dtft properties - the signal and image processing...
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1Copyright © 2001, S. K. Mitra
DTFT PropertiesDTFT Properties
• Example - Determine the DTFT of
• Let• We can therefore write
• From Table 3.1, the DTFT of x[n] is givenby
1],[)1(][ <αµα+= nnny n
1],[][ <αµα= nnx n
][][][ nxnxnny +=
ω−ω
α−= j
j
eeX
11)(
)( ωjeY
2Copyright © 2001, S. K. Mitra
DTFT PropertiesDTFT Properties• Using the differentiation property of the
DTFT given in Table 3.2, we observe thatthe DTFT of is given by
• Next using the linearity property of theDTFT given in Table 3.2 we arrive at
][nxn
2)1(11)(
ω−
ω−
ω−
ω
α−α=
α−ω=
ω j
j
j
j
ee
eddj
dedXj
22 )1(1
11
)1()( ω−ω−ω−
ω−ω
α−=
α−+
α−α= jjj
jj
eeeeeY
3Copyright © 2001, S. K. Mitra
DTFT PropertiesDTFT Properties• Example - Determine the DTFT of
the sequence v[n] defined by
• From Table 3.1, the DTFT of is 1• Using the time-shifting property of the
DTFT given in Table 3.2 we observe thatthe DTFT of is and the DTFTof is][ 1−nv
]1[][]1[][ 1010 −δ+δ=−+ npnpnvdnvd][nδ
]1[ −δ n ω− je
)( ωjeV
)( ωω− jj eVe
4Copyright © 2001, S. K. Mitra
DTFT PropertiesDTFT Properties• Using the linearity property of Table 3.2 we
then obtain the frequency-domainrepresentation of
as
• Solving the above equation we get
]1[][]1[][ 1010 −δ+δ=−+ npnpnvdnvd
ω−ωω−ω +=+ jjjj eppeVedeVd 1010 )()(
ω−
ω−ω
++= j
jj
eddeppeV
10
10)(
5Copyright © 2001, S. K. Mitra
Energy Density SpectrumEnergy Density Spectrum
• The total energy of a finite-energy sequenceg[n] is given by
• From Parseval’s relation given in Table 3.2we observe that
∑=∞
−∞=ng ng 2][E
∫ ωπ
=∑=π
π−
ω∞
−∞=deGng j
ng
22
21 )(][E
6Copyright © 2001, S. K. Mitra
Energy Density SpectrumEnergy Density Spectrum
• The quantity
is called the energy density spectrum• The area under this curve in the range
divided by 2π is the energy ofthe sequence
π≤ω≤π−
2)()( ω=ω j
gg eGS
7Copyright © 2001, S. K. Mitra
Energy Density SpectrumEnergy Density Spectrum• Example - Compute the energy of the
sequence
• Here
where
∞<<∞−πω= nn
nnh cLP ,sin][
∫ ωπ
=∑π
π−
ω∞
−∞=deHnh j
LPn
LP22 )(
21][
π≤ω<ωω≤ω≤
=ω
c
cjLP eH
,00,1
)(
8Copyright © 2001, S. K. Mitra
Energy Density SpectrumEnergy Density Spectrum
• Therefore
• Hence, is a finite-energy sequence
∞<πω=∫ ω
π=∑
ω
ω−
∞
−∞=
c
nLP
c
c
dnh21][ 2
][nhLP
9Copyright © 2001, S. K. Mitra
DTFT Computation UsingDTFT Computation UsingMATLABMATLAB
• The function freqz can be used tocompute the values of the DTFT of asequence, described as a rational function inin the form of
at a prescribed set of discrete frequencypoints
NjN
j
MjM
jj
ededdepeppeX ω−ω−
ω−ω−ω
++++++=
........)(
10
10
lω=ω
10Copyright © 2001, S. K. Mitra
DTFT Computation UsingDTFT Computation UsingMATLABMATLAB
• For example, the statementH = freqz(num,den,w)
returns the frequency response values as avector H of a DTFT defined in terms of thevectors num and den containing thecoefficients and , respectively at aprescribed set of frequencies between 0 and2π given by the vector w
}{ ip }{ id
11Copyright © 2001, S. K. Mitra
DTFT Computation UsingDTFT Computation UsingMATLABMATLAB
• There are several other forms of thefunction freqz
• The Program 3_1 in the text can be used tocompute the values of the DTFT of a realsequence
• It computes the real and imaginary parts,and the magnitude and phase of the DTFT
12Copyright © 2001, S. K. Mitra
DTFT Computation UsingDTFT Computation UsingMATLABMATLAB
• Example - Plots of the real and imaginaryparts, and the magnitude and phase of theDTFT
are shown on the next slide
ω−ω−
ω−ω−
ω−ω−
ω−ω−
ω
+++++−
+−
=
43
2
43
2
41.06.17.237.21
008.0033.005.0033.0008.0
)(
jj
jj
jj
jj
j
eeee
eeee
eX
13Copyright © 2001, S. K. Mitra
DTFT Computation UsingDTFT Computation UsingMATLABMATLAB
0 0.2 0.4 0.6 0.8 1-1
-0.5
0
0.5
1Real part
ω/π
Am
plitu
de
0 0.2 0.4 0.6 0.8 1-1
-0.5
0
0.5
1Real part
ω/π
Am
plitu
de
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1Magnitude Spectrum
ω/π
Mag
nitu
de
0 0.2 0.4 0.6 0.8 1-4
-2
0
2
4Phase Spectrum
ω/π
Pha
se, r
adia
ns
14Copyright © 2001, S. K. Mitra
DTFT Computation UsingDTFT Computation UsingMATLABMATLAB
• Note: The phase spectrum displays adiscontinuity of 2π at ω = 0.72
• This discontinuity can be removed using thefunction unwrap as indicated below
0 0.2 0.4 0.6 0.8 1-7
-6
-5
-4
-3
-2
-1
0Unwrapped Phase Spectrum
ω/π
Pha
se, r
adia
ns
15Copyright © 2001, S. K. Mitra
Linear Convolution UsingLinear Convolution UsingDTFTDTFT
• An important property of the DTFT is givenby the convolution theorem in Table 3.2
• It states that if y[n] = x[n] h[n], then theDTFT of y[n] is given by
• An implication of this result is that thelinear convolution y[n] of the sequencesx[n] and h[n] can be performed as follows:
*)( ωjeY
)()()( ωωω = jjj eHeXeY
16Copyright © 2001, S. K. Mitra
Linear Convolution UsingLinear Convolution UsingDTFTDTFT
• 1) Compute the DTFTs andof the sequences x[n] and h[n], respectively
• 2) Form the DTFT• 3) Compute the IDFT y[n] of
)()()( ωωω = jjj eHeXeY
)( ωjeX )( ωjeH
)( ωjeY
×x[n]
h[n]y[n]
DTFT
DTFTIDTFT
)( ωjeY)( ωjeX
)( ωjeH
17Copyright © 2001, S. K. Mitra
Discrete Fourier TransformDiscrete Fourier Transform• Definition - The simplest relation between a
length-N sequence x[n], defined for , and its DTFT is
obtained by uniformly sampling onthe ω-axis between at ,
• From the definition of the DTFT we thus have
10 −≤≤ Nn
10 −≤≤ Nk
)( ωjeX)( ωjeX
π≤ω≤ 20 Nkk /2π=ω
,][)(][1
0
/2/2
∑==−
=
π−π=ω
ω N
n
NkjNk
j enxeXkX
10 −≤≤ Nk
18Copyright © 2001, S. K. Mitra
Discrete Fourier TransformDiscrete 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 sequencex[n]
• Using the notation theDFT is usually expressed as:
NjN eW /2π−=
10,][][1
0−≤≤∑=
−
=NkWnxkX
N
n
nkN
19Copyright © 2001, S. K. Mitra
Discrete Fourier TransformDiscrete Fourier Transform• The inverse discrete Fourier transform
(IDFT) is given by
• To verify the above expression we multiplyboth sides of the above equation byand sum the result from n = 0 to 1−= Nn
nNW l
10,][1][1
0−≤≤∑=
−
=
− NnWkXN
nxN
k
knN
20Copyright © 2001, S. K. Mitra
Discrete Fourier TransformDiscrete Fourier Transform
resulting in
∑ ∑∑−
=
−
=
−−
=
=
1
0
1
0
1
0
1N
n
nN
N
k
knN
N
n
nN WWkX
NWnx ll ][][
∑ ∑−
=
−
=
−−=1
0
1
0
1 N
n
N
k
nkNWkX
N)(][ l
∑ ∑−
=
−
=
−−=1
0
1
0
1 N
k
N
n
nkNWkX
N)(][ l
21Copyright © 2001, S. K. Mitra
Discrete Fourier TransformDiscrete Fourier Transform
• Making use of the identity
we observe that the RHS of the lastequation is equal to
• Hence
=∑
−
=
−−1
0
N
n
nkNW )( l
,,
0N ,rNk =− lfor
otherwiser an integer
][lX
][][ ll XWnxN
n
nN =∑
−
=
1
0
22Copyright © 2001, S. K. Mitra
Discrete Fourier TransformDiscrete Fourier Transform
• Example - Consider the length-N sequence
• Its N-point DFT is given by
11001−≤≤
=Nn
n,,
=][nx
10 01
0=== ∑
−
=N
N
n
knN WxWnxkX ][][][
10 −≤≤ Nk
23Copyright © 2001, S. K. Mitra
Discrete Fourier TransformDiscrete Fourier Transform
• Example - Consider the length-N sequence
• Its N-point DFT is given by=][ny 11100
1−≤≤+−≤≤
=Nnmmn
mn,,
,
kmN
kmN
N
n
knN WWmyWnykY === ∑
−
=][][][
1
010 −≤≤ Nk
24Copyright © 2001, S. K. Mitra
Discrete Fourier TransformDiscrete Fourier Transform
• Example - Consider the length-N sequencedefined for
• Using a trigonometric identity we can write10),/2cos(][ −≤≤π= NrNrnng
( )NrnjNrnj eeng /2/221][ π−π +=
( )rNN
rNN WW += −
21
10 −≤≤ Nn
25Copyright © 2001, S. K. Mitra
Discrete Fourier TransformDiscrete Fourier Transform• The N-point DFT of g[n] is thus given by
∑−
==
1
0
N
n
knNWngkG ][][
,21 1
0
)(1
0
)(
∑+∑=
−
=
+−
=
−− N
n
nkrN
N
n
nkrN WW
10 −≤≤ Nk
26Copyright © 2001, S. K. Mitra
Discrete Fourier TransformDiscrete Fourier Transform
• Making use of the identity
we get
−==
=otherwise0for2for2
,,/,/
][ rNkNrkN
kG
=∑
−
=
−−1
0
N
n
nkNW )( l
,,
0N ,rNk =− lfor
otherwiser an integer
10 −≤≤ Nk
27Copyright © 2001, S. K. Mitra
Matrix RelationsMatrix Relations• The DFT samples defined by
can be expressed in matrix form as
where
10,][][1
0−≤≤∑=
−
=NkWnxkX
N
n
knN
xDX N=
[ ]TNXXX ][.....][][ 110 −=X
[ ]TNxxx ][.....][][ 110 −=x
28Copyright © 2001, S. K. Mitra
Matrix RelationsMatrix Relations
and is the DFT matrix given byND NN ×
=
−−−
−
−
21121
1242
121
1
11
1111
)()()(
)(
)(
NN
NN
NN
NNNN
NNNN
N
WWW
WWWWWW
L
MOMMM
L
LL
D
29Copyright © 2001, S. K. Mitra
Matrix RelationsMatrix Relations
• Likewise, the IDFT relation given by
can be expressed in matrix form as
where is the IDFT matrix
10,][][1
0−≤≤∑=
−
=
− NnWkXnxN
k
nkN
NN ×1−ND
XDx 1−= N
30Copyright © 2001, S. K. Mitra
Matrix RelationsMatrix Relationswhere
• Note:
=
−−−−−−
−−−−
−−−−
−
21121
1242
121
1
1
11
1111
)()()(
)(
)(
NN
NN
NN
NNNN
NNNN
N
WWW
WWWWWW
L
MOMMM
L
LL
D
*NN N
DD 11 =−
31Copyright © 2001, S. K. Mitra
DFT Computation UsingDFT Computation UsingMATLABMATLAB
• The functions to compute the DFT and theIDFT are fft and ifft
• These functions make use of FFTalgorithms which are computationallyhighly efficient compared to the directcomputation
• Programs 3_2 and 3_4 illustrate the use ofthese functions
32Copyright © 2001, S. K. Mitra
DFT Computation UsingDFT Computation UsingMATLABMATLAB
• Example - Program 3_4 can be used tocompute the DFT and the DTFT of thesequence
as shown below150),16/6cos(][ ≤≤π= nnnx
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
Normalized angular frequency
Mag
nitu
de
indicates DFT samples
33Copyright © 2001, S. K. Mitra
DTFT from DFT byDTFT from DFT byInterpolationInterpolation
• The N-point DFT X[k] of a length-Nsequence x[n] is simply the frequencysamples of its DTFT evaluated at Nuniformly spaced frequency points
• Given the N-point DFT X[k] of a length-Nsequence x[n], its DTFT can beuniquely determined from X[k]
)( ωjeX
)( ωjeX
10,/2 −≤≤π=ω=ω NkNkk
34Copyright © 2001, S. K. Mitra
DTFT from DFT byDTFT from DFT byInterpolationInterpolation
• Thus
∑=−
=
ω−ω 1
0][)(
N
n
njj enxeX
njN
n
N
k
nkN eWkX
Nω−−
=
−
=
−∑
∑=
1
0
1
0][1
∑ ∑=−
=
−
=
π−ω−1
0
1
0
)/2(][1 N
k
N
n
nNkjekXN
S
35Copyright © 2001, S. K. Mitra
DTFT from DFT byDTFT from DFT byInterpolationInterpolation
• To develop a compact expression for thesum S, let
• Then• From the above
1111 −+=−+∑= −=
NNNn
n rSrr)/2( Nkjer π−ω−=
∑= −=
10S N
nnr
11S 111 −+∑+=∑= −==
NNn
nNn
n rrrr
1111 −+=−+∑= −=
NNNn
n rSrr
36Copyright © 2001, S. K. Mitra
DTFT from DFT byDTFT from DFT byInterpolationInterpolation
• Or, equivalently,
• Hence
Nrrr −=−=− 1S)1(SS
)]/2([
)2(
11
11S Nkj
kNjN
ee
rr
π−ω−
π−ω−
−−=
−−=
]2/)1)][(/2[(
22sin
22sin
−π−ω−⋅
π−ω
π−ω
= NNkje
NkN
kN
37Copyright © 2001, S. K. Mitra
DTFT from DFT byDTFT from DFT byInterpolationInterpolation
• Therefore
∑ ⋅
π−ω
π−ω
=−
=
−π−ω−1
0
]2/)1)][(/2[(
22sin
22sin
][1 N
k
NNkje
NkN
kN
kXN
)( ωjeX
38Copyright © 2001, S. K. Mitra
Sampling the DTFTSampling the DTFT
• Consider a sequence x[n] with a DTFT• We sample at N equally spaced points
, developing the Nfrequency samples
• These N frequency samples can beconsidered as an N-point DFT Y[k] whose N-point IDFT is a length-N sequence y[n]
)( ωjeX)( ωjeX
Nkk /2π=ω 10 −≤≤ Nk)}({ kjeX ω
39Copyright © 2001, S. K. Mitra
Sampling the DTFTSampling the DTFT
• Now
• Thus
• An IDFT of Y[k] yields
∑=∞
−∞=
ω−ω
l
ll jj exeX ][)(
)()(][ /2 Nkjj eXeXkY k πω ==
∑=∑=∞
−∞=
∞
−∞=
π−
l
l
l
l ll kN
Nkj Wxex ][][ /2
∑=−
=
−1
0][1][
N
k
nkNWkY
Nny
40Copyright © 2001, S. K. Mitra
Sampling the DTFTSampling the DTFT
• i.e.
• Making use of the identity
∑ ∑−
=
−∞
−∞==
1
0
1 N
k
nkN
kN WWx
Nny l
l
l][][
= ∑∑
−
=
−−∞
−∞=
1
0
1 N
k
nkNW
Nx )(][ l
l
l
=∑
−
=
−−1
0
1 N
n
rnkNW
N)(
,,
01 mNnr +=for
otherwise
41Copyright © 2001, S. K. Mitra
Sampling the DTFTSampling the DTFTwe arrive at the desired relation
• Thus y[n] is obtained from x[n] by addingan infinite number of shifted replicas ofx[n], with each replica shifted by an integermultiple of N sampling instants, andobserving the sum only for the interval
10 −≤≤+= ∑∞
−∞=NnmNnxny
m],[][
10 −≤≤ Nn
42Copyright © 2001, S. K. Mitra
Sampling the DTFTSampling the DTFT• To apply
to finite-length sequences, we assume thatthe samples outside the specified range arezeros
• Thus if x[n] is a length-M sequence with , then y[n] = x[n] for
10 −≤≤+= ∑∞
−∞=NnmNnxny
m],[][
NM ≤ 10 −≤≤ Nn
43Copyright © 2001, S. K. Mitra
Sampling the DTFTSampling the DTFT• If M > N, there is a time-domain aliasing of
samples of x[n] in generating y[n], and x[n]cannot be recovered from y[n]
• Example - Let
• By sampling its DTFT at , and then applying a 4-point IDFT to
these samples, we arrive at the sequence y[n]given by
}{]}[{ 543210=nx↑
)( ωjeX 4/2 kk π=ω30 ≤≤ k
44Copyright © 2001, S. K. Mitra
Sampling the DTFTSampling the DTFT
,• i.e.
{x[n]} cannot be recovered from {y[n]}
][][][][ 44 −+++= nxnxnxny 30 ≤≤ n
}{]}[{ 3264=ny↑
45Copyright © 2001, S. K. Mitra
Numerical Computation of theNumerical Computation of theDTFT Using the DFTDTFT Using the DFT
• A practical approach to the numericalcomputation of the DTFT of a finite-lengthsequence
• Let be the DTFT of a length-Nsequence x[n]
• We wish to evaluate at a dense gridof frequencies , ,where M >> N:
)( ωjeX
)( ωjeXMkk /2π=ω 10 −≤≤ Mk
46Copyright © 2001, S. K. Mitra
Numerical Computation of theNumerical Computation of theDTFT Using the DFTDTFT Using the DFT
• Define a new sequence
• Then
∑=∑=−
=
π−−
=
ω−ω 1
0
/21
0][][)(
N
n
MknjN
n
njj enxenxeX kk
−≤≤−≤≤
=1,0
10],[][
MnNNnnx
nxe
∑=−
=
π−ω 1
0
/2][)(M
n
Mknjj enxeX k
47Copyright © 2001, S. K. Mitra
Numerical Computation of theNumerical Computation of theDTFT Using the DFTDTFT Using the DFT
• Thus is essentially an M-point DFT of the length-M sequence
• The DFT can be computed veryefficiently using the FFT algorithm if M isan integer power of 2
• The function freqz employs this approachto evaluate the frequency response at aprescribed set of frequencies of a DTFTexpressed as a rational function in
)( kjeX ω
][kXe ][nxe][kXe
ω− je
48Copyright © 2001, S. K. Mitra
DFT PropertiesDFT Properties
• Like the DTFT, the DFT also satisfies anumber of properties that are useful insignal processing applications
• Some of these properties are essentiallyidentical to those of the DTFT, while someothers are somewhat different
• A summary of the DFT properties are givenin tables in the following slides
49Copyright © 2001, S. K. Mitra
Table 3.5:Table 3.5: General Properties General Propertiesof DFTof DFT
50Copyright © 2001, S. K. Mitra
Table 3.6:Table 3.6: DFT Properties: DFT Properties:Symmetry RelationsSymmetry Relations
x[n] is a complex sequence
51Copyright © 2001, S. K. Mitra
Table 3.7:Table 3.7: DFT Properties: DFT Properties:Symmetry RelationsSymmetry Relations
x[n] is a real sequence
52Copyright © 2001, S. K. Mitra
Circular Shift of a SequenceCircular Shift of a Sequence
• This property is analogous to the time-shifting property of the DTFT as given inTable 3.2, but with a subtle difference
• Consider length-N sequences defined for
• Sample values of such sequences are equalto zero for values of n < 0 and Nn ≥
10 −≤≤ Nn
53Copyright © 2001, S. K. Mitra
Circular Shift of a SequenceCircular Shift of a Sequence
• If x[n] is such a sequence, then for anyarbitrary integer , the shifted sequence
is no longer defined for the range• We thus need to define another type of a
shift that will always keep the shiftedsequence in the range
][][ onnxnx −=1
on
10 −≤≤ Nn
10 −≤≤ Nn
54Copyright © 2001, S. K. Mitra
Circular Shift of a SequenceCircular Shift of a Sequence
• The desired shift, called the circular shift,is defined using a modulo operation:
• For (right circular shift), the aboveequation implies
][][ Noc nnxnx ⟩−⟨=
0>on
+−−=
],[],[
][ nnNxnnxnxo
oc
oo
nnNnn<≤
−≤≤0for
1for
55Copyright © 2001, S. K. Mitra
Circular Shift of a SequenceCircular Shift of a Sequence
• Illustration of the concept of a circular shift
][nx ]1[ 6⟩−⟨nx
]5[ 6⟩+⟨= nx ]2[ 6⟩+⟨= nx
]4[ 6⟩−⟨nx
56Copyright © 2001, S. K. Mitra
Circular Shift of a SequenceCircular Shift of a Sequence
• As can be seen from the previous figure, aright circular shift by is equivalent to aleft circular shift by sample periods
• A circular shift by an integer numbergreater than N is equivalent to a circularshift by
ononN −
on
Non ⟩⟨
57Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution
• This operation is analogous to linearconvolution, but with a subtle difference
• Consider two length-N sequences, g[n] andh[n], respectively
• Their linear convolution results in a length- sequence given by)( 12 −N
2201
0−≤≤−= ∑
−
=Nnmnhmgny
N
mL ],[][][
][nyL
58Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution• In computing we have assumed that
both length-N sequences have been zero-padded to extend their lengths to
• The longer form of results from thetime-reversal of the sequence h[n] and itslinear shift to the right
• The first nonzero value of is , and the last nonzero value
is
12 −N
][nyL
][nyL
][nyL][][][ 000 hgyL =
][][][ 1122 −−=− NhNgNyL
59Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution
• To develop a convolution-like operationresulting in a length-N sequence , weneed to define a circular time-reversal, andthen apply a circular time-shift
• Resulting operation, called a circularconvolution, is defined by
10],[][][1
0−≤≤∑ ⟩−⟨=
−
=Nnmnhmgny
N
mNC
][nyC
60Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution
• Since the operation defined involves twolength-N sequences, it is often referred to asan N-point circular convolution, denoted as
y[n] = g[n] h[n]• The circular convolution is commutative,
i.e.g[n] h[n] = h[n] g[n]
N
N N
61Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution
• Example - Determine the 4-point circularconvolution of the two length-4 sequences:
as sketched below
},{]}[{ 1021=ng }{]}[{ 1122=nh↑ ↑
n3210
1
2 ][ng
n3210
1
2 ][nh
62Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution• The result is a length-4 sequence
given by
• From the above we observe
][nyC
,][][][][][3
04∑ ⟩−⟨==
=mC mnhmgnhngny 4
30 ≤≤ n
∑ ⟩⟨−==
3
04][][]0[
mC mhmgy
]1[]3[]2[]2[]3[]1[]0[]0[ hghghghg +++=
621101221 =×+×+×+×= )()()()(
63Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution• Likewise ∑ ⟩−⟨=
=
3
04]1[][]1[
mC mhmgy
][][][][][][][][ 23320110 hghghghg +++=711102221 =×+×+×+×= )()()()(
∑ ⟩−⟨==
3
04]2[][]2[
mC mhmgy
][][][][][][][][ 33021120 hghghghg +++=
611202211 =×+×+×+×= )()()()(
64Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolutionand
• The circular convolution can also becomputed using a DFT-based approach asindicated in Table 3.5
521201211 =×+×+×+×= )()()()(][][][][][][][][ 03122130 hghghghg +++=
∑ ⟩−⟨==
3
04]3[][]3[
mC mhmgy
][nyC
65Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution• Example - Consider the two length-4
sequences repeated below for convenience:
• The 4-point DFT G[k] of g[n] is given by
n3210
1
2 ][ng
n3210
1
2 ][nh
4210 /][][][ kjeggkG π−+=4644 32 // ][][ kjkj egeg ππ −− ++
3021 232 ≤≤++= −− kee kjkj ,// ππ
66Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution• Therefore
• Likewise,
,][ 21212 −=−−=G,][ jjjG −=+−= 1211
jjjG +=−+= 1213][
,][ 41210 =++=G
4210 /][][][ kjehhkH π−+=4644 32 // ][][ kjkj eheh ππ −− ++
3022 232 ≤≤+++= −−− keee kjkjkj ,// πππ
67Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution
• Hence,
• The two 4-point DFTs can also becomputed using the matrix relation givenearlier
,][ 611220 =+++=H
,][ 011222 =−+−=H,][ jjjH −=+−−= 11221
jjjH +=−−+= 11223][
68Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution
+−−=
−−−−
−−=
=
j
j
jj
jj
gggg
GGGG
12
14
1021
111111
111111
3210
3210
4
][][][][
][][][][
D
+
−=
−−−−
−−=
=
j
j
jj
jj
hhhh
HHHH
10
16
1122
111111
111111
3210
3210
4
][][][][
][][][][
D
is the 4-point DFT matrix4D
69Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution
• If denotes the 4-point DFT ofthen from Table 3.5 we observe
• Thus30 ≤≤= kkHkGkYC ],[][][
][kYC ][nyC
−=
=
20
224
33221100
3210
j
j
HGHGHGHG
YYYY
CCCC
][][][][][][][][
][][][][
70Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution• A 4-point IDFT of yields][kYC
=
][][][][
*
][][][][
3210
41
3210
4
CCCC
CCCC
YYYY
yyyy
D
=
−
−−−−−−=
5676
20
224
111111
111111
41
j
j
jj
jj
71Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution
• Example - Now let us extended the twolength-4 sequences to length 7 byappending each with three zero-valuedsamples, i.e.
≤≤≤≤= 640
30nnngnge ,
],[][
≤≤≤≤= 640
30nnnhnhe ,
],[][
72Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution
• We next determine the 7-point circularconvolution of and :
• From the above
60,][][][6
07 ≤≤∑ ⟩−⟨=
=nmnhmgny
mee
][nge ][nhe
][][][][][ 61000 eeee hghgy +=
][][][][][][][][ 16253443 eeeeeeee hghghghg ++++
22100 =×== ][][ hg
73Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution
• Continuing the process we arrive at,)()(][][][][][ 6222101101 =×+×=+= hghgy
][][][][][][][ 0211202 hghghgy ++=,)()()( 5202211 =×+×+×=][][][][][][][][][ 031221303 hghghghgy +++=,)()()()( 521201211 =×+×+×+×=
][][][][][][][ 1322314 hghghgy ++=,)()()( 4211012 =×+×+×=
74Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution
• As can be seen from the above that y[n] isprecisely the sequence obtained by alinear convolution of g[n] and h[n]
,)()(][][][][][ 1111023325 =×+×=+= hghgy
111336 =×== )(][][][ hgy
][nyL
][nyL
75Copyright © 2001, S. K. Mitra
Circular ConvolutionCircular Convolution• The N-point circular convolution can be
written in matrix form as
• Note: The elements of each diagonal of the matrix are equal
• Such a matrix is called a circulant matrixNN ×
−
−−−
−−−
=
−
]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
ggg
hNhNhNh
hhhhhNhhhhNhNhh
Ny
yyy
C
CCC
ML
MOMMMLLL
M