DSP First, 2/e READING ASSIGNMENTSdspfirst.gatech.edu/archives/lectures/DSPFirst-L17-pp4.pdf · DSP First, 2/e Lecture 17 DFT: Discrete Fourie r Transform READING ASSIGNMENTS This

DSP First, 2/e

Lecture 17DFT: Discrete Fourier

Transform

READING ASSIGNMENTSREADING ASSIGNMENTS

This Lecture: Chapter 8, Sections 8-1, 8-2 and 8-4

Aug 2016 © 2003-2016, JH McClellan & RW Schafer 3

LECTURE OBJECTIVESLECTURE OBJECTIVES Discrete Fourier Transform Discrete Fourier Transform

1

)/2(][][N

nkNjenxkX

1

)/2(][1][N

nkNjekXnx

DFT from DTFT by frequency samplingfrequency sampling

0

][][n

enxk 0

][][kN

y q y p gq y p g

DFT computation (FFT)

DFT pairs and propertiesPeriodicity in DFT (time & frequency)


Periodicity in DFT (time & frequency)

Sample the DTFT DFTSample the DTFT DFT Want computable Fourier transformWant computable Fourier transform Finite signal length (L) Finite number of frequenciesq

njj enxeX ˆˆ ][)(

1

ˆˆ ][)(L

njj kk enxeX n

0n

1,2,1,0,)/2(ˆ NkkNk

][][)()(:Periodic ˆ)2ˆ( kXNkXeXeX jj

k is the frequency index


][][)()(:Periodic )( kXNkXeXeX jj

Want a Computable INVERSE Fourier TransformINVERSE Fourier Transform Write the inverse DTFT as a finite Riemann sum:

1

0

)/2(2

ˆˆ )(lim][N

k

nNkjj

NeeXnx k

Note that

0k

NN 1

2/2

2ˆ )()(

Propose:

)(][where,][1][ ˆ1

)/2( kjN

nNkj eXkXekXN

nx

This is the inverse Discrete Fourier Transform (IDFT)

)(][,][][0kN

1N

Aug 2016 6© 2003-2016, JH McClellan & RW Schafer

DFTforward be will,][][1

0

)/2(

N

k

nNkjenxkX

Inverse DFT when L=N (proof)(proof) Complex exponentials are ORTHOGONALComplex exponentials are ORTHOGONAL

][1][1

0

)/2(ekXN

nxN

k

nkNj

][1 1)/2(

1)/2(

0

eemxN

NN

nkNjN

mkNj

k

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

0 0

eemx

NN N

nkNjmkNj

k m

][][1

][

1 1)()/2(

0 0

nxemx

eemxN

N NmnkNj

m k

mnmnN

mnN0

][


][][0 0

)()( nxemxN m k

j

Orthogonality of Complex ExponentialsExponentials

The sequence set: 110for1)/2(

NneNknNj The sequence set:

11 1)()/2(

1)/2()/2( eee

NmknNj

NmnNjknNj

1,,1,0for 0 Nne k

,111 )(2

00

mke

eN

eeN

mkj

nn

otherwise ,01 )()/2(eN mkNj

b Nk0 Nk ||because ,

and

Nmk ,0

Ne lj

21lim

Nmk ||


and Ne lNjl

)/2(0 1

lim

4-pt DFT: Numerical Examplep p

Take the 4 pt DFT of the following signal Take the 4-pt DFT of the following signal]1[][][ nnnx ]0,0,1,1[]}[{ nx

20011]3[]2[]1[]0[]0[ 0000 jjjj exexexexX

2/32/22/0 ]3[]2[]1[]0[]1[ jjjjX 4/

2/32/22/0

21

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

j

jjjj

ej

exexexexX

320 jjjj 00011]3[]2[]1[]0[]2[ 320 jjjj exexexexX

2/932/30 ]3[]2[]1[]0[]3[ jjjj exexexexX


4/21

][][][][][jej

N-pt DFT: Numerical Examplep p

Take the N pt DFT of the impulse Take the N-pt DFT of the impulse][][ nnx ]0,0,0,1[]}[{ nx

1

)/2(][][N

nkNjenkX

0

)/2(

0

1][ nkNj

n

en 0

1][n

en

]1,,1,1,1[]}[{ kX


],,,,[]}[{

4-pt iDFT: Numerical Example4 pt iDFT: Numerical Example


Matrix Form for N-pt DFTMatrix Form for N pt DFT In MATLAB, NxN DFT matrix is dftmtx(N), ( )

• Obtain DFT by X = dftmtx(N)*x• Or, more efficiently by X = fft(x,N)

• Fast Fourier transform (FFT) algorithm later

]0[1111]0[ xX

]2[]1[

11

]2[]1[

/)1(4/8/4

/)1(2/4/2

xx

eeeeee

XX

NNjNjNj

NNjNjNj

]1[1]1[ /)1)(1(2/)1(4/)1(2 NxeeeNX NNNjNNjNNj

Signal


DFT matrixSignalvector

FFT: Fast Fourier TransformFFT: Fast Fourier Transform

FFT is an algorithm for computing the DFT FFT is an algorithm for computing the DFT

N log2N versus N2 operationsg2 p Count multiplications (and additions) For example, when N = 1024 = 210

≈10,000 ops vs. ≈1,000,000 operations ≈1000 times faster

What about N=256, how much faster?


Zero-Padding gives denser FREQUENCY SAMPLINGFREQUENCY SAMPLING

W t l f DTFT Want many samples of DTFT WHY? to make a smooth plot Finite signal length (L) Finite number of frequencies (N)

ˆj Thus, we need

1

ˆˆL

jj

)(][,, ̂jeXkXNNL

0

][)(n

njj kk enxeX

1,2,1,0,)/2(ˆ NkkNk


1,2,1,0,)/2( NkkNk

Zero-Padding with the FFTZero-Padding with the FFT

Get many samples of DTFT Get many samples of DTFT Finite signal length (L)

Fi it b f f i (N) Finite number of frequencies (N) Thus, we need )(][,, ̂jeXkXNNL

In MATLAB• Use X = fft(x,N)

Wi h l h N• With L=length(x) less than N• Define xpadtoN = [x,zeros(1,N-L)];• Take the N pt DFT of xpadtoN


• Take the N-pt DFT of xpadtoN

DFT periodic in k (frequency domain)(frequency domain) Since DTFT is periodic in frequency the DFT Since DTFT is periodic in frequency, the DFT

must also be periodic in k)(][ )/2( kNjeXkX

What about Negative indices and Conjugate)()()(][

)(][)/2())/2())(/2())(/2( kNjNNkNjNkNj eXeXeXNkX

eXkX

What about Negative indices and Conjugate Symmetry?

)()( )/2()/2( XX kNjkNj ]1[]31[32

XXN

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

kXkXeXeX kNjkNj

]2[]30[]1[]31[

XXXX


][][ kXkNX ]3[]29[ XX

DFT Periodicity in Frequency IndexFrequency Index

)()(][ )/2(ˆ eXeXkX kNjj k

1,2,1,0)()(][ )(

NkeXeXkX jj k

)()(][][ ˆ)2ˆ( jj eXeXkXNkX )()(][][ eXeXkXNkX

]2[]2[],[][

XNXkXkNX


]2[]2[,e.g. XNX

DFT pairs & propertiesDFT pairs & properties

Recall DTFT pairs because DFT is sampled DTFT Recall DTFT pairs because DFT is sampled DTFT See next two slides

DFT acts on a finite-length signal so we can useDFT acts on a finite-length signal, so we can use DTFT pairs & properties for finite signals

Want DFT properties related to computation

A d ill t t i And, we will concentrate on one more pair: DTFT and DFT of finite sinusoid (or cexp)

Length-L signal Length-L signal N-pt DFT


Th 3 i l h i fi it l thThese 3 signals have infinite length


Summary of DTFT PairsSummary of DTFT Pairs


These 3 properties involve circular indexingThese 3 properties involve circular indexing



Convolution Property not the samethe same Almost true for DFT:Almost true for DFT: Convolution maps to multiplication of transforms

Need a different kind of convolution Need a different kind of convolution CIRCULAR CONVOLUTION

LATER i d d DSP LATER in an advanced DSP course

Like ise for Time Shifting Likewise, for Time-Shifting Has to be circular Because the “n” domain is also periodic


Delay Property of DFTDelay Property of DFT

Recall DTFT propert for time shifting Recall DTFT property for time shifting:dnjjj

d eeXeYnnxny ˆˆˆ )()(][][

Expected DFT property via frequency sampling

I di h t b l t d

dnNkjd ekXkYnnxny )/2(][][][][

Indices such as must be evaluated modulo-N because

dnn dd nNkjNnNkj ee )/2())(/2(


DTFT of a Length-L PulseDTFT of a Length L Pulse

Know DTFT of finite rectangular pulse Know DTFT of finite rectangular pulse Dirichlet form and a linear phase term

1 )ˆsin(01 LLn 2/)1(ˆ

21

21

ˆ

)ˆsin()sin()(

otherwise001

][

Ljj eLeXLn

nx

)ˆsin()ˆsin()ˆ(

21

21

LDL

Use frequency-sampling to get DFT)1)(/(2/)1)(/2(2

1)/2())/2(sin(][ LNkjLNkj eNkDeNkLkX


21

)/2())/2(sin(

][ L eNkDeNk

kX

DTFT of a Finite Length Complex Exponential (1)Complex Exponential (1) Know DTFT of finite rectangular pulseKnow DTFT of finite rectangular pulse Dirichlet form and a linear phase term

2/)1(ˆ21

ˆ )ˆsin(01 Ljj LLn 2/)1(

21

2

)ˆsin()sin()(

otherwise00

][

Ljj eLeXn

nx

Use frequency shift property)ˆsin()ˆsin()ˆ(

21

21

LDL

Use frequency-shift property2/)1)(ˆˆ(

102

1ˆ

ˆ0

0

))ˆˆ(i ())ˆˆ(sin()(

th i00

][

Ljjnj

eLeYLne

ny


021 ))(sin(otherwise0

DTFT of a Finite Length Complex Exponential (2)Complex Exponential (2) Know DTFT, so we can sample in frequencyKnow DTFT, so we can sample in frequency

2/)1)(ˆˆ(

021

021

ˆˆ

00

))ˆˆ(sin())ˆˆ(sin()(

otherwise00

][

Ljjnj

eLeYLne

ny

Thus, the N-point DFT is02 ))((otherwise0

ˆ j ˆ0

00

][

nj

NnLLne

ny

Dirichlet Function

2/)1)(ˆ(02

21

2ˆ

02))ˆ(sin(][

ˆat)(][

LjNk

Nkj

Nk

eLkY

eYkY

)ˆsin()ˆsin()ˆ(

21

21

LDL


02

21 ))ˆ(sin(

][

N

k ekY

20-pt DFT of Complex ExponentialExponential

)5.2(ˆ 20 N

21ˆ20 ))ˆ(sin(0 knj kLLne 2/)1)(ˆ(

02

21

021

110

20

))ˆ(sin())(sin(][

otherwise00

][

Lj

NkN

jN

k

eLkXLne

nx


form Dirichletshifteda is outline theso)ˆsin()ˆsin()ˆ(

21

21

LDL

20-pt DFT of Complex Exp: “on the grid” the grid

)2(ˆ 20 N

/))(ˆ(21ˆ

20 ))ˆ(sin(0 knj kLLne 2/)1)(ˆ(

02

21

021

220

2

))ˆ(sin())(sin(][

otherwise00

][

Lj

NkN N

k

eLkXLne

nx

Aug 2016 © 2003-2016, JH McClellan & RW Schafer 29form Dirichletshifteda is outline theso)ˆsin()ˆsin()ˆ(

21

21

LDL

50-pt DFT of Sinusoid: zero paddingpadding

50,202

NL

4.0)10(ˆ 20 N

?

04 otherwise0

0)ˆcos(][

LnnA

nx


2/)1)(ˆ(0

2212/)1)(ˆ(

02

21

40

20

2

)ˆ()ˆ(][ LjN

kL

LjN

kL

Nk

Nk

eADeADkX

RECALL: BandPass Filter (BPF)( )

fFrequency shiftingup and down is doneby cosine multiplicationin the time domain

BPF i f hift d

in the time domain

BPF is frequency shiftedversion of LPF (below)

)ˆsin()ˆcos(2][ diff2

1

midBP nnnnh

ˆ|ˆ|ˆ1ˆ|ˆ|0

)(21

1

coco

coˆ

BPj

DTFTeH


|ˆ|ˆ0

2co

50-pt DFT of Sinusoid: zero paddingpadding

50,202

NL

4.0)10(ˆ 20 N

?

Zero-crossings of Dirichlet ?Width of Dirichlet ?

Thus we havea simple BPF


Density of frequency samples?

DSP First, 2/e READING ASSIGNMENTSdspfirst.gatech.edu/archives/lectures/DSPFirst-L17-pp4.pdf · DSP First, 2/e Lecture 17 DFT: Discrete Fourie r Transform READING ASSIGNMENTS This

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