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<p class="MsoNormal"><span style="font-size:12.0pt; line-height:107%; font-family:"Arial",sans-serif">Achim said:<span style="">
</span><span style=""> </span></span><span style="font-size:12.0pt; line-height:107%; font-family:"Arial",sans-serif; color:black">“An EXOR is simply multiplication (given the right encoding).”</span></p>
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<span style="font-size:12.0pt; font-family:"Arial",sans-serif; color:black">True and widely understood.
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<span style="font-size:12.0pt; font-family:"Arial",sans-serif; color:black">Achim followed with: “That is what allows an LFSR to implement a matrix multiplication with such low hardware complexity.”</span></p>
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<span style="font-size:12.0pt; font-family:"Arial",sans-serif; color:black">I have no idea what that means – please elaborate.</span></p>
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<span style="font-size:12.0pt; font-family:"Arial",sans-serif; color:black">Matthew then replied:<span style="">
</span><span style=""> </span>“It's also addition, in a different encoding - and that's the reason for the LFSR to be called linear.<span style="">
</span>All these "polynomials" are polynomials defined over the field GF(2), in which the addition operation is the same thing as XOR on bits.”</span></p>
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<span style="font-size:12.0pt; font-family:"Arial",sans-serif">EXOR is still non-linear in the PRBS structure – the particular application under discussion here.
<span style=""> </span>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).<span style="">
</span>This IS linear, and K-S, generalized, IS a perfectly understood linear filter in a feedback loop.
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<span style="font-size:12.0pt; font-family:"Arial",sans-serif">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:
<span style=""> </span><span style=""> </span><span style=""> </span>“</span><span style="font-size:12.0pt; font-family:"Arial",sans-serif; color:black">Yes.<span style="">
</span>That works because whatever the original distribution(s), mixing them will (rather quickly under easily met conditions) converge towards a Gaussian </span><span style="font-family: Arial, sans-serif; font-size: 12pt;">one (says the central limit theorem).”</span></p>
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<span style="font-size:12.0pt; font-family:"Arial",sans-serif">I was saying that you MULTIPLY the two sequences by EXOR.<span style="">
</span>The little guy simply “chops up” any misbehavior of the long sequence. <span style="">
</span>They are not added so the CLT does NOT apply.<span style=""> </span>The EXORed output still has a uniform (1 or 0) non-Gaussian distribution.
<span style=""> </span>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].</span></p>
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<span style="font-size:12.0pt; font-family:"Arial",sans-serif">One last thing:<span style="">
</span>By chance today I same across another PRBS reference that mentions several interesting things – like the “heartbeat” issue just mentioned.<span style="">
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<span style="font-size:12.0pt; font-family:"Arial",sans-serif"><a href="http://electronotes.netfirms.com/Noise2.pdf" id="LPlnk179753" class="OWAAutoLink" previewremoved="true">http://electronotes.netfirms.com/Noise2.pdf</a></span></p>
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<span style="font-size:12.0pt; font-family:"Arial",sans-serif">-Bernie</span></p>
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