Discrete Fourier Transform Dr Yvan Petillotceeyrp/WWW/Teaching/B39SE1/DSP2.pdf · Discrete Fourier Transform Dr Yvan Petillot ... Discrete Linear convolution Discrete Fourier Transform

1

Discrete Fourier Transform 3.1

Discrete Fourier Transform

Dr Yvan Petillot


Section Contents

• LTIs and DFT

• Discrete Fourier Transform

• Relation between DFT and Fourier Transform

• Effects of various parameters on DFT

• Summary

2


Fourier Transform

• Used for spectral analysis

• Core to filtering (Convolution theorem)

• Core to signal modeling

• The Basic tool of digital signal processing


LTIs

h(t)x(n) y(n)= λ x(n)

Eigenfunctions of LTIs?eigenvalue

Are

∞<<∞= n- ,e)n(x jnw

As:

∑

∑

∑∑

∞

−∞=

−

∞

−∞=

−

∞

−∞=

−∞

−∞=

=

λ===

=−==

k

jwkjw

jwnjwjwn

k

jwkjwn

k

)kn(jw

k

e)k(h)e(H

e)e(Hee)k(he

e)k(h)kn(x)k(h)n(x*)n(h)n(y

Discrete Time Fourier Transform

3



Reminder

|A|

-fa fa -fa fa-fa fa-fs fs

fs,fat

|A|

FT

discrete signal

periodic spectra

|A|

-fa fa

fs,fat

|A|

FT

Periodic signal

Discrete spectrum


Fourier Series Construction

Original Signal (Sum of Sinusoidal Components)

Individual Pure Sinusoidal Components

4



Discrete Fourier seriesGive representation for periodic signals

Real signal:Non periodic, finite length. We cannot use Fourier Series!

a non-periodic finite sequence can be considered as one period of a periodic

infinite sequence!

signalperiodic the of period the is T whereT/2w

dte)t(xT1

)n(X wheree)n(X)t(x

0

2/T

2/T

)tjnw(

k

tjww 00

π=

== ∫∑−

−∞

−∞=



Example ∞≤≤∞≤≤=+ k- 1,-Nn0 ),n(x)kNn(xp

≤≤

= elsewhere 0

1-Nn0 ),n(x)n(x p

5



Discrete Fourier Transform:Discrete Fourier Series of the periodised signal xp(n)

Use only one period of the resulting sequence for x(n)

Definition of the DFT

1-Nn0 ,e )k(XN1

x(n)

1-Nk0 ,e )n(x X(k)

1-Nk0 ,e )n(x)k(X

N

kn2j1N

0k

N

kn2j1N

0n

N

kn2j1N

0np

≤≤=

≤≤=

≤≤=

π−

=

π−−

=

π−−

=

∑

∑

∑

Inverse DFT



Example

Calculate the DFT

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Discrete Fourier Transform / Fourier Transform

Discrete Time Fourier Transform

θ−−

=

θ ∑=jn1N

0n

j e )n(x)e(Xn)

N

k2(j1N

0n

e )n(xX(k)

π−−

=∑=


θN

k2π


Record length/Frequency resolution/Sampling frequency

∑∑−

=

∞

−∞=

=−δ=1N

0ks

kss )kT(x)kTt()t(x)k(x

nkN2

j1N

0k

nkN

2j-1N

0n

e )fk(XN1

)nT(x

e )nT(x)fk(X

π−

=

π−

=

∑

∑

∆=

=∆

Ts is the sampling frequency

∆f is the frequency spacing (resolution) of the DFT coefficients

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Example: )t600cos(B)t200cos(A)t(x π+π=

Sampled at 1kHz.

Find the period of the resulting sequence x(nT), i.e. N

Find the frequency resolution



Frequency resolution:

If x(nT) is made of frequency components spaced by less than ∆f Hertz, the DFT will NOT represent them

duration signal:T

T1

N

ff

0

0

s ==∆

Same example but k = 0,1,…,9

Notice folding around k = 5 = fs/2

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An analogue signal of 200ms length is sampled at 2.5 kHz.

What is the maximum frequency present is aliasing is to be avoided?

What is the frequency resolution of the DFT?

What analogue frequencies are represented by the DFT?



Solution: An analogue signal of 200ms length is sampled at 2.5 kHz.

What is the maximum frequency present is aliasing is to be avoided?

Nyquist: max frequency = fs/2 = 1.25kHz

What is the frequency resolution of the DFT?

What analogue frequencies are represented by the DFT?

0, 5, 10, 15 ,…, 1250, -1245, …, -5 Hz

Hz52.0

1T1

Nf

f0

s ====∆

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Examples of applications

Hold N constant with two sampling periods Ta, Tb=2 Ta.

Effect?

2f

NT21

NT1

T1

f ,NT

1T1

f a

ab0b

a0a

∆====∆==∆


Examples of applications

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DFT analysis


DFT analysis example

nk)6

2(j5

0n

e )n(xX(k)

π−

=∑=

Ts= 0.01sN 0 1 2 3 4 5x(n) 5 -1.5 6.5 -3 6.5 -1.5

Frequency content of the sequence?

Frequency resolution?

11


Properties of the DFT

Linearity:

[ ] )k(bX)k(aXe (n)]bx)n(ax[(n)bx(n)axDFT 21

nk)N

2(j1N

0n2121 +=+=+

π−−

=∑

Symmetry

if x(n) is a sequence of real numbers:

[ ] [ ]

[ ] [ ]

)kN(X)k(X

k)-X(NX(k)

odd N ,2

1N1,2,..., k ,)kN(XIm)k(XIm

even N ,12N

1,2,..., k ,)kN(XRe)k(XRe

−−∠=∠

=

−=−−=

−=−=



Circular shift

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Circular shift

kmN

2j

12

12

e)k(X)k(X

)mn(x)n(xπ

=

+=

Demonstration

kmN

2j

12

112

e)k(X)k(X

)mn(*)n(x)mn(x)n(xπ

=

+δ=+=

Direct demonstration left as an exercise



Alternate IDFT

*

N

kn2j1N

0k

*

N

kn2j1N

0k

e )k(XN1

x(n)

1-Nn0 ,e )k(XN1

x(n)

=

≤≤=

π−−

=

π−

=

∑

∑

Duality PARSEVAL THEOREM

)k(x)N(XN1

)k(X x(n)

−↔

↔

? N1

Why ∑∑∞

−∞=

∞

−∞=

=k

2

n

2)k(X

N1

x(n)

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Example:

Complete the relationship using the duality principle:

elsewhere 0 x(n)

1-N 0,1,...,n n/N),cos(2 x(n)

==π=

?X(k) n/N)cos(2 x(n) =↔π=

? x(-k)? X(n)N1 =↔=



Example:

elsewhere 0 x(n)

1-N 0,1,...,n n/N),cos(2 x(n)

==π=

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Discrete convolution

• Remember: Filtering is a linear convolution!

• Infinite signals uncommon!

• Can we do the same with finite, periodic signals?

∑∞

−∞=

−=∗=m

2121 )mn(x)m(x)n(x)n( x c(n) Discrete Linear convolution


Discrete convolution

Linear Convolution

∑∞

−∞=

−=∗=m

2121 )mn(x)m(x)n(x)n( x c(n) Linear convolution

Periodic Convolution

)k(X)k(X)k(X

e (n)xN1

(k)X

e (n)xN1

(k)X

p2p1p3

N

kn2j1N

0kp22p

N

kn2j1N

0kp11p

=

=

=

π−−

=

π−−

=

∑

∑)n)(

Nk2

(j1N

0mp3p3

1N

0n

)mn)(N

k2(j1N

0m2pp1p2p1p3

e )n(x)k(X

e (m) x)n(x)k(X)k(X)k(X

π−−

=

−

=

+π−−

=

∑

∑∑

=

==

∑−

=

−=1N

0mp2p1p3 )mn(x)m(x)n(x

Periodic convolution

Periodic Convolution: Same as linear convolution but over one period

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Periodic Convolution

Example: Find the periodic convolution of sequence below

)k(X)k(X)k(X p2p1p3 =∑−

=

−=1N


DFT



Final result:

A bit tedious?

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Other solution:



Summary:

Periodic convolution of two sequences can be obtained by:

• Remaining in the time domain and using the convolution sum directly.

• Moving into the frequency domain using the following scheme:

x1p(n) DFT

x2p(n) DFTx x3p(n)= x1p(n)* x2p(n)IDFT

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Circular convolution

New Problem:

Can we perform linear convolution with finite length signals using DFTs ?

If we use DFT, we are dealing with sampled spectrum which implies … periodic signals!

Why not use the sum definition and forget about DFTs?



A Parte:

Direct DFT Radix 2 FFT Direct Sum and addsconvolution

Numberof points

Complexmultiplies

Complexadditions

Complexmultiplies

Complexadditions

Complexmultiplies

Complexadditions

N N2 N2-N (N/2)log2(N)

Nlog2(N) 2 N2 N2

4 16 12 4 8 32 16

16 256 240 32 64 512 256

64 4096 4032 192 384 8192 4096

256 65536 65280 1024 2048 131072 65536

1024 1048576 1047552 5120 10240 2097152 1048576

DFTs in FFTs form are good for you!

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Definition:

Consider x1(n) and x2(n) and x1p(n) and x2p(n) their periodic equivalent.

∑−

=

−=1N


)n(x)n(x)mn(x)m(x)n(x 21

period one

1N

0mp2p13 ∗=

−= ∑

−

=Circular convolution

)k(X)k(X)k(X p2p1p3 =

x1(n) DFT

x2(n) DFTx x3(n)= x1(n)* x2(n)IDFT



Is it what we want?

Different values

Different length

Linear convolution:

N1+N2-1 length

Circular convolution:

max(N1,N2).

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Solution?Forcing linear and Circular

Convolution to be equivalent


Frequency Convolution

Use duality properties

)k(X)k(XN1

)n(x)n(x

)k(X)k(X)n(x (n)x

2121

2121

∗↔

↔∗

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Correlation

• Measure of similarity

• Target detection & classification

• Noise rejection

• Signal Modeling

• Related to the Power Spectral Density (PSD)

• Related to statistical description of signals (noise)


Correlation

Same as convolution but with no time reversal.

∑∞

−∞=

+=⊗=m

2121xx )mn(x)m(x)n(x (n)x(n)R21

∑∞

−∞=

+=⊗=m

11111xx )mn(x)m(x)n(x (n)x(n)R1

Crosscorrelation

Autocorrelation

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Correlation

Example


Correlation

Example

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Correlation

Properties

(-n)R(n)R1111 xxxx =

(-n)R(n)R1221 xxxx =

)n(x)n(x(p)R

)n(x)n(x(p)R

21xx

21xx

12

21

∗−=

−∗=

Convolution

Autocorrelation is always even

Cross correlation case


Correlation and DFTs

Circular correlation:

∑∑−

=

−

=

+=

+=

1N

0m21

period one

1N

0mp2p1xx )mn(x)m(x)mn(x)m(x(n)R~

21

23


Correlation and DFTs

Can we perform circular correlation with DFTs ?

)n(x)n(x)mn(x)m(x)n(R~

21

period one

1N

0mp2p1xx 21

⊗=

+= ∑

−

=Circular correlation

[ ] )k(X)k(X)n(R~

DFT p2p1*

xx 21=

x1(n) DFT

x2(n) DFT*x IDFT )n(R

~21xx


Linear Correlation and DFTs

24


Example of application

0 50 100 150 200 250 300-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

original signal0 100 200 300 400 500 600

-30

-20

-10

0

10

20

30

40

50

60

autocorrelation

0 50 100 150 200 250 300-4

-3

-2

-1

0

1

2

3

4

noisy signal (0db SNR) 0 100 200 300 400 500 600-30

-20

-10

0

10

20

30

40

50

60

cross-correlation


Learning outcomes

Discrete Fourier Transform, definition and properties

DFT analysis

Influence of parameters on frequency estimation

Discrete convolution: linear / circular

Discrete correlation: linear / circular

Discrete Fourier Transform Dr Yvan Petillotceeyrp/WWW/Teaching/B39SE1/DSP2.pdf · Discrete Fourier Transform Dr Yvan Petillot ... Discrete Linear convolution Discrete Fourier Transform

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