A Practical Guide to Troubleshooting LMS Filter Adaptation

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

2. FIR Filters Can Have a Dramatic Effect on Signal Samples 25dB SNR signal, before FIR Same signal, after FIR

3. 5-Tap FIR Structure Used to Filter the Noisy Samples D D D D k-2 k-3 k-4 k k-1 h0 h1 h-1 h-2 h2  yk • A common FIR architecture is the tapped delay line • FIRs may have from 3 to over 100 taps

4. 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. 5-Tap FIR with LMS Adaptation k-2 k-3 k-4 k k-1 D D D D h-2 h-1 h0 h1 h2 ek yk tk 

6. Important Considerations for Proper LMS Adaptation • Initial tap weight setting, • Error determination • Data-directed • Decision-directed • Spectral content of input signal • LMS update parameter, m, also called the step size

7. Generic Initial Tap Weights Don’t Always Work

8. Choosing the Right Initial Tap Weights Requires Insight

9. 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. How is the Error Term in LMS Determined? • Error term: ek = yk - tk • How do we know the target? • Data-directed target determination • Requires that the input signal is known (“training sequence”) • Ensures that ek 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. Adapting on Known Data Yields “Initial” Tap Weights

12. 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. Spectral Content of the Input Signal: No Noise Means No Noise Enhancement Penalty

14. With Noise, These Tap Weights are No Longer Good!

15. Spectral Content of the Input Signal: LMS Adjusts the FIR Differently, Based on Single-Frequencies

16. Choosing the LMS Update Parameter (m) • Small m • 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 m • Faster adaptation • Typically more noise at output of FIR • Coarser determination of coefficient values • Possibly unstable

17. Small m Very Low MSE (TSE)

18. Large m Noisy Adaptation

19. Very Large m Unstable Adaptation!

20. The Best of Both Worlds: The Gearshift Algorithm • Gearshift Algorithm • “Acquisition” • Larger m for quicker adaptation • “Tracking” • Smaller m for more accurate tap weights • Smaller m for lower squared error at the filter output • Rule-of-Thumb for determining m • For known channels • m is based on the eigenvalues of the autocorrelation matrix of the input • For unknown channels • m <  1/{(number of taps)(average power in the input signal)}

21. 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. 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. 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 m • 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. Acknowledgement and References The author thanks ChannelScience.com for providing the PRMLproTM 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, 2nd Edition, Kluwer Academic Publishers, 1994. [6] PRMLproTM, available for download at www.ChannelScience.com