[sdiy] Long LFSRs (Was Psych Tone)

[Bernard Arthur Hutchins, Jr]

Achim said:     “An EXOR is simply multiplication (given the right encoding).”

True and widely understood.

Achim followed with: “That is what allows an LFSR to implement a matrix multiplication with such low hardware complexity.”

I have no idea what that means – please elaborate.

Matthew then replied:    “It's also addition, in a different encoding - and that's the reason for the LFSR to be called linear.  All these "polynomials" are polynomials defined over the field GF(2), in which the addition operation is the same thing as XOR on bits.”

EXOR is still non-linear in the PRBS structure – the particular application under discussion here.   Compare, for example, the marvelous Karplus-Strong “plucked string algorithm” which is a similar-looking long shift register with two weighted feedback taps at the very end back to the input (typically both being ½, and ADDED).  This IS linear, and K-S, generalized, IS a perfectly understood linear filter in a feedback loop.

Next, with regard to my suggesting using a shorter PRBS sequence to correct for the “tricks” played by a longer ones (blundering into audible special cases that linger), Achim said:     “Yes.  That works because whatever the original distribution(s), mixing them will (rather quickly under easily met conditions) converge towards a Gaussian one (says the central limit theorem).”

I was saying that you MULTIPLY the two sequences by EXOR.  The little guy simply “chops up” any misbehavior of the long sequence.   They are not added so the CLT does NOT apply.  The EXORed output still has a uniform (1 or 0) non-Gaussian distribution.  The spectrum is AS white as the PRBS’s themselves. [Meaning: technically the PRBS takes on the spectrum (sync low-pass roll-off) of the sample-and-hold that is inherent in the shift register].

One last thing:  By chance today I same across another PRBS reference that mentions several interesting things – like the “heartbeat” issue just mentioned.

http://electronotes.netfirms.com/Noise2.pdf

-Bernie

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