Additive synthesis involves the summation of multiple sinusoidal signals to form a more complex signal. While this approach appears quite simple, the synthesis of time-varying signals using additive techniques requires the control and careful manipulation of many sinusoidal parameters.

## Fourier Analysis

• By Fourier theory, any complex waveform can be decomposed into a (possibly infinite) sum of sinusoids, each with its own amplitude, frequency, and phase parameters.

• In general, waveforms with time-domain discontinuities require an infinite number of sinusoids to be perfectly reconstructed.

• In digital audio contexts, however, we are only concerned with sinusoidal components that have frequencies up to half the sample rate.

• A square wave and its associated magnitude spectrum are shown in Fig. 1 below.

• A sawtooth wave and its associated magnitude spectrum are shown in Fig. 2 below.

• A triangular wave and its associated magnitude spectrum are shown in Fig. 3 below.

## 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:

• The previous example demonstrated additive synthesis'' for a fixed, periodic waveform.

• In most synthesis contexts, the desired result is a signal that varies with time. In this case, it is necessary to develop a method for estimating the parameters of each sinusoidal component over time.

## FFT Synthesis/Resynthesis

• Additive synthesis parameters in a discrete-time implementation can be determined using the Fast Fourier Transform (FFT).

• The analyzed time-domain signal is split into blocks or frames'', each of which is processed using the FFT (referred to as the Short-Time Fourier Transform (STFT).

• The STFT provides a means for joint time-frequency analysis.

• As well, a time-domain signal can be resynthesized using the Inverse Fast Fourier Transform (IFFT). The resulting IFFT frames are assembled'' using overlap-add techniques.

• With improvements in computer processing speed, it is now possible to perform IFFT resynthesis in real time.

• FFT/IFFT synthesis lends itself well to sound transformations, such as time-stretching and pitch scaling.

## Sines + Noise: SMS Synthesis

• One particular approach to analysis/resynthesis is called spectral modeling synthesis (SMS).

• The SMS technique seeks to reduce the spectral data by extracting only a specific, relatively small number of spectral peaks from each STFT representation.

• The spectral energy that remains once these peaks are identified and removed is approximated by linearly-enveloped noise.

• SMS examples: http://mtg.upf.edu/technologies/sms

## Audio Compression

• Many audio compression strategies are based on a data reduction approach like that of SMS.

• MPEG coders incorporate perceptual masking information to reduce the number of spectral peaks necessary for reconstruction.