Decimation in time and frequency

DECIMATION IN DECIMATION IN TIME AND FREQUENCYTIME AND FREQUENCY

Dr. C. SarithaDr. C. Saritha

Lecturer in ElectronicsLecturer in Electronics

SSBN Degree & PG CollegeSSBN Degree & PG College

ANANTAPUR ANANTAPUR

INDEX

INTRODUCTION TO FFTINTRODUCTION TO FFT DECIMATION IN TIME(DIT)DECIMATION IN TIME(DIT) DECIMATION IN FREQUENCY(DIF)DECIMATION IN FREQUENCY(DIF) DIFFERENCES AND SIMILARITIESDIFFERENCES AND SIMILARITIES

Fourier TransformFourier Transform A fourier transform is an useful A fourier transform is an useful

analytical tool that is important for many analytical tool that is important for many fields of application in the digital signal fields of application in the digital signal processing.processing.

In describing the properties of the In describing the properties of the fourier transform and inverse fourier fourier transform and inverse fourier transform, it is quite convenient to use transform, it is quite convenient to use the concept of time and frequency.the concept of time and frequency.

In image processing applications it plays In image processing applications it plays aa

critical role. critical role.

Fast fourier transformFast fourier transform

Fast fourier transform proposed by Fast fourier transform proposed by Cooley and Tukey in 1965.Cooley and Tukey in 1965.

The fast fourier transform is a highly The fast fourier transform is a highly efficient procedure for computing the efficient procedure for computing the DFT of a finite series and requires DFT of a finite series and requires less number of computations than less number of computations than that of direct evaluation of DFT.that of direct evaluation of DFT.

The FFT is based on decomposition The FFT is based on decomposition and breaking the transform into and breaking the transform into smaller transforms and combining smaller transforms and combining them to get the total transform.them to get the total transform.

DiscrDiscrete Fourier Transformete Fourier Transform The DFT pair was given asThe DFT pair was given as

Baseline for computational complexity: Baseline for computational complexity:

Each DFT coefficient requiresEach DFT coefficient requiresN complex multiplicationsN complex multiplicationsN-1 complex additionsN-1 complex additions

All N DFT coefficients requireAll N DFT coefficients requireNN22 complex multiplications complex multiplicationsN(N-1) complex additionsN(N-1) complex additions

1N

0k

knN/2jekXN1

]n[x

1

0

/2][N

n

knNjenxkX

What is FFT?What is FFT? The fast fourier is an algorithm used The fast fourier is an algorithm used

to compute the DFT. It makes use of to compute the DFT. It makes use of the symmetry and periodicity the symmetry and periodicity properties of twiddle factor wproperties of twiddle factor wN N to to effectively reduce the DFT effectively reduce the DFT computation time.computation time.

It is based on the fundamental It is based on the fundamental principle of decomposing the principle of decomposing the computation of DFT of a sequence of computation of DFT of a sequence of length N into successively smaller length N into successively smaller DFT.DFT.

Symmetry and periodicitySymmetry and periodicity

Symmetry

Periodicity

)()(

)()(

)(

kNnN

nNkN

knN

nNkN

NnkN

knN

knN

knN

WWW

WWW

WW

kN

N/kN

N/N

mnkmN

nkN

mnkmN

nkN

WWW

WWWW

)2(2

//

,1

,

FFT algorithm provides speed FFT algorithm provides speed increase factors, when compared increase factors, when compared with direct computation of the DFT, with direct computation of the DFT, of approximately 64 and 205 for 256 of approximately 64 and 205 for 256 point and 1024 point transforms point and 1024 point transforms respectively.respectively.

The number of multiplications and The number of multiplications and additions required to compute N-additions required to compute N-point DFT using radix-2 FFT are point DFT using radix-2 FFT are NlogNlog22N and N/2 logN and N/2 log22N respectively.N respectively.

Example:Example:

The number of complex multiplications The number of complex multiplications required using direct computation isrequired using direct computation is

NN22=64=6422 =4096 =4096

The number of complex multiplications The number of complex multiplications required using FFT is required using FFT is

N/2logN/2log2 2 N=64/2logN=64/2log2 2

64=19264=192

Speed improvement factor Speed improvement factor =4096/192= 21.33.=4096/192= 21.33.

FFT AlgorithmsFFT Algorithms

There are basically two types of FFT There are basically two types of FFT algorithms. algorithms.

They are:They are:

1.1. Decimation in Time Decimation in Time

2.2. Decimation in frequency Decimation in frequency

Decimation in timeDecimation in time DIT algorithm is used to calculate the DIT algorithm is used to calculate the

DFT of a N-point sequence.DFT of a N-point sequence. The idea is to break the N-point The idea is to break the N-point

sequence into two sequences, the DFTs sequence into two sequences, the DFTs of which can be obtained to give the of which can be obtained to give the DFT of the original N-point sequence.DFT of the original N-point sequence.

Initially the N-point sequence is divided Initially the N-point sequence is divided into N/2-point sequences xinto N/2-point sequences xee(n) and (n) and xx00(n) ,(n) ,

which have even and odd numbers of which have even and odd numbers of x(n) respectively.x(n) respectively.

The N/2-point DFTs of these two The N/2-point DFTs of these two sequences are evaluated and sequences are evaluated and combined to give the N-point DFT.combined to give the N-point DFT.

Similarly the N/2-point DFTs can be Similarly the N/2-point DFTs can be expressed as a combination of N/4-expressed as a combination of N/4-point DFTs.point DFTs.

This process is continued until we are This process is continued until we are left with two point DFT.left with two point DFT.

This algorithm is called decimation-This algorithm is called decimation-in-time because the sequence x(n) is in-time because the sequence x(n) is often split into smaller sequences.often split into smaller sequences.

Radix-2 DIT- FFT AlgorithmRadix-2 DIT- FFT Algorithm

Radix-2: the sequence length N satisfied:L is an integer

LN 2

To decompose an N point time domain signal into N signals each containing a single point. Each decomposing stage uses an interlace decomposition, separating the even- and odd-indexed samples;

To calculate the N frequency spectra corresponding to these N time domain signals.


0 4 8 12 2 6 10 14 1 5 9 13 3 7 11 15

0 8 4 12 2 10 6 14 1 9 5 13 3 11 7 15

0 8 4 12 2 10 6 14 1 9 5 13 3 11 7 15

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 signal of 16 points

0 2 4 6 8 10 12 14 1 3 5 7 9 11 13 152 signals of 8 points

4 signals of 4 points

8 signals of 2 points

16 signals of 1 point


Algorithm principle

To divide N-point sequence x(n) into two N/2-point sequence x1(r) and x2(r)

12

,2,1,0 , )12()( );2()( 21 N

rrxrxrxrx To compute the DFT of x1(r) and x2(r)

)12

~0( )12()()(

)12

~0( )2()()(

12

0 2

12

0 2

22

12

0 2

12

0 2

11

NkWrxWrxkX

NkWrxWrxkX

N

r

rkN

N

r

rkN

N

r

rkN

N

r

rkN

To compute the DFT of N-point sequence x(n)

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

)()(

)12()2(

)()()()(

21

12

0 2

2

12

0 2

1

12

0

)12(

12

0

2

1

)(0

1

)(0

1

0

NkkXWkX

WrxWWrx

WrxWrx

WnxWnxWnxkX

kN

N

r

rkN

kN

N

r

rkN

N

r

krN

N

r

rkN

N

oddn

nkN

N

evenn

nkN

N

n

nkN

)12

,1,0( )()(

)2

()2

()2

(

)12

,1,0( )()()(

21

2

)2

(

1

21

NkkXWkX

NkXW

NkX

NkX

NkkXWkXkX

kN

Nk

N

kN

)(nx

)(1 rx

)(2 rx

)(1 kX

)(2 kX

)(kX

Butterfly computation flow graph

)12

,1,0( )()()2

(

)12

,1,0( )()()(

21

21

NkkXWkX

NkX

NkkXWkXkX

kN

kN

)(1 kX

)(2 kXk

NW

)()( 21 kXWkX kN

)()( 21 kXWkX kN

1

There are 1 complex multiplication and 2 complex additions

N/2-point

DFT

N/2-point

DFT

)0(1X

)1(1X

)2(1X

)3(1X

)0(2X

)1(2X

)2(2X

)3(2X

0NW

1NW

2NW

3NW

)0()0(1 xx

)2()1(1 xx

)4()2(1 xx

)6()3(1 xx

)1()0(2 xx

)3()1(2 xx

)5()2(2 xx

)7()3(2 xx

)(1 rx

)(2 rx

)4(X1

)5(X1

)6(X1

)7(X1

)0(X

)1(X

)2(X

)3(X

N-point DFT


for 32N

)(nx2-point

DFT

2-pointDFT

2-pointDFT

2-pointDFT

Synthesize the 2-point DFTs into a 4-point DFT



)(kX

3-stage synthesize, each has N/2 butterfly computation

The computation complexity


•At the end of computation flow graph at any stage, output variables can be stored in the same registers previously occupied by the corresponding input variables.

•This type of memory location sharing is called in-place computation which results in significant saving in overall memory requirements.

The distance between two nodes in a butterfly

For there are L stagesLN 2

Stage Stage DistanceDistance

stage 1stage 1 11

stage 2stage 2 22

stage 3stage 3 44

stage stage LL

12 L


Bit-reversed order

In the DFT computation scheme, the DFT samples X(k) appear at the output in a sequential order while the input samples x(n) appear in a different order: a bit-reversed order.

Thus, a sequentially ordered input x(n) must be reordered appropriately before the fast algorithm can be implemented.

Let m, n represent the sequential and bit-reversed order in binary forms respectively, then:m: 000 001 010 011 100 101 110 111

n: 000 100 010 110 001 101 011 111

Why is the input bit-reversed order

1n 2n

)( 012 nnnx

0n

0

1

0

1

0

10

1

0

1

0

10

1

)000(x

)100(x

)010(x

)110(x

)001(x

)101(x

)011(x

)111(x

)(0x

)(4x

)(2x

)(6x

)(1x

)(5x

)(3x

)(7x

How to get the bit-reversed order

Let represent the natural order, the represent the bit-reversed order, then:

n n̂

)ˆ()(ˆ nxnxnn , if

)0(A )1(A )2(A )3(A )4(A )5(A )6(A )7(A

)0(x )1(x )2(x )3(x )4(x )5(x )6(x )7(xn

n̂ )0(x )7(x)1(x)4(x )6(x)2(x )3(x)5(x

Decimation-In-FrequencyDecimation-In-Frequency

It is a popular form of FFT algorithm.It is a popular form of FFT algorithm. In this the output sequence x(k) is In this the output sequence x(k) is

divided into smaller and smaller divided into smaller and smaller subsequences, that is why the name subsequences, that is why the name decimation in frequency,decimation in frequency,

Initially the input sequence x(n) is Initially the input sequence x(n) is divided into two sequences x1(n) and divided into two sequences x1(n) and x2(n) consisting of the first n/2 x2(n) consisting of the first n/2 samples of x(n) and the last n/2 samples of x(n) and the last n/2 samples of x(n) respectivelysamples of x(n) respectively

Radix-2 DIF- FFT AlgorithmRadix-2 DIF- FFT Algorithm

Algorithm principle

To divide N-point sequence x(n) into two N/2-point sequence

12

0 ),2

(

12

0 ),(

Nn

Nnx

NnnxThe former N/2-point

The latter N/2-point

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

0 1 2 3 0 1 2 3

butterfly computation

0 1 2 3 0 1 2 3

butterfly computation butterfly computation

0 1 0 1 0 1 0 1

butterfly butterfly butterfly butterfly

0 4 2 6 1 5 3 7

0 1 0 1 0 10 1

)(nx

)(kX

To compute the DFT of N-point sequence x(n)

)1,1,0( )2

()1()(

)2

()(

)2

()(

)()()()(

12

0

12

0

2

12

0

)2

(1

2

0

1

2

12

0

1

0

NkWNnxnx

WNnxWnx

WNnxWnx

WnxWnxWnxkX

N

n

nkN

k

N

n

nkN

kN

N

N

n

kNn

N

N

n

nkN

N

Nn

nkN

N

n

nkN

N

n

nkN


To separate the even and odd numbered samples of X(k)

)12

,,1,0 ( ,12 ,2let Nrrkrk

)12

,1,0( )2

()( )2(1

2

0 2

NrWNnxnxrX

N

n

nrN

)12

,1,0( )2

()( )12(1

2

0 2

NrWWNnxnxrX

N

n

nrN

nN


)12

,1,0( )( )12(

)12

,1,0( )( )2(

12

0 22

12

0 21

NrWnxrX

NrWnxrX

N

n

nrN

N

n

nrN

12

,1,0 )

2()()(

)2

()()(let

2

1

N

nW

Nnxnxnx

Nnxnxnx

nN


)(nx

)2

( Nnx n

NW

1

)2

()()(1Nnxnxnx

nNW

Nnxnxnx )

2()()(2

Butterfly computation flow graph

There are 1 complex multiplication and 2 complex additions

for 32N

N/2-point

DFT

N/2-point

DFT

)0(X

)2(X

)4(X

)6(X

)1(X

)3(X

)5(X

)7(X

)0(1x

)1(1x

)2(1x

)3(1x

)0(x

)1(x

)2(x

)3(x

)4(x

)5(x

)6(x

)7(x)3(2x3

NW1

)2(2x2NW

1

)1(2x1NW

1

)0(2x0NW

1

for 32N

)0(x

)2(x

)1(x

)3(x

)4(x

)6(x

)5(x

)7(x

)0(X

)4(X

)2(X

)6(X

)1(X

)5(X

)3(X

)7(X

1

1

1

1

0NW

1NW

2NW

3NW

1

1

1

1

1

0NW

1

0NW

0NW

2NW

1

0NW

1

0NW

0NW

2NW


The comparison of DIT and DIF

The order of samplesDIT-FFT: the input is bit- reversed order and the output is natural orderDIF-FFT: the input is natural order and the output is bit- reversed order

The butterfly computationDIT-FFT: multiplication is done before additionsDIF-FFT: multiplication is done after additions


Both DIT-FFT and DIF-FFT have the identical computation complexity. i.e. for , there are total L stages and each has N/2 butterfly computation. Each butterfly computation has 1 multiplication and 2 additions.

LN 2

Both DIT-FFT and DIF-FFT have the characteristic of in-place computation.

A DIT-FFT flow graph can be transposed to a DIF-FFT flow graph and vice versa.

Decimation in time and frequency

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