- Fourier Analysis
- Summation of Sines
- FFT Synthesis/Resynthesis
- Sines + Noise: SMS Synthesis
- Audio Compression

- 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.

- 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.

- 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.

- 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

- 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.

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