band limited oscillators

[Chris Strellis]

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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