Where is the Aliasing?

• Given the infinite summations of the Fourier series “recipes” above, one should expect there to be aliasing in the associated spectra. However, this is not apparent in Figs. 6 and 7.

• It turns out that when the waveform period in samples is an integer (as is the case with the waveforms plotted in Figs. 6 and 7), the aliased frequency components “reflect back” on top of non-aliased components. The aliasing is happening but it is not obvious in this case.

• In a discrete-time synthesis context, the period $P$ (in samples) for a periodic signal can be determined from its frequency ($f$) as $P = f_s / f$, where $f_s$ is the sample rate.

• As an example, suppose we have a sawtooth waveform with $P = 50$ and a sample rate $f_s = 44100$ Hz. The fundamental, or first partial, frequency of the signal would be 882 Hz. The 24th, 25th and 26th partial frequencies would be 21168, 22050 and 22932 Hz. This last component, being greater than half the sample rate by 882, would alias back to 21168 Hz, which is the frequency of the 24th partial.

• The aliasing in this case is not obvious because no new spectral components result, though there is an associated timbral modification.

• In most synthesis contexts, $P$ is rarely an integer. In order to compute a signal with the correct frequency, it is then necessary to maintain an accurate “internal” time index and truncate/round/interpolate it to determine the waveform output at a given time step.

• No matter the computational technique used, when $P$ is not an integer, the aliased spectral components will fall between non-aliased components and be clearly perceived, as shown in Fig. 8 for a sawtooth spectrum.

  Figure 8: The frequency magnitude response for a sawtooth waveform generated from a non-bandlimited sawtooth wave table using a non-integer sample period.

  

• This can also be viewed as the addition of noise to the waveform signal introduced by a pitch-period jitter in the sampling process.