(We = Kenji Rikitake)

15-OCT-2019

LMS convergence condition: 0 < mu < (2 / (N * (A^2)))
                           
where 
* N: filter order
* A: maximum amplitude aka complex abs for the signal
* alpha: step size
* mu: alpha / N

Therefore: A < sqrt(2 / alpha) when alpha is given

For example, if alpha = 0.1, A < sqrt(20), so A < 4.47

25-OCT-2019

More conservative condition of LMS convergence: 0 < mu < (1 / (3 * tr[R]))
where tr[R] represents the total average input signal power,
also tr[R] = N * (A^2) [1]

Therefore: 0 < mu < (1 / (3 * (N * (A^2))))
which is equivalent to: 0 < alpha < (1 / (3 * (A^2)))
so the more conservative estimation is: : A < sqrt((1/3) * (1/alpha))

For example, if alpha = 0.1, A < sqrt(10/3), so A < 1.82

Reference:
[1]: A. Feuer and E. Weinstein, "Convergence analysis of LMS filters with uncorrelated Gaussian data," in IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 33, no. 1, pp. 222-230, February 1985.
doi: 10.1109/TASSP.1985.1164493
http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1164493&isnumber=26189
See expression (35)

18-SEP-2020

We've tried to use a variable step size algorithm [2], but the experimentation
result didn't show improvement (i.e., lower level) of QMM output. We concluded
that the variable step size was redundant and not necessary for FM stereo
broadcast reception, since we don't need faster convergence speed than the
pseudo-NLMS algorithm with the nearly-optimal parameter setting descrived above.

[2] D. He, M. Wang, Y. Han and S. Hui, "Variable Step Size LMS Adaptive Algorithm Based on Exponential Function," 2019 IEEE 2nd International Conference on Information Communication and Signal Processing (ICICSP), Weihai, China, 2019, pp. 473-477, doi: 10.1109/ICICSP48821.2019.8958492.
Note on [2]: the relationship of mu(n) and e(n) in this paper is inaccurate, though the curvature pattern of the relationship is worth considering.

19-SEP-2020

We evaluated the Complex Normalized LMS [3] (Complex NLMS) with measuring the
signal square norm processed in the multipath filter, which resulted in an
improved reception condition under weak signals (e.g., IF ~= -55dB). We
decided to implement NLMS for v0.9.0.

[3] B. Widrow, J. McCool and M. Ball, "The complex LMS algorithm," in Proceedings of the IEEE, vol. 63, no. 4, pp. 719-720, April 1975, doi: 10.1109/PROC.1975.9807.

19-SEP-2020 part 2

One of the most impressive papers of surveying LMS algorithms [4] tells that
for adaptive noise cancellation NLMS is still the best VS-LMS algorithm, shown
as:

"[...] we come to a surprising conclusion that none of the VS-LMS algorithms
performed better than the NLMS algorithm — this is clear from both the sum of
MSE and the Final MSE." ([4, Section 4.3])

[4] D. Bismor, K. Czyż, Z. Ogonowski, Review and Comparison of Variable
Step-Size LMS Algorithms, International Journal of Acoustics and Vibration,
Vol. 21, No 1, pp. 24-39, 2016. URL:
https://iiav.org/ijav/content/volumes/21_2016_590031458046128/vol_1/835_fullpaper_1207561458214850.pdf
DOI: https://doi.org/10.20855/ijav.2016.21.1392

Also, our estimation that the square norm value of the input signal for the
filter order N is N itself was a good estimation if the reception signal is
strong and the IF AGC output was stabilzed. For NHK-FM Tokyo 82.5MHz where the
input level was ~-23dB on Airspy HF+, the measured square norm was between in
350~450, when the filter order was 401, in +- 13% error. OTOH, for a much
weaker station such as NHK-FM Yokohama 81.9MHz, whose level was ~-55dB, the
measured square norm was between in 100~1500, in much wider range, and caused
the coefficient instability.
