Computation of the DFT of Real...

45
1 Copyright © 2001, S. K. Mitra Computation of the DFT of Computation of the DFT of Real Sequences Real Sequences In most practical applications, sequences of interest are real In such cases, the symmetry properties of the DFT given in Table 3.7 can be exploited to make the DFT computations more efficient

Transcript of Computation of the DFT of Real...

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1Copyright © 2001, S. K. Mitra

Computation of the DFT ofComputation of the DFT ofReal SequencesReal Sequences

• In most practical applications, sequences ofinterest are real

• In such cases, the symmetry properties ofthe DFT given in Table 3.7 can be exploitedto make the DFT computations moreefficient

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NN-Point -Point DFTsDFTs of Two of Two LengthLength--NNReal SequencesReal Sequences

• Let g[n] and h[n] be two length-N realsequences with G[k] and H[k] denoting theirrespective N-point DFTs

• These two N-point DFTs can be computedefficiently using a single N-point DFT

• Define a complex length-N sequence

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

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NN-Point -Point DFTsDFTs of Two Length- of Two Length-NNReal SequencesReal Sequences

• Let X[k] denote the N-point DFT of x[n]• Then, from Table 3.6 we arrive at

• Note that

]}[*][{][21

NkXkXkG ⟩⟨−+=

]}[*][{][21

NjkXkXkH ⟩⟨−−=

][*][* NN kNXkX ⟩−⟨=⟩⟨−

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NN-Point-Point DFTs DFTs of Two Length- of Two Length-NNReal SequencesReal Sequences

• Example - We compute the 4-point DFTs ofthe two real sequences g[n] and h[n] givenbelow

• Then is given by

}{]}[{},{]}[{ 11221021 == nhng↑ ↑

]}[{]}[{]}[{ nhjngnx +=

}{]}[{ jjjjnx +++= 12221↑

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NN-Point-Point DFTs DFTs of Two Length- of Two Length-NNReal SequencesReal Sequences

• Its DFT X[k] is

• From the above

• Hence

+

=

+

++

−−−−

−−=

22

264

1

2221

111111

111111

3210

j

j

jjjj

jj

jj

XXXX

][][][][

][][* 22264 jjkX −−−=

]22264[]4[* 4 −−−=⟩−⟨ jjkX

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NN-Point-Point DFTs DFTs of Two Length- of Two Length-NNReal SequencesReal Sequences

• Therefore

verifying the results derived earlier

}{]}[{ jjkG +−−= 1214

}{]}[{ jjkH +−= 1016

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22NN-Point DFT of a Real-Point DFT of a RealSequence Using an Sequence Using an NN-point DFT-point DFT

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

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

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

Nnnvnhnvng ≤≤+== 0122 ],[][],[][

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22NN-Point DFT of a Real-Point DFT of a RealSequence Using an Sequence Using an NN-point DFT-point DFT

• Define a length-N complex sequence

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

]}[{]}[{]}[{ nhjngnx +=

]}[*][{][21

NkXkXkG ⟩⟨−+=

]}[*][{][21

NjkXkXkH ⟩⟨−−=

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22NN-Point DFT of a Real-Point DFT of a RealSequence Using an Sequence Using an NN-point DFT-point DFT

• Now ∑−

==

12

02

N

n

nkNWnvkV ][][

∑ ∑−

=

=

+++=1

0

1

0

122

22 122

N

n

N

n

knN

nkN WnvWnv )(][][

∑ ∑−

=

=+=

1

0

1

02

N

n

N

n

kN

nkN

nkN WWnhWng ][][

∑ ∑−

=

=−≤≤+=

1

0

1

02 120

N

n

N

n

nkN

kN

nkN NkWnhWWng ,][][

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22NN-Point DFT of a Real-Point DFT of a RealSequence Using an Sequence Using an NN-point DFT-point DFT

• i.e.,

• Example - Let us determine the 8-pointDFT V[k] of the length-8 real sequence

• We form two length-4 real sequences asfollows

120],[][][ 2 −≤≤⟩⟨+⟩⟨= NkkHWkGkV NkNN

}{]}[{ 11102221=nv↑

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22NN-Point DFT of a Real-Point DFT of a RealSequence Using an Sequence Using an NN-point DFT-point DFT

• Now

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

}{]}[{]}[{ 10212 == nvng↑

}{]}[{]}[{ 112212 =+= nvnh↑

70],[][][ 484 ≤≤⟩⟨+⟩⟨= kkHWkGkV k

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22NN-Point DFT of a Real-Point DFT of a RealSequence Using an Sequence Using an NN-point DFT-point DFT

1064000 =+=+= ][][][ HGV][][][ 111 1

8 HWGV +=41422111 4 .)()( / jjej j −=−+−= − π

202222 228 −=⋅+−=+= − /][][][ πjeHWGV

][][][ 333 38 HWGV +=

41420111 43 .)()( / jjej j −=+++= − π

264004 48 −=⋅+=+= − πjeHWGV ][][][

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22NN-Point DFT of a Real-Point DFT of a RealSequence Using an Sequence Using an NN-point DFT-point DFT

][][][ 115 58 HWGV +=

41420111 45 .)()( / jjej j +=−+−= − π

202226 2368 −=⋅+−=+= − /][][][ πjeHWGV

][][][ 337 78 HWGV +=

41422111 47 .)()( / jjej j +=+++= − π

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Linear Convolution Using theLinear Convolution Using theDFTDFT

• Linear convolution is a key operation inmany signal processing applications

• Since a DFT can be efficiently implementedusing FFT algorithms, it is of interest todevelop methods for the implementation oflinear convolution using the DFT

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Linear Convolution of TwoLinear Convolution of TwoFinite-Length SequencesFinite-Length Sequences

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

• Denote• Define two length-L sequences

1−+= MNL

−≤≤−≤≤= 10

10LnNNnngnge ,

],[][

−≤≤−≤≤= 10

10LnM

Mnnhnhe ,],[][

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Linear Convolution of TwoLinear Convolution of TwoFinite-Length SequencesFinite-Length Sequences

• Then

• The corresponding implementation schemeis illustrated below

][][][][][][ nhngnynhngny CL === * L

Zero-paddingwith

zeros1)( −N point DFT

Zero-paddingwith

zeros1)( −M−−+ )( 1MN

point DFT×

g[n]

h[n]

Length-N

][nge

][nhe −−+ )( 1MN

−−+ )( 1MNpoint IDFT

][nyL

Length-M Length- )( 1−+ MN

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Linear Convolution of a Finite-Linear Convolution of a Finite-Length Sequence with anLength Sequence with anInfinite-Length SequenceInfinite-Length Sequence

• We next consider the DFT-basedimplementation of

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

][][][][][ nxnhnxhnyM∑−

==−=

1

0l

ll *

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Overlap-Add MethodOverlap-Add Method• We first segment x[n], assumed to be a

causal sequence here without any loss ofgenerality, into a set of contiguous finite-length subsequences of length N each:

where

][nxm

∑∞

=−=

0mm mNnxnx ][][

−≤≤+= otherwise0

10,

],[][ NnmNnxnxm

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Overlap-Add MethodOverlap-Add Method

• Thus we can write

where

• Since h[n] is of length M and is oflength N, the linear convolutionis of length

][nxm][][ nxnh m*

][][][ nxnhny mm = *

∑∞

=−==

0mm mNnynxnhny ][][][][ *

1−+MN

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Overlap-Add MethodOverlap-Add Method• As a result, the desired linear convolution

has been broken up into asum of infinite number of short-lengthlinear convolutions of lengtheach:

• Each of these short convolutions can beimplemented using the DFT-based methoddiscussed earlier, where now the DFTs (andthe IDFT) are computed on the basis of

points

][][][ nxnhny = *

1−+MN

)( 1−+MN

][][][ nhnxny mm = L

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Overlap-Add MethodOverlap-Add Method

• There is one more subtlety to take care ofbefore we can implement

using the DFT-based approach• Now the first convolution in the above sum,

, is of lengthand is defined for

∑∞

=−=

0mm mNnyny ][][

1−+MN20 −+≤≤ MNn

][][][ 00 nxnhny = *

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Overlap-Add MethodOverlap-Add Method• The second short convolution

, is also of lengthbut is defined for

• There is an overlap of samplesbetween these two short linear convolutions

• Likewise, the third short convolution , is also of length

but is defined for

1−+MN22 −+≤≤ MNnN

20 −+≤≤ MNn

1−M

][][ nxnh 2*

][][ nxnh 1*

=][2 ny

=][1 ny

1−+MN

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Overlap-Add MethodOverlap-Add Method

• Thus there is an overlap of samplesbetween and

• In general, there will be an overlap ofsamples between the samples of the shortconvolutions andfor

• This process is illustrated in the figure onthe next slide for M = 5 and N = 7

][][ nxnh r 1−* ][][ nxnh r*

1−M

1−M][][ nxnh 1* ][][ nxnh 2*

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Overlap-Add MethodOverlap-Add Method

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Overlap-Add MethodOverlap-Add MethodAdd

Add

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Overlap-Add MethodOverlap-Add Method• Therefore, y[n] obtained by a linear

convolution of x[n] and h[n] is given by],[][ nyny 0=

],[][][ 710 −+= nynyny],[][ 71 −= nyny

],[][][ 147 21 −+−= nynyny],[][ 142 −= nyny

•••

60 ≤≤ n107 ≤≤ n1311 ≤≤ n1714 ≤≤ n2018 ≤≤ n

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Overlap-Add MethodOverlap-Add Method

• The above procedure is called the overlap-add method since the results of the shortlinear convolutions overlap and theoverlapped portions are added to get thecorrect final result

• The function fftfilt can be used toimplement the above method

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Overlap-Add MethodOverlap-Add Method• Program 3_6 illustrates the use of fftfilt

in the filtering of a noise-corrupted signalusing a length-3 moving average filter

• The plots generated by running this programis shown below

0 10 20 30 40 50 60-2

0

2

4

6

8

Time index n

Am

plitu

de

s[n]y[n]

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Overlap-Save MethodOverlap-Save Method• In implementing the overlap-add method

using the DFT, we need to compute two -point DFTs and one -

point IDFT since the overall linearconvolution was expressed as a sum ofshort-length linear convolutions of length

each• It is possible to implement the overall linear

convolution by performing instead circularconvolution of length shorter than

)( 1−+MN )( 1−+MN

)( 1−+MN

)( 1−+MN

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Overlap-Save MethodOverlap-Save Method

• To this end, it is necessary to segment x[n]into overlapping blocks , keep theterms of the circular convolution of h[n]with that corresponds to the termsobtained by a linear convolution of h[n] and

, and throw away the other parts ofthe circular convolution

][nxm

][nxm

][nxm

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Overlap-Save MethodOverlap-Save Method

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

• Let denote the result of a linearconvolution of x[n] with h[n]

• The six samples of are given by

][nyL

][nyL

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Overlap-Save MethodOverlap-Save Method][][][ 000 xhyL =

][][][][][ 01101 xhxhyL +=][][][][][][][ 0211202 xhxhxhyL ++=][][][][][][][ 1221303 xhxhxhyL ++=

][][][][][ 22314 xhxhyL +=][][][ 325 xhyL =

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Overlap-Save MethodOverlap-Save Method• If we append h[n] with a single zero-valued

sample and convert it into a length-4sequence , the 4-point circularconvolution of and x[n] is givenby

][][][][][][][ 2231000 xhxhxhyC ++=][][][][][][][ 3201101 xhxhxhyC ++=]0[]2[]1[]1[]2[]0[]2[ xhxhxhyC ++=][][][][][][][ 1221303 xhxhxhyC ++=

][nhe][nhe][nyC

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Overlap-Save MethodOverlap-Save Method• If we compare the expressions for the

samples of with the samples of ,we observe that the first 2 terms of donot correspond to the first 2 terms of ,whereas the last 2 terms of areprecisely the same as the 3rd and 4th termsof , i.e.,

][nyL

][nyL

][nyL

][nyC][nyC

][nyC

],[][ 00 CL yy ≠ ][][ 11 CL yy ≠

],[][ 22 CL yy = ][][ 33 CL yy =

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Overlap-Save MethodOverlap-Save Method

• General case: N-point circular convolutionof a length-M sequence h[n] with a length-Nsequence x[n] with N > M

• First samples of the circularconvolution are incorrect and are rejected

• Remaining samples correspondto the correct samples of the linearconvolution of h[n] with x[n]

1−M

1+−MN

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Overlap-Save MethodOverlap-Save Method

• Now, consider an infinitely long or verylong sequence x[n]

• Break it up as a collection of smaller length(length-4) overlapping sequences as

• Next, form

][nxm∞≤≤≤≤+= mnmnxnxm 0302 ,],[][

][][][ nxnhnw mm = 4

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Overlap-Save MethodOverlap-Save Method• Or, equivalently,

• Computing the above for m = 0, 1, 2, 3, . . . ,and substituting the values of wearrive at

][][][][][][][ 2231000 mmmm xhxhxhw ++=

][][][][][][][ 3201101 mmmm xhxhxhw ++=

][][][][][][][ 0211202 mmmm xhxhxhw ++=

][][][][][][][ 1221303 mmmm xhxhxhw ++=

][nxm

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Overlap-Save MethodOverlap-Save Method][][][][][][][ 22310000 xhxhxhw ++=][][][][][][][ 32011010 xhxhxhw ++=

][][][][][][][][ 202112020 yxhxhxhw =++=][][][][][][][][ 312213030 yxhxhxhw =++=

←Reject

←Reject

←Save

←Save

][][][][][][][ 42512001 xhxhxhw ++=][][][][][][][ 52213011 xhxhxhw ++=

][][][][][][][][ 422314021 yxhxhxhw =++=][][][][][][][][ 532415031 yxhxhxhw =++=

←Reject

←Reject

←Save

←Save

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Overlap-Save MethodOverlap-Save Method

][][][][][][][ 62514002 xhxhxhw ++= ←Reject

][][][][][][][ 72415012 xhxhxhw ++= ←Reject

][][][][][][][][ 642516022 yxhxhxhw =++= ←Save

][][][][][][][][ 752617032 yxhxhxhw =++= ←Save

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Overlap-Save MethodOverlap-Save Method

• It should be noted that to determine y[0] andy[1], we need to form :

and compute forreject and , and saveand

][nx 1−,][,][ 0100 11 == −− xx

][][][ nxnhnw 11 −− = 4 30 ≤≤ n][01−w ][11−w ][][ 021 yw =−

][][ 131 yw =−

][][],[][ 1302 11 xxxx == −−

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Overlap-Save MethodOverlap-Save Method

• General Case: Let h[n] be a length-Nsequence

• Let denote the m-th section of aninfinitely long sequence x[n] of length Nand defined by

with M < N

10)],1([][ −≤≤+−+= NnmNmnxnxm

][nxm

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Overlap-Save MethodOverlap-Save Method

• Let• Then, we reject the first samples of

and “abut” the remaining samples of to form , the linear convolution of

h[n] and x[n]• If denotes the saved portion of ,

i.e.

][nwm

][nwm

][][][ nxnhnw mm = N

1−M1+−MN

][nyL

][nym ][nwm

−≤≤−−≤≤= 21],[

20,0][ NnMnwMnny

mm

Page 43: Computation of the DFT of Real Sequencessip.cua.edu/res/docs/courses/ee515/chapter03/ch3-3.pdfOverlap-Save Method • To this end, it is necessary to segment x[n] into overlapping

43Copyright © 2001, S. K. Mitra

Overlap-Save MethodOverlap-Save Method

• Then

• The approach is called overlap-savemethod since the input is segmented intooverlapping sections and parts of the resultsof the circular convolutions are saved andabutted to determine the linear convolutionresult

11],[)]1([ −≤≤−=+−+ NnMnyMNmny mL

Page 44: Computation of the DFT of Real Sequencessip.cua.edu/res/docs/courses/ee515/chapter03/ch3-3.pdfOverlap-Save Method • To this end, it is necessary to segment x[n] into overlapping

44Copyright © 2001, S. K. Mitra

Overlap-Save MethodOverlap-Save Method• Process is illustrated next

Page 45: Computation of the DFT of Real Sequencessip.cua.edu/res/docs/courses/ee515/chapter03/ch3-3.pdfOverlap-Save Method • To this end, it is necessary to segment x[n] into overlapping

45Copyright © 2001, S. K. Mitra

Overlap-Save MethodOverlap-Save Method