(x)0(V

0

222

0

222

−=−=

=+=

10/5/2012 80S. THAI SUBHA CHAPTER-III

Page 41: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

41

1

1x(0)

x(4)

1x(2)

x(6) 1

1

1x(1)

x(5)

1x(3)

x(7)1

-1

-1

-1

-1

7

-3

7

1

9

-7

15

-3

1

1

-j

-j

0

4W

1

4W

0

4W

1

4W

-1

-1

-1

-1

14

-3-j

0

-3+j

24

-7+j3

-7-j3

-6

0

2W

0

2W

0

2W

0

2W

10/5/2012 81S. THAI SUBHA CHAPTER-III

• Third stage output:

07.6j828.5)3(FW)3(F)7(X

6j)2(FW)2(F)6(X

807.0j172.0)1(FW)1(F)5(X

10)0(FW)0(F)4(X

807.0j172.0)3(FW)3(F)3(X

6j)2(FW)2(F)2(X

07.6j828.5)1(FW)1(F)1(X

38)0(FW)0(F)0(X

2

3

81

2

2

81

2

1

81

2

0

81

2

3

81

2

2

81

2

1

81

2

0

81

−−=−=

−=−=

−−=−=

−=−=

+−=+=

=+=

+−=+=

=+=

j3)1(VW)1(V)3(F

0)0(VW)0(V)2(F

j3)1(VW)1(V)1(F

14)0(VW)0(V)0(F

12

1

4111

12

0

4111

12

1

4111

12

0

4111

+−=−=

=−=

−−=+=

=+=

3j7)1(VW)1(V)3(F

6)0(VW)0(V)2(F

3j7)1(VW)1(V)1(F

24)0(VW)0(V)0(F

22

1

4212

22

0

4212

22

1

4212

22

0

4212

−−=−=

−=−=

+−=+=

=+=

10/5/2012 82S. THAI SUBHA CHAPTER-III

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10/5/2012

42

1

1x(0)

x(4)

1x(2)

x(6) 1

1

1x(1)

x(5)

1x(3)

x(7)1

-1

-1

-1

-1

2

0

4

0

4

0

2

0

1

1

-j

-j

0

4W

1

4W

0

4W

1

4W

-1

-1

-1

-1

6

0

-2

0

6

0

0

2

X(k) = {38, -5.828+j6.07, j6, -0.172+j8.07, -10,

-0.172-j8.07, -j6, -5.828-j6.07}10/5/2012 83S. THAI SUBHA CHAPTER-III

0

8W

1

8W

2

8W

3

8W

1

-j

0.707 - j 0.707

- 0.707 - j 0.707

X(0)=38

X(1)=

-5.828+j6.07

X(3)

=- 0.172+j8.07

X(5)=

- 0.172-j8.07

X(7)=

-5.828-j6.07

X(4)= -10

X(2) = j6

X(6)= -j6

-1

-1

-1

-1

10/5/2012 84S. THAI SUBHA CHAPTER-III

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43

1

1x(0)

x(4)

1x(2)

x(6) 1

1

1x(1)

x(5)

1x(3)

x(7)1

-1

-1

-1

-1

7

-3

7

1

9

-7

15

-3

1

1

-j

-j

0

4W

1

4W

0

4W

1

4W

-1

-1

-1

-1

14

-3-j

0

-3+j

24

-7+j3

-7-j3

-6

0

2W

0

2W

0

2W

0

2W

0

8W

1

8W

2

8W

3

8W

1

-j

0.707 - j 0.707

- 0.707 - j 0.707

X(0)=38

X(1)=

-5.828+j6.07

X(3)

=- 0.172+j8.07

X(5)=

- 0.172-j8.07

X(7)=

-5.828-j6.07

X(4)= -10

X(2) = j6

X(6)= -j6

-1

-1

-1

-1

X(k) = {38, -5.828+j6.07, j6, -0.172+j8.07, -10,

-0.172-j8.07, -j6, -5.828-j6.07}10/5/2012 85S. THAI SUBHA CHAPTER-III

Difference between DIT and DIF FFT algorithmDifference between DIT and DIF FFT algorithm

• In this algorithm, the N-point time domain sequence is converted to two groups of N/2 point sequences, then each N/2 point sequence is converted to two groups of N/4-point sequences.

• This process continues until we get N/2 groups of 2-point sequences. Finally 2-point DFT of each 2-point sequence is computed.

• In DIF, the input is in natural order and the output is in bit reversed form.

• Complex multiplication takes place after addition /subtraction

10/5/2012 86S. THAI SUBHA CHAPTER-III

Page 44: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

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44

Compute

two groups

of 4-point

DFT

Compute 2-

point

sequence

Compute 2-

point

sequence

Compute 2-

point

sequence

Compute 2-

point

sequence

Compute

two groups

of 2-point

DFT

Compute

two groups

of 2-point

DFT

x(0)

x(4)

x(2)

x(6)

x(5)

x(1)

x(3)

x(7)

X(0)

X(1)

X(2)

X(3)

X(4)

X(5)

X(6)

X(7)

g1(0)

g1(1)

g1(3)

g1(2)

g2(0)

g2(1)

g2(2)

g2(3)

d11(0)

d11(1)

d12(0)

d12(1)

d21(0)

d21(1)

d22(0)

d22(1)

10/5/2012 87S. THAI SUBHA CHAPTER-III

• First stage output:

[ ]

[ ]

[ ]

[ ] 3

82

2

82

1

82

0

82

1

1

1

1

W)7(x)3(x)3(g

W)6(x)2(x)2(g

W)5(x)1(x)1(g

W)4(x)0(x)0(g

)7(x)3(x)3(g

)6(x)2(x)2(g

)5(x)1(x)1(g

)4(x)0(x)0(g

−=

−=

−=

−=

+=

+=

+=

+=

10/5/2012 88S. THAI SUBHA CHAPTER-III

Page 45: Fourier transform of discrete time signals Discrete … 6 Discrete Fourier Transform • DFT of a discrete time signal x(n) is a finite duration discrete frequency sequence. • DFT

10/5/2012

45

• Second stage output:

• Third stage output:

[ ]

[ ] 1

41112

0

41112

1111

1111

W)3(g)1(g)1(d

W)2(g)0(g)0(d

)3(g)1(g)1(d

)2(g)0(g)0(d

−=

−=

+=

+=

[ ]

[ ] 1

42222

0

42222

2221

2221

W)3(g)1(g)1(d

W)2(g)0(g)0(d

)3(g)1(g)1(d

)2(g)0(g)0(d

−=

−=

+=

+=

[ ] 0

21111

1111

W)1(d)0(d)4(X

)1(d)0(d)0(X

−=

+=

[ ] 0

21212

1212

W)1(d)0(d)6(X

)1(d)0(d)2(X

−=

+=

[ ] 0

22121

2121

W)1(d)0(d)5(X

)1(d)0(d)1(X

−=

+=

[ ] 0

22222

2222

W)1(d)0(d)7(X

)1(d)0(d)3(X

−=

+=

10/5/2012 89S. THAI SUBHA CHAPTER-III

• Compute 8-point DFT using DIF FFT

algorithm for the following sequence:

x(n)={1, 2, 3, 4, 4, 3, 2, 1}

10/5/2012 90S. THAI SUBHA CHAPTER-III

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46

0

8W

1

8W

2

8W

3

8W

1

-j

0.707 - j 0.707

- 0.707 - j 0.707

x(0)

x(1)

x(2)

x(3)

x(4)

x(5)

x(6)

x(7)

1

1

-j

-j

0

4W

1

4W

0

4W

1

4W

1

1

1

1

1

1

1

1

5

-3

5

5

5

-j

-0.707 + j 0.707

-2.121 – j2.121

10

10

0

0

-2.828 -j 1.414

2.828 -j 1.414

-3-j

-3+j

20

0

0

0

-5.828

-j 2.414

-0.172

+j 0.414

-0.172

-j 0.414

-5.828

+j 2.414

x(n)={1, 2, 3, 4, 4, 3, 2, 1}

X(k) = {20, -5.828-j2.414, 0, -0.172-j0.414, 0, -0.172+j0.414,

0, -5.828+j2.414}10/5/2012 91S. THAI SUBHA CHAPTER-III