Periodic Waveforms

From Fourier theory, we know that any reasonably smooth periodic waveform can be decomposed into a (possibly infinite) sum of sinusoids, each with its own amplitude, frequency, and phase parameters.
The frequency components of a steady periodic waveform are always harmonic.
The number of sinusoids needed to perfectly reconstruct a periodic waveform is dependent on the “smoothness” of the waveform. Waveforms with time-domain discontinuities, such as the sawtooth wave shown in Fig. 1 below, require an infinite number of sinusoids to be perfectly reconstructed.

Figure 1: One period of a sawtooth wave and its associated frequency magnitude response.
$\begin{figure}\begin{center} \epsfig{file=figures/saw.eps, width=3.5in} \end{center} \end{figure}$
This is also clear from the Fourier series representation of a sawtooth wave of fundamental frequency , which is given by:
$\displaystyle \frac{1}{2} - \frac{1}{\pi} \sum_{k=1}^{\infty} \frac{1}{k} \sin(2 \pi f k t) \hspace{0.2in} \mbox{for } k = 1, 2, 3, \ldots$
In digital audio contexts, however, we are only interested in keeping sinusoidal components that have frequencies up to half the sample rate.

**Figure 1:** One period of a sawtooth wave and its associated frequency magnitude response.
$\begin{figure}\begin{center} \epsfig{file=figures/saw.eps, width=3.5in} \end{center} \end{figure}$

An additive synthesis approach for a sawtooth signal, using the spectral recipe above, is:

fs = 44100;                    % sampling rate
T = 1/fs;                      % sampling period
t = 0:T:0.1;                   % time vector
N = 12;                        % number of sinusoid components to sum

f = 50;                        % fundamental frequency
omega = 2*pi*f;                % angular frequency
phi = -2*pi*0.25;              % 1/4 cycle phase offset

x = 0;
for n = 1:N
  x = x + cos(n*omega*t + phi) ./ n;
end

plot(t, x);
xlabel('Time (seconds)');
ylabel('Signal Amplitude');
s = sprintf('Sum of %d Sinusoidal Components', N);
title(s)

Sawtooth signals resulting from 6 and 100 sinusoidal components are shown in Fig. 2:

Figure 2: Sawtooth signals constructed from 6 and 100 sinusoidal components.
$\begin{figure}\begin{center} \epsfig{file=figures/sawsignals.eps, width=5in} \end{center} \end{figure}$

**Figure 2:** Sawtooth signals constructed from 6 and 100 sinusoidal components.
$\begin{figure}\begin{center} \epsfig{file=figures/sawsignals.eps, width=5in} \end{center} \end{figure}$

For steady, periodic signals, a wavetable approach can also be used:

fs = 44100;                    % sampling rate
N = 12;                        % number of sinusoid components to sum
M = 1024;                      % table size in samples

t = zeros(1, M);
for n = 1:N,
  t = t + sin(2*pi*n*(0:M-1)/M) ./ n;
end

plot(t);
xlabel('Table Index (samples)');
ylabel('Signal Amplitude');
s = sprintf('Sum of %d Sinusoidal Components', N);
title(s)

A more efficient (and general) wavetable approach could use a table with one period of a sinusoid and as many pointers as desired sinusoidal components, each of which is incremented according to its respective frequency.