spectrum estimation.ppt

7/25/2019 spectrum estimation.ppt

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Spectrum EstimationDr. Hassanpour Payam Masoumi

Mariam ZabihiAdvanced Digital Signal Processing Seminar Department of Electronic Engineering

oushirvani !niversity


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"ourse #utlines

$%ntroduction&ourier Series and 'ransform

'ime(&re)uency *esolutions

Autocorrelation + spectrum estimation

$ on,parametric MethodsPeriodogram

Modified Periodogram

-artlett s Method

/elch s Method

-lac0man,'u0ey Method

$Parametric Methods


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&ourier Series and 'ransform

&ourier basis functionsreal and imaginar parts of a comple2 sinusoidvector representation of a comple2 e2ponential.

t jk e 3

*e

%m

t

4sin5 3t k 4cos5 3t k


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&ourier Series :

==

k

t jk k ect x 345

= 6

63

3

3

3451 T

T

t jk k dt et x

T

c

k=,-1,0,1,

k t

45t x 45k c

7

n3T

ff T 3off on T T T +=3 3

1

T


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t k

= dt et x f X ft j 64545

45t x 45 f X

= df e f X t x ft j 64545

7

&ourier 'ransform

= ff T 3


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Discrete Fourier Transform (DFT)435 X 415 X

415 N x 415 N X

435 x

415 x

=

=1

3

6

4545 N

m

N

km j

em xk X

7....17371....7745451

3

6

==

=k ek X m x

N

m

N km

j

D&'


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Autocorrelation + Spectrum estimation

Autocorrelation

=

=k

jk x

jk x ek r e P

4545

4548594516

1:lim k r n xk n x N x

N

N n N

=++ =

Po;er spectrum

Spectrum estimation is a problem that involvesestimating from finite number of noisymeasurements of 25n4.

45 t j x e P


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onparametric methods

$Peroidogram

$Modified periodogram

$-artlett method

$/elch method

$-lac0man,'u0ey method


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'he periodogram

=

+=k N

n x n xk n x

N k r

1

3

459451

45?66

45n x N 6=.=

1 N


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'he estimated autocorrelation se)uence

/hite noise po;er spectrum

'he periodogram of ;hite noise cont.


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Periodogram of sinusoid in noise

454sin545 3 nvn An x ++=

4561

45 366 += Ae P v

j x

3

6v

6

6

1 A


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Periodogram of sinusoid in noise cont.

415win 41545< 11 = N z e P j X

45n x

6=.=1

N

451 n z 451 n y

41545< 66 = N z e P j

X 6

=.=1

N

456 n z 456 n y

41545< = N z e P L j X L 6=.=

1 N

45n z L45n y L

465win

45 Lwin


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Periodogram -ias

454561

485


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454sin545 3 nvn An x ++=

6

A1

A

3

Periodogram of sinusoid in noise cont.

4B545CA1

485


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25n4 1sin53.A n 4 v5n4= + +E2ample

16D= N E16= N


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Periodogram *esolution

454sin54sin545 661111 nvn An An x ++++=

4561

4561

45 66

616

16 ++= A Ae P v

j x

4B545C1

4B545C1

485


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E2ample

16D= N E16= N

1 625n4 1sin53.A n 4 1sin53.AE n 4 v5n4= + + + +


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Properties of the periodogram

-ias

*esolution

ariance

61

3

45451

45<

=

=n

n

jn R

j per enwn x N

e P

454561

485


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Modified Periodogram6

6 45451

=45=1

45< +

=

==n

jn R

j N

j per enwn x N

e X N

e P

/ould there be any benefit in replacing the rectangular ;indo;

;ith other ;indo;sI 5for e2ample triangular ;indo;4

645945

61

45<

j R

j x

j per eW e P N

e P = 641546sin546sin5

457 = N j j R e

N eW

6

45451

4516

*ectangular /indo;

>16

Hamming /indo;


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Properties of the M,periodogram

-ias

*esolution ;indo; dependent

ariance

64545

61

485


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-artlett s method 5periodogram averaging4

k ien x L

e P L

n

jni

ji per ...77671J45

145

spectrum estimation.ppt

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