dtft properties - the signal and image processing...

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


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


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


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)(


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



• 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


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

)(



• Therefore

• Hence, is a finite-energy sequence

∞<πω=∫ ω

π=∑

ω

ω−

∞

−∞=

c

nLP

c

c

dnh21][ 2

][nhLP


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



• 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



• 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



• 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



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



• 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


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


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


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


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


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


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



• 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



• 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



• 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



• 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


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




we get

−==

=otherwise0for2for2

,,/,/

][ rNkNrkN

kG

=∑

−

=

−−1

0

N

n

nkNW )( l

,,

0N ,rNk =− lfor

otherwiser an integer

10 −≤≤ Nk


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


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


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


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 =−


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


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


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



• 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



• 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



• 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



• Therefore

∑ ⋅

π−ω

π−ω

=−

=

−π−ω−1

0

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

22sin

22sin

][1 N

k

NNkje

NkN

kN

kXN

)( ωjeX


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



• 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



• i.e.


∑ ∑−

=

−∞

−∞==

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


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


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


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



,• i.e.

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

][][][][ 44 −+++= nxnxnxny 30 ≤≤ n

}{]}[{ 3264=ny↑


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



• 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



• 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


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


Table 3.5:Table 3.5: General Properties General Propertiesof DFTof DFT


Table 3.6:Table 3.6: DFT Properties: DFT Properties:Symmetry RelationsSymmetry Relations

x[n] is a complex sequence


Table 3.7:Table 3.7: DFT Properties: DFT Properties:Symmetry RelationsSymmetry Relations

x[n] is a real sequence


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



• 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



• 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



• Illustration of the concept of a circular shift

][nx ]1[ 6⟩−⟨nx

]5[ 6⟩+⟨= nx ]2[ 6⟩+⟨= nx

]4[ 6⟩−⟨nx



• 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 ⟩⟨


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 lengthsequence given by)( 12 −N

2201

0−≤≤−= ∑

−

=Nnmnhmgny

N

mL ],[][][

][nyL


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



• 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



• 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



• 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


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 =×+×+×+×= )()()()(


Circular ConvolutionCircular Convolution• Likewise ∑ ⟩−⟨=

=

3

04]1[][]1[

mC mhmgy

][][][][][][][][ 23320110 hghghghg +++=711102221 =×+×+×+×= )()()()(

∑ ⟩−⟨==

3

04]2[][]2[

mC mhmgy

][][][][][][][][ 33021120 hghghghg +++=

611202211 =×+×+×+×= )()()()(


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


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


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



• Hence,

• The two 4-point DFTs can also becomputed using the matrix relation givenearlier

,][ 611220 =+++=H

,][ 011222 =−+−=H,][ jjjH −=+−−= 11221

jjjH +=−−+= 11223][



+−−=

−−−−

−−=

=

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



• 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

][][][][][][][][

][][][][


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



• Example - Now let us extended the twolength-4 sequences to length 7 byappending each with three zero-valuedsamples, i.e.

≤≤≤≤= 640

30nnngnge ,

],[][

≤≤≤≤= 640

30nnnhnhe ,

],[][



• 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



• Continuing the process we arrive at,)()(][][][][][ 6222101101 =×+×=+= hghgy

][][][][][][][ 0211202 hghghgy ++=,)()()( 5202211 =×+×+×=][][][][][][][][][ 031221303 hghghghgy +++=,)()()()( 521201211 =×+×+×+×=

][][][][][][][ 1322314 hghghgy ++=,)()()( 4211012 =×+×+×=



• 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


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

dtft properties - the signal and image processing...

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