LMS Adaptive Filter for system identification

i am currently attempting system identification using the LMS algorithm. The input and the output data are available and are very noisy and consists of multiple frequencies. The input and the output data are shown below.

The LMS algorithm fails to converge, i.e., adapt to the desired output signal. See images below.

However, passing the signals through a low-pass filter first and then implementing the LMS for low frequency converges the filter output to the reference and minimizes the filter error. See image below.

Could anyone explain me how and why LMS performs better for low frequecy data. Also, how should I move forward? Is it ok to pass the data through LPF first and apply the LMS filter. Would it cause loss of information and produce wrong system identification? How can I implement LMS algorithm for high frequency signal.

I thank all the experts on any assistance on this.

Best regards,

• Could you share the data (Samples, System Coefficients, etc...) as a MAT file?
– Royi
Oct 29, 2022 at 9:13

In general for a standard LMS you can only ensure convergence if the stepsize $$µ < 1 / (2p\sigma^2)$$ . With p being the filter order and $$\sigma^2$$ the variance of the input signal x. Therefore if you first low-pass filter your signal, the variance is reduced and the possible save range for µ increases.