band limited oscillators

Fri Feb 6 12:57:40 CET 2009

Thanks Chris, that’s a really interesting executive summary !

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 ?

M.

  _____  

From: music-bar-bounces at lists.music-bar.org
[mailto:music-bar-bounces at lists.music-bar.org] On Behalf Of Chris Strellis
Sent: vendredi 6 février 2009 9:25
To: Music-bar
Subject: RE: band limited oscillators

I can offer some tips sent into the SDIY list recently from the great Antti
Huovilainen

http://antti.smartelectronix.com/

also http://www.diy.synth.net/gallery/main.php?g2_itemId=1078  Synth DIY UK
2006

Some DSP theory links here:
http://www.chameleon.synth.net/english/links.shtml

> I'm curious, what approach are you using to get 'alias free' oscillators? 

> Simply using a much higher internal sampling rate and then a low pass 

> FIR filter?  Or something more sophisticated than this.

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.

1) Trivial saw with oversampling

Pros: Easy, can do any waveshape, allows simple sync and FM

Cons: Requires massive (64..256x) oversampling to sound good

2) Sum of sines

Sum nyquist/freq number of sines to produce exactly bandlimited sawtooth.

Pros: No aliasing

Cons: Too slow to be of use in practise.

3a) Differentiated parabole wave

Synthesize parabole (diff(phase^2)*1/freq for -1 <= phase < 1). Aliasing
falls at 12dB/oct (compared to 6dB/oct for trivial saw).

Pros: Almost as easy as trivial saw. 1/freq can be derived from interpolated
table lookup (store 1/freq for each note)

Cons: diff(phase^2) can get very small for low frequencies requiring 24 or

32 bit resolution. Requires 1.5-2x oversampling to avoid annoying warble
between 10-20 kHz.

3b) Slewrate limited saw

Use a trivial saw-tri pwm oscillator with the pulse width set to exactly one
sample. Can be shown to be equivalent to 2a.

Pros: Doesn't require frequency dependent scaling or high resolution
computations.

Cons: Same as 3a

3c) Other waveshaping methods

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.

4) Mipmapped wavetables

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.

Pros: Good quality with higher order interpolator or oversampling mipmaps. 

Can do arbitrary waveforms. Easy FM. Easy phase distortion.

Cons: Needs lots of memory. Number of harmonics limited for low notes. 

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

octave) to avoid missing frequencies between 15-20 kHz.

5a) BandLimited Impulse Trains (BLIT)

Synthesize bandlimited impulse train and integrate that to produce saw.

Pros: Good quality. No oversampling required.

Cons: Complicated, slow, has numerical issues. Difficult to do FM, PWM or
sync.

5b) BandLimited StEps (BLEP)

For each oscillator reset, sum a bandlimited step with the trivial saw. 

The steps are precalculated and stored in a table (can be quite short when
interpolation is used between two phases.

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.

Cons: Requires a divide per cycle. Can be complicated: calculating required
table entry is not trivial when using sync or pwm.

HTH

Chris

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