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<span style="font-size:12.0pt;font-family:"Courier New";mso-fareast-font-family:"Times New Roman";
color:black"><o:p>A Good Project ? </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Courier New";mso-fareast-font-family:"Times New Roman";
color:black"><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif;mso-fareast-font-family:
"Times New Roman";color:black">I am not an abstract mathematician although I do understand “engineering math” pretty well, and have thought a good deal how this all applies
to a particular “application”; a particular “receiving system” – the human ear/brain.<span style="mso-spacerun:yes">
</span>So to be more productive, let’s transition from abstract math to engineering math (with its familiar terminology) and thereafter to perceptual and aesthetic aspects of PRBS generators (the design aspects being fairly clear).<span style="mso-spacerun:yes">
</span>To discuss one notion of some importance to us, what does the “spectrum” look like?<span style="mso-spacerun:yes">
</span>Is it “white”? <o:p></o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif;mso-fareast-font-family:
"Times New Roman";color:black"><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif;mso-fareast-font-family:
"Times New Roman";color:black">Any perceived spectrum must pass the physical windows of being limited in frequency to about 20 Hz to 15 kHz, and in time to a duration of perhaps
100 milliseconds per “frame” (“time constant of the ear”). <span style="mso-spacerun:yes">
</span>Recognizing that reasonable length PRBS sequences (say N=24 so that 2^N-1 approaches 17 million), and that a frame may need to be, say, 4410 samples for 100 ms, we usually achieve a reasonably white hiss for each frame.<span style="mso-spacerun:yes">
</span>It works.<span style="mso-spacerun:yes"> </span>[There is a slight roll-off (reaching -4db) at half the sampling frequency due to the hold time of the shift register clockings.
<span style="mso-spacerun:yes"> </span><span style="mso-spacerun:yes"> </span><span style="mso-spacerun:yes"> </span><o:p></o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">Problems are expected if the shift register is too short (say only N=5 so that 2^N-1 is only 31 and thus repeats at something like a kHz – strongly pitched.<span style="mso-spacerun:yes">
</span>Or, as Richie’s N=127 demonstrated, apparently it can be way too long to avoid very audible traps.<span style="mso-spacerun:yes">
</span>And any remaining problems can likely be handled by EXORing the longer sequence with a shorter one, as discussed above.<span style="mso-spacerun:yes">
</span>So the problem of getting a reliable white noise generator is under control, and this noise can be additionally filtered (perhaps with a bandpass VCF) as desired.<o:p></o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">NOW........<o:p></o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">That’s HALF the story:
<span style="mso-spacerun:yes"> </span>– white noise as raw material for processing into musical SOUND.<span style="mso-spacerun:yes">
</span>The other application is the use of a PRBS to generate “random tunes” (melodies) like the Psych Tone.<span style="mso-spacerun:yes">
</span>It is unlikely that we are interested in melodies of only two pitches, nor of sequences as long or longer than about a dozen.<span style="mso-spacerun:yes">
</span>As importantly, we should be concerned with the DEGREE OF CORRELATION BETWEEN SUCCESSIVE PITCHES<span style="mso-spacerun:yes">
</span>- what is “traditional”?<span style="mso-spacerun:yes"> </span>Thus, several relevant factors come in:<o:p></o:p></span></p>
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<span class="MsoHyperlink"><span style="font-size:12.0pt;font-family:"Arial",sans-serif"><a href="http://electronotes.netfirms.com/EN76.pdf" id="LPlnk849693" class="OWAAutoLink" previewremoved="true">http://electronotes.netfirms.com/EN76.pdf</a></span></span><span style="font-family: Arial, sans-serif; font-size: 12pt;"> </span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">First we probably use a much much slower PRBS clocking rate, since we probably want notes coming out (likely as control voltages to a VCO) at perhaps from about 2 to about 10 tones per second.<span style="mso-spacerun:yes">
</span>And we want many more possible pitch levels (perhaps 40 or so – half a piano keyboard if conventionally configured).<span style="mso-spacerun:yes">
</span>And, as suggested, a total trial sequence length of perhaps at most a dozen tones – AND we prefer it to be exactly reproducible.<o:p></o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">Getting more than just two output levels is almost certainly a matter of adding up multiple register stages.<span style="mso-spacerun:yes">
</span><span style="mso-spacerun:yes"> </span>For example, we could sum 12 stages (12 “taps”).<span style="mso-spacerun:yes">
</span>[This we CAN do, perhaps offering nothing really new, even if the sequence length is less than 12 by extending the shift register in feed-forward beyond the lasts tap fed back – just delayed values of the output.]<span style="mso-spacerun:yes">
</span>The output is thus numbered as the sum of from 1 to 12 (possibly 0 to 12) ones, not just a 0 or a 1. This “running sum” produces tones approaching a Gaussian distribution and one that is very HIGHLY CORRELATED (low- passed).
<span style="mso-spacerun:yes"> </span>This sum is NOT white (it is often termed “Red Noise”).
<span style="mso-spacerun:yes"> </span>In fact, for the next output value, the SUMMED bits on the registers can either stay the same, go up by 1, or down by 1.<span style="mso-spacerun:yes">
</span>No big jumps (melodically just arpeggio phrases).<span style="mso-spacerun:yes">
</span>Likely too boring.<span style="mso-spacerun:yes"> </span><span style="mso-spacerun:yes"> </span><o:p></o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">Alternatives to uniforms tap weights should be considered.<span style="mso-spacerun:yes">
</span>Here it may be useful to point out that we are talking about the “impulse-response” of a digital FIR [Finite (length) Impulse Response] filtering. For example, binary weighing (resistors in powers of 2 summing – a somewhat standard “D/A”<span style="mso-spacerun:yes"> </span>converter)
can be used, but why not experiment?<span style="mso-spacerun:yes"> </span>With this weighting, a particular 1 might enter the converter as a MSB (big change) and leave as the LSB (small change). The distribution still approaches Gaussian and is still not
nearly white, a degree of low-passing correlation still apparent. <o:p></o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">Neuvo and Ku (IEEE reference in link above) used an all-digital similar approach to this summing issue.
<span style="mso-spacerun:yes"> </span>They added up a “frame” of M samples of the PRBS generator, latched and converted this as a “word” and then went on to the next M bits, keeping none of the previous sample.
<span style="mso-spacerun:yes"> </span>Thus the successive words were uncorrelated and the generator as a whole was white.<span style="mso-spacerun:yes">
</span>This was a noise generator and was a cement-block-like wire-wrapped TTL tour de force. Similar Psych Tone speed approaches (and more!) in software seem quite doable.<o:p></o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">Another way to avoid the excessive correlation of the moving sum (or even the binary weighting would be to spread the taps further apart and perhaps more randomly. Also, it is unclear what the use
of “non-minimal length” sequences might do to correlation. But just what are we looking for?<span style="mso-spacerun:yes">
</span><o:p></o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">Well first keep in mind that we want some sort of weighting to give many possible pitch levels.<span style="mso-spacerun:yes">
</span>This we get, approaching Gaussian, through the summation and courtesy of the Central Limits Theorem.
<o:p></o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">Correlation is a separate issue.<span style="mso-spacerun:yes">
</span>“White” means noncorrelated.<span style="mso-spacerun:yes"> </span>That is, without correlation a particular current pitch offers no clue as to what pitch is coming next.
<span style="mso-spacerun:yes"> </span>Each new pitch is a surprise and all melodic intervals are about equally likely.
<span style="mso-spacerun:yes"> </span>Filtering (making the spectrum non-white), as with the “running sum” above, gives us next pitches that are either the same as the current one or one step up or one step down.
<span style="mso-spacerun:yes"> </span>Neither of these contrasting cases resembles our ingrained experience with musical melody which is a balanced mixture of fulfilled expectation and surprise.<span style="mso-spacerun:yes">
</span><o:p></o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">Voss and Clarke [J. Acoust. Soc. Amer., “ ‘1/f Noise’ in Music: Music from 1/f Noise”, Vol. 63 (1978)] and subsequent authors indicated that musical tradition with regard to melody (and many other
things) showed an inherent 1/f POWER spectrum. <span style="mso-spacerun:yes"> </span>That is, it was neither flat (white - uncorrelated) nor was it the 1/f (Amplitude – 1/f^2 in Power) typical of an integrator/summation; but rather 1/sqrt(f) usually called
“Pink Noise”.<span style="mso-spacerun:yes"> </span>So with regard to a secession of tones acceptable as a melody we prefer to have expectation sometimes fulfilled as well as numerous surprises.<o:p></o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif">A search of the Internet will produce a good deal about 1/f and pink sequence and generation algorithms.<span style="mso-spacerun:yes">
</span>Any person experimenting with a software version of Psych-Tone-like structures might do well to consider how PRBS sequences might relate to 1/f.<span style="mso-spacerun:yes">
</span><span style="mso-spacerun:yes"> </span><o:p></o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p>- Bernie</o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
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<span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
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<div id="divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" style="font-size:11pt" color="#000000"><b>From:</b> Bernard Arthur Hutchins, Jr<br>
<b>Sent:</b> Monday, January 7, 2019 7:51:53 PM<br>
<b>To:</b> synth-diy@synth-diy.org<br>
<b>Subject:</b> [sdiy] Long LFSRs (Was Psych Tone)</font>
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<p class="x_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="x_OWAAutoLink">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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