chapter 5 finite-length discrete transform. §5.2.1 the discrete fourier transform definition - the...

Chapter 5

Finite-Length DiscretFinite-Length Discrete Transforme Transform

§5.2.1 The Discrete Fourier Transform

Definition - The simplest relation between a length-N sequence x[n], defined for 0≤n≤N-1, and its DTFT X(ejω) is obtained by uniformly sampling X(ejω) on the ω-axis between 0≤ω≤2π at ωk= 2πk/N, 0≤k≤N-1

From the definition of the DTFT we thus have

10,][)(][1

0

/2

/2

NkenxeXkX

N

n

Nkj

Nk

j


Note: X[k] is also a length-N sequence in the frequency domain

The sequence X[k] is called the discrete Fourier transform (DFT) of the sequence x[n]

Using the notation WN= e-j2π/N the DFT is usually expressed as:

10,][][1

0

NkWnxkX

N

n

nkN


The inverse discrete Fourier transform

(IDFT) is given by

To verify the above expression we multiply both sides of the above equation by and sum the result from n = 0 to n=N-1

Wn

N

10,][1

][1

0

NnWkXN

nxN

k

knN


resulting in

1

0

1

0

)(

1

0

1

0

)(

1

0

1

0

1

0

][1

][1

][1

][

N

k

N

k

nKN

N

n

N

k

nKN

N

n

nN

N

k

knN

N

n

nN

WkXN

WkXN

WWkXN

Wnx


Making use of the identity

we observe that the RHS of the last

equation is equal to X[l]

][][ XWnxN

n

nN

1

0

otherwise,0integeran,for,1

0

)( rrNkNW

N

n

nkN

Hence


Example – Consider the length-N sequence

11001

Nn

n,,

][nx

101]0[][][ 01

0

NkWxWnxkX N

N

n

knN ，

Its N-point DFT is given by


Example – Consider the length-N sequence

11,10,0

,1][

Nnmmn

mnny

10

][][][1

0

Nk

WWmyWnykY kmN

kmN

N

n

knN ，

Its N-point DFT is given by


Example – Consider the length-N sequencedefined for 0≤n≤N-1

10),/2cos(][ NrNrnng

)(

2

12

1][ /2/2

rnN

rnN

NrnjNrnj

WW

eeng

Using a trigonometric identity we can write


The N-point DFT of g[n] is thus given by

102

1

][][

1

0

1

0

)()(

1

0

Nk

WW

WngkG

N

n

N

n

nkrN

nkrN

N

n

knN



otherwise,0integeran,for,1

0

)( rrNkNW

N

n

nkN

we get

10otherwis0for,2/for,2/

][

Nk

rNkNrkN

kG，

§5.2.2 Matrix Relations The DFT samples defined by

10,][][1

0

NkWnxkXN

n

knN

T

T

NxxxxNXXXX

]1[]1[]]0[[]]1[]1[]0[[

can be expressed in matrix form asX=Dnx

where

§5.2.2 Matrix Relations

and DN is the N×N DFT matrix given by

2)1()1(2)1(

)1(242

)1(21

1

1

1

1111

NN

NN

NN

NNNN

NNNN

N

WWW

WWW

WWW

D

§5.2.2 Matrix Relations Likewise, the IDFT relation given by

where is the N×N DFT matrixDN

1

10,][1

][1

0

NnWKXN

nxN

k

knN

XDx N1

can be expressed in matrix form as

§5.2.2 Matrix Relationswhere

Note:DD NN N

*1 1

2)1()1(2)1(

)1(242

)1(21

1

1

1

1

1111

NN

NN

NN

NNNN

NNNN

N

WWW

WWW

WWW

D

§5.2.3 DFT Computation UsingMATLAB

The functions to compute the DFT and the

IDFT are fft and ifft These functions make use of FFT algorithm

s which are computationally highly efficient compared to the direct computation

Programs 5_1.m and 5_2.m illustrate the use of these functions

§5.2.3 DFT Computation UsingMATLAB

Example –a Program 5_3.m can be used to compute the DFT and the DTFT of the sequence x[n]=cos(6πn/16), 0≤n≤15as shown below

indicates DFT samples

§5.3.3 DTFT from DFT byInterpolation

The N-point DFT X[k] of a length-N

sequence x[n] is simply the frequency samples of its DTFT X(ejω) evaluated at N uniformly spaced frequency points

10,/2 NkNkk Given the N-point DFT X[k] of a length-N s

equence x[n], its DTFT X(ejω) can be uniquely determined from X[k]


Thus

S

N

n

nNkjN

k

njN

n

kn

N

N

k

N

n

njj

e

eW

ee

kXN

kXN

nxX

1

0

)/2(1

0

1

0

1

0

1

0

][1

]][1

[

][)(


To develop a compact expression for the sum S, let

)/2( Nkjer Then From the above

1

0

N

n

nrS

11

111

0

1

1

1

NNN

n

n

NN

n

N

n

nn

rSrr

rrrrS


Or, equivalently,

Hence

]2/)1)][(/2[

)]/2([

)2(

22

sin

22

sin

1

1

1

1

NNkj

Nkj

kNjN

e

NkN

kNe

e

r

rS

NrSrrSS 1)1(


therefore

]2/)1)][(/2[1

0

22

sin

22

sin][

1

)(

NNkjN

k

j

e

NkN

kN

kXN

eX

§5.3.4 Sampling the DTFT

Consider a sequence x[n] with a DTFT X(ejω)

We sample X(ejω) at N equally spaced points ωk=2πk/N, 0≤k≤N-1 deleloping the N frequency samples

These N frequency samples can be considered as an N-point DFT Y[k] whose N- point IDFT is a length-N sequence y[n]

)}({ e kjX


Now

＝－

jj exeX ][)(

＝－＝－

kN

Nkj

Nkjj

Wxex

eXeXkY k

][][

)()(][

/2

/2

knN

N

k

WkYN

ny

1

0

][1

][＝

Thus

An IDFT of Y[k] yields


i.e.

1

0

)(

1

0

1][

][1

][

N

k

nkN

knN

kN

N

k

WN

x

WWxN

ny

＝


otherwise,0for,11 1

0

)( mNnrW

N

N

n

rnkN

＝


we arrive at the desired relation

m

NnmNnxny 10,][][

Thus y[n] is obtained from x[n] by adding an infinite number of shifted replicas of x[n], with each replica shifted by an integer multiple of N sampling instants, and observing the sum only for the interval 0≤n≤N-1

§5.3.4 Sampling the DTFT To apply

10,][][

NnmNnxnym

to finite-length sequences, we assume that the samples outside the specified range are zeros

Thus if x[n] is a length-M sequence with M≤N, then y[n]= x[n] , for 0≤n≤N-1

§5.3.4 Sampling the DTFT If M > N, there is a time-domain aliasing of s

amples of x[n] in generating y[n], and x[n] cannot be recovered from y[n]

Example – Let {x[n]}={0 1 2 3 4 5} ↑

By sampling its DTFT X(ejω) at ωk=2πk/4, 0≤k≤3 and then applying a 4-point IDFT to these samples, we arrive at the sequence y[n] given by


y[n]= x[n]+ x[n+4]+x[n-4] 0≤n≤3 i.e. {y[n]}={4 6 2 3}

↑ {x[n]} cannot be recovered from {y[n]}

Numerical Computation of theDTFT Using the DFT

A practical approach to the numerical computation of the DTFT of a finite-length sequence

Let X(ejω) be the DTFT of a length-N sequence x[n]

We wish to evaluate X(ejω) at a dense grid of frequencies ωk=2πk/M, 0≤k≤M-1,where M >> N:


Define a new sequence

MknjN

n

njN

n

j enxenxeX kk /21

0

1

0

][][)(

MknjM

ne

j enxeX k /21

0

][)(

1,0

10,][][

MnN

Nnnxnxe

Then


The DFT Xe[k] can be computed very efficiently using the FFT algorithm if M is an integer power of 2

The function freqz employs this approach to evaluate the frequency response at a prescribed set of frequencies of a DTFT expressed as a rational function in e-jω

Thus is essentially an M-point DFT Xe[k] of the length-M sequence xe[n]

)(e kjX

DFT Properties Like the DTFT, the DFT also satisfies a num

ber of properties that are useful in signal processing applications

Some of these properties are essentially identical to those of the DTFT, while some others are somewhat different

A summary of the DFT properties are given in tables in the following slides

Table 5.1: DFT Properties: Symmetry Relations

x[n] is a complex sequence

Table 5.2: DFT Properties: Symmetry Relations

x[n] is a real sequence

Table 5.3: General Properties of DFT

§5.4.1 Circular Shift of a Sequence

This property is analogous to the time- shifting property of the DTFT as given in Table 3.4, but with a subtle difference

Consider length-N sequences defined for

0≤n≤N-1 Sample values of such sequences are equal

to zero for values of n<0 and n≥N


If x[n] is such a sequence, then for any arbitrary integer n0, the shifted sequence

x1[n]= x(n-n0)

is no longer defined for the range 0≤n≤N-1 We thus need to define another type of a shi

ft that will always keep the shifted

sequence in the range 0≤n≤N-1


The desired shift, called the circular shift, is defined using a modulo operation:

][][ 0 Nc nnxnx

00

00

0,][

1,][][

nnfornnNx

Nnnfornnxnxc

For n0>0 (right circular shift), the above equation implies


Illustration of the concept of a circular shift

]5[]1[

6

6

nxnx －

]2[]4[

6

6

nxnx －][nx


As can be seen from the previous figure, a right circular shift by n0 is equivalent to a left circular shift by N-n0 sample periods

A circular shift by an integer number n0 greater than N is equivalent to a circular shift by

Nn0

§5.4.2 Circular Convolution

This operation is analogous to linear convolution, but with a subtle difference

Consider two length-N sequences, g[n] and

h[n], respectively Their linear convolution results in a length-(2

N-1) sequence yL[n] given by

2201

0

Nnmnhmgny

N

mL ],[][][


In computing yL[n] we have assumed that both length-N sequence have been zero-padded to extend their lengths to 2N-1

The longer form of yL[n] results from the time-reversal of the sequence h[n] and its linear shift to the right

The first nonzero value of yL[n] is

yL[0]=g[0]h[0] ,and the last nonzero value is

yL[2N-2]=g[N-1]h[N-1]

§5.4.2 Circular Convolution To develop a convolution-like operation resul

ting in a length-N sequence yC[n] , we need to define a circular time-reversal, and then apply a circular time-shift

Resulting operation, called a circular convolution, is defined by

10],[][][1

0

Nnmnhmgny

N

mNC

§5.4.2 Circular Convolution Since the operation defined involves two leng

th-N sequences, it is often referred to as an N-point circular convolution, denoted as

The circular convolution is commutative, i.e.

][][][][ ngnhnhng NN

][][][ nhngny N


Example – Determine the 4-point circular convolution of the two length-4 sequences:

as sketched below

}1122{]}[{},1021{]}[{ nhng


The result is a length-4 sequence yC[n] given by

6)21()10()12()21(

]1[]3[]2[]2[]3[]1[]0[]0[

][][]0[ 4

3

0

hghghghg

mhmgm

Cy

,][][][][][3

04

mC mnhmgnhngny 4

30 n From the above we observe


Likewise

7)11()10()22()21(]2[]3[]3[]2[]0[]1[]1[]0[

]1[][]1[ 4

3

0

hghghghg

mhmgm

Cy

6)11()20()22()11(]3[]3[]0[]2[]1[]1[]2[]0[

]2[][]2[ 4

3

0

hghghghg

mhmgm

Cy


The circular convolution can also be computed using a DFT-based approach as indicated in Table 5.3

5)21()20()12()11(

]0[]3[]1[]2[]2[]1[]3[]0[

]3[][]3[ 4

3

0

hghghghg

mhmgm

Cy

yC[n]

and

§5.4.2 Circular Convolution Example – Consider the two length-4

sequences repeated below for convenience:

The 4-point DFT G[k] of g[n] is given by

G[k]= g[0]+g[1]e-j2πk/4 +g[2]e-j4πk/4 +g[3]e-j6πk/4

=1+2e-jπk/2 +e-j3πk/2 , 0≤k≤3

§5.4.2 Circular Convolution Therefore G[0]=1+2+1=4,

G[1]=1-j2+j=1-j,

G[2]=1-2-1=-2,

G[3]= 1+j2-j=1+j Likewise,

H[k]=h[0]+h[1]e-j2πk/4 +h[2]e-j4πk/4

+h[3]e-j6πk/4

=2+2e-jπk/2 +e-jπk +e-j3πk/2 , 0≤k≤3


Hence ， H[0]=2+2+1+1=6,

H [1]=2-j2-1+j=1-j,

H [2]=2-2+1-1=0,

H [3]=2+j2-1-j=1+j The two 4-point DFTs can also be computed

using the matrix relation given earlier


j

j

jj

jj

g

g

g

g

D

G

G

G

G

1

2

1

4

1

0

2

1

11

1111

11

1111

]3[

]2[

]1[

]0[

]3[

]2[

]1[

]0[

4

j

j

jj

jj

h

h

h

h

D

H

H

H

H

1

0

1

6

1

1

2

2

11

1111

11

1111

]3[

]2[

]1[

]0[

]3[

]2[

]1[

]0[

4

D4 is the 4-point DFT matrix


If YC[k] denotes the 4-point DFT of yC[n] then from Table 3.5 we observe

YC[k]= G[k] H[k] , 0≤k≤3

2

0

2

24

]3[]3[

]2[]2[

]1[]1[

]0[]0[

]3[

]2[

]1[

]0[

j

j

HG

HG

HG

HG

YYYY

C

C

C

C

Thus


A 4-point IDFT of YC[k] yields

5

6

7

6

2

0

2

24

11

1111

11

1111

4

1

]3[

]2[

]1[

]0[

4

1

]3[

]2[

]1[

]0[

*

4

j

j

jj

jj

YYYY

D

yyyy

C

C

C

C

C

C

C

C


Example – Now let us extended the two length-4 sequences to length 7 by appending each with three zero-valued samples, i.e.

64,0

30,][][

64,0

30,][][

n

nnhnh

n

nngng

e

e

§5.4.2 Circular Convolution We next determine the 7-point circular

convolution of ge[n] and he[n] :

From the above

y[0]=ge[0]he[0]+ge[1]he[6]+ge[3]he[4] +ge

[4]he[3]+ge[5]he[2]+ge[6]he[1]

=ge[0]he[0]=1×2=2

60,][][][6

07

nmnhngnym

ee

§5.4.2 Circular Convolution Continuing the process we arrive

y[1]=g[0]h[1]+g[1]h[0]=(1×2)+(2×2)=6 ，y[2]=g[0]h[2]+g[2]h[0]

=(1×2)+(2×2)+(0×2)=5 ，y[3]=g[0]h[3]+g[1]h[2]+g[2]h[1]+g[3]h[0]

=(1×1)+(2×1)+(0×2)+(1×2)=5 ，y[4]=g[1]h[3]+g[2]h[2]+g[3]h[1]

=(2×1)+(0×1)+(1×2)=4 ，


y[5]=g[2]h[3]+g[3]h[2]=(0×1)+(1×1)=1,

y[6]=g[3]h[3]=(1×1)=1 As can be seen from the above that y[n] is p

recisely the sequence yL[n] obtained by a linear convolution of g[n] and h[n]

§5.4.2 Circular Convolution The N-point circular convolution can be writt

en in matrix form as

]1[

]2[

]1[

]0[

]0[]3[]2[]1[

]3[]0[]1[]2[

]2[]1[]0[]1[

]1[]2[]1[]0[

]1[

]2[

]1[

]0[

Ng

g

g

g

hNhNhNh

hhhh

hNhhh

hNhNhh

Ny

y

y

y

C

C

C

C

Note: The elements of each diagonal of the N×N matrix are equal

Such a matrix is called a circulant matrix


First, the samples of the two sequences are multiplied using the conventional multiplication method as shown on the next slide

Consider the evaluation of y[n]=h[n] g[n] where {g[n]} and {h[n]} are length-4 sequences

＊

Tabular Method We illustrate the method by an example


The partial products generated in the 2nd, 3rd, and 4t

h rows are circularly shifted to the left as indicated above

§5.4.2 Circular Convolution The modified table after circular shifting is sho

wn below

The samples of the sequence { yc[n]} are obtained by adding the 4 partial products in the column above of each sample


Thus

yc[0]=g[0]h[0]+ g[3]h[1]+ g[2]h[2]+ g[1]h[3]

yc[1]=g[1]h[0]+ g[0]h[1]+ g[3]h[2]+ g[2]h[3]

yc[2]=g[2]h[0]+ g[1]h[1]+ g[0]h[2]+ g[3]h[3]

yc[3]=g[3]h[0]+ g[2]h[1]+ g[1]h[2]+ g[0]h[3]

§5.9 Computation of the DFT of Real Sequences

In most practical applications, sequences of interest are real

In such cases, the symmetry properties of the DFT given in Table 5.2 can be exploited to make the DFT computations more efficient

§5.9.1 N-Point DFTs of Two Length-N Real Sequences

Let g[n] and h[n] be two length-N real sequences with G[k] and H[k] denoting their respective N-point DFTs

These two N-point DFTs can be computed efficiently using a single N-point DFT

Define a complex length-N sequence

x[n]=g[n]+jh[n] Hence, g[n]=Re{x[n]} and h[n]=Im{x[n]}


Let X[k] denote the N-point DFT of x[n] Then, from Table 5.1 we arrive at

][][

2

1][

][][2

1][

*

*

N

N

kXkXj

kH

kXkXkG

][][ **NN kNXkX

Note that for 0≤k≤N-1,


Example – We compute the 4-point DFTs of the two real sequences g[n] and h[n] given below

1122][,1021][ nhng

Then {x[n]}={g[n]}+j{h[n]} is given by

jjjjnx 12221][


Its DTF X[k] is

2

2

2

64

1

22

21

11

1111

11

1111

]3[

]2[

]1[

]0[

j

j

j

j

j

j

jj

jj

X

X

X

X

From the above X*[k]=[4-j6 2 -2 -j2]

Hence

X*[<4-k>4]=[4-j6 -j2 -2 2]


Therefore

{G[k]}=[4 1-j -2 1+j]

{G[k]}=[6 1-j 0 1+j]

verifying the results derived earlier

§5.9.2 2N-Point DFT of a RealSequence Using an N-point DFT

Let v[n] be a length-2N real sequence with an 2N-point DFT V[k]

Define two length-N real sequences g[n]and h[n] as follows:

Nnnvnhnvng 0,]12[][,]2[][

Let G[k] and H[k] denote their respective N- point DFTs


Define a length-N complex sequence

{x[n]}={g[n]}+j{h[n]}

with an N-point DFT X[k] Then as shown earlier

]}[][{21][

]}[][{21][

N

N

kXkXj

kH

kXkXkG


Now

120,][][

][][

]12[]2[

][][

1

02

1

0

2

1

0

1

0

)12(

2

1

0

2

2

1

0

2

12

0

Nknhng

nhng

nvnv

nvkV

WWW

WWW

WW

W

nk

N

N

n

k

N

nk

N

N

n

k

N

nk

N

N

n

nk

N

N

n

kn

N

N

n

nk

N

N

n

nk

N

N

n


i.e., 120,][][][ 2 NkkHWkGkV N

kNN

}11102221{]}[{ nv

We form two length-4 real sequences as follows

Example – Let us determine the 8-point DFT V[k] of the length-8 real sequence


Now 70,][][][ 484 kkHWkGkV k

Substituting the values of the 4-point DFTs G[k] and H[k] computed earlier we get

}1122{]}12[{]}[{

}1021{]}2[{]}[{

nvnh

nvng


264]0[]0[]4[4142.01)1()1(

]3[]3[]3[202]2[]2[]2[

4142.21)1()1(]1[]1[]1[

1064]0[]0[]0[

48

4/3

38

2/28

4/

18

j

j

j

j

eHWGVjjej

HWGVeHWGV

jjejHWGV

HGV


4142.21)1()1(]3[]3[]7[

202]2[]2[]6[4142.01)1()1(

]1[]1[]5[

4/7

78

2/368

4/5

58

jjejHWGV

eHWGVjjej

HWGV

j

j

j

§5.10 Linear Convolution Usingthe DFT

Linear convolution is a key operation in many signal processing applications

Since a DFT can be efficiently implemented using FFT algorithms, it is of interest to develop methods for the implementation of linear convolution using the DFT

§5.10.1 Linear Convolution of TwoFinite-Length Sequences

Let g[n] and h[n] be two finite-length sequences of length N and M, respectively

Denote L=N+M-1 Define two length-L sequences

1,010,][

][

1,010,][

][

LnMNnnh

nh

LnNNnng

ng

e

e

§5.10.1 Linear Convolution of TwoFinite-Length Sequences

The corresponding implementation scheme is illustrated below

Then

Zero-paddingwith

(M-1) zeros

Zero-paddingwith

(N-1) zeros

(N+M-1)point DFT

(N+M-1)point DFT

(N+M-1)point IDFT

g[n]

h[n]

ge[n]

he[n]

yL[n]Length-N

Length-M Length-(N+M-1)

LyL[n]=g[n] h[n]=yC[n]=ge[n] he[n] ﹡

§5.10.2 Linear Convolution of a Finite- Length Sequence with an Infinite-

Length Sequence We next consider the DFT-based implement

ation of

][][][][][1

0

nxnhnxhnyM

where h[n] is a finite-length sequence of length M and x[n] is an infinite length (or a finite length sequence of length much greater than M)

§5.10.2 Overlap-Add Method

We first segment x[n], assumed to be a causal sequence here without any loss of generality, into a set of contiguous finite-length subsequences xm [n] of length N each:

0

][][m

m mNnxnx

where

otherwise,0

10,][][

NnmNnxnxm

§5.10.2 Overlap-Add MethodThus we can write

where

0

][][][][m

m mNnynxnhny ﹡

Since h[n] is of length M and xm [n] is of length N, the linear convolution is of length N+M-1

﹡ ][][ nxnh m

][][][ nxnhny mm ﹡

§5.10.2 Overlap-Add Method As a result, the desired linear convolution

has been broken up into a sum of infinite number of short-length linear convolutions of length N+M-1each: ][][][ nhnxny mm ﹡

][][][ nxnhny ﹡

Each of these short convolutions can be implemented using the DFT-based method discussed earlier, where now the DFTs (and the IDFT) are computed on the basis of N+M-1 points

§5.10.2 Overlap-Add Method There is one more subtlety to take care of

before we can implement

0

][][m

m mNnyny

Now the first convolution in the above sum, , is of length N+M-1 and is defined for 0≤ N+M-2

﹡ ][][][ 00 nxnhny

using the DFT-based approach


There is an overlap of M-1samples between these two short linear convolutions

The second short convolution , is also of length N+

M-1 but is defined for N≤n≤2N+M-2][][][ 11 nxnhny ﹡

Likewise, the third short convolution

, is also of length N+M-1 but is defined for 2N≤n≤3N+M-2

﹡ ][][][ 22 nxnhny


This process is illustrated in the figure on the next slide for M=5 and N=70

In general, there will be an overlap of M-1 samples between the samples of the short convolutions h[n] xr-1[n] and h[n] xr [n] for (r-1)N≤n≤rN+M-2

﹡﹡

Thus there is an overlap of M-1 samples between h[n] x1[n] and h[n] x2[n] ﹡﹡

§5.10.2 Overlap-Add Method Therefore, y[n] obtained by a linear convolu

tion of x[n] and h[n] is given by

y[n]=y0[n], 0≤n≤6

y[n]=y0[n]+ y1[n-7], 7≤n≤10

y[n]=y1[n-7], 11≤n≤13

y[n]= y1[n-7]+ y2[n-14], 14≤n≤17

y[n]= y2[n-14], 18≤n≤20

§5.10.2 Overlap-Add Method The above procedure is called the overlap-

add method since the results of the short linear convolutions overlap and the overlapped portions are added to get the correct final result

The function fftfilt can be used to implement the above method

§5.10.2 Overlap-Add Method Program 5_5 illustrates the use of fftfilt in the

filtering of a noise-corrupted signal using a length-3 moving average filter

The plots generated by running this program is shown below

§5.10.2 Overlap-Save Method In implementing the overlap-add method usi

ng the DFT, we need to compute two (N+M-1)-point DFTs and one (N+M-1)-point IDFT since the overall linear convolution was expressed as a sum of short-length linear convolutions of length (N+M-1) each

It is possible to implement the overall linear convolution by performing instead circular

convolution of length shorter than (N+M-1)

§5.10.2 Overlap-Save Method

To this end, it is necessary to segment x[n]

into overlapping blocks xm[n], keep the terms of the circular convolution of h[n] with xm[n] that corresponds to the terms obtained by a linear convolution of h[n] and xm[n], and throw the other parts of the circular convoluition


To understand the correspondence between the linear and circular convolutions, consider a length-4 sequence x[n] and a length-3 sequence h[n]

Let yL[n] denote the result of a linear convolution of x[n] with h[n]

The six samples of yL[n] are given by


yL[0]=h[0]x[0]

yL[1]=h[0]x[1]+h[1]x[0]

yL[2]=h[0]x[2]+h[1]x[1]+h[2]x[0]

yL[3]=h[0]x[3]+h[1]x[2]+h[2]x[1]

yL[4]=h[1]x[3]+h[2]x[2]

yL[5]=h[2]x[3]


If we append h[n] with a single zero-valued sample and convert it into a length-4 sequence he[n], the 4-point circular convolution yC[n] of he[n] and x[n] is given by

yC[0]=h[0]x[0]+h[1]x[3]+h[2]x[2]

yC[1]=h[0]x[1]+h[1]x[0]+h[2]x[3]

yC[2]=h[0]x[2]+h[1]x[1]+h[2]x[0]

yC[3]=h[0]x[3]+h[1]x[2]+h[2]x[1]

§5.10.2 Overlap-Save Method If we compare the expressions for the sample

s of yL[n] with the samples of yC[n],

we observe that the first 2 terms of yC[n] do not correspond to the first 2 terms of yL[n],

whereas the last 2 terms of yC[n] are precisely the same as the 3rd and 4th terms of yL[n], i.e.,

yL[0]≠yC[0], yL[1]≠yC[1]

yL[2]≠yC[2], yL[3]≠yC[3]

§5.10.2 Overlap-Save Method General case: N-point circular convolution of

a length-M sequence h[n] with a length-N sequence x[n] with N > M

First M-1samples of the circular convolution are incorrect and are rejected

Remaining N - M+1 samples correspond to the correct samples of the linear convolution of h[n] with x[n]

§5.10.2 Overlap-Save Method Now, consider an infinitely long or very long

sequence x[n] Break it up as a collection of smaller length

(length-4) overlapping sequences xm[n] as xm

[n]= x [n+2m], 0≤n≤3, 0≤m≤∞ Next,form

wm[n]=h[n] xm[n] 4

§5.10.2 Overlap-Save Method Or, equivalently,

wm[0]=h[0]xm[0]+h[1]xm[3]+h[2]xm[2]

wm[1]=h[0]xm[1]+h[1]xm[0]+h[2]xm[3]

wm[2]=h[0]xm[2]+h[1]xm[1]+h[2]xm[0]

wm[3]=h[0]xm[3]+h[1]xm[2]+h[2]xm[1] Computing the above for m = 0, 1, 2, 3, . . . , an

d substituting the values of xm[n] we arrive at

§5.10.2 Overlap-Save Methodw0[0]=h[0]x[0]+h[1]x[3]+h[2]x[2] ← Reject

w0[1]=h[0]x[1]+h[1]x[0]+h[2]x[3] ← Reject

w0[2]=h[0]x[2]+h[1]x[1]+h[2]x[0]=y[2]← Save

w0[3]=h[0]x[3]+h[1]x[2]+h[2]x[1]=y[2]← Save

w1[0]=h[0]x[2]+h[1]x[5]+h[2]x[4] ← Reject

w1[1]=h[0]x[3]+h[1]x[2]+h[2]x[5] ← Reject

w1[2]=h[0]x[4]+h[1]x[3]+h[2]x[2]=y[4]← Save

w1[3]=h[0]x[5]+h[1]x[4]+h[2]x[3]=y[5]← Save


w2[0]=h[0]x[4]+h[1]x[5]+h[2]x[6] ← Reject

w2[1]=h[0]x[5]+h[1]x[4]+h[2]x[7] ← Reject

w2[2]=h[0]x[6]+h[1]x[5]+h[2]x[4]=y[6]← Save

w2[3]=h[0]x[7]+h[1]x[6]+h[2]x[5]=y[7]← Save


It should be noted that to determine y[0] and y[1], we need to form x-1[n] :

x-1[0]=0, x-1[1]=0,

x-1[2]=x[0], x-1[3]=x[1] ][][][ 11 nxnhnw 4and compute for 0≤n≤3

reject w-1[0] and w-1[1] ,and save w-1[2]= y[0]

and w-1[3]= y[1]

§5.10.2 Overlap-Save Method General Case: Let h[n] be a length-N seque

nce Let xm[n] denote the m-th section of an infinit

ely long sequence x[n] of length N and defined by

xm[n]= x [n+m(N-m+1)], 0≤n≤N-1

with M<N


Then, we reject the first M-1 samples of wm[n] and “abut” the remaining N-M+1 samples of wm[n] to form yL[n], the linear convolution of h[n] and x[n]

If ym[n] denotes the saved portion of wm[n], i.e.

Let wm[n]=h[n] xm[n] N

21,][

20,0][

NnMnw

Mnny

mm


Then

yL [n+m(N-M+1)]=ym[n] , M-1≤n≤N-1 The approach is called overlap-save method

since the input is segmented into overlapping sections and parts of the results of the circular convolutions are saved and abutted to determine the linear convolution result

§5.10.2 Overlap-Save Method Process is illustrated next

chapter 5 finite-length discrete transform. §5.2.1 the discrete fourier transform definition - the...

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