Today ESE 53: Digital Signal Processing! IIR Filter Design " Lec 8: March 30, 207 IIR Filters and Adaptive Filters " Bilinear Transformation! Transformation of DT Filters! Adaptive Filters! LMS Algorithm Penn ESE 53 Spring 207 Khanna 2 IIR Filter Design! Transform continuous-time filter into a discretetime filter meeting specs! Want to implement continuous-time system in discrete-time " Pick suitable transformation from s (Laplace variable) to z (or t to n) " Pick suitable analog H c (s) allowing specs to be met, transform to H(z)! We ve seen this before impulse invariance 3 4! With H c (jω) bandlimited, choose! With H c (jω) bandlimited, choose H(e jω ) = H c ( j ω T ), ω < π H(e jω ) = H c ( j ω T ), ω < π! With the further requirement that T be chosen such that! With the further requirement that T be chosen such that H c ( jω) = 0, Ω π / T H c ( jω) = 0, Ω π / T h[n] = Th c (nt ) 5 6
IIR by! If H c (jω) 0 for ω d > π/t, no aliasing and H(e jω ) = H(jω/T), ω<π! To get a particular H(e jω ), find corresponding H c and T d for which above is true (within specs)! Note: T d is not for aliasing control, used for frequency scaling. 7 8 e at L s a 9 0 jω 2 2
! Let, h[n] = T h c (nt )! If sampling at Nyquist Rate then H(e jω ) = T ω H T c j T 2πk T k= H c ( jω) = 0, Ω π / T H(e jω T ) = T H j ω c, ω <π T! Sampling the impulse response is equivalent to mapping the s-plane to the z-plane using: " z = e st0 = r e jω! The entire Ω axis of the s-plane wraps around the unit circle of the z-plane an infinite number of times! The negative half s-plane maps to the interior of the unit circle and the RHP to the exterior! This means stable analog filters (poles in LHP) will transform to stable digital filters (poles inside unit circle)! This is a many-to-one mapping of strips of the s-plane to regions of the z-plane " Not a conformal mapping " The poles map according to z = e st0, but the zeros do not always 3 4 Mapping Mapping 5 6 Bilinear Transformation! Limitation of : overlap of images of the frequency response. This prevents it from being used for high-pass filter design! The technique uses an algebraic transformation between the variables s and z that maps the entire jω-axis in the s-plane to one revolution of the unit circle in the z-plane. 7 8 3
Bilinear Transformation Bilinear Transformation! Substituting s = σ+ j Ω and z = e jω 9 20 Bilinear Transformation : Notch Filter! The continuous time filter with: s 2 2 + Ω H a (s) = 0 s 2 2 + Bs + Ω 0 2 22 Transformation of DT Filters Transformation of DT Filters H lp (Z) Z = G(z ) H lp (Z) Z = G(z )! Z complex variable for the LP filter! z complex variable for the transformed filter! Map Z-plane#z-plane with transformation G! Map Z-plane#z-plane with transformation G 23 24 4
: :! Lowpass#highpass " Shift frequency by π! Lowpass#highpass " Shift frequency by π Z = z Z = z 25 26 2: 2:! Lowpass#bandpass! Lowpass#bandpass Z = z 2 H lp (z) = H az bp (z) = + az 2 Z = z 2 H lp (z) = H az bp (z) = + az 2 Pole at z=a Pole at z=±j a Pole at z=a Pole at z=±j a! Lowpass#bandstop 27 Z = z 2 H lp (z) = H az bs (z) = az 2 Pole at z=± a 28 Transformation Constraints on G(z - ) Transformation Constraints on G(z - )! If H lp (Z) is the rational system function of a causal and stable system, we naturally require that the transformed system function H(z) be a rational function and that the system also be causal and stable.! Respective unit circles in both planes " G(Z -l ) must be a rational function of z - " The inside of the unit circle of the Z-plane must map to the inside of the unit circle of the z-plane " The unit circle of the Z-plane must map onto the unit circle of the z-plane. 29 30 5
Transformation Constraints on G(z - )! Respective unit circles in both planes Transformation Constraints on G(z - )! Respective unit circles in both planes Z = G(z ) e jθ = G(e jω ) e jθ = G(e jω ) e j G(e jω ) Z = G(z ) e jθ = G(e jω ) e jθ = G(e jω ) e j G(e jω ) = G(e jω ) θ = G(e jω ) 3 32 Transformation Constraints on G(z - ) General Transformation! General form that meets all constraints:! Lowpass#lowpass " a k real and a k <! Changes passband/stopband edge frequencies 33 34 General Transformation General Transformation! Lowpass#lowpass! Lowpass#lowpass! Changes passband/stopband edge frequencies! Changes passband/stopband edge frequencies 35 36 6
General Transformations Reminder: Simple Low Pass Filter H(e jω ) (db - Log scale) 2 ω c π ω 37 Penn ESE 53 Spring 207 Khanna Adapted from M. Lustig, EECS Berkeley 38 The Filtering Problem Adaptive Filters! Filters may be used for three information-processing tasks " Filtering! An adaptive filter is an adjustable filter that processes in time " It adapts " Smoothing " Prediction x[n] Adaptive Filter y[n] _ d[n] + e[n]=d[n]-y[n]! Given an optimality criteria we often can design optimal filters " Requires a priori information about the environment! Adaptive filters are self-designing using a recursive algorithm " Useful if complete knowledge of environment is not available a priori Update Coefficients Penn ESE 53 Spring 207 Khanna 39 40 Adaptive Filter Applications Adaptive Filter Applications! System Identification! Identification of inverse system 4 42 7
Adaptive Filter Applications Adaptive Filter Applications! Adaptive Interference Cancellation! Adaptive Prediction 43 44 Stochastic Gradient Approach Least-Mean-Square (LMS) Algorithm! Most commonly used type of Adaptive Filters! Define cost function as mean-squared error! The LMS Algorithm consists of two basic processes " Filtering process " Difference between filter output and desired response! Based on the method of steepest descent " Move towards the minimum on the error surface to get to minimum " Calculate the output of FIR filter by convolving input and taps " Calculate estimation error by comparing the output to desired signal " Adaptation process " Adjust tap weights based on the estimation error " Requires the gradient of the error surface to be known 45 46 Adaptive FIR Filter: LMS Adaptive FIR Filter: LMS 47 48 8
Adaptive FIR Filter: LMS LMS Algorithm Steps! Filter output! Estimation error y n = M x n k h * n k! Tap-weight adaptation h k n + = h n k + µx n k k=0 e n = d n y n n e* Input Signal x[i] x[i-] x[i-m+] x[i-m] h 0* [n] h * [n] h M-* [n] h M* [n] 49 50 Adaptive Filter Applications Adaptive Interference Cancellation! Adaptive Interference Cancellation 5 52 Stability of LMS Admin! The LMS algorithm is convergent in the mean square if and only if the step-size parameter satisfy! HW 7 posted after class tonight " Due 4/6 2 0 < µ < λmax! Here λ max is the largest eigenvalue of the correlation matrix of the input data! More practical test for stability is 0 < µ < input 2 signal power! Larger values for step size " Increases adaptation rate (faster adaptation) " Increases residual mean-squared error 53 Penn ESE 53 Spring 207 Khanna Adapted from M. Lustig, EECS Berkeley 54 9