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<p style="margin: 0in 0in 8pt; line-height: 107%; font-size: 11pt; font-family: Calibri, sans-serif">
<span style="font-size: 14pt; line-height: normal;">When we, as musical engineers, think of some notion such as “pitch” we usually have it from several points of view.</span><span style="font-size: 14pt; line-height: normal;">
</span><span style="font-size: 14pt; line-height: normal;">From the highest level (engineering-mathematics) we obtain pitch from the Continuous-Time Fourier Transform (CTFT), but this is usually not obtainable (the infinite limits, and usually a mathematical
expression for a musical sound s(t) is completely missing, which absolutely prevents us from doing the integral!).</span></p>
<p style="margin: 0in 0in 8pt; line-height: 107%; font-size: 11pt; font-family: Calibri, sans-serif">
<span style="font-size: 14pt; line-height: normal;">At a second level, computation IS possible with a sampled signal s(nT), using the Discrete Fourier Transform (DFT=FFT), which also self-windows in time, but which has (at least) uncertainty (Heisenberg sense
for FT) and is in general potentially problematic.</span></p>
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<span style="font-size: 14pt; line-height: normal;">At a third level (psychoacoustics) we insist on answers that agree with what a human listener hears.
</span><span style="font-size: 14pt; line-height: normal;"> </span><span style="font-size: 14pt; line-height: normal;">What the modled pitch “should be” is irrelevant if it is wrong or unstable.</span></p>
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<span style="font-size: 14pt; line-height: normal;">At the fourth level, we need to have a scheme that is practical in terms of construction/programmability as an end goal.
</span><span style="font-size: 14pt; line-height: normal;"> </span><span style="font-size: 14pt; line-height: normal;">At this level, all sorts of “rule breaking” may appear and be applauded if they just WORK.
</span></p>
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<span style="font-size: 14pt; line-height: normal;">Do we find the same four levels if we are considering obtaining an ENVELOPE?</span><span style="font-size: 14pt; line-height: normal;">
</span><span style="font-size: 14pt; line-height: normal;">Yes and NO.</span><span style="font-size: 14pt; line-height: normal;">
</span><span style="font-size: 14pt; line-height: normal;">The first two levels (Hilbert transform) are so infrequently discussed:</span></p>
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<a href="http://electronotes.netfirms.com/Hilbert.pdf"><span style="font-size: 14pt; line-height: normal;">http://electronotes.netfirms.com/Hilbert.pdf</span></a><span style="font-size: 14pt; line-height: normal;">
</span></p>
<p style="margin: 0in 0in 8pt; line-height: 107%; font-size: 11pt; font-family: Calibri, sans-serif">
<a href="http://electronotes.netfirms.com/AN355.pdf"><span style="font-size: 14pt; line-height: normal;">http://electronotes.netfirms.com/AN355.pdf</span></a></p>
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<span style="font-size: 14pt; line-height: normal;">that they do not come readily to mind.</span><span style="font-size: 14pt; line-height: normal;">
</span><span style="font-size: 14pt; line-height: normal;">So it is not apparent that there is any possibility of an “above all” engineering notion of a correct answer to compare to.</span><span style="font-size: 14pt; line-height: normal;">
</span><span style="font-size: 14pt; line-height: normal;">Instead, we have generally relied on levels three and four (what does the ear think is right; what can we build), which at least has problem with time-averaging.</span></p>
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<span style="font-size: 14pt; line-height: normal;">To get the “envelope” of a signal s(t), first form the “analytic signal” S(t) = s(t) + js’(t) where s’(t) is the</span><span style="font-size: 14pt; line-height: normal;"> </span><span style="font-size: 14pt; line-height: normal;">Hilbert
Transform of s(t).</span><span style="font-size: 14pt; line-height: normal;"> </span>
<span style="font-size: 14pt; line-height: normal;">Then take the magnitude of S(t) as the envelope.</span><span style="font-size: 14pt; line-height: normal;">
</span><span style="font-size: 14pt; line-height: normal;">The HT, like the CTFT is an all-time integral transform, and is largely unobtainable at level 1 for the same reasons.</span><span style="font-size: 14pt; line-height: normal;">
</span><span style="font-size: 14pt; line-height: normal;"> </span><span style="font-size: 14pt; line-height: normal;">At level 2 (computability) however, there is a discrete-time HT that is a largely trivial standard FIR digital filter (see pages 10-12) of:</span></p>
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<a href="http://electronotes.netfirms.com/EN199.pdf"><span style="font-size: 14pt; line-height: normal;">http://electronotes.netfirms.com/EN199.pdf</span></a></p>
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<span style="font-size: 14pt; line-height: normal;">which is basically a 90 degree phase shifter: s’(nT) from a windowed s(nT).</span></p>
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<span style="font-size: 14pt; line-height: normal;">Believing is perhaps a first step in understanding.
</span><span style="font-size: 14pt; line-height: normal;"> </span><span style="font-size: 14pt; line-height: normal;">What if you had s(t) = A Sin(t)?
</span><span style="font-size: 14pt; line-height: normal;"> </span><span style="font-size: 14pt; line-height: normal;">Shifting 90 degrees s’(t) = A cos(t), so S(t) = A Sin(t) + jA Cos(t), is the analytic signal and its magnitude (square root of sum of squares)
is A, which everyone agrees is the correct answer and involves no low-pass filtering complications.
</span><span style="font-size: 14pt; line-height: normal;"> </span><span style="font-size: 14pt; line-height: normal;">This example is non-linear!
</span><span style="font-size: 14pt; line-height: normal;"> </span><span style="font-size: 14pt; line-height: normal;">You do need a full broadband HT.
</span><span style="font-size: 14pt; line-height: normal;"> </span><span style="font-size: 14pt; line-height: normal;">So here, it is just the simplest example.
</span><span style="font-size: 14pt; line-height: normal;"> </span><span style="font-size: 14pt; line-height: normal;">MANY years ago Hal Chamberlin pointed out that the above example while correct did not validate the general expanded scheme!</span></p>
<p style="margin: 0in 0in 8pt; line-height: 107%; font-size: 11pt; font-family: Calibri, sans-serif">
<span style="font-size: 14pt; line-height: normal;">Just a few things to keep in mind.</span></p>
<p style="margin: 0in 0in 8pt; line-height: 107%; font-size: 11pt; font-family: Calibri, sans-serif">
<span style="font-size: 14pt; line-height: normal;">-Bernie</span></p>
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