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<p class=MsoNormal><font size=2 color=navy face=Arial><span lang=EN-GB
style='font-size:10.0pt;font-family:Arial;color:navy'>Thanks Chris, that’s
a really interesting executive summary !<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span lang=EN-GB
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=EN-GB
style='font-size:10.0pt;font-family:Arial;color:navy'>How’s the list, I
always thought it was mainly hw based but it seems there’s some software
one too… what’s the ratio between the two ? Should I join ?<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span lang=EN-GB
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=EN-GB
style='font-size:10.0pt;font-family:Arial;color:navy'>M.<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span lang=EN-GB
style='font-size:10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<div>
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face="Times New Roman"><span lang=EN-US style='font-size:12.0pt'>
<hr size=2 width="100%" align=center tabindex=-1>
</span></font></div>
<p class=MsoNormal><b><font size=2 face=Tahoma><span lang=EN-US
style='font-size:10.0pt;font-family:Tahoma;font-weight:bold'>From:</span></font></b><font
size=2 face=Tahoma><span lang=EN-US style='font-size:10.0pt;font-family:Tahoma'>
music-bar-bounces@lists.music-bar.org
[mailto:music-bar-bounces@lists.music-bar.org] <b><span style='font-weight:
bold'>On Behalf Of </span></b>Chris Strellis<br>
<b><span style='font-weight:bold'>Sent:</span></b> vendredi 6 février
2009 9:25<br>
<b><span style='font-weight:bold'>To:</span></b> Music-bar<br>
<b><span style='font-weight:bold'>Subject:</span></b> RE: band limited oscillators</span></font><span
lang=EN-US><o:p></o:p></span></p>
</div>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color="#1f497d" face=Calibri><span lang=EN-GB
style='font-size:11.0pt;font-family:Calibri;color:#1F497D'>I can offer some
tips sent into the SDIY list recently from the great Antti Huovilainen<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color="#1f497d" face=Calibri><span lang=EN-GB
style='font-size:11.0pt;font-family:Calibri;color:#1F497D'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color="#1f497d" face=Calibri><span lang=EN-GB
style='font-size:11.0pt;font-family:Calibri;color:#1F497D'><a
href="http://antti.smartelectronix.com/">http://antti.smartelectronix.com/</a><o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color="#1f497d" face=Calibri><span lang=EN-GB
style='font-size:11.0pt;font-family:Calibri;color:#1F497D'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color="#1f497d" face=Calibri><span lang=EN-GB
style='font-size:11.0pt;font-family:Calibri;color:#1F497D'>also <a
href="http://www.diy.synth.net/gallery/main.php?g2_itemId=1078">http://www.diy.synth.net/gallery/main.php?g2_itemId=1078</a>
Synth DIY UK 2006<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color="#1f497d" face=Calibri><span lang=EN-GB
style='font-size:11.0pt;font-family:Calibri;color:#1F497D'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color="#1f497d" face=Calibri><span lang=EN-GB
style='font-size:11.0pt;font-family:Calibri;color:#1F497D'>Some DSP theory
links here: http://www.chameleon.synth.net/english/links.shtml<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color="#1f497d" face=Calibri><span lang=EN-GB
style='font-size:11.0pt;font-family:Calibri;color:#1F497D'><o:p> </o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>> I'm curious, what approach are you using to get
'alias free' oscillators? <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>> Simply using a much higher internal sampling rate
and then a low pass <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>> FIR filter? Or something more sophisticated
than this.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'><o:p> </o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Since this question gets asked a lot, I'll list some
of the common methods. Roughly from easy to hard. Oversampling here means
proper oversampling with high quality lowpass filtering before decimating to
target samplerate. Simply averaging N samples will not work.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'><o:p> </o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>1) Trivial saw with oversampling<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Pros: Easy, can do any waveshape, allows simple sync
and FM<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Cons: Requires massive (64..256x) oversampling to
sound good<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'><o:p> </o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>2) Sum of sines<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Sum nyquist/freq number of sines to produce exactly
bandlimited sawtooth.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Pros: No aliasing<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Cons: Too slow to be of use in practise.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'><o:p> </o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>3a) Differentiated parabole wave<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Synthesize parabole (diff(phase^2)*1/freq for -1 <=
phase < 1). Aliasing falls at 12dB/oct (compared to 6dB/oct for trivial
saw).<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'><o:p> </o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Pros: Almost as easy as trivial saw. 1/freq can be
derived from interpolated table lookup (store 1/freq for each note)<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Cons: diff(phase^2) can get very small for low
frequencies requiring 24 or<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>32 bit resolution. Requires 1.5-2x oversampling to
avoid annoying warble between 10-20 kHz.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'><o:p> </o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>3b) Slewrate limited saw<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Use a trivial saw-tri pwm oscillator with the pulse
width set to exactly one sample. Can be shown to be equivalent to 2a.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'><o:p> </o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Pros: Doesn't require frequency dependent scaling or
high resolution computations.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Cons: Same as 3a<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'><o:p> </o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>3c) Other waveshaping methods<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Several other methods can be used to sample a smooth
function and then warp the spectrum to resemble saw. Generally slower and more
complicated than 2a or 2b.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'><o:p> </o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>4) Mipmapped wavetables<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Precalculate a version (mipmap) for each octave (or
half octave) with exact number of harmonics. Select nearest mipmap and
interpolate the stored waveform on playback.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'><o:p> </o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Pros: Good quality with higher order interpolator or
oversampling mipmaps. <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Can do arbitrary waveforms. Easy FM. Easy phase
distortion.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Cons: Needs lots of memory. Number of harmonics
limited for low notes. <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Requires oversampling the mipmaps (using longer table
than strictly required by the number of stored harmonics) or using high order
(FIR) interpolator. Requires oversampling or more mipmaps (half or quarter<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>octave) to avoid missing frequencies between 15-20
kHz.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'><o:p> </o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>5a) BandLimited Impulse Trains (BLIT)<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Synthesize bandlimited impulse train and integrate
that to produce saw.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'><o:p> </o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Pros: Good quality. No oversampling required.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Cons: Complicated, slow, has numerical issues.
Difficult to do FM, PWM or sync.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'><o:p> </o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>5b) BandLimited StEps (BLEP)<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>For each oscillator reset, sum a bandlimited step with
the trivial saw. <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>The steps are precalculated and stored in a table (can
be quite short when interpolation is used between two phases.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'><o:p> </o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Pros: Very good quality. No oversampling required. Can
do bandlimited FM, PWM and sync. Probably the only method that can do audio
rate PWM and sync.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'>Cons: Requires a divide per cycle. Can be complicated:
calculating required table entry is not trivial when using sync or pwm.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face=Consolas><span lang=EN-GB
style='font-size:10.5pt'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color="#1f497d" face=Calibri><span lang=EN-GB
style='font-size:11.0pt;font-family:Calibri;color:#1F497D'>HTH<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=2 color="#1f497d" face=Calibri><span lang=EN-GB
style='font-size:11.0pt;font-family:Calibri;color:#1F497D'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color="#1f497d" face=Calibri><span lang=EN-GB
style='font-size:11.0pt;font-family:Calibri;color:#1F497D'>Chris<o:p></o:p></span></font></p>
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