1. Lecture 13 Outline: FIR Filter Design: Freq. Response Matching, Causal Design, Windowing Announcements: Reading: “4: FIR Discrete-Time Filters” pp. 1-18 (except 12.5-13.5, i.e. skip Direct Form realization and Stability) Some typos in HW 4: see email list or Piazza MT poll extended (May 4-6 am/pm. Class conflicts only). Tentatively May 6 9:20am-11:20am if no other options. Will cover through FIR Filter Design Review of Last Lecture Frequency Response Matching Causal Design The Art and Science of Windowing

2. Review of Last Lecture MQAM modulation sends independent bit streams on cosine and sine carriers where baseband signals have L=  M levels Leads to data rates of log 2 M/T s bps (can be very high) FIR filter design entails approximating an ideal discrete or continuous filter with a discrete filter of finite duration Impulse response matching minimizes the time domain error  between desired filter and its FIR approximation Optimal filter has M+1 of original discrete-time impulse response values Sharp windowing causes “Gibbs” phenomenon (wiggles)    j a e H   0

3. Frequency Response Matching Given a desired frequency response H d (e j  ) Objective: Find FIR approximation h a [n]: h a [n]=0 for |n|>M/2 that minimizes error of freq. response Set and By Parseval’s identity: Time-domain error and frequency-domain error equal Optimal filter same as in impulse response matching and

4. Causal Design and Group Delay Can make h a [n] causal by adding delay of M/2 Leads to causal FIR filter design If  H a (e j  ) constant,  H(e j  ) linear in  with slope -.5M Most filters do not have a linear phase, which corresponds to a constant delay for all . Group delay defined as Constant for linear phase filters Piecewise constant for piecewise linear phase filters Group delay that is not constant can introduce distortion

5. Art and Science of Windowing Window design is created as an alternative to the sharp time-windowing in h a [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 H a (e j  ) Design often trades “wiggles” in main vs. sidelobes

6. Example: Window for ideal LPF 00.511.522.53 -0.2 0 0.2 0.4 0.6 0.8 1  W ( e j  ) M = 16 Boxcar Triangular 00.511.522.53 -0.2 0 0.2 0.4 0.6 0.8 1  W ( e j  ) M = 16 Hamming Hanning

8. We are given a desired response h d [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  Design a length M+1 window function w[n], real and even, to mitigate Gibbs while keeping good approximation to h d [n] Calculate the noncausal FIR approximation h a [n] Calculate the noncausal windowed FIR approximation h w [n] Add delay of M/2 to h w [n] to get h[n] Summary of FIR Design

9. Main Points Frequency response matching minimizes frequency domain error; same noncausal design as IR matching Optimal filter has M+1 of original discrete-time impulse response values Sharp windowing causes “Gibbs” phenomenon (wiggles) Can make IR/FR response matching filters causal by introducing a delay (linear phase shift) Filters with non-constant group delay cause distortion Refine design to using a smooth windowing to mitigate Gibbs phenomenon Goal is to approximated desired filter without Gibbs/wiggles Design tradeoffs involve main lobe vs. sidelobe sizes