Why does my signal magnitude increase after adaptive filtering?

Question

I am using a series-cascade of multiple NLMS adaptive filters each with step size 0.0040, leakage factor 1.0, and 100 filter taps. My signal gains magnitude at each step of the filtering, say the peak magnitude increases from 0.2 originally to 2.5 after using the first adaptive filter to 12.5 after the using the second adaptive filter on the output of the first one and finally 30 after using the third adaptive filter on the output of the second one. Why is this happening?

I have tweaked the step size and leakage factor but that did not help. Isn't my filter supposed to reduce the magnitude of my signal?

Answer 1

I don’t have the specific details for your filter but with digital filters in general it is typical for the filter to grow the signal in band in contrast to analog filters that shrink the signal out of band. It is all just a matter of scaling. Consider the simple case of a moving average FIR filter consisting of the summation of the previous N samples; such a filter will grow any low frequency signal within its passband, specifically growing a DC component by a factor of N. (We could then divide the result by N to get the actual average of try samples, but the primary filter itself here is growing the signal).

This approach of considering digital filters as "growing a signal in band" is particularly important for noise considerations in fixed point design, where you should typically avoid scaling the signal prior to the filter or the filters coefficients in order to normalize the result, but always allow the filter to grow the signal and then scale afterward. This is the reason for extended precision accumulators, and the reason is rather straightforward: if you scale the signal prior to the filter in a fixed point design you are effectively increasing the quantization noise. This quantization noise is typically modeled as independent white noise from sample to sample so in the filter will increase at a rate of $10Log_{10}(N)$ for the $N$ taps in the filter. If you scale after you are only adding this higher quantization noise level once. Scaling the coefficients can be shown to accumulate quantization noise contributions in a similar way. This is also a reason why equalizers and adaptive filters should not be any longer than they need to be (to avoid noise enhancement).