Summation of Sines

Knowing the spectral recipe of a given waveform, we can attempt to recreate it using sinusoidal oscillators (which could be implemented with wave tables):

fs = 44100;                    % sampling rate
T = 1/fs;                      % sampling period
t = [0:T:0.1];                 % time vector
N = 6;                         % 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 = 1:2:2*N,
  x = x + cos(n*omega*t + phi) ./ n;
end

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

Waveforms resulting from 6 and 200 sinusoidal components are shown in Figs. 4 and 5:

Figure 4: A waveform constructed from six sinusoidal signals.
$\begin{figure}\begin{center} \epsfig{file=figures/squareadd6.eps, width=3.5in} \end{center} \end{figure}$

Figure 5: A waveform constructed from 200 sinusoidal signals.
$\begin{figure}\begin{center} \epsfig{file=figures/squareadd200.eps, width=3.5in} \end{center} \end{figure}$
The previous example demonstrated “additive synthesis” for a fixed, periodic waveform.
In most synthesis contexts, the desired result is a signal whose frequency content varies with time. In this case, it is necessary to develop a method for estimating the parameters of each sinusoidal component over time.