chapter 5 finite-length discrete transform. §5.2.1 the discrete fourier transform definition - the...
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![Page 1: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/1.jpg)
Chapter 5
Finite-Length DiscretFinite-Length Discrete Transforme Transform
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§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
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§5.2.1 The Discrete Fourier Transform
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
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§5.2.1 The Discrete Fourier Transform
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
![Page 5: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/5.jpg)
§5.2.1 The Discrete Fourier Transform
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
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§5.2.1 The Discrete Fourier Transform
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
![Page 7: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/7.jpg)
§5.2.1 The Discrete Fourier Transform
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
![Page 8: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/8.jpg)
§5.2.1 The Discrete Fourier Transform
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
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§5.2.1 The Discrete Fourier Transform
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
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§5.2.1 The Discrete Fourier Transform
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
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§5.2.1 The Discrete Fourier Transform
Making use of the identity
otherwise,0integeran,for,1
0
)( rrNkNW
N
n
nkN
we get
10otherwis0for,2/for,2/
][
Nk
rNkNrkN
kG,
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§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
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§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
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§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
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§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
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§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
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§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
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§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]
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§5.3.3 DTFT from DFT byInterpolation
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
[
][)(
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§5.3.3 DTFT from DFT byInterpolation
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
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§5.3.3 DTFT from DFT byInterpolation
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(
![Page 22: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/22.jpg)
§5.3.3 DTFT from DFT byInterpolation
therefore
]2/)1)][(/2[1
0
22
sin
22
sin][
1
)(
NNkjN
k
j
e
NkN
kN
kXN
eX
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§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
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§5.3.4 Sampling the DTFT
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
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§5.3.4 Sampling the DTFT
i.e.
1
0
)(
1
0
1][
][1
][
N
k
nkN
knN
kN
N
k
WN
x
WWxN
ny
=
Making use of the identity
otherwise,0for,11 1
0
)( mNnrW
N
N
n
rnkN
=
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§5.3.4 Sampling the DTFT
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
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§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
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§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
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§5.3.4 Sampling the DTFT
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]}
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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:
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Numerical Computation of theDTFT Using the DFT
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
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Numerical Computation of theDTFT Using the DFT
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
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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
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Table 5.1: DFT Properties: Symmetry Relations
x[n] is a complex sequence
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Table 5.2: DFT Properties: Symmetry Relations
x[n] is a real sequence
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Table 5.3: General Properties of DFT
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§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
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§5.4.1 Circular Shift of a Sequence
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
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§5.4.1 Circular Shift of a Sequence
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
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§5.4.1 Circular Shift of a Sequence
Illustration of the concept of a circular shift
]5[]1[
6
6
nxnx -
]2[]4[
6
6
nxnx -][nx
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§5.4.1 Circular Shift of a Sequence
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
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§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 ],[][][
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§5.4.2 Circular Convolution
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]
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§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
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§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
![Page 46: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/46.jpg)
§5.4.2 Circular Convolution
Example – Determine the 4-point circular convolution of the two length-4 sequences:
as sketched below
}1122{]}[{},1021{]}[{ nhng
![Page 47: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/47.jpg)
§5.4.2 Circular Convolution
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
![Page 48: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/48.jpg)
§5.4.2 Circular Convolution
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
![Page 49: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/49.jpg)
§5.4.2 Circular Convolution
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
![Page 50: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/50.jpg)
§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
![Page 51: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/51.jpg)
§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
![Page 52: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/52.jpg)
§5.4.2 Circular Convolution
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
![Page 53: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/53.jpg)
§5.4.2 Circular Convolution
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
![Page 54: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/54.jpg)
§5.4.2 Circular Convolution
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
![Page 55: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/55.jpg)
§5.4.2 Circular Convolution
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
![Page 56: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/56.jpg)
§5.4.2 Circular Convolution
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
![Page 57: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/57.jpg)
§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
![Page 58: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/58.jpg)
§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 ,
![Page 59: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/59.jpg)
§5.4.2 Circular Convolution
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]
![Page 60: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/60.jpg)
§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
![Page 61: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/61.jpg)
§5.4.2 Circular Convolution
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
![Page 62: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/62.jpg)
§5.4.2 Circular Convolution
The partial products generated in the 2nd, 3rd, and 4t
h rows are circularly shifted to the left as indicated above
![Page 63: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/63.jpg)
§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
![Page 64: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/64.jpg)
§5.4.2 Circular Convolution
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]
![Page 65: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/65.jpg)
§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
![Page 66: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/66.jpg)
§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]}
![Page 67: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/67.jpg)
§5.9.1 N-Point DFTs of Two Length-N Real Sequences
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,
![Page 68: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/68.jpg)
§5.9.1 N-Point DFTs of Two Length-N Real Sequences
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][
![Page 69: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/69.jpg)
§5.9.1 N-Point DFTs of Two Length-N Real Sequences
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]
![Page 70: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/70.jpg)
§5.9.1 N-Point DFTs of Two Length-N Real Sequences
Therefore
{G[k]}=[4 1-j -2 1+j]
{G[k]}=[6 1-j 0 1+j]
verifying the results derived earlier
![Page 71: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/71.jpg)
§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
![Page 72: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/72.jpg)
§5.9.2 2N-Point DFT of a RealSequence Using an N-point DFT
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
![Page 73: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/73.jpg)
§5.9.2 2N-Point DFT of a RealSequence Using an N-point DFT
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
![Page 74: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/74.jpg)
§5.9.2 2N-Point DFT of a RealSequence Using an N-point DFT
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
![Page 75: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/75.jpg)
§5.9.2 2N-Point DFT of a RealSequence Using an N-point DFT
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
![Page 76: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/76.jpg)
§5.9.2 2N-Point DFT of a RealSequence Using an N-point DFT
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
![Page 77: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/77.jpg)
§5.9.2 2N-Point DFT of a RealSequence Using an N-point DFT
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
![Page 78: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/78.jpg)
§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
![Page 79: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/79.jpg)
§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
![Page 80: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/80.jpg)
§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] ﹡
![Page 81: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/81.jpg)
§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)
![Page 82: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/82.jpg)
§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
![Page 83: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/83.jpg)
§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 ﹡
![Page 84: Chapter 5 Finite-Length Discrete Transform. §5.2.1 The Discrete Fourier Transform Definition - The simplest relation between a length- N sequence x[n],](https://reader035.vdocuments.site/reader035/viewer/2022062301/56649ec05503460f94bcae79/html5/thumbnails/84.jpg)
§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
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§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
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§5.10.2 Overlap-Add Method
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
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§5.10.2 Overlap-Add Method
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] ﹡﹡
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§5.10.2 Overlap-Add Method
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§5.10.2 Overlap-Add Method
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§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
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§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
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§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
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§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)
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§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
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§5.10.2 Overlap-Save Method
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
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§5.10.2 Overlap-Save Method
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]
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§5.10.2 Overlap-Save Method
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]
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§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]
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§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]
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§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
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§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
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§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
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§5.10.2 Overlap-Save Method
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
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§5.10.2 Overlap-Save Method
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]
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§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
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§5.10.2 Overlap-Save Method
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
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§5.10.2 Overlap-Save Method
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
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§5.10.2 Overlap-Save Method Process is illustrated next
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§5.10.2 Overlap-Save Method