musicdsp.org source code archive - WSInf

Bandlimited waveforms synopsis. (click this to go back to the index)
References : Joe Wright
Linked file : waveforms.txt (this linked file is included below)
Notes :
(see linkfile)
Comments
from : dflatccrmadotstanforddotedu
comment : The abs(sin) method from the Lane CMJ paper is not bandlimited! It's basically just a crappy method for BLIT.
You forgot to mention Eli Brandt's minBLEP method. It's the best! You just have to know how to properly generate a nice minblep table... (slightly
dilated, see Stilson and Smith BLIT paper, at the end regarding table implementation issues)
Linked files
From: "Joe Wright"
To:
Subject: Re: waveform discussions
Date: Tue, 19 Oct 1999 14:45:56 +0100
After help from this group and reading various literature I have finished my
waveform engine. As requested, I am now going to share some of the things
are have learnt from a practical viewpoint.
Problem:
The waveforms of interest are sawtooth, square and triangle.
The waveforms must be bandlimited (i.e. fitting under Nyquist). This
precludes simple generation of the waveforms. For example, the
analogue/continuous formular (which has infinite harmonics):
s(t) = (t*f0) mod 1 (t=time, f0=frequency of note)
procudues aliasing that cannot be filtered out when converted to the
digital/discrete form:
s(n) = (f0*n*Ts) mod 1 (n=sample, Ts equals 1/sampling rate)
The other condition of this problem is that the waveforms are generatable in
real-time. Additionally, bonuses are given for solutions which allow
pulse-width modulation and triangle asymmetry.
The generation of these waves is non-triaval and below is discussed three
techniques to solve the problem - wavetables, approximation through
processing of |sinewave| and BLIT integration. BLIT integration is
discussed in depth.
Wavetables:
You can generate a wavetable for the waveform by summing sine waves up to
the Nyquist frequency. For example, the sawtooth waveform can be generated
by:
s(n) = Sum[k=1,n] (1/k *sin(2PI*f0*k*Ts)) where f0*n < Ts/2
The wavetable can then be played back, pitched up or down subject that
pitched f*n < Ts/2. Anything lower will not alias but it may lack some
higher harmonics if pitched too low.
To cover these situations, use multiple wavetables describing different
frequency ranges within which it is fine to pitch up or down.
You may need to compromise between number of wavetables and accuracy because
of memory considerations (especially if over-sampling). This means some
wavetables will have to cover larger ranges than they should. As long as
the range is too far in the lower direction rather than higher, you will not
alias (you will just miss some higher harmonics).
With wavetables you can add a sawtooth to an inverted sawtooth offset in
time, to produce a pulse/square wave. Vary the offset to vary the pulse
width. Asymmetry of triangle waves is not possible (as far as I know)
though.