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