One common low-pass filter is the moving average filter. It's easy to build and intuitive to understand, but its characteristics in the frequency domain are actually not that great. A much better low-pass filter is the Butterworth filter. Let's explore these two filters in a bit of detail.
The moving average filter is not circularly symmetric, which causes its frequency domain performance to vary according to direction (it's anisotropic). The Butterworth filter specified in eq. 4-117 is circularly symmetric, so ideally its performance does not depend on direction (it's isotropic). Since these filters are implemented in discrete form, the act of sampling usually introduces some small amount of deviation from pure isotropic behavior.
For these problems, you are asked to compute and plot the frequency domain characteristics of several filters. Plot these as 1-D cross sections between -Tt and t.
1. Construct the following moving average filters:
a. 10 pts Low-pass filter: ham,n) of size 5x5, each coefficient = 1/25
Compute H1a(u,v) and plot its 1-D magnitude cross section, |H1a(u,0)|
b. 10 pts Low-pass filter: hbm,n of size 15x15, each coefficient = 1/225
Compute Hb(u,v) and plot its 1-D magnitude cross section, |Hb(u,0)|
c. 10 pts What do you notice about the frequency domain performance of these two filters? How different are they? Explain the reasons for this difference.
d. 10 pts Now plot their 1-D magnitude cross sections at an angle of /445 degrees). That is, plot |H1a(u,u)| and |Hb(u,u)|. Be careful to plot at the same distance scale as before - the distance from (0,0) to (0,0) is not the same as the distance from (0,0) to (,). What can you observe about how these filters change characteristics as the orientation angle changes by 45 degrees? Explain the reasons for this change.
