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<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>I built a frequency shifter following the
Bode plan. This is frequency shifting by manifesting certain
trigonometric product-to-sum formulae using electronic circuits:<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span lang=PT-BR
style='font-size:10.0pt;font-family:Arial;color:navy'>sin u sin v = 0.5 [cos(u –
v) – cos (u + v)]<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span lang=PT-BR
style='font-size:10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span lang=PT-BR
style='font-size:10.0pt;font-family:Arial;color:navy'>cos u cos v = 0.5 [cos(u –
v) + cos (u + v)]<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span lang=PT-BR
style='font-size:10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>So, if you have two signals with their
90-degree quadrature signals (say, u is the audio you want to shift and v is
the on-board quadrature oscillator), then if you multiply the signals together
(using a four-quadrant multiplier) and also multiply their quadrature signals
together (using a second four-quadrant multiplier) then you can sum the
multiplier outputs together, and you will get the following (by adding the two
equations together):<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>cos (u – v)<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>This represents the audio signal u which
has been frequency-shifted downward by the frequency of the oscillator signal
v. Of course, the audio signal probably has many frequencies u occurring
simultaneously, and they will all be shifted down by v. That’s what
makes frequency shifting sound so alien and weird.<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>By being clever with the summations of the
multiplier output signals (based on a little bit of algebra), you can also recover
the up-shifted audio:<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>cos (u + v) <o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>With both the down- and up-shifted signals,
you can get a stereo effect.<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>The entire circuit consists of a 90-degree
phase displacement network to generate the cosine of the incoming audio (I designed
mine with 12 stages from 15 Hz to 15 kHz using a little thing I found on the
internet called QuadNet), a quadrature oscillator to generate both sine and
cosine waves at frequency v (mine is TZFM and consists of two Rubicon cores
with sine shapers, with one syncing the other in such a way that the two are
always 90 degrees out of phase), two four-quadrant multipliers (I built a dual
unit from a single 2164 chip – two linearized VCAs), and a couple of
output amplifier stages for doing the summing. The key to success is to
AC couple the signals into the multipliers to eliminate DC offsets in the
incoming signal, which is the single largest source of error in the circuit.
If that is done properly, the multipliers require no trimming (if accurate
summing resistors are chosen).<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'>The circuit works great and sounds
super freaky. I’m going to be building another one for one of our
members here shortly.<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
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<p class=MsoNormal><b><font size=2 face=Tahoma><span style='font-size:10.0pt;
font-family:Tahoma;font-weight:bold'>From:</span></font></b><font size=2
face=Tahoma><span style='font-size:10.0pt;font-family:Tahoma'> Synth-diy
[mailto:synth-diy-bounces@synth-diy.org] <b><span style='font-weight:bold'>On
Behalf Of </span></b>ColinMuirDorward<br>
<b><span style='font-weight:bold'>Sent:</span></b> Tuesday, March 10, 2020 7:01
PM<br>
<b><span style='font-weight:bold'>To:</span></b> *SYNTH DIY<br>
<b><span style='font-weight:bold'>Subject:</span></b> [sdiy] frequency shifter</span></font><o:p></o:p></p>
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<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'><o:p> </o:p></span></font></p>
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<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'>Hi, I got a little lost trying to understand what a frequency shifter is.
I mean the pre-digital method used by Moog (I think?).<o:p></o:p></span></font></p>
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<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'>I recently built a 4pole APF, and was really impressed with some of the
pitching effects I could achieve with it. I'm guessing this is an entirely
different method than the frequency shifters like Moog and JH have done.<o:p></o:p></span></font></p>
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<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'>Is the APF method used by anyone? What are its limitations, and what is
it even doing? <o:p></o:p></span></font></p>
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<div>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'>Well, I guess I'm just looking for some conversation on the topic of
analog frequency/pitch shifting methods. If anyone has any thoughts/experience
they'd like to share.<o:p></o:p></span></font></p>
</div>
<div>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'>Cheers,<o:p></o:p></span></font></p>
</div>
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<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'>Colin<o:p></o:p></span></font></p>
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<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'><br>
-- <o:p></o:p></span></font></p>
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<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'><a href="https://www.instagram.com/colinmuirdorward/" target="_blank">https://www.instagram.com/colinmuirdorward/</a><o:p></o:p></span></font></p>
</div>
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<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'><a href="https://www.instagram.com/colinmuirdorward/" target="_blank">-<br>
</a><o:p></o:p></span></font></p>
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<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'><a href="https://www.instagram.com/ssdp_synthesis/" target="_blank">https://www.instagram.com/ssdp_synthesis/</a><o:p></o:p></span></font></p>
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