Licenciado

Alias-Free Digital Synthesis of Classic Analog Waveforms
Tim Stilsonstilti@ccrma.stanford.edu Julius Smithjos@ccrma.stanford.edu
CCRMA http:
www-ccrma.stanford.edu 

Music Department, Stanford University

Abstract
Techniques are reviewed and presented for alias-free digital synthesis of classical analog synthesizer waveforms such as pulse train and sawtooth waves. Techniquesdescribed include summation of bandlimited primitive waveforms as well as table-lookup techniques. Bandlimited pulse and triangle waveforms are obtained by integrating the difference of two out-of-phase bandlimited impulse trains. Bandlimited impulse trains are generated as a superposition of windowed sinc functions. Methods for more general bandlimited waveform synthesis are also reviewed. Methods areevaluated from the perspectives of sound quality, computational economy, and ease of control.

1

Introduction
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impulse train

Any analog signal with a discontinuity in the waveform (such as pulse train or sawtooth) or in the waveform slope (such as triangle wave) must be bandlimited to less than half the sampling rate before sampling to obtain a correspondingdiscrete-time signal. Simple methods of generating these waveforms digitally contain aliasing due to having to round off the discontinuity time to the nearest available sampling instant. The signals primarily addressed here are the impulse train, rectangular pulse, and sawtooth waveforms. Because the latter two signals can be derived from the first by integration, only the algorithm for the impulsetrain is developed in detail.

10

15

20

Box Train, and Sample Positions 1 0.8 0.6 0.4

2

Why Simple Discrete-Time Pulse Trains are Aliased

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Rounded−Time Pulse Train 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20

The “obvious” way to generate a discrete-time version of an impulse train is to approximate it by a unitsample-pulse train. The unit sample pulse δn is defined as1; n = 0 ∆ δn = 0; jnj = 1; 2; 3; : : : The unit-sample pulse is only defined for integer n, so we have a problem: Suppose the desired impulse-train frequency is f1 = 1=T1, then the period in samples has to be P = T1 =Ts = Fs = f1 , where Fs = 1=Ts is the sampling rate, and P is rarely an integer. Because pitch perception

Figure 1: Rounded-Time Impulse Train as a Sampled Version of an IdealRectangular Pulse Train

Stilson and Smith

1

Alias-Free Synthesis

is so accurate , it does not work to round M to the nearest integer, except at frequencies so low that the error is on the order of a tenth of a percent (around 100 Hz or below for a 50 kHz sampling rate). Therefore, to make the pitch right, it is necessary to compute the impulse arrival times very accurately and roundeach arrival time to the nearest sample instant. This process can be modeled as pitch-period jitter which obviously adds noise to the signal. It can also be modeled as a uniform sampling of a periodic sequence of rectangular pulses one sample wide (Figure 1): pt  =
∆

Y e jωTs  ∝
=

k=,∞ ∞ k=,∞ ∞

∑ ∑

∞

X ω + k2πFs Pω + k2πFs δω + k2πFs , lω1 

∑

l =,∞

1; jt j  Ts=2 0; jt j Ts =2

The Fourier transform of this pulse is a sinc function Pω =
∆

Z ∞

,∞

pt e, jωt dt = Ts sinc f Ts 

where ω = 2π f , and sincx =
∆

sinπx : πx

A periodic sequence of these rectangular pulses is constructed as xt  =
∆

The desired pulse-train spectrum is only the k = 0 term above. Each nonzero k term contributes a string of aliased of harmonicsacross the entire frequency band. The amount of aliasing is highly significant and audible, as can be predicted from looking at Fig. 6. The only frequency which does not suffer aliasing is DC since the sinc spectral envelope goes through zero at all multiples of the sampling rate. For this reason, the aliasing is reduced at very low frequencies relative to the sampling rate. This provides another...