vs. flatness of its PSD Convergence rate of filter coefficients toward their optimum value, at a given frequency depends on the value of the power spe...

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Introduction about Adaptive Filters Introduction about Blind Adaptive Filters Convergence analysis of LMS algorithm How transforms improve the convergence rate of LMS Why Wavelet transform? Our blind equalization approaches Simulation results

Equalization {an }

Pulse Shape Modulator

{an }

Pulse Shape Modulator

{aˆ n } Channel

equalizer

MF

{aˆ n } Channel

Composite Channel h[n]

MF

Equalizer w[n]

h[n] * w[n] = δ [n] ⇒ w[ n ] =

N

∑wh

k =− N

N

k n−k

1 n = 0 = 0 n = ±1, ± 2, ...

∑ w δ [n − i]

i=− N

i

Example : h0 h  1 h2  h3 h  4

h−1 h− 2 h−3 h− 4   w− 2  0      h0 h−1 h− 2 h−3   w−1  0   h1 h0 h−1 h− 2   w0  = 1      h2 h1 h0 h−1   w1  0 h3 h2 h1 h0   w+ 2  0

xn

Delay

Input xn Signal

Channel

en Random noise Generator (2) en = a n − y n , error = E [ en2 ] = E [ a n + ( ∑ wk x n − k ) 2 − 2 a n ∑ wk x n − k ] 2

200 180 160 140 120 100 80 60 40 20 0 20

15

10

5

0

5

10

15

20

yn =

N

∑w x

k =− N

k

n−k

en = an − yn error = E[en2 ]

1 ∂ error wk (n + 1) = wk (n)an − µ 2 ∂ wk

∂ error ∂ en = 2 E[en ] ∂ wk ∂ wk ∂ yn = −2 E[en ] = −2 E[en xn − k ] ∂ wk

X n = [ xn + N ,..., xn +1 , xn , xn −1 ,..., xn − N ]

T

Wn = [ w− N ,..., w−1 , w0 , w1 ,..., wN ]T yn =

N

∑ wk xn−k

k =− N

en = an − yn

Yn = X nT Wn en = an − yn

1 ∂ error wk (n + 1) = wk (n)an − µ 2 ∂ wk

∂ error = −2 E[en xn − k ] ∂ wk

wn +1 = wn + µ en xn

Equalization, Deconvolution, System compenstation

System Identification

Noise Cancellation

Prediction

Sidelobe cancellation x1 x2 y

xN

y(t)=Σ xi(t)

a1

y

a2

aN

x1 x2

xN

y(t)=Σai xi(t)

Sound Clip  NORMA LIZED INPUT VOIC E S IGNA L (*.W AV) 1 0.8 0.6

Amplitude of Signal

0.4



0.2 0



-0.2 -0.4 -0.6 -0.8 -1 1000

2000

3000

4000 5000 6000 Number of Samples

7000

8000

9000 10000

Graphs – w/o and w/ equalization W ITH NO EQUALIZATION

W ITH E QUA LIZA TION

1

1 Voice S ignal, W ith No Equalization Original Voice Signal

0.6

0.6

0.4

0.4

0.2 0 -0.2

0.2 0 -0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1 1000

2000

3000

4000 5000 6000 Number of Samples

7000

8000

9000 10000

V oice S ignal, With Equalization Original V oice S ignal

0.8

A mplitude of S ignal

A mplitude of Signal

0.8

-1 1000

2000

3000

4000 5000 6000 Number of S amples

7000

8000

9000 10000

Blind Equalization Delay

Input Signal

Channel

Blind equalization approaches 







Stochastic gradient descent approach that minimizes a chosen cost function over all possible choices of the equalizer coefficients in an iterative fashion Higher Order Statistics (HOS) method that is using the higher order cumulants spread of the underlying process, and hence to the flatness Approaches that exploit statistical cyclostationarity information coefficients toward their optimum value, at a given frequency Algorithms that are based on the maximum likelihood criterion. depends on the value of the power spectral density of the

Blind Equalization: HOS C x(1) = E{ x (t )} = ∫ x (t ) f X ( x ) dx

: First Order Statistics

Second Order Statistics : C x( 2) (τ ) = E{x(t ) x(t + τ )} = ∫ x(t ) x(t + τ ) f X ( x(t ), x(t + τ )) dx Third Order Statistics : C x(3) (τ 1 ,τ 2 ) = E{x(t ) x(t + τ 1 ) x(t + τ 2 )} = ∫ x(t ) x(t + τ 1 ) x(t + τ 2 ) f X ( x(t ), x(t + τ 1 ), x(t + τ 2 )) dx

Blind Equalization: HOS White Noise

x[n]

y[n]

Channel

AWGN C

( 2) y

⇒ S y ( f ) = H ( f ) S input ( f ) + S n ( f )

C

( 2) y

⇒ Sy ( f ) = H( f ) K

2

2

Blind Equalization: HOS White Noise

x[n]

y[n]

Channel

AWGN 

For Gaussian signals: C x( n ) = 0

for n > 2

C y( 4 ) = C h( 4 ) + C n( 4 )

Fractionally Spaced Equalizer h1[n]

t=nT/P

Y(m)

y (m) = ∑ W (l )h(m − lP ) + n(m) l

h1[n]

t=T/P

h2[n]

t=2T/P

hi[n]

t=iT/P

hN[n]

t=NT/P

y1[m] y2[m] yi[m] yN[m]

yi (m) = ∑ W (l )hi (m − l ) + ni (m) l

YN (n) = HWN + Lh (n) + N N (n)

Fractionally Spaced Equalizer Fractionally Spaced Sampling ⇒ Cyclostationary Output

YN (n) = HW N + Lh (n) + N N (n)

Formula: 

Channel model



Cost function definition



Updating equalizer coefficients



∇Jˆ (T ) = YN (n)YN′ (n) g 0 − σ W2 H (:,1)

Convergence rate of LMS algorithm 

It is well known that the convergence behavior of conventional LMS algorithm depends on the eigenvalue spread of input process

Faster convergence rate of LMS algorithm

Smaller EIG-spread of input correlation matrix

EIG-spread of input correlation matrix (R) vs. flatness of its PSD 

Convergence rate of filter coefficients toward their optimum value, at a given frequency depends on the value of the power spectral density of the underlying process at that frequency relative to all other frequencies. Smaller EIG-spread of input correlation matrix

PSD flatness of input signal

EIG-spread of R vs. shape of error surface: Example 1, EIG = 1.22

Example 2, EIG = 3

Example 3, EIG = 100

Summary: Better convergence rate of LMS algorithm

Shape (circularity) of error surface

Smaller EIG-spread of input correlation matrix

PSD flatness of input signal

How transforms improve the convergence rate of LMS? 







Band-partitioning property of Wavelet transform  Transformed elements are (at least) approximately uncorrelated with one another  Correlation matrix is closer to a diagonal matrix  An appropriate normalization can convert the result to a normalized matrix whose EIG spread will be much smaller

Advantages of using Wavelet transform 





Efficient transform algorithms exist (e.g. the Mallat algorithm) Transforms can be implemented as filter banks with FIR filters Strong mathematical foundations allow the possibility of custom designing the wavelets e.g. the lifting scheme

Wavelet transform algorithm:

Matrix form implementation of Wavelet transform Low Pass



As mentioned in the previous High Pass slide, Wavelet transform Data Width consists of two low-pass and high-pass filters

c0  c3    .  .     c2 c 1

c1 c2 c3 −c2 c1 −c0 c0 c1 c2 c3 c3 −c2 c1 −c0 . .

.. .. c0 c1 c2 c3 −c2 c1

c3 −c0

c0 c3

          c3  −c0   c1  −c2 

Why Wavelet transform? 

wavelet analysis filters are ”constant-Q” filters; i.e., the ratio of the bandwidth to the center frequency of the band is constant

band-partitioning property of Daubechies filters

After transformation, each coefficient shows the amount of energy passed from one of above filters

The bandwidth of the filters in low frequencies is narrow compared to the bandwidth of the filters in higher frequencies

It’s more probable that the output of filters contain the same amount of energy

Most communication signals have a low-pass nature more likely to obtain a flat spectrum.

PSD of a typical communication signal after different transforms

Effect of Wavelet transform on error surface

MSE of a TD-Godard algorithm vs. WD-Godard algorithm

Formula: 

Channel model



Cost function definition



Updating equalizer coefficients



MSE of a TD-FSE algorithm compared with WDFSE algorithm

References 



Simon Haykin



Prentice Hall

