Introduction to Adaptive Digital Filters Algorithms V.Majidzadeh Advisor: Dr.Fakhraei v.majidzadeh@ece.ut.ac.ir

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

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

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

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

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.

Basic Principles of Adaptive Filtering The general structure of an adaptive filter

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

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:

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

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.

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

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

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.

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 .

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.

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)

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

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

Case Study Simulation results

Simulation results

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.

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.