ssp (meng pfiev) chapter3

8/13/2019 Ssp (Meng Pfiev) Chapter3

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Dept. of Telecomm. Eng,Faculty of EEE

SSP2013DHT, HCMUT

Chapter 3:

Spectrum Analysis


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References

[1] Simon Haykin,Adaptive Filter Theory, Prentice Hall,1996 (3rdEd.), 2001 (4thEd.).

[2] Steven M. Kay, Fundamentals of Statistical Signal Processing:

Estimation Theory, Prentice Hall, 1993.

[3] Alan V. Oppenheim, Ronald W. Schafer,Discrete-Time Signal

Processing, Prentice Hall, 1989.

[4] Athanasios Papoulis,Probability, Random Variables, andStochastic Processes, McGraw-Hill, 1991 (3rdEd.), 2001 (4thEd.).


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3. Power Spectrum (1)

( ) ( ) ,j tX x t e dt +

=

For a deterministic signalx(t), the spectrum is well defined: IfX()

represents its Fourier transform, i.e., if

then |X() |2 represents its energy spectrum. This follows from Parsevals

theorem since the signal energy is given by

Thus, |X() |2 represents the signal energy in the band (, + ).

2 2

12( ) | ( ) | .x t dt X d E + +

= =

t0

( )X t

0

2| ( )|X Energy in ( , ) +

+

(3-1)

(3-2)


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( ) ( )T j t

T TX X t e dt

=

However for stochastic processes, a direct application of (3-1) generates

a sequence of random variables for every . Moreover, for a stochastic

process,E{|X(t) |2} represents the ensemble average power (instantaneousenergy) at the instant t.

To obtain the spectral distribution of power versus frequency for stochastic

processes, it is best to avoid infinite intervals to begin with, and start with a

finite interval ( T, T ) in (3-1). Formally, partial Fourier transform of a

processX(t) based on ( T, T ) is given by

so that

represents the power distribution associated with that realization based

on ( T, T ). Notice that (3-4) represents a random variable for every

and its ensemble average gives, the average power distribution based on

( T, T ). Thus

2 2| ( ) | 1( )

2 2

T j tT

T

XX t e dt

T T

=

(3-3)

(3-4)


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1 2

1 2

2( )*

1 2 1 2

( )1 2 1 2

| ( ) | 1( ) { ( ) ( )}

2 2

1( , )

2

T

XX

T T j t tT

T T

T T j t t

T T

XP E E X t X t e dt dt

T T

R t t e dt dtT

= =

=

1 2 ( )

1 2 1 2

1( ) ( ) .

2T XX

T T j t t

T TP R t t e dt dt

T

=

2

2

2 | |

22

1( ) ( ) (2 | |)

2

( ) (1 ) 0

T XX

XX

T j

T

T j

TT

P R e T d T

R e d

=

=

(3-5)

represents the power distribution ofX(t) based on ( T, T ). IfX(t) is

assumed to be w.s.s, thenRXX(t1, t2) =RXX(t1- t2), and (3-5) simplifies to

Let = t1 t2 , we get

to be the power distribution of the w.s.s. processX(t) based on ( T, T ).

Finally letting T in (3-6), we obtain

(3-6)


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( ) lim ( ) ( ) 0XX T XX

j

TS P R e d

+

= =

F T( ) ( ) 0.XX XX

R S

12( ) ( )XX XX

jR S e d

+

=

212 ( ) (0) {| ( ) | } ,XX XXS d R E X t P the total power. +

= = =

to be thepower spectral densityof the w.s.s processX(t). Notice that

i.e., the autocorrelation function and the power spectrum of a w.s.s process

form a Fourier transform pair, a relation known as the Wiener-Khinchin

Theorem. From (3-8), the inverse formula gives

(3-7)

and in particular for = 0, we get

From (3-10), the area under SXX() represents the total power of the

processX(t), and hence SXX() truly represents the power spectrum.

(3-8)

(3-9)

(3-10)


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( )* *

1 1 1 1

2

1

12

12

( ) ( )

( ) 0.

i j

XX XX

iXX

n n n nj t t

i j i j i ji j i j

n j t

ii

a a R t t a a S e d

S a e d

+

= = = =

+

=

=

=

+ 0

represents the powerin the band( , ) +

( )XXS ( )XXS

( ) ( ) 0.XX XX

R nonnegative - definite S

The nonnegative-definiteness property of the autocorrelation functiontranslates into the nonnegative property for its Fourier transform

(power spectrum)

(3-11)

From (3-11), it follows that

(3-12)


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0

( ) ( )

( )cos

2 ( )cos ( ) 0

XX XX

XX

XX XX

jS R e d

R d

R d S

+

+

=

=

= =

IfX(t) is a real w.s.s process, thenRXX() =RXX(-), so that

(3-13)

so that the power spectrum is an even function, (in addition to being realand nonnegative).


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3. Power Spectra and Linear Systems (1)

* *( ) ( ) ( ), ( ) ( ) ( ) ( ).XY XX YY XX

R R h R R h h = =

( ) ( ), ( ) ( )f t F g t G

( ) ( ) ( ) ( )f t g t F G

{ ( ) ( )} ( ) ( ) j tf t g t f t g t e dt

+

=

h(t)X(t) Y(t)If a w.s.s processX(t) with autocorrelation

functionRXX() SXX() is applied to alinear system with impulse response h(t), then

the cross correlation functionRXY() and the output autocorrelation function

RYY() can be determined. From there

(3-14)

If (3-15)

then(3-16)

since

(3-17){ }

( )

{ ( ) ( )}= ( ) ( )

= ( ) ( ) ( )

= ( ) ( ).

j t

j j t

f t g t f g t d e dt

f e d g t e d t

F G

+ +

+ +


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* *

( ) { ( ) ( )} ( ) ( )XY XX XXS R h S H = =

( )* * *

( ) ( ) ( ),j j th e d h t e dt H

+ +

= =

( ) ( ) j tH h t e dt

+

=

Using (3-15)-(3-17) in (3-14) we get

2

( ) { ( )} ( ) ( )

( ) | ( ) | .

YY YY XY

XX

S R S H

S H

= =

=

(3-18)since

where

(3-19)

represents the transfer function of the system, and

(3-20)

From (3-18), the cross spectrum need not be real or nonnegative. However

the output power spectrum is real and nonnegative and is related to the

input spectrum and the system transfer function as in (3-20). Eq. (3-20) can

be used for system identification as well.

Example of Thermal noise (Example 11.1, p. 351, [4])


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( ) ( ) ( ) .WW WW R q S q = =

W.S.S White Noise Process: If W(t) is a w.s.s white noise process, then

Thus the spectrum of a white noise process is flat, thus justifying its

name. Notice that a white noise process is unrealizable since its total

power is indeterminate.

From (3-20), if the input to an unknown system is a white noise process,

then the output spectrum is given by

2( ) | ( ) |YY

S q H =

Notice that the output spectrum captures the system transfer function

characteristics entirely, and for rational systems Eq (3-22) may be used

to determine the pole/zero locations of the underlying system.

(3-21)

(3-22)


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2 , | | / 2( ) | ( ) | .0, | | / 2

XX

q BS q H

B

= =

>

/ 2 / 2

/ 2 / 2

( ) ( )

sin( / 2)sinc( / 2)

( / 2)

XX XX

B Bj j

B BR S e d q e d

BqB qB B

B

= =

= =

2| ( )|H

/ 2B / 2B

1

qB

( )XX

R

Example 3.1: A w.s.s white noise process W(t) is passed through a low pass

filter (LPF) with bandwidthB/2. Find the autocorrelation function of the

output process.

Solution: LetX(t) represent the output of the LPF. Then from (3-22)

(3-23)

Inverse transform of SXX() gives the output autocorrelation function to

be(3-24)


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1

2( ) ( )

t T

t TTY t X d

+

=

( ) ( ) ( ) ( ) ( )Y t h t X d h t X t

+

= =

2( ) ( ) | ( ) | .YY XX

S S H =

Eq (3-23) represents colored noise spectrum and (3-24) its autocorrelation

function.

Example 3.2: Let

T Tt

( )h t

1 / 2T

1

2( ) sinc( )

T j t

TTH e dt T

+

= =

represent a smoothing operation using a moving window on the input

processX(t). Find the spectrum of the output Y(t) in term of that ofX(t).

(3-25)

Solution: If we define an LTI system with impulse

response h(t) as in the figure, then in term of h(t),

(3-25) reduces to

(3-26)

so that (3-27)

where (3-28)


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2( ) ( ) sinc ( ).

YY XX S S T =

( )XX

S 2sinc ( )T

T

( )YY

S

so that

(3-29)

Notice that the effect of the smoothing operation in (3-25) is to suppress

the high frequency components in the input (beyond / T), and the

equivalent linear system acts as a low-pass filter (continuous-time moving

average) with bandwidth 2/ T in this case.


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3. Discrete-Time Processes (1)

( ) ( ) ( ),n

X t X nT t nT=

{ } ,kr +

( ) ( ).XX k

k

R r kT +

=

=

( ) 0,XX

j T

kk

S r e

+

=

=

( ) ( 2 / )XX XXS S T = +

22 .B

T

=

For discrete-time w.s.s stochastic processesX(nT) with autocorrelation

sequence (proceeding as above) or formally defining a continuous

time process we get the corresponding

autocorrelation function to be

Its Fourier transform is given by

(3-30)

and it defines the power spectrum of the discrete-time processX(nT).

From (3-30),

(3-31)

so that SXX() is a periodic function with period

(3-32)


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3. Discrete-Time Processes (2)

1

( )2 XX

B jk T

k Br S e d B

= 2

0

1{| ( ) | } ( )

2 XX

B

Br E X nT S d

B

= =

*( ) ( ) ( )XY XX

jS S H e

=

2( ) ( ) | ( ) |YY XX

jS S H e

=

( ) ( )j j nT

n

H e h nT e +

=

=

This gives the inverse relation

and

represents the total power of the discrete-time processX(nT). The

input-output relations for discrete-time system h(nT) translate into

(3-33)

(3-34)

(3-35)and

(3-36)

where (3-37)

represents the discrete-time system transfer function.


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3. Matched Filter (1)

0( ) ( ) ( ), 0r t s t w t t t = + < 0. ThusH(s) and its inverse 1/H(s)

can be chosen to be stableand causalin (3-58). Such a filter is known asthe Wiener factor, and since it has all its poles and zeros in the left half

plane, it represents a minimum phase factor. In the rational case, ifX(t)

represents a real process, then SXX() is even and hence (3-58) reads


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2 20 ( ) ( ) | ( ) ( ) | .XX XX s j s jS S s H s H s = = = =

2 2 2

4

( 1)( 2)( )

( 1)XX

S

+ =

+

2 2 22

4

(1 )(2 )( ) .

1XX

s sS s

s

+ =

+

Example 18.3: Consider the spectrum

which translates into

2s j=

2s j=

1s=+1s =

1

2

js

+= 1

2

js

+=

1

2

js

=

1

2

js

=

The poles () and zeros ( ) of this

function are shown in the figure. From there to maintain the symmetry

condition in (3-59), we may group together the left half factors as

2

21 1

2 2

( 1)( 2 )( 2 ) ( 1)( 2)( )

2 1j js s

s s j s j s sH s

s s

+ + +

+ + + += =

+ +

(3-59)


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( )( ) ( ) jH A e =

0ln ( ) ln ( ) ( ) ( ) .j tH A j b t e dt

+ = =

and it represents the Wiener factor for the spectrum SXX() above.

Observe that the poles and zeros (if any) on thej-axis appear in even

multiples in SXX() and hence half of them may be paired withH(s) (andthe other half withH( s)) to preserve the factorization condition in (3-58).

Notice thatH(s) is stable, and so is its inverse.

More generally, ifH(s) is minimum phase, then lnH(s) is analytic on

the right half plane so that

gives

0

0

ln ( ) ( )cos

( ) ( )sin

t

t

A b t t dt

b t t dt

=

=

Thus

(3-60)


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( ) {ln ( )}.A = H

( ) .

XXS d

Since cost and sint are Hilbert transform pairs, it follows that the phase

function () in (3-60) is given by the Hilbert transform of lnA(). Thus

Eq. (3-60) may be used to generate the unknown phase function of a

minimum phase factor from its magnitude.

For discrete-time processes, the factorization conditions take the form

and

In that case 2( ) | ( ) |XX

jS H e =

0

( ) ( ) k

k

H z h k z

=

=

(3-61)

(3-62)

(3-63)

where the discrete-time system


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2( ) ( ) | | ( ) | ( ).WWs jL s L s L j S = = =

is analytic together with its inverse in |z| >1. This unique minimum phase

function represents the Wiener factor in the discrete-case.

Matched Filter in Colored Noise: Returning back to the matched filter

problem in colored noise, the design can be completed as shown in figure

below

h0(t)=sg(t0 t)1( ) ( )G j L j =

0t t=

( ) ( )g

s t n t +( ) ( ) ( )r t s t w t = +

Whitening Filter Matched Filter

where G(j) represents the whitening filter associated with the noise

spectral density SWW() as in (3-55)-(3-58). Notice that G(s) is the inverse

of the Wiener factorL(s) corresponding to the spectrum SWW() i.e.,

(3-64)

The whitened output sg(t) + n(t) in the figure is similar to (3-38), and from

(3-45) the optimum receiver is given by 0 0( ) ( )gh t s t t =


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1( ) ( ) ( ) ( ) ( ) ( ).g gs t S G j S L j S

= =where

If we insist on obtaining the receiver transfer functionH() for the original

colored noise problem, we can deduce it easily from the below figure

( )H 0

t t=

L-1(s) L(s) ( )H 0

t t=

( )r t( )r t

where0

0

1 1 *0

1 1 *

( ) ( ) ( ) ( ) ( )

( ){ ( ) ( )}

j t

g

j t

H L j H L S e

L L S e

= =

=(3-65)

0()H


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3. AM/FM Noise Analysis (1)

0( ) ( ) cos( ) ( ),X t m t t n t = + +

0( ) cos( ( ) ) ( ),X t A t t n t = + + +

Consider the noisy AM signal

and the noisy FM signal

where

0( ) FM

( ) ( ) PM.

t

c m d

t cm t

=

(3-66)

(3-67)

(3-68)

Here m(t) represents the message signal and a random phase jitter in

the received signal. In the case of FM, (t) = (t) = c m(t) so that the

instantaneous frequency is proportional to the message signal. We will

assume that both the message process m(t) and the noise process n(t) arew.s.s with power spectra Smm() and Snn() respectively. We wish to

determine whether the AM and FM signals are w.s.s, and if so their

respective power spectral densities.


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0

1( ) ( ) cos ( )2

XX mm nnR R R = +

Solution: AM signal: In this case from (3-66), if we assume U(0, 2),

then

so that (see figure below)

0 0( ) ( )( ) ( ).2

XX XX

XX nn

S SS S

+ += +

( )mmS

0

00

( )XX

S 0

( )mm

S

(a) (b)

(3-69)

(3-70)

Thus AM represents a stationary process under the above conditions.


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FM signal: In this case (suppressing the additive noise component in (3-67),

we obtain2

0

0

2

0

0

2

0

( / 2, / 2) {cos( ( / 2) ( / 2) )

cos( ( / 2) ( / 2) )}

{cos[ ( / 2) ( / 2)]2

cos[2 ( / 2) ( / 2) 2 ]}

[ {cos( ( / 2) ( / 2))}cos2

{sin( ( / 2) ( / 2))

XXR t t A E t t

t t

AE t t

t t t

AE t t

E t t

+ = + + + +

+ +

= + +

+ + + + +

= +

+ 0}sin ]

since

0

0

0

{cos(2 ( / 2) ( / 2) 2 )}

{cos(2 ( / 2) ( / 2))} {cos 2 }

{sin(2 ( / 2) ( / 2))} {sin 2 } 0.

E t t t

E t t t E

E t t t E

+ + + +

= + + +

+ + + =

0

0

(3-71)


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2

0 0( / 2, / 2) [ ( , ) cos ( , ) sin ]2XXA

R t t a t b t + =

( , ) {cos( ( / 2) ( / 2))}a t E t t = +

( , ) {sin( ( / 2) ( / 2))}b t E t t = +

Eq (3-71) can be rewritten as

where

and

22

2

( )( ) ( )mm

d RR c R

d

= =

In general, a(t,) and b(t,) depend on both tand so that noisy FM is not

w.s.s in general, even if the message process m(t) is w.s.s. In the special

case when m(t) is a stationary Gaussian process, from (3-68), (t) is also a

stationary Gaussian process with autocorrelation function

for the FM case.

(3-72)

(3-73)

(3-74)

(3-75)


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2 2( (0) ( )).Y

R R

=

22 2 ( (0) ( ))/ 2{ } Y R Rj Y

E e e e = =

{ } {cos } {sin } ( , ) ( , ),jYE e E Y jE Y a t jb t = + = +

In that case the random variable2( / 2) ( / 2) ~ (0, )

YY t t N = +

where

Hence its characteristic function is given by

which for = 1 gives

where we have made use of (3-76) and (3-73)-(3-74). On comparing

(3-79) with (3-78), we get( (0) ( ))

( , ) R R

a t e

=

and

(3-76)

(3-77)

(3-78)

(3-79)

(3-80)

( , ) 0b t (3-81)


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so that the FM autocorrelation function in (3-72) simplifies into2

( (0) ( ))

0( ) cos .2XX

R RAR e

=

Notice that for stationary Gaussian message input m(t) (or (t)), the

nonlinear outputX(t) is indeed strict sense stationary with autocorrelation

function as in (3-82).

FindRXX() for narrowband FM and wideband FM?

(3-82)


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Chapter 4:

Eigenanalysis


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4. Eigenanalysis: Definitions (1)

Let Rdenote (MM)-correlation matrix of wide-senseDTStP represented by (M1)-observation vector u(n).Finding (M1)-vector qsuch that:

for some constant , rewriting in form:

(4.2) has nonzero solution in vector qif:

Equation (4.3) called characteristic equation, whose

roots 1, 2,, Mcalled eigenvalues.General, (4.3) has

Mdistinct roots Msolutions in the vector q.

qRq = (4.1)

( ) 0qIR = (4.2)

( ) 0det = IR (4.3)


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4. Eigenanalysis: Definitions (2)

Let I denote i-th eigenvalue of matrix R,qibe a

nonzero vector such that:

Vector qicalled eigenvectorassociated with i

Example: see pp. 161-162, [1]

iii qRq = (4.4)


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4. Eigenanalysis: Properties (1)

1, 2,, Mare eigenvalues of correlation matrix R, theneigenvalues of correlation matrix Rkequal 1

k, 2k,, M

kfor any k > 0.

Let q1, q2,, qMbe eigenvector corresponding to distincteigenvalues 1, 2,, Mof (MM)-correlation matrix R.Then, q1, q2,, qMare linearly independent.

Let1, 2,, Mbe eigenvalues of (MM)-correlationmatrix R. Then all these eigenvalues are real andnonnegative.

Let q1, q2,, qMbe eigenvector corresponding to distincteigenvalues 1, 2,, Mof (MM)-correlation matrix R.Then, q1, q2,, qMare orthogonal to each other.


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Let q1, q2,, qMbe eigenvector corresponding to distincteigenvalues 1, 2,, Mof (MM)-correlation matrix R.

Define (MM)-matrix:

where

Define (MM)-diagonal matrix:

Then, original matrix Rmay be diagonalized as:

where Qis nonsingular with Q-1=QH Qis unitarymatrix.

[ ]MqqqQ ,,, 21 =

==

ji

jij

H

i

,0

,1qq

),,,(diag 21 M =

RQQ =H (4.5)


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(4.5) can be written:

Let1, 2,, Mbe eigenvalues of (MM)-correlation

matrix R. Then sum of these eigenvalues equals thetrace ofR:

traceof a square matrix defined as sum of diagonalelements of the matrix.

=

==M

i

H

iii

H

1

qqQQR (4.6)

[ ] =

=M

i

i

1

tr R (4.7)

4 i i i (4)


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Condition number = eigenvalue spread = eigenvalue

factor:

Eigenvalues of correlation matrix of a DTStP are

bounded by min and max values of power spectraldensity of the process:

Condition number is bounded by:

min

max)(

=R (4.8)

MiSS i ,...,2,1,maxmin = (4.9)

min

max

min

max)(S

S=

R

4 Ei l i P i (5)


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Minimax Theorem: Let (MM)-correlation matrix Rhave eigenvalues1, 2,, Mthat are arranged in

decreasing order as:

Then:

where Cis subspace of vector space of all(M1)-complex vector; dim(C): dimension of subspace C.

Subspace of dimension kdefined as set of complexvectors that can be written as a linear combination ofq1,q2,,qk

M 21

MkH

H

CkCk ,,2,1,maxmin)dim( ==

= xx

Rxx

0xx (4.10)

4 Ei l i P i (6)


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Karhunen-Loeve expansion: Let (M1)-vector u(n)drawn from a wide-sense stationary process of zero

mean and correlation matrix R. Letq1, q2,, qMbe

eigenvector corresponding toMeigenvalues of R.

Vectoru(n) may be expanded as linear combination of

these eigenvector:

Coefficients of the expansion are zero-mean,

uncorrelated random variables defined by:

=

=M

i

ii ncn1

)()( qu (4.11)

)()( nnc Hii uq=(4.12)

4 Ei l i P ti (7)


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where

Physical interpretation: viewing eigenvectors q1,q2,,qM as coordinates of anM-dimensional space,thus representing random vector u(n) by set of itsprojections c1(n), c2(n), , cM(n) on these axes.

From (4.11):

From (4.12), (4.13):

MincE i ,,2,1,0)]([ ==

==

jijincncE iji

0)]()([ (4.13)

2

1

2)()( nnc

M

i

i u==

MincE ii ,,2,1,)(2

==

(4.14)

(4.15)

4 Ei l i L R k M d li (1)


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Faculty of EEE

SSP2013

DHT, HCMUT48

4. Eigenanalysis: Low-Rank Modeling (1)

Dimensionality reduction: Transformation in a way that adata vector(in data space) can be represented by a reduced

number of effective feature (infeature space) and yetretain most of intrinsic content of input data.

ConsiderM-dimensional data vector u(n) (representingrealization of a wide sense stationary process) that may be

expanded as a linear combination of eigenvectors q1, q2,,qMcorresponding to distinct eigenvalues 1, 2,, Mof(MM)-correlation matrix R of u(n). See (4.11). If knowing

prior knowledgethat (M-p) eigenvalues p+1,, Mare allvery small, then truncating (4.11) at term i=p. Then, an

approximate reconstructionof data vector u(n) can bedefined:

4 Ei l i L R k M d li (2)


49/53


Faculty of EEE

SSP2013

DHT, HCMUT49


Vector u(n) has rankp


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Faculty of EEE

SSP2013

DHT, HCMUT50


Reconstruction error vector:

Mean-square error:

Error is small good data reconstruction ifeigenvalues

p+1

,, M

are all very small.

Example: Application of Low-Rank Modeling (see

pp.179-180, [1])

+===M

pi

ii ncnnn1

)()()()( quue (4.17)

[ ] +===M

pi

inE1

2)( e (4.18)

4 Ei l i Ei filt (1)


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Faculty of EEE

SSP2013

DHT, HCMUT51

4. Eigenanalysis: Eigenfilters (1)

Design optimum FIR filter with optimization criterion

of maximizing output signal-to-noise ratio

4 Ei l i Ei filt (2)


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Faculty of EEE

SSP2013

DHT, HCMUT52


Effects of signal only: Average power of signal at

filter output:

Effects of noise only: Average power of noise at filter

output:

Output signal-to-noise ratio:

RwwH

oP = (4.18)

wwHoN2= (4.19)

ww

RwwH

H

o

ooN

PSNR 2)( == (4.20)

4 Eigenanal sis: Eigenfilters (3)


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Dept of Telecomm Eng SSP2013


Optimum problem: max (SNR)osubject to wHw=1.

Applying minimax theorem (4.10):

Optimum FIR filterthat produces (4.21) having

coefficient vectorwo=qmax is called eigenfilter,where qmax: eigenvector associated with max.

Note: Eigen filtermaximizes the output SNR for a

random signal(w.s.s) in additive white noise,while a matched filtermaximizes the output SNR fora known signalin additive white noise.

2max

max,)(

=oSNR (4.21)

ssp (meng pfiev) chapter3

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