1. 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 (i.e. read all of 1-24 except , 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

2. 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

3. 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

4. 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

5. 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]

6. 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

7. Example: Lowpass Filter
Ideal LPF with cutoff Wc:

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

11. 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