Linear Filter Design ESE 531: Digital Signal Processing ! Used to be an art " Now, lots of tools to design optimal filters ! For DSP there are two common classes Lec 17: March 26, 2019 " Infinite impulse response IIR Design of FIR Filters " Finite impulse response FIR ! Both classes use finite order of parameters for design ! Today we will focus on FIR designs Penn ESE 531 Spring 2019 – Khanna Penn ESE 531 Spring 2019 – Khanna Adapted from M. Lustig, EECS Berkeley 3 What is a Linear Filter? Filter Specifications ! Attenuates certain frequencies ! Passes certain frequencies ! Affects both phase and magnitude ! IIR " Mostly non-linear phase response " Could be linear over a range of frequencies ! FIR " Much easier to control the phase " Both non-linear and linear phase Penn ESE 531 Spring 2019 – Khanna Adapted from M. Lustig, EECS Berkeley 4 Penn ESE 531 Spring 2019 - Khanna 5 Impulse Invariance Bilinear Transformation ! Let, ! The technique uses an algebraic transformation Hc ( jΩ) = 0, Ω ≥ π / T between the variables s and z that maps the entire h[n] = Thc (nT ) jΩ-axis in the s-plane to one revolution of the unit ! If sampling at Nyquist Rate then circle in the z-plane. ∞ j ⎡ ⎛ω 2πk ⎞⎤ H(e ω ) H j = ∑ c ⎢ ⎜ − ⎟⎥ k=−∞ ⎣ ⎝T T ⎠⎦ jω ⎛ ω ⎞ H(e ) = Hc ⎜ j ⎟, |ω|<π ⎝ T ⎠ Penn ESE 531 Spring 2019 - Khanna 6 Penn ESE 531 Spring 2019 - Khanna 7 1 Transformation of DT Filters CT Filters ! Butterworth CT-Time Lowpass " Monotonic in pass and stop bands ! Chebyshev, Type I Z −1 = G(z−1) " Equiripple in pass band and monotonic in stop band Hlp(Z) ! Chebyshev, Type II " Monotonic in pass band and equiripple in pass band ! Elliptic ! Z – complex variable for the LP filter " Equiripple in pass and stop bands ! z – complex variable for the transformed filter ! Appendix B in textbook ! Map Z-plane#z-plane with transformation G Penn ESE 531 Spring 2019 - Khanna 8 Penn ESE 531 Spring 2019 - Khanna 9 Design Comparison Butterworth ! Design specifications ! Butterworth " passband edge frequency ωp = 0.5π " Monotonic in pass and stop bands " stopband edge frequency ωs = 0.6π " maximum passband gain = 0 dB " minimum passband gain = -0.3dB " maximum stopband gain =-30dB ! Use bilinear transformation to design DT low pass filter for each type Penn ESE 531 Spring 2019 - Khanna 10 Penn ESE 531 Spring 2019 - Khanna 11 Chebyshev Elliptic ! Type I ! Type II ! Elliptic " Equiripple in pass band and " Monotonic in pass band and " Equiripple in pass and stop bands monotonic in stop band equiripple in pass band Penn ESE 531 Spring 2019 - Khanna 12 Penn ESE 531 Spring 2019 - Khanna 13 2 Comparisons What is a Linear Filter? ! Attenuates certain frequencies ! Passes certain frequencies ! Affects both phase and magnitude ! IIR " Mostly non-linear phase response " Could be linear over a range of frequencies ! FIR " Much easier to control the phase " Both non-linear and linear phase Penn ESE 531 Spring 2019 – Khanna Penn ESE 531 Spring 2019 - Khanna 14 Adapted from M. Lustig, EECS Berkeley 15 FIR GLP: Type I and II FIR GLP: Type III and IV Penn ESE 531 Spring 2019 – Khanna Penn ESE 531 Spring 2019 – Khanna Adapted from M. Lustig, EECS Berkeley 16 Adapted from M. Lustig, EECS Berkeley 17 FIR Design by Windowing Example: Moving Average jω ! Given desired frequency response, Hd(e ) , find an impulse response sin (N +1 2)ω w[n]↔W (e jω ) = ( ) sin(ω 2) ! Obtain the Mth order causal FIR filter by truncating/windowing it 1 e− jωM 2 sin (M 2 +1 2)ω w[n − M 2]↔W (e jω ) = ( ) M +1 M +1 sin(ω 2) Penn ESE 531 Spring 2019 – Khanna Penn ESE 531 Spring 2019 – Khanna Adapted from M. Lustig, EECS Berkeley 18 Adapted from M. Lustig, EECS Berkeley 19 3 Example: Moving Average FIR Design by Windowing ! We already saw this before, ! For Boxcar (rectangular) window Penn ESE 531 Spring 2019 – Khanna Penn ESE 531 Spring 2019 – Khanna Adapted from M. Lustig, EECS Berkeley 20 Adapted from M. Lustig, EECS Berkeley 21 FIR Design by Windowing FIR Design by Windowing Penn ESE 531 Spring 2019 – Khanna Penn ESE 531 Spring 2019 - Khanna 22 Adapted from M. Lustig, EECS Berkeley 23 Tapered Windows Tradeoff – Ripple vs. Transition Width Penn ESE 531 Spring 2019 – Khanna Penn ESE 531 Spring 2019 – Khanna Adapted from M. Lustig, EECS Berkeley 24 Adapted from M. Lustig, EECS Berkeley 25 4 Kaiser Window Kaiser Window ! Near optimal window quantified as the window ! M=20 maximally concentrated around ω=0 ! Two parameters – M and β ! α=M/2 ! – th I0(x) zero order Bessel function of the first kind Penn ESE 531 Spring 2019 - Khanna 26 Penn ESE 531 Spring 2019 - Khanna 27 Kaiser Window Kaiser Window ! M=20 ! β=6 Penn ESE 531 Spring 2019 - Khanna 28 Penn ESE 531 Spring 2019 - Khanna 29 Approximation Error FIR Filter Design jω ! Choose a desired frequency response Hd(e ) " non causal (zero-delay), and infinite imp. response " If derived from C.T, choose T and use: ! Window: " Length M+1 ⇔ affects transition width " Type of window ⇔ transition-width/ ripple Penn ESE 531 Spring 2019 – Khanna Penn ESE 531 Spring 2019 - Khanna 30 Adapted from M. Lustig, EECS Berkeley 31 5 FIR Filter Design FIR Filter Design jω ! Choose a desired frequency response Hd(e ) ! Determine truncated impulse response h1[n] " non causal (zero-delay), and infinite imp. response " If derived from C.T, choose T and use: ! ! Window: Apply window " Length M+1 ⇔ affects transition width " Type of window ⇔ transition-width/ ripple ! Check: " jω Modulate to shift impulse response " Compute Hw(e ), if does not meet specs increase M or " Force causality change window Penn ESE 531 Spring 2019 – Khanna Penn ESE 531 Spring 2019 – Khanna Adapted from M. Lustig, EECS Berkeley 32 Adapted from M. Lustig, EECS Berkeley 33 Example: FIR Low-Pass Filter Design Example: FIR Low-Pass Filter Design ! The result is a windowed sinc function ! High Pass Design: " Design low pass n " Transform to hw[n](-1) ! General bandpass " Transform to 2hw[n]cos(ω0n) or 2hw[n]sin(ω0n) Penn ESE 531 Spring 2019 – Khanna Penn ESE 531 Spring 2019 – Khanna Adapted from M. Lustig, EECS Berkeley 34 Adapted from M. Lustig, EECS Berkeley 35 Lowpass Filter w/ Kaiser Window Lowpass Filter w/ Kaiser Window ! Filter specs: ! Filter specs: " passband edge frequency ωp = 0.4π " passband edge frequency ωp = 0.4π " stopband edge frequency ωs = 0.6π " stopband edge frequency ωs = 0.6π " Max ripple (δ) = .001 " Max ripple (δ) = .001 Penn ESE 531 Spring 2019 - Khanna 36 Penn ESE 531 Spring 2019 - Khanna 37 6 Lowpass Filter w/ Kaiser Window Lowpass Filter w/ Kaiser Window Penn ESE 531 Spring 2019 - Khanna 38 Penn ESE 531 Spring 2019 - Khanna 39 Lowpass Filter w/ Kaiser Window Characterization of Filter Shape Penn ESE 531 Spring 2019 – Khanna Penn ESE 531 Spring 2019 - Khanna 40 Adapted from M. Lustig, EECS Berkeley 41 Time Bandwidth Product Design through FFT ! To design order M filter: ! Over-Sample/discretize the frequency response at P points where P >> M (P=15M is good) ! Sampled at: ! Compute h1[n] = IDFTP(H1[k]) ! Apply M+1 length window: Penn ESE 531 Spring 2019 – Khanna Penn ESE 531 Spring 2019 – Khanna Adapted from M. Lustig, EECS Berkeley 42 Adapted from M. Lustig, EECS Berkeley 43 7 Example Example ! For M+1=14 " P = 16 and P = 1026 Penn ESE 531 Spring 2019 – Khanna Penn ESE 531 Spring 2019 – Khanna Adapted from M. Lustig, EECS Berkeley 44 Adapted from M. Lustig, EECS Berkeley 45 Optimal Filter Design Optimality ! Window method " Design Filters heuristically using windowed sinc functions ! Optimal design jω " Design a filter h[n] with H(e ) jω " Approximate Hd(e ) with some optimality criteria - or satisfies specs. ! Least Squares: ! Variation: Weighted Least Squares: Penn ESE 531 Spring 2019 – Khanna Penn ESE 531 Spring 2019 – Khanna Adapted from M. Lustig, EECS Berkeley 46 Adapted from M. Lustig, EECS Berkeley 47 Optimality Example of Complex Filter ! Chebychev Design (min-max) ! Larson et. al, “Multiband Excitation Pulses for Hyperpolarized 13C Dynamic Chemical Shift Imaging” JMR 2008;194(1):121-127 ! Need to design 11 taps filter with following frequency response: " Parks-McClellan algorithm - equiripple " Also known as Remez exchange algorithms (signal.remez) " Can also use convex optimization Penn ESE 531 Spring 2019 – Khanna Penn ESE 531 Spring 2019 – Khanna Adapted from M. Lustig, EECS Berkeley 48 Adapted from M. Lustig, EECS Berkeley 49 8 Least-Squares v.s. Min-Max Admin ! HW 7 " Out now " Due Sunday 3/31 " More Matlab " Less book Penn ESE 531 Spring 2019 – Khanna Penn ESE 531 Spring 2019 – Khanna Adapted from M. Lustig, EECS Berkeley 50 Adapted from M. Lustig, EECS Berkeley 51 9 .