Lecture 13 Outline: Windowing in FIR Filter Design Lecture 13 Outline: Windowing in FIR Filter Design. Design Summary and Examples Announcements: Midterm May 11 in class. Will cover through FIR Filter Design. More details on Wednesday HW 4 posted, 50% longer than usual, 4 extra days to complete (due May 8) There will be no additional HW during the week of the MT Extra credit for practice MT (posted shortly) My Ohs Wednesday will be before class (12:30-1:30 outside classroom) Review of Last Lecture The Art and Science of Windowing Summary of FIR Filter Design FIR Realization: Direct Form Examples: LPF, Differentiator, Hilbert

Review of Last Lecture Impluse and freq. response matching minimizes the time/freq. domain error e between desired filter and its FIR approximation By Parseval’s, errors are the same; IR matching yields 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 IR Match: FR Match:

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 (ppt only, see reader for details) Ideal Differentiator: imaginary and odd hd[n] obtained via integration by parts

Example: Hilbert Transform (ppt only, see reader for details) Ideal Hilbert Transform (cts time): Translate to discrete time: Imaginary and odd in W Use IDFT to get IR Integrate negative/positive halves real and odd in n.

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 (next topic) Example designs for LPF, Differentiator, and Hilbert Transformer Hamming smooths out wiggles from rectangular window Introduces more distortion at transition frequencies than rectangular window Homework covers differentiator in more detail, and a high pass filter