**The least mean squares (LMS) adaptive filter is an amazingly simple
filter with an equally amazing ability to filter noise from a signal even
when the noise is only partially represented by the reference signal. Even
when no phase shift or amplitude change is present, the adaptive filter
may offer computational efficincies over standard FIR filters, and certainly
reduce the design costs of producing a filter.**

**The ability of the filter to operate sucessfully lies in its ability
to adapt to changes in phase an amplitude. This adaptation can only occur
if there are sufficient samples in the filter such that the maximum phase
shift at every frequency can be compensated for by the filter.**

**The LMS filter does not respond well to very rapid changes in noise.
Large changes result in very steep gradients which result in very large
filter weight adaptations. The next few samples of input usually result
in equally large gradients in the opposite direction. Samples processed
with such widely varying filkter weights tend to pop. This behavior is
evidenced by the last few seconds of our noise cancellation simulation,
where the cymbals are removed imcompletely and leave an artifact as well.**

**LMS adaptive filters work well for two of the four filter systems
described above. By extension the Type II Inverse Filter should work as
well as the Type I filter. We made no attempt to test the Type III Predictive
Filter System since the literature indicated that the LMS filter is unsuitable
for such application.**

**We find that there are several more aspects of LMS adaptive filters
worth exploring. First and foremost is the selection of the optimal filter
length. While we know it is related to phase shift and the total number
of frequencies that must be dealt with, we have not established even a
good rule of thumb for determining filter length. Similarly, we were only
to obtain values of mu through trial and error: We made it as large as
possible until our results blew up, then backed off.**

**Another area worth investigating is the degree of non-correlation
between the input signal and noise. Because each of our experiments used
quite different noise and input signals, we merely operated under the assumption
that noise and input were statistically uncorrelated. Luckily, we turned
out to be right. Experiments could be devised to determine at what point
the LMS filter begins to be unable to tell the difference between signal
and noise.**

**Finally, it would be interesting so see how small the signal to noise
ratio can be before the LMS filter fails. The largest differential we used
was 1 to 3.**

*10/31/96*

**Examination of a DSP application is a vital and important project
if one expects to get a decent grade out of this class.**