1. A Practical Guide to Troubleshooting LMS Filter Adaptation Prepared by Charles H. Sobey, Chief Scientist ChannelScience.com June 30, 2000

2. A Practical Guide to Troubleshooting LMS Filter Adaptation 2 Prepared by ChannelScience.com FIR Filters Can Have a Dramatic Effect on Signal Samples 25dB SNR signal, before FIR Same signal, after FIR

3. A Practical Guide to Troubleshooting LMS Filter Adaptation 3 Prepared by ChannelScience.com 5-Tap FIR Structure Used to Filter the Noisy Samples A common FIR architecture is the tapped delay line FIRs may have from 3 to over 100 taps DDDD h -2 h -1 h0h0 h1h1 h2h2           yy

4. A Practical Guide to Troubleshooting LMS Filter Adaptation 4 Prepared by ChannelScience.com How Are the Optimal Tap Weights Determined? The least mean square (LMS) algorithm adjusts the tap weights such that the mean squared error at the output of the FIR is minimized. LMS update equation: Does not minimize bit error rate (BER) Can be UNSTABLE, even when used with FIR filters!

5. A Practical Guide to Troubleshooting LMS Filter Adaptation 5 Prepared by ChannelScience.com 5-Tap FIR with LMS Adaptation DDDD h -2           yy tktk h -1 h0h0 h1h1 h2h2 ekek

6. A Practical Guide to Troubleshooting LMS Filter Adaptation 6 Prepared by ChannelScience.com Important Considerations for Proper LMS Adaptation Initial tap weight setting, Error determination –Data-directed –Decision-directed Spectral content of input signal LMS update parameter, , also called the step size

7. A Practical Guide to Troubleshooting LMS Filter Adaptation 7 Prepared by ChannelScience.com Generic Initial Tap Weights Don’t Always Work

8. A Practical Guide to Troubleshooting LMS Filter Adaptation 8 Prepared by ChannelScience.com Choosing the Right Initial Tap Weights Requires Insight

9. A Practical Guide to Troubleshooting LMS Filter Adaptation 9 Prepared by ChannelScience.com Options for Choosing Initial Tap Weights Calculate them, based on a known input signal –Impractical in production environment –Impractical when the channel (signal) is unknown a priori Guess. That is, use an average value based on past experience –Limits the range of inputs that can be successfully filtered Let LMS determine the starting values –What are the starting values for this?

10. A Practical Guide to Troubleshooting LMS Filter Adaptation 10 Prepared by ChannelScience.com How is the Error Term in LMS Determined? Error term: e k = y k - t k How do we know the target? –Data-directed target determination Requires that the input signal is known (“training sequence”) Ensures that e k is always correct Useful when the tap weights are not close to the correct values, such as during initialization procedures –Decision-directed target determination Works on unknown data sequences Useful when the tap weights are close the to correct values

11. A Practical Guide to Troubleshooting LMS Filter Adaptation 11 Prepared by ChannelScience.com Adapting on Known Data Yields “Initial” Tap Weights

12. A Practical Guide to Troubleshooting LMS Filter Adaptation 12 Prepared by ChannelScience.com Decision-directed Target Determination Slicer, a simple threshold device –Fast –Cheap? –Makes more errors Viterbi Algorithm –Decisions are delayed –Expensive? –Often the best detector

13. A Practical Guide to Troubleshooting LMS Filter Adaptation 13 Prepared by ChannelScience.com Spectral Content of the Input Signal: No Noise Means No Noise Enhancement Penalty

14. A Practical Guide to Troubleshooting LMS Filter Adaptation 14 Prepared by ChannelScience.com With Noise, These Tap Weights are No Longer Good!

15. A Practical Guide to Troubleshooting LMS Filter Adaptation 15 Prepared by ChannelScience.com Spectral Content of the Input Signal: LMS Adjusts the FIR Differently, Based on Single-Frequencies

16. A Practical Guide to Troubleshooting LMS Filter Adaptation 16 Prepared by ChannelScience.com Choosing the LMS Update Parameter (  ) Small  –Slower adaptation –Typically less noise at output of the FIR –More accurate determination of coefficient values –Likely to be stable –Can get hung in local minima in decision-directed mode Large  –Faster adaptation –Typically more noise at output of FIR –Coarser determination of coefficient values –Possibly unstable

17. A Practical Guide to Troubleshooting LMS Filter Adaptation 17 Prepared by ChannelScience.com Small   Very Low MSE (TSE)

18. A Practical Guide to Troubleshooting LMS Filter Adaptation 18 Prepared by ChannelScience.com Large   Noisy Adaptation

19. A Practical Guide to Troubleshooting LMS Filter Adaptation 19 Prepared by ChannelScience.com Very Large   Unstable Adaptation!

20. A Practical Guide to Troubleshooting LMS Filter Adaptation 20 Prepared by ChannelScience.com The Best of Both Worlds: The Gearshift Algorithm Gearshift Algorithm –“Acquisition” Larger  for quicker adaptation –“Tracking” Smaller  for more accurate tap weights Smaller  for lower squared error at the filter output Rule-of-Thumb for determining  –For known channels  is based on the eigenvalues of the autocorrelation matrix of the input –For unknown channels  1/{(number of taps)(average power in the input signal)}

21. A Practical Guide to Troubleshooting LMS Filter Adaptation 21 Prepared by ChannelScience.com Constrained Adaptation Limited range of FIR tap weight values Quantization of FIR tap weights Simplifications of the LMS algorithm (signed LMS) Interaction with other feedback control loops –Automatic Gain Control (AGC) –Phase-Locked Loop (PLL) –Often addressed by holding one or two taps constant

22. A Practical Guide to Troubleshooting LMS Filter Adaptation 22 Prepared by ChannelScience.com Other Important Considerations Minimizing MSE does not always minimize the bit error rate Additional taps can improve filtering at the expense of –Die area ($) –Power –Delay –Time needed to optimize the taps In general, the FIR input must be sampled with a different phase if the number of FIR taps is odd or even Other optimization algorithms –Recursive Least Squares (RLS) Faster convergence (exponential weighting) But more complex (matrix inversion) –Custom algorithms that are driven by other signal characteristics

23. A Practical Guide to Troubleshooting LMS Filter Adaptation 23 Prepared by ChannelScience.com Summary of LMS Guidelines Conditions –Unknown signal in noise, sampled at the correct phase (PLL) –Size of FIR (number of taps) is pre-determined –Training pattern with appropriate spectral characteristics is available Initialize –Use the rule-of-thumb to determine  –Determine initial FIR tap weights Set to...0 0 1 0 0..., or other appropriate value if your situation is more predictable Use data-directed adaptation on a known training pattern On unknown data –Input signal must have broad, representative spectral content –Use decision-directed adaptation Need good decisions with small delay

24. A Practical Guide to Troubleshooting LMS Filter Adaptation 24 Prepared by ChannelScience.com Acknowledgement and References The author thanks ChannelScience.com for providing the PRMLpro TM software that was used to create the examples for this presentation. [1] Simon Haykin, Adaptive Filter Theory, Publisher Prentice-Hall, Inc., 1991. [2] R.D. Cideciyan, et al., “A PRML System for Digital Magnetic Recording,” IEEE Journal on Selected Areas in Communications, Vol. 10, No. 1, January 1992, pp. 38-56. [3] H.K. Thapar and A.M. Patel, “A Class of Partial Response Systems for Increasing Storage Density in Magnetic Recording,” IEEE Trans. Magn., Vol. MAG-23, No. 5, September 1987, pp. 3666-3668. [4] P.Kabal and S.Pasupathy, “Partial-Response Signaling,” IEEE Trans on Comm., pp. 921-934, September, 1975. [5] Edward A. Lee and David G. Messerschmitt, Digital Communication, 2 nd Edition, Kluwer Academic Publishers, 1994. [6] PRMLpro TM, available for download at www.ChannelScience.com