Lecture 14 Outline: Windowing in FIR Filter Design Lecture 14 Outline: Windowing in FIR Filter Design. Design Summary and Examples Announcements: Reading: “4: FIR Discrete-Time Filters” pp. 14-24 (i.e. read all of 1-24 except 13-13.5, i.e. skip Stability: read Direct Form Realization on pg. 12) Midterm May 6 9:20am-11:20am. Academic conflicts will be accommodated (let us know of any by this Friday). Will cover through FIR Filter Design. More details on MT, review sessions, and extra OHs on Monday HW 5 posted today, short HW, extra credit for practice MT (posted shortly) Guest lecture Friday by Gordon Wetzstein on Virtual/Augmented Reality I will not have OHs Friday, extra OHs next week Review of Last Lecture The Art and Science of Windowing Summary of FIR Filter Design FIR Realization: Direct Form Examples: LPF and Differentiator

Review of Last Lecture Freq. response matching minimizes the frequency domain error e between desired filter and its FIR approximation By Parseval’s, same as time-domain error; so filter same as IR matching Sharp windowing causes “Gibbs” phenomenon (wiggles) Causal Design Can make ha[n] causal by adding delay of M/2 Group delay defined as Often constant (1 delay) or piecewise constant (1 delay per freq. group) Group delay that is not constant can introduce distortion

Art and Science of Windowing Window design is created as an alternative to the sharp time-windowing in ha[n] Used to mitigate Gibbs phenomenon Window function (w[n]=0, |n|>M/2) given by Windowed noncausal FIR design: Frequency response smooths Gibbs in Ha(ejW) Design often trades “wiggles” in main vs. sidelobes

Typical Window Designs 0.5 1 1.5 2 2.5 3 -0.2 0.2 0.4 0.6 0.8 W ( e j ) M = 16 Boxcar Triangular Hamming Hanning

Summary of FIR Design We are given a desired response hd[n] which is generally noncausal and IIR Examples are ideal low-pass, bandpass, highpass filters May be derived from a continuous-time filter Choose a filter duration M+1 for M even Larger M entails more complexity/delay, less approximation error e Design a length M+1 window function w[n], real and even, to mitigate Gibbs while keeping good approximation to hd[n] Calculate the noncausal FIR approximation ha[n] Calculate the noncausal windowed FIR approximation hw[n] Add delay of M/2 to hw[n] to get h[n]

FIR Realization: Direct Form Consists of M delay elements and M+1 multipliers Can introduce different delays at different freq. components of x[n] Will discuss more when we cover z transforms Efficient implementation using Discrete-Fourier Transform (DFT) Next class topic

Example: Lowpass Filter Ideal LPF with cutoff Wc:

Example: Differentiator Ideal Differentiator: imaginary and odd hd[n] obtained via integration by parts

Main Points FIR design entails choice of window function to mitigate Gibbs Goal is to approximated desired filter without Gibbs/wiggles Design tradeoffs involve main lobe vs. sidelobe sizes Typical windows: rectangle (boxcar), triangle, Hanning, and Hamming FIR design for desired hd[n] entails picking a length M, setting ha[n]=hd[n], |n|M/2, choosing window w[n] with hw[n]=h[n]w[n]to mitigate Gibbs, and setting h[n]=hw[n-M/2] to make design causal FIR implemented directly using M delay elements and M+1 multipliers Can introduce group delay Efficiently implemented with DFT Example designs for LFP and Differentiator Hamming smooths out wiggles from rectangular window Introduces more distortion at transition frequencies than rectangular window