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Spectrum Estimation

Dr. HassanpourPayam MasoumiMariam ZabihiAdvanced Digital Signal Processing SeminarDepartment of Electronic EngineeringNoushirvani University

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Course Outlines

•Introduction

Fourier Series and Transform

Time/Frequency Resolutions

Autocorrelation & spectrum estimation

•Non-parametric Methods

Periodogram

Modified Periodogram

Bartlett’s Method

Welch’s Method

Blackman-Tukey Method

•Parametric Methods

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Fourier Series and Transform

Fourier basis functions:

real and imaginar parts of a complex sinusoid

vector representation of a complex exponential.

tjke 0

Re

Im

t

)sin( 0tk )cos( 0tk

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Fourier Series:

k

tjkkectx 0)(

2

20

0

0

0)(1 T

T

tjkk dtetx

Tc

k=…,-1,0,1,…

kt

)(tx )(kc

,

n0T

ffT0offon TTT 0 0

1

T

5

t k

dtetxfX ftj 2)()(

)(tx )( fX

dfefXtx ftj 2)()(

,

Fourier Transform:

ffT0

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Discrete Fourier Transform (DFT)

)0(X)1(X

)1( Nx )1( NX

)0(x

)1(x

1

0

2

)()(N

m

N

kmj

emxkX

,....1,0,1....,,)()(1

0

2

kekXmxN

m

N

kmj

DFT

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Autocorrelation & Spectrum estimation

Autocorrelation:

k

jkx

jkx ekreP )()(

)()}(*)(12

1{lim krnxknx

N x

N

NnN

Power spectrum :

Spectrum estimation is a problem that involves

estimating from finite number of noisy

measurements of x(n).

)( tjx eP

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Nonparametric methods

•Peroidogram

•Modified periodogram

•Bartlett method

•Welch method

•Blackman-Tukey method

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The periodogram

kN

nx nxknx

Nkr

1

0

)(*)(1

)(ˆ

k

jkx

jkper ekreP )(ˆ)(ˆ

Estimated autocorrelation:

Estimated power spectrum or periodogram:

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2|)(|1

)()(1

)(ˆ jN

jN

jN

jper eX

NeXeX

NeP

The periodogram cont.

)(nx

)()()( nxnwnx RN

x N N N Nn

1 1r̂ (k) x (n k)x (n) x (k) x ( k)

N N

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The periodogram of white noise

)(nx

DFT )(kX N22 |)(|

1)(ˆ kX

NeP N

Nkjper

: white noise with a variance , length N=32 2

)(nxN2|.|

1

N

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The estimated autocorrelation sequence

White noise power spectrum

The periodogram of white noise cont.

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Periodogram of sinusoid in noise

)()sin()( 0 nvnAnx

)(2

1)( 0

22 AeP vj

x

0

2v

2

2

1A

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Periodogram of sinusoid in noise cont.

)1(win )1()(ˆ1

1 NzeP jX

)(nx

2|.|1

N

)(1 nz)(1 ny

)1()(ˆ2

2 NzeP jX

2|.|1

N

)(2 nz)(2 ny

)1()(ˆ NzeP Lj

XL2|.|

1

N

)(nzL)(nyL

)2(win

)(Lwin

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Periodogram Bias

)()(2

1)}(ˆ{

j

Bj

xj

per eWePePE

{}E)(ˆ krx )()(1 1

0

krN

kNkr

N x

kN

nx

B

N | k || k | 0

w (k) N0 o.w

jkBx

jper ekwkrePE _)()()}(ˆ{

)()}(ˆ{lim jx

jper

NePePE

Thus, the bias is deference between estimated and actual Power spectrum.

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)()sin()( 0 nvnAnx

2

4

1A

0

Periodogram of sinusoid in noise cont.

)]()([4

1)}(ˆ{ )()(22 00 j

Bj

Bvj

per eWeWAePE

2

kNk

20

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x(n) 1sin(0.4 n ) v(n) Example:

128N 512N

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Periodogram Resolution

)()sin()sin()( 221111 nvnAnAnx

)(2

1)(

2

1)( 2

221

21

2 AAeP vj

x

)]()([4

1

)]()([4

1)}(ˆ{

)()(22

)()(21

2

22

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jB

jB

jB

jBv

jper

eWeWA

eWeWAePE

NeP j

per

289.0)](ˆ[Res

Set equal to the width of main lobe of the spectral windowat it’s half power or 6dB point.

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Example:

128N 512N

1 2x(n) 1sin(0.4 n ) 1sin(0.45 n ) v(n)

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Properties of the periodogram

Bias:

Resolution:

Variance:

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0

)()(1

)(ˆ

n

n

jnR

jper enwnx

NeP

)()(2

1)}(ˆ{

j

Bj

xj

per eWePePE

NeP j

per

289.0)](ˆ[Res

)()}(ˆ{ 2 jx

jper ePePVar

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Modified Periodogram2

2 )()(1

|)(|1

)(ˆ

n

jnR

jN

jper enwnx

NeX

NeP

Would there be any benefit in replacing the rectangular window

with other windows? (for example triangular window)

2)(*)(

2

1)(ˆ

j

Rj

xj

per eWePN

eP 2)1(

)2sin(

)2sin()(,

Njj

R eN

eW

2

)()(1

)(ˆ

n

jnR

jM enwnx

NUeP

deWN

nwN

U jN

n

221

0

)(2

1)(

1

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1 2x(n) 0.1sin(0.2 n ) 1sin(0.3 n ) v(n) Example:

N=128Rectangular Window

N=128Hamming Window

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Properties of the M-periodogram

Bias:

Resolution: window dependent

Variance:

2)()(

2

1)}(ˆ{

jj

xj

M eWePNU

ePE

)()}(ˆ{ 2 jx

jM ePePVar

2

)()(1

)(ˆ

n

jnR

jM enwnx

NUeP

21

0

)(1

N

n

nwN

U

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Bartlett’s method (periodogram averaging)

kienxL

ePL

n

jni

jiper ...,,2,1;)(

1)(ˆ

21

0

....

PointsL PointsL PointsL

)(1 nx )(2 nx )(nxk

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Properties of Bartlett’s method

Bias:

Resolution:

Variance:

1

0

21

0

)(1

)(ˆk

i

L

n

jnjB eiLnx

NeP

)()(2

1)}(ˆ{

j

Bj

xj

B eWePePE

NkeP j

B

289.0)](ˆ[Res

)(1

)}(ˆ{ 2 jx

jper eP

kePVar

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1 2x(n) 1sin(0.25 n ) 3sin(0.45 n ) v(n) Example:

1

512

k

N

4

512

k

N

16

512

k

N

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1 2x(n) 1sin(0.25 n ) 3sin(0.45 n ) v(n) Example:

4

128

k

N

4

512

k

N

4

1024

k

N

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Welch’s method (M-periodogram averaging)

....

1,.....,1,0;)()( LniDnxnxi Overlap = L-D

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Properties of Welch’s method

1

0

21

0

)(1

)(ˆk

i

L

n

jnjW eiDnx

KLUeP

1

0

21

0

)( )(1

,)(ˆ1)(ˆ

L

n

k

i

jiM

jW nw

LUeP

LeP

Bias

Resolution Window dependent

Variance

2)()(

2

1)}(ˆ{

jj

xj

B eWePLU

ePE

overlapwithePN

LePVar j

xj

per %50)(16

9)}(ˆ{ 2

304

512

k

N

hamming,%50

128,512

overlap

LN

)()25.0sin(3)2.0sin(1)( 21 nvnx Example:

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1 2x(n) 1sin(0.2 n ) 3sin(0.25 n ) v(n) Resolution:

hamming,%50

64,512

overlap

LN

hamming,%50

128,512

overlap

LN

hamming,%50

256,512

overlap

LN

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1 2x(n) 1sin(0.2 n ) 3sin(0.25 n ) v(n)

rRectangula,%50

128,512

overlap

LN

Bartlett,%50

128,512

overlap

LN

windowing:

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1 2x(n) 1sin(0.2 n ) 3sin(0.25 n ) v(n) windowing:

Hanningoverlap

LN

,%50

128,512

Hamming,%50

128,512

overlap

LN

Blackman,%50

128,512

overlap

LN

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Blackman-Tukey’s method (Periodogram smoothing)

•Note: Bartlett & Welch are design to reduce the variance if the priodogram

by averaging and modified it.

•Periodogram is computed by taking the Fourier transform of a consistent

estimate of the auto correlation sequence.

•For any finite data record of length N, the variance of will be large

for values of k that are close to N. for example:

)(ˆ krx

1)0()1(1

)1(ˆ nklagatxNxN

Nrx

•In Bartlett & Welch, the variance is decreased by reducing the variance

of autocorrelation estimate by averaging.

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Blackman-Tukey’s method cont.

•In the Blackman-Tukey method, the variance is decreased by applying a

window to in order to decrease the contribution of the unreliable

estimates to the periodogram.

Specifically, the Blackman-Tukey spectrum estimation is:

)(ˆ krx

M

Mk

jkx

jBT ekwkreP )()(ˆ)(ˆ

•For example, if w(k) is a rectangular window extending from –M to M

with M<N-1 , then having the largest variance are set to zero and

consequently, the power spectrum estimation will have a smaller variance.

)(ˆ krx

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Properties of B-T’s method

Bias

Resolution Window dependent

Variance

)()(2

1)}(ˆ{

jj

xj

BT eWePePE

M

Mk

jkx

jBT ekwkreP )()(ˆ)(ˆ

M

M

jx

jBT kw

NePePVar )(

1)()}(ˆ{ 22

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)()25.0sin(1)2.0sin(3)( 21 nvnx windowing:

Hanning

MN 128,512

rRectangula

128,512 MN

Bartlett

MN 128,512

Blackman

128,512 MN

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Performance comparisons

•We can summarized the performance of each technique in terms of two criteria.

(I) Variability (which is a normalized variance)

(II) Figure of merit

)}(ˆ{

)}(ˆ{2

j

x

jx

ePE

ePVar

•That is approximately the same for all of the nonparametric methods

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Summeryvariability Resolution Figure of merit

Periodogram

Bartlett

Welch

BlackmanTukey

1

k

1

k

1

8

9

N

M

3

2

N

289.0

Nk

289.0

L

228.1

M

264.0

N

289.0

N

289.0

N

272.0

N

243.0

***50% overlap and the Bartlett window***