Introduction to Adaptive Digital Filters Algorithms V.Majidzadeh Advisor: Dr.Fakhraei [email protected]

Introduction to Adaptive DigitalFilters Algorithms

V.Majidzadeh

Advisor: Dr.Fakhraei

[email protected]

2

Outlines

Basic Principles of Adaptive Filtering

Analytical Framework for developing Adaptive Algorithms

Algorithms for Adaptive FIR Filters

Case Study (Adaptive Digital Correction of Analog Errors in Delta-Sigma-Pipeline ADC Architecture

Conclusion

References

3


The Need for Adaptive Filtering (An Intuitive Example)Air is a cost effective communication channel

Wave scattering limits capacity and reliability of communication

200-600 Km

Tra

nsm

eter

Rec

ieve

r

Turbulance

Wave Scattering

4


The Need for Adaptive Filtering (An Intuitive Example)The received signal is the sum of individual components

S(t): Transmited signal

gi(t): gain of the propagation path I

h(t,τ): Time varying channel impulse response

L

ii tgitStr

1

)()()(

)(*),()( tSthtr

L

ii ittgth

1

)()(),(

5


The Need for Adaptive Filtering (An Intuitive Example)

Transmitted signalS (t)

Transmiter Time-varying channel

Inverse time-varying

Filter

Equalized Signal

Received signalr (t)

Time-varying channel model

Equalized time-varying channel

Delay DelayDelay

Received Signal

+

h(t,1) h(t,2) h(t,L)h(t,3)

Transmiter

6


The general structure of an adaptive filterDigital Filter

A conventional digital filter with updateable coefficients.

Quality AssessmentAssess the quality of the filter and generate error signal.

Depends on the adaptive filter application.

Adaptation algorithmThe way in witch the quality assessment is converted into parameter

adjustment.

The parameters available for adjustment might be the impulse response sequence value or more complicated function of the filter’s frequency response.

7


The general structure of an adaptive filter

Filter OutputFilter InputDigital Filter

Adaptation Algorithm

Quality Assessment

E(n)

adp

Filter Output

Desired Output

E(n)

_

+

Filter Output

Test signal

E(n)Xcorr

1Z

W0 W1 WN-1

adp

Filter Output

Filter Input1Z 1Z

8


Useful notations and assumptions:For simplicity taped-delay-line FIR filter used to develop formulas

Filter tap length is assumed to be N with weights Wi where i= 0…N-1.

Filter produce output according to the convolution sum

To facilitate our development define input vector X(k) and weight vectors W as

1

0

)()(Nl

llwlkxky

t)]. X(k-N) ) x(k--[x(k) x(kX(k) 121

] …. W W W = [WW Nt

1210

(k) W X(k) = Xy (k) = W tt

9


Basic FormulationAssume that the filter desired output signal d(k) is available

Use L samples of the input sequence where L>N

Construct summed square error function as below:

1

1

2)()(

Lk

Nk

kykdJ

1

1

** )]()()][()([Lk

Nk

kykdkykdJ

1

1

*1

1

*1

1

*1

1

2)()()()()()()(

Lk

Nk

htLk

Nk

hLk

Nk

tLk

Nk

WkXkXWWkXkdkdkXWkdJ

1

1

2)(

L

Nk

kdD

1

1

* )()(Lk

Nk

kdkXP

1

1

)()(Lk

Nk

h kXkXR

10


Basic Formulation

WSS minimize J if and only if [Nobel 1977]

Evaluate gradient of J:

**)( RWWPWPWDJ ttt *)Re(2 RWWPWD tt

W

WssWW

H

J 0

the second derivative is positive definite

RWPJW 22

PWR SS .

11


Two Solution Techniques

Direct SolutionIf matrix R can be inverted then the normal equations can be used to

find WSS .

Computation complexity is high.

Iterative ApproximationIteratively estimate WSS making use of initial value for WSS and try

to improve it in each iteration step.

PRWSS10

12


The Gradient Search Approach[2]

Two presume on WSS :

The optimal solution WSS is unique.

Any difference between the actual weight vector W and the optimal one, WSS, leads to increase in performance function, J.

C is a small positive constant Jmin

)1( lW )(lW

)(lWdW

dJ

)(

)()1(lWdW

dJClWlW

0WFilter Coefficient (W)

Pe

rfo

rman

ce

Fu

nct

ion

(J

)

One dimensional gradient search on performance function J

13


LMS Algorithm[1],[2]:

)().(2)]([)( 2 kekekekG WW

)().(2

)}()({)(2

kXke

kXWkdke tW

)()()()1(

)()()(

)()(

kXkekWkW

kykdke

kXWky t

Filter output

Error formation

Weight vector update

14


Properties of the LMS :Bounds on the adaptive constant

Modifying the recursive LMS equations in terms of eigenvalue of matrix R results:

Convergence region when :

or

Adaptive time constant:Number of iterations required for any transient to decay to 1/e(37%) of

its initial value.

)0()1(}.)1({)(1

0i

li

l

ni

nii WplW

11 i

ii

1

l

max

20

15


Relative LMS Algorithms:Complex LMS,[1]:

Input, output, and weight vectors are complex.

Normalized LMS,[1]:Find a safe margin for to assure stability.

Increase computation complexity .

N

ii

t kXkXAvrg1

)}().({ max)}().({ kXkXAvrg t

)().(

)().(.)()1(

kXkX

kXkelWlW

t

)()()()1(

)()()(

)()(

* kXkekWkW

kykdke

kXWky t

16


Relative LMS Algorithms:Sign-Error-LMS,[2]:

Sign-Data-LMS,[2]:

Sign-Sign-LMS,[2]:

Multiplier less implementation achieves with noisy gradient estimate.

Convergence may be problem in Sign-Sign-LMS.

)()}.({.)()1( kXkesignkWkW

)}({).(.)()1( kXsignkekWkW

)}({)}.({.)()1( kXsignkesignkWkW

17


Griffiths Algorithm,[1]The reference signal d(k) is not available

Pm can be determined using stochastic solutions to circumvent the need for d(k)

mPkXkyEkWEkWE

kXkdkXkykWkW

.)}()({.)}({)}1({

)().()()(.)()1(

18

Case Study

Adaptive Digital Correction of Analog Errors in Delta-Sigma-Pipeline ADC Architecture.[3],[4]

Out

z

1

Sine Wave

Random-sequence

1

Noise-Shaping-FilterM-Bit-Pipeline

Input

error

output

LMS

[RS]

b2/b1

a2

b1

a1

4*1.81

4

[RS]

z

1

z

1

z

1B-Bit-Flash

1-1.81z +z -1 -2

XCORR

Correlation

HP

-Filt

er

H(z)

E1

E2

V(z)

W(z)

a1 a2 b1 b2

1.81 1 1.81 1

Table 1 Modulator Coefficients

19

Case Study

Where:

Output of the modulator can be written as below:

Excess term:

)(..81.1)(

)().()()(

11

1

zEzzW

zNTFzEzXzV a

2181.11)( zzzNTFa

)}()(.{.)()}.()(4{)( 12 zVzXzGzNTFzEzWzOut d

)(.).()().()(.. 11

21 zEzGNTFNTFzNTFzEzXzG dad

)(.).( 11 zEzGNTFNTF da

20

Case Study

Simulation results

Modulator output spectrum with mismatch

Frequency MHz

Ma

gn

itu

de

dB

-160

-140

-120

-100

-80

-60

-40

-20

10-2

10-1

100

101

SNDR Including 0.8% Coefficients mismatch =78.8 dB

-1.58 dBFS sin input

16384 pts FFT

Ideal SNDR=90.2 dB

Coefficients Mismatch (%)

SN

DR

(d

B)

SNDR performance versus coefficients mismatch

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 180

82

84

86

88

90

92

21

Simulation results

10-2

10-1

100

101

-160

-140

-120

-100

-80

-60

-40

-20

0Modulator output spectrum with mismatch

Frequency MHz

Mag

nit

ud

e d

B

SNDR Including 0.8% mismatch and adaptive digital correction=86.3 dB

-1.58 dBFS sin input

16384 pts FFT

Ideal SNDR=90.2 dB

Number of iteration

SN

DR

(d

B)

SNDR performance versus iteration number

65

70

75

80

85

90

0 2000 4000 6000 8000 10000 12000

22

Conclusion

Adaptive algorithms can be used to estimate unknown system.

Adaptive filters usually includes three main modules, digital filter, quality assessment, and adaptation algorithm.

The parameters available for adjustment might be the impulse response sequence value or more complicated function of the filter’s frequency response.

There is a trade off between adaptation speed and accuracy. Higher speeds leads to noisy adaptation.

23

References

[1] M.G.Larimore, “theory and design of adaptive filters”, John Wiley & Sons, 1987. [2]Widrow, and McCool, “a comparison of adaptive algorithms based on the methods

of steepest descent and random search”, IEEE.Trans. Of Antennas and propagation, vol.AP-24,pp.615-636,september 1986.

[3] P. Kiss et al., “Adaptive Digital Correction of Analog Errors in MASH ADC’s-Part II: Correction Using Test-Signal Injection,” IEEE Trans. Circuits Syst. II, vol. 47, no. 7,

pp. 629-638, July, 2000. [4] A. Bosi, A. Panigada, G. Cesura, and R.Castello, “An 80MHz 4 Oversampled

Cascaded -pipelined ADC with 75dB DR and 87dB SFDR,” ISSCC 2005, Session 9, Switched-Capacitor Modulators, 9.5.

Documents

Introduction to Adaptive Digital Filters Algorithms V.Majidzadeh Advisor: Dr.Fakhraei [email protected]