1. Introduction to Adaptive Digital Filters Algorithms
V.Majidzadeh
Advisor: Dr.Fakhraei

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. Basic Principles of Adaptive Filtering
The Need for Adaptive Filtering (An Intuitive Example)
Air is a cost effective communication channel
Wave scattering limits capacity and reliability of communication

4. Basic Principles of Adaptive Filtering
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

5. Basic Principles of Adaptive Filtering
The Need for Adaptive Filtering (An Intuitive Example)

6. Basic Principles of Adaptive Filtering
The general structure of an adaptive filter
Digital Filter
A conventional digital filter with updateable coefficients.
Quality Assessment
Assess the quality of the filter and generate error signal.
Depends on the adaptive filter application.
Adaptation algorithm
The 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. Basic Principles of Adaptive Filtering
The general structure of an adaptive filter

8. Analytical Framework for developing Adaptive Algorithms
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

9. Analytical Framework for developing Adaptive Algorithms
Basic Formulation
Assume 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:

10. Analytical Framework for developing Adaptive Algorithms
Basic Formulation
WSS minimize J if and only if [Nobel 1977]
Evaluate gradient of J:
the second derivative is positive definite

11. Analytical Framework for developing Adaptive Algorithms
Two Solution Techniques
Direct Solution
If matrix R can be inverted then the normal equations can be used to find WSS .
Computation complexity is high.
Iterative Approximation
Iteratively estimate WSS making use of initial value for WSS and try to improve it in each iteration step.

12. Algorithms for Adaptive FIR Filters
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

13. Algorithms for Adaptive FIR Filters
LMS Algorithm[1],[2]:
Filter output
Error formation
Weight vector update

14. Algorithms for Adaptive FIR Filters
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.

15. Algorithms for Adaptive FIR Filters
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 .

16. Algorithms for Adaptive FIR Filters
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.

17. Algorithms for Adaptive FIR Filters
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)

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

19. Case Study Where: Output of the modulator can be written as below:
Excess term:

20. Case Study
Simulation results

21. Simulation results

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 ,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 , 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.