dsp projects call 9952749533

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1/84

EXPERT SYSTEMS AND SOLUTIONS

Email: [email protected]

[email protected]

Cell: 9952749533www.researchprojects.info

PAIYANOOR, OMR, CHENNAI

Call For Research Projects Final

year students of B.E in EEE, ECE, EI,

M.E (Power Systems), M.E (Applied

Electronics), M.E (Power Electronics)

Ph.D Electrical and Electronics.

Students can assemble their hardware in our

Research labs. Experts will be guiding theprojects.


2/84

Discrete Fourier Transform

&Fast Fourier Transform


3/84


4/84

Review To use Matlab, we have to truncate sequences and

then evaluate the expression at finitely many points.

The evaluation were obviously approximations to theexact calculations.

In other words, the DTFT and the z-transform are not

numerically computable transform.


5/84

Introduction Therefore we turn our attention to a numerically computable

transform.

It is obtained bysamplingthe DTFT transform in the

frequency domain (or the z-transform on the unit circle).

We develop this transform by analyzingperiodic sequences.

From FT analysis we know that aperiodic function can always

be represented by a linear combination of harmonically related

complex exponentials (which is form of sampling).

This give us the Discrete Fourier Series representation.

We extend the DFS to finite-duration sequences, which leads

to a new transform, called the Discrete Fourier Transform.


6/84

Introduction The DFT avoids the two problems mentioned above

and is a numerically computable transform that is

suitable for computer implementation. The numerical computation of the DFT for long

sequences isprohibitively time consuming.

Therefore several algorithms have been developed to

efficiently compute the DFT.

These are collectively calledfast Fourier transform

(or FFT) algorithms.


7/84

The Discrete Fourier Series Definition: Periodic sequence

N: thefundamental periodof the sequences

From FT analysis we know that theperiodic functions can besynthesized as a linear combination of complex exponentialswhose frequencies are multiples (orharmonics) of the

fundamental frequency (2pi/N). From the frequency-domain periodicity of the DTFT, we

conclude that there are a finite number of harmonics; thefrequencies are {2pi/N*k,k=0,1,,N-1}.

knkNnxnx ,),(~)(~ !


8/84

The Discrete Fourier Series A periodic sequence can be expressed as

.,1,0,)(1

)(1

0

2

s!!

!

nekXnxk

knj T

},1,0),(~

{ .s!kK are called the discrete Fourier seriescoefficients, which are given by

.,1,0,)(~)(~ 1

0

2

s!!

!

kenxk

N

n

knjNT

The discrete Fourier series representation of periodic sequences


9/84

The Discrete Fourier Series X(k) is itself a (complex-valued)periodic

sequence with fundamental period equal to N.

!

!

!!

!!

!

1

0

1

0

)(1)]([I FS)(

)()]([DFS)(

Let2

N

k

nk

N

N

n

nk

N

j

N

WkXN

kXnx

WnxnxkX

eW NT

Example 5.1


10/84

To perform frequency analysis on a discrete-time signal,x(n),

need to convert the time-domain to an equivalent frequency-domainrepresentation. In order to this, need to use a powerful computationaltool to perform frequency analysis called

Discrete Fourier Transform or DFT.

The continuous Fourier Transform is defined as below:

However, this integral equation of Fourier Transform is not suitable toperform frequency analysis due to this 2 reasons:

Continuous nature can be handled by Computer

The limits of integration cannot be from minus infinity to infinity.There should be a finite length sequences that can be handled by

computer.

DFT : DEFINATION


11/84


12/84

DFT : PROPERTIES

The are 4 properties ofDFT:

1. Periodicity

If X(k) is the N-point DFT ofx(k),

x(n+N) =x(n), for all nX(k+N) = X(k), for all k

It shows that DFT is periodic with period N, also known

as Cyclic property of the DFT

2. Linearity

If X1(k) and X2(k) are the N-point DFTofx1(n) andx2(n),

ax1(n) + bx2(n)DFT aX1(k) + bX2(k)


13/84

3. Circular ShiftingLetx(n) be a sequence of length N and X(k) is N

-point DFT, thus the sequence,x,

(n) obtained from

x(n) by shiftingx(n) cyclically by m units. Then,

x,(n) DFT X(k)e-j2Tkm/N

4. Parsevals Theorem

ifx(n)DFT

X(k) andy(n) DFT Y(k)

thus, y*(n) = 1/N Y*(k)

!

1

0

)(N

n

nx

!

1

0

)(N

k

k


14/84

DFT : RELATIONSHIP WITH z-TRANSFORM

The z-transform of the sequence,x(n) is given by:

X(z) = z-n

, ROC include unit circle

by defining zk= ej2Tk/N, k = 0,1, 2,, N-1

X(k) = X(z)|zk = e

j2Tk/N , k = 0,1,2,, N-1

= e-j2Tnk/N

where k= 2Tk/N, k = 0,1,2,,N-1

g

g!n nx )(

g

g!n

nx )(


15/84

DFT : EXAMPLES

EXAMPLE 1:EXAMPLE 1:

Find the DFT for the following finite length sequence,

x(n) = { , , }

Solution :Solution :

1. Determine the sequence length, NN = 3, k = 0,1,2

2. Use DFT formula to determine X(k)

X(k) = e-j2Tnk/N , k = 0,1, 2

X(0) = + + =

X(1) = + e-j2T/3 + e-j4T/3

= + [cos (2T/3) jsin(2T/3) + [cos (4T/3) jsin(4T/3)= + [-0.5 j0.866] + [ -0.5 + j0.866]

= + [-1] = 0

!

1

0

)(N

n

nx


16/84

Continued from Examples 1:Examples 1:

X(2) = + e-j4T/3 + e-j8T/3

= + [cos (4T/3) jsin(4T/3)] + [cos (8T/3) jsin(8T/3)]

= + [-0.5 +j0.866] + [-0.5

j0.866]

= 0

Thus,

X(k) = { , 0, 0}


17/84

Examples 2:Examples 2:

Given the following the finite length sequences,

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

Perform DFT for this sequences.

Solution :Solution :1. Determine the sequence length, N = 6.

2. Use DFT formula to determine X(k).

X(k) = e-j2Tnk/N , k = 0,1,2,3,4,5

X(0) = 12, X(1) = -1.5 + j2.598

X(2) = -1.5 + j0.866, X(3) = 0

X(4) = -1.5 j0.866, X(5) = -1.5 j2.598

Thus,

X(k) = {12, -

1.5 + j2.598, -

1.5 + j0.866, 0, -

1.5 j0.866,

-1.5 j2.598}

!

1

0

)(N

n

nx


18/84

Examples 3:Examples 3:

Find the DFT for the convolution of 2 sequences :

x1(n) = {2,1, 2,1} &x2(n) = {1, 2, 3, 4}

Solution :Solution :

1. Determine the sequence length for eachsequence, N = 4. Thus, k = 0,1,2,3

2. Perform DFT for each sequences,

(i) X1(0) = 6, X1(1) = 0, X1(2) = 2, X2(3) = 0

X1(k) = {6,0,2,0}

(ii) X2(0) = 10, X2(1) = -2+j2, X2(2) = -2, X2(3) = -2-j2X2(k) = {10,-2+j2,-2,-2-j2}

3. Perform Convolution by :

X3(k) = X1(k) X2(k)

= {60, 0, -4, 0}


19/84

IDFT : DEFINATION

IDFT is the inverse Discrete Fourier Transform.

The finite length sequence can be obtained from the Discrete Fourier

Transform by performing IDFT.

The IDFT is defined as :

x(n) = 1/N e-j2Tnk/N,

where n = 0,1,, N-1

!

1

0

)(N

k

kX


20/84


21/84

IDFT :EXAMPLES

EXAMPLES 5:EXAMPLES 5:

Obtain the finite length sequence,x(n) from the DFT sequence inExample 3.

Solution :

1. The sequence in Example 3 is :

X3(k) = {60, 0, -4, 0}

2. Use IDFT formula to obtainx(n):

x3(n) = 1/4 e-j2Tnk/4,

x3(0) = 14, x3(1) = 16, x3(2) = 14, x3(3) = 16

Thus the finite length sequences are :

x3

(k) = {14,16,14,16}

!

3

0

)(

k

kX


22/84

DFT & IDFT : COMPLEXITY OF DFT

A Large number of multiplications and additions are required to compute

DFT.

To compute the 8-point ofDFT of the sequence,x(n),

The X(k) will be the summation ofx(0)e-j2T(0)k/8 untilx(7)e-j2T(7)k/8 For

the eight terms, there will be 64 multiplication (82) and 56 addition (8 x

(8-1))

Hence, for N-point DFT, there will be N2 multiplication and N(N-1)

addition.


23/84

Thus, need one algorithm to reduce the number of calculation and

speeds up the computation ofDFT. The algorithm is called Fast

Fourier Transform (FFT). It utilizes special properties of the DFT to

construct a computational procedure that requires Nlog2N complex

multiplication and Nlog2N complex addition to carry out N-point DFT.

The MATLAB command use to perform DFT & IDFT is:

1. fft2 () - for DFT

2. ifft2 () - forIDFT


24/84

FFT : DEFINATION

FFT is the algorithm that efficiently computes theDFT.

In applying FFT, the DFT expression can bewritten as :

X(k) = WNnk , k = 0,1,,N-1

where,WN = e-j2T/N

!

1

0

)(N

n

nx


25/84

IFFT :DEFINATION

The FFT forIDFT can be defined as:

x(n) = 1/N WNnk

WWNN22

= (e= (e--j2j2TT/N/N

))22

= e= e--j2j2TT2/N2/N

= W= WN/2N/2

!

1

0

)(N

k

kX


26/84

DFT & IDFT : SUMMARY

The DFT & IDFT can be summarized below:

1. It is a powerful method to perform frequency

analysis which are used widely in digital

image processing including blurring and enhancing. 2. Since the DFT & IDFT will become tedious

when the length of the sequence become

big, one algorithm is develop to overcome

this problem. 3. The algorithm can be found in MATLAB. The function

are :

1. FFT2 = to perform DFT

2.IFFT2 = to perform IDFT


27/84

Ex5.2 DFS of square wave seq.

Note

1. Envelope like the sinc function;

2. Zeros occur at N/L (reciprocal of duty cycle);3. Relation of N to density of freq. Samples;

ss!

!

)/sin(

)/sin(

,2,,0,

)( /)1(

Nk

NkL

e

NNkL

kXNLj

T

TT

.


28/84

Relation to the z-transform

!

! ee

!1

0

)()(else here,0

10,Nonzero)(

N

n

nznxzXNn

nx

? A

!

! ee

!1

0

2

)()(else here,0

10),()(

N

n

nkj

NenxkXNnnx

nxT

kNjezzX

kX T

2|)()(

~

!!

The DFS X(k) represents N evenly spaced samples of the z-transform X(z) around the unit circle.


29/84

Relation to the DTFT

k

j

N

n

j nN

n

j nj

NeXkX

enxenxeX

T2|)()(

)()()(1

0

1

0

!

!

!

!

!!

)()()(

2,

2

1

11

jkj

k

eXeXkX

kkN

wandN

wLet

k

!!

!!!TT

The DFS is obtained by evenly sampling the DTFT at w1 intervals.

The interval w1 is thesampling intervalin the frequency domain. It is called

frequency resolutionbecause it tells us how close are the frequency samples.


30/84

Sampling and construction in the

z-domain

g

g!

g

g!

!

!!

ss!!

m

km

N

m

kmj

ez

Wmxemx

kzXkX

N

kN

j

)()(

,2,1,0,|)()(~

2

2

T

T .

g

g!

g

g!

g

g!

g

g!

g

g!

g

g!

!

!

g

g!

!

!!

!!

!!

rr

r

N

k

nkN

N

k

kn

N

m

km

N

N

k

kn

N

rNnxrNmnmx

rNmnmxWN

mx

WWmxN

WkXN

nx

)()()(

)()(1)(

)(1

)(1

)(

1

0

)(

1

0

1

0

H

H

DFS & z-transform

IDFS


31/84

Comments

When wesample X(z) on the unit circle, we

obtain aperiodic sequence in the time domain.

This sequence is a linear combination of theoriginal x(n) and its infinite replicas, each

shifted by multiples of N or N.

If x(n)=0 for n=N, then there will beno overlap or aliasingin the time domain.


32/84

Comments

ee

!!

ee!

else

NnnxnRnxnx

Nnfornxnx

N

0

101)()()()(

10)()(

RN(n) is called a rectangular window of length N.

THEOREM1: Frequency Sampling

If x(n) is time-limited (finite duration) to [0,N-1], then N

samples of X(z) on the unit circle determine X(z) for all z.


33/84

Reconstruction Formula

Let x(n) be time-limited to [0,N-1]. Then from Theorem 1 we should be able to

recover the z-transform X(z) using its samples X~(k).

!

!

!

!

!

!

!

!

!

!

!

!

!!!

1

01

1

0

1

0

11

0

1

0

1

0

1

0

1

0

1

0

1

1

)(

~1

)(~1

)(~1

)(

~1

)(~

)()(

N

kk

N

NkN

N

N

k

N

n

nk

N

N

k

N

n

nkn

N

N

n

nN

k

kn

N

N

n

nN

n

n

zW

zW

kX

N

zWkXN

zWkXN

zW

kX

NznxznxzX

WN-kN

=1

!

!

1

011

)(~

1)(

N

kk

N

N

zW

kX

N

zzX


34/84

The DTFT Interpolation Formula

!

!

!

!

1

0/2

1

0/2 1

1)(

1

)(1)(

N

kjwNkj

jwNN

kjwNkj

jwNjw

eeN

ekX

ee

kX

N

eeX

TT

2

1

2

2

sin

sin)(

!*Njw

w

wN

eN

w An interpolation polynomial

!

*!1

0

2

)()(

N

kN

kjww

kXeX

TThis is the DTFT interpolation

formula to reconstruct X(e

jw

) fromits samples X~(k)

Since , we have that X(ej2pik/N)=X~(k), whichmeans that the interpolation is exact at sampling points.

1)0( !*


35/84

The Discrete Fourier Transform

The discrete Fourier series provided us a mechanismfornumerically computingthe discrete-time Fouriertransform.

It also alert us to a potential problem ofaliasingin thetime domain.

Mathematics dictates that the sampling of the discrete-time Fourier transform result in aperiodic sequences

x~(n).

But most of the signals in practice are not periodic.They are likely to be offinite duration.


36/84


Theoretically, we can take care of this problem bydefining aperiodic signal whoseprimary shape is thatof the finite duration signal and then using the DFS on

this periodic signal. Practically, we define a new transform called theDiscrete Fourier Transform (DFT), which is theprimary period of the DFS.

This DFT is the ultimate numerically computableFourier transform for arbitrary finite durationsequences.


37/84


First we define a finite-duration sequence x(n) that has N

samples over 0


38/84


The Discrete Fourier Transform of an N-point sequence isgiven by

10,)()(

)()(0

10)()]([)(

1

0

ee!

!

ee

!!

!

NkWnxkX

nRkXelse

NkkXnxDkX

N

n

nk

N

N

Note that the DFT X(k) is also an N-point sequence, that is, it is notdefined outside of 0


39/84

Matlab Implementation

XW

N

x

xWX

N

N

*1!

!

? A

-

!ee!

2)1()1(

)1(1

1

1

111

1,0

N

N

N

N

N

NNkn

NN

WW

WWNnkWW

.

/1//

.

.


40/84


41/84


42/84

Properties of the DFT

3. Conjugation:N

kXnxD ))(()]([** !

4. Symmetry properties for real sequences:

Letx(n) be a real-valued N-point sequence

NkXkX ))(()(* !

N

N

N

N

kXkX

kXkX

sequenceoddcircularkNXkX

sequenceevencircularkXkX

))(()(

|))((||)(|

:]))((Im[)](Im[

:]))((e[)](e[

!

!!

!


43/84

Comments:

Circular symmetry

Periodic conjugate symmetry

About 50% savings in computation as well as in storage.

X(0) is a real number: the DC frequency

X(N/2)(N is even) is also real-valued:Nyquist component

Circular-even and circular-odd components:

]))(()([2

1)(

]))(()([21)(

Noc

Nec

nxnxnx

nxnxnx

!

!

The real-valued signals

]))((Im[)](Im[)]([

]))((Re[)](Re[)]([

Noc

Nec

kXkXnxDFT

kXkXnxDFT

!!

!!

Function, p143


44/84

Properties

5. Circular shift of a sequence

)()]())(([ kXWnRmnxD kmNNN

!

6. Circular shift in the frequency domain

)())(()]([ kRlkXnxWDNN

nl

N!

7. Circular convolution**

10,))(()()()(1

02121 ee!

!

NnmnxmxnxnxN

mN

)()()]()([ 2121 kXkXnxnxDFT !


45/84

Properties

8. Multiplication: )()(1

)]()([ 2121 kXkXN

nxnxD !

9. Parsevals relation:

!

!

!!1

0

21

0

2 |)(|1

|)(|N

k

N

nx kX

NnxE

Energy spectrumPower spectrum 2

2

)(

|)(|

N

kX

N

kX


46/84

Linear convolution using the DFT

In general, the circular convolution is an aliasedversion of the

linear convolution.

If we make both x1(n) and x2(n) N=N1+N2-1 point sequences by

padding an appropriate number of zeros, then the circular

convolution is identicalto the linear convolution.

)()()()()(

)()()(

)())(()()()()(

3

1

021

1

021

1

021214

1

nrNnxnrNknxkx

nrNknxkx

nknxkxnxnxnx

Nr

Nr

N

m

N

N

m r

N

N

mN

-

!

-

!

-

!

-

!!

g

g!

g

g!

!

!

g

g!

!


47/84

Error Analysis

When N=max(N1,N2) is chosen for circular convolution, then

the first (M-1) samples are in error, where M=min(N1,N2).

10)(

)()]()([)(

),max()()(

)()()()(

3

33

210

3

3334

ee!

!

u

-

!

-

!!

{

g

g!

NnNnx

nNnxNnxne

NNNnrNnx

nxnrNnxxxne

N

Nr

Nr

X3(n) is also causaln=0,1,(N1+N2-1)-N


48/84

Block Convolution

Segment the infinite-length input sequence into smaller sections (or blocks),

process each section using the DFT, and finally assemble the output sequence

from the outputs of each section. This procedure is called a block convolution

operation.

Let us assume that the sequence x(n) is sectioned into N-point sequence and

that the impulse response of the filter is an M-point sequence, where M


49/84

The Fast Fourier Transform

Although the DFT is computable transform, the

straightforward implementation is very inefficient,

especially when the sequence length N is large.

In 1965, Cooley and Tukey showed the a procedure to

substantially reduce the amount of computations

involved in the DFT.

This led to the explosion of applications of the DFT. All these efficient algorithms are collectively known

asfast Fourier transform (FFT) algorithms.


50/84

The FFT

Using the Matrix-vector multiplication toimplement DFT:

X=WNx (WN: N*N, x: 1*N, X: 1*N) takes NN multiplications and (N-1)N

additions of complex number.

Number ofcomplex mult. CN=O(N2)

A complex multiplication requires 4 realmultiplications and 2 real additions.


51/84

Goal of an Efficient computation

The total number of computations should be linearrather than quadratic with respect to N.

Most of the computations can be eliminated using the

symmetry and periodicityproperties

CN=Nlog2N

If N=2^10, CN=will reduce to 1/100 times.

kn

N

Nkn

N

nNk

N

Nnk

N

kn

N

WW

WWW

!

!!

2/

)()(

Decimation-in-time: DIT-FFT, decimation-in-frequency: DIF-FFT


52/84


53/84

A4-point DFTFFT example

neededationmultiplicnoWAorAWso

WW

WW

Wwhere

Wxx

xxW

j

Whh

gg

jW

jhh

gg

XX

XX

22

1*12

0*12

1*0

2

0*0

22

22

221

212

21

21

**

1111

*)3()2(

)1()0(**.

1

11

**.1

11*

)3()2(

)1()0(

-

!

-

!

-

-

!

-

-

!

-

!

-


54/84

Divide-and-combine approach

To reduce the DFT computations quadratic dependence on N,

one must choose a composite numberN=LM since L2+M2


55/84


DFTpoM

M

m

mq

M

DFTpoL

L

l

lp

L

mp

N

M

m

Lmq

N

L

l

Mlp

N

mp

N

M

m

L

l

LqpmMl

N

WWmlxW

WWmlx

W

WmlxqpX

int

1

0

int

1

0

1

0

1

0

1

0

1

0

))((

),(

),(

),(),(

!

!

!

!

!

!

-

!

-

!

!

Three-step procedure: P155

Twiddle factor


56/84


The total number of complex multiplications

for this approach can now be given by

CN=ML2+N+LM2


57/84

A 8-point DFTFFT example

1. two 4-point DFT for m=1,2

-

-

!

-

!

-

-

!

vvvv

vvvv

vvvv

vvvv

!

! !

)1,3()0,3(

)1,2()0,2(

)1,1()0,1(

)1,0()0,0(

*),(

),(),(

334

234

134

034

324

224

124

024

314

214

114

014

304

204

104

004

3

0

2

1

0

3

0

48

xx

xx

xx

xx

WWWW

WWWW

WWWW

WWWW

mp

WmlxWWqpX

l

mq

m l

plpm

/


58/84

? A

-

-

!

!

v

!

vv

vv

!

112

012

10

2

00

2

2

1

0

8

8

*

)1,3()0,3(

)1,2()0,2()1,1()0,1(

)1,0()0,0(

),(),(

2.3

8.24

),(),(.2

WW

WW

GG

GG

GG

GG

WmpFWqp

withDFTXntpoi

multnscplxofmultdotmatrixais

mpFWmpG

mq

m

pm

pm


59/84

Number of multiplications

A 4-point DFT is divided into two 2-point

DFTs, with one intermedium matrix mult.

number of multiplications=

44cplx 21+ 14 cplx 16 6

A 8-point DFT is divided into two 4-point

DFTs, with one intermedium matrix mult.

8826 + 24 64 20

For16-point DFT:

1616220 + 28 256 56


60/84

Radix-2 FFT Algorithms

Let N=2v; then we choose M=2 and L=N/2 and divide x(n) into

two N/2-point sequence.

This procedure can be repeatedagain and again. At each stage

the sequences are decimatedand the smallerDFTs combined.This decimation ands afterv stages when we have N one-point

sequences, which are also one-point DFTs.

The resulting procedure is called the decimation-in-time FFT

(DIF-FFT) algorithm, for which the total number of complexmultiplications is: CN=Nv= N*log2N;

using additional symmetries: CN=Nv= N/2*log2N

Signal flowgraph in Figure 5.19


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Decimation-in-frequency FFT

In an alternate approach we choose L=2, M=N/2 and

follow the steps in (5.49).

We can get the decimation-frequency FFT (DIF-FFT)algorithm.

Its signal flowgraph is a transposed structure of the

DIT-FFT structure.

Its computational complexity is also equal to CN=Nv=N/2*log2N


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Matlab Implementation

Function: X = fft(x,N)

If length(x)


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Fast Convolutions

Use the circular convolution to implement the linear

convolution and the FFT to implement the circular

convolution.

The resulting algorithm is called a fast convolution algorithm.

If we choose N=2v and implement the radix-2 FFT, then the

algorithm is called a high-speed convolution.

If x1(n) is N1-point, x2(n) is N2-point, then

Compare the linear convolution and the high-speed conv.

)1(log 2122! NNN


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High-speed Block Convolution

Overlap-and-save method

We can now replace the DFT by the radix-2

FFT algorithm to obtain a high-speed overlap-and-save algorithm.


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Discrete Fourier transform

DFT

110 ,...,, Nxxx

110

,...,,

NXXX

!

!1

0

2N

i

ikN

j

ik

exXT

Maps a set of input

points to anotherset of outputpoints.The operation isreversible.


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Roots of the unity

real

imaginary

(1, 0)

(0, j)

(-1, 0)

(0, -j)

What are the Nth roots of unity?If N = 8 then we have

78

26

8

2

58

24

8

2

38

22

8

2

1

8

20

8

2

,

,

,

,

TT

TT

TT

TT

jj

jj

jj

jj

ee

ee

ee

ee

Define Nj

N eW

T2

!


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Calculating the DFT

!

!

!!1

0

1

0

2 N

i

ik

i

N

i

ikN

j

ikN

WxexX

T

!

1

3

2

1

0

321

3963

2642

1321

1

3

2

1

0

.

...1

...

...1

...1

...1

1...1111

...

N

N

N

N

N

N

N

N

NNNN

N

NNNN

N

NNNN

N x

x

x

x

x

WWWW

WWWW

WWWW

WWWW

X

X

X

X

X

How many arithmetic (+ and *) operations do we need to

calculate the DFT?


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E l h N 8


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Example when N=8Objective: Compute X0, X1, X7 given x0, x1, , x7

magicbox

magicbox

x0

x2

x4

x6

x1

x3

x5

x7

eX0eX1eX2eX3

oX0oX1o

X2oX3

1

8W

0

8WX0

2

8W

38W

4

8W

5

8W

6

8W

7

8W

X1

X2

X3

X4

X5

X6

X7

k

N

Nk

N WW ! 2/Note

that


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Now lets apply the idea recursively

x0

x4

x2

x6

x1

x5

x3

x7

eX0

eX1

eX

2e

X3

oX0

oX

1o

X2

oX3

1

8W

0

8WX0

2

8W

3

8W

4

8W

5

8W6

8W

7

8W

X1

X2

X3

X4

X5

X6

X7

eeX0

eeX1

eoX0

eoX1

oeX0

oeX1

ooX0

ooX1

0

4W

1

4W

2

4W

3

4W

0

4W

1

4W

2

4W

3

4W


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One more timex0

x4

x2

x6

x1

x5

x3

x7

eX0e

X1e

X2

eX

3

oX0

oX1

oX

2o

X3

1

8W

0

8WX0

2

8W

3

8W

4

8W

5

8W

6

8W

7

8W

X1

X2

X3

X4

X5

X6

X7

ee

X0

eeX1

eoX0

eoX1

oeX0

oeX1

ooX

0

ooX1

0

4W

1

4W

2

4W

3

4W

0

4W

1

4W

2

4W

3

4W

How many operations do we need now? What is the execution time on a general purpose CPU?

What is the execution time on a FPGA? How many resources u need?


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Another way to visualize FFT computations

How can we determine the order of the first inputs?

x0

x4

x2

x6

x1

x5

x3

x7

Butter

flyButte

rfly

Butterfly

Butter

fly

Butter

flyButter

fly

Butter

flyButter

fly

Butter

flyButter

fly

Butter

flyButte

r

fly

X0X4

X1X5

X2X6

X3X7


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Savings so far We have split the DFT computation into two halves:

Have we gained anything? Consider the nominal number of multiplications

for

Original form produces multiplications

New form produces multiplications

So were already ahead .. Lets keep going!!

X[k] !

k!0

N1

x[n]WNnk

!n!0

(N/ 2 )1

x[2r]WN / 2rk

WNk

n!0

(N/ 2 )1

x[2r1]WN / 2rk

N! 8

2(42 ) 8 ! 40

82 ! 4


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Signal flowgraph notation

In generalizing this formulation, it is most convenient to adopt a graphic

approach

Signal flowgraph notation describes the three basic DSP operations:

Addition

Multiplication by a constant

Delay

x[n]

y[n]x[n]+y[n]

x[n]a

ax[n]

x[n] x[n-1]z-1

Si l fl h t ti


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Signal flowgraph representation

of 8-point DFT

Recall that the DFT is now of the form

The DFT in (partial) flowgraph notation:

X[k] ! G[k] WN

kH[k]


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The complete decomposition into


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The complete decomposition into

2-point DFTs

Now lets take a closer look at


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Now lets take a closer look at

the 2-point DFT

The expression for the 2-point DFT is:

Evaluating for we obtain

which in signal flowgraph notation looks like ...

X[k] !

n!0

1

x[n] 2nk!

n!0

1

x[n]ej 2Tnk/ 2

k! 0,

X[0] ! x[0] x[1]

X[1] ! x[0] ej2T1 / 2x[1] ! x[0] x[1]

This topology is referred to as the

basic butterfly

The complete 8 point


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The complete 8-point

decimation-in-time FFT


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Comparing processing with and


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Slow DFT requiresNmults; FFT requiresNlog2(N) mults

Filtering using FFTs requires 3(Nlog2(N))+2Nmults

Let

N E1

E

216 .25 .8124

32 .156 .50

64 .0935 .297

128 .055 .171256 .031 .097

1024 .0097 .0302

Comparing processing with and

without FFTs

Note: 1024-point FFTs

accomplish speedups of 100

for filtering, 30 for DFTs!

E1 ! Nlog2 (N) / N2; E2 ! [ (Nlog2 (N)) N] / N

2


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Additional timesavers: reducing

multiplications in the basic butterfly

As we derived it, the basic butterfly is of the form

Since we can reducing computation by 2 by

premultiplying by

W/

! 1

WNr

WNr

WN

rN /

WNr

1

1


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dsp projects call 9952749533

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