FFT and Convolution Bibliography

Works Cited:

Computer Music by Charles Dodge and Thomas A. Jeres:
DFTs are computationally intensive. Cooley and Tukey reduced the number of computations significantly and produced what is now known as FFT. (55) “The input to a particular FFT is… a windowed portion of the signal because it represent a snap shot of the sound. (244)
Convolution – “…a dynamic filtering operation.” (327) “…one way to conceptualize convolution is as a filtering process in which one signal is passed through a filter whose frequency response is the spectrum of the other signal.” (328)

Elements of Computer Music by F. Richard Moore:
“A Fourier transformation can be used to associate a unique spectrum with any waveform. The spectrum shows, in effect, how to construct the analyzed waveform out of a set of sinusoidal harmonics…” (29)

The Computer Music Tutorial by Curtis Roads:
“if the input sounds are instrumental, the result may sound as if one instrument is “playing” the other”

output[n]=a[n]*unit[n]=a[n] convolution with the unit impulse is said to be an identity operation

convolution involves multiplication, but the convolution of two signals is different from multiplication of two signals a is multiplied by every sample of b which makes an array of samples of length b for every sample of a

A fundamental canon of signal processing is that the convolution of two waveforms is equal to the multiplication of their spectra

“Convolution in the time domain is equal to multiplication in the frequency domain and vice versa”

convolution is directly related to filtering

Many practical applications of convolution use a method called fast convolution. Fast convolution for long sequences takes advantage of the fact that the product of two n-point discrete Fourier transforms is equal to the DFT of the convolution of two n-point sequences.

Musical significance of convolution

Various sonic transformations can be explained as convolutions, including filtration, temporal effects, and modulation

Filtering as convolution
Filtering is multiplication of spectra but goes beyond simple filtering and becomes cross synthesis

Temporal effects of convolution
Convolution also induces time-domain effects like echo, time smearing and reverberation

Modulation as convolution
AM and RM call for multiplication of time–domain waveforms – the law of convolution states that multiplication of two wave forms convolves their spectral convolution accounts for the sidebands that result from these modulations

Convolution with grains and pulsars
Involves convolutions of sounds with clouds of sonic grains

Linear convolution is direct convolution length(output)=length(a)+length(b)-1

Circular convolution an anomaly that can occur when convolution is implemented with the FFT

Deconvolution
Separation of excitation (glottal pulses) —autoregressive and the resonance (vocal tract formants) –homomorphic deconvolution —of vocal sounds

Baron de Fourier (1768 – 1830)
“Arbitrarily complicated Periodic signals could be represented as a sum of many simultaneous simple signals”

Time and Frequency Domains
“Convolution in the time domain is equal to multiplication in the frequency domain and vice versa”

All excerpts taken from pages (419-432)

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