1. 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

2. 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:

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

10. 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.

12. 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