Electronic Music

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Electronic Music

• Dr Ian Drumm

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Phase Vocoder

• Aims
• To introduce the concept of analysis synthesis
  methods
• Learning Outcomes
• Short-Time Fourier transform (STFT)
• The application of the phase vocoder

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Introduction

• The phase vocoder is an analysis synthesis
  technique that can also be realised as a
  sophisticated effects box
• Transforms signal into time-varying spectra.
  These can be altered and a new signal synthesised.

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Basic Concept

• The Discrete Fourier Transform
• The Discrete Inverse Fourier Transform

is a time series
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Phase Vocoder

• Take a signal (i.e. a signal indexed by k)
• Apply short time, windowed DFTs (or FFTs) to
  successive sections of length M
• Hop every by s samples after each section
• Store DFTs an array index by integers m,l

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Short Time Fourier Transform

• This gives the Short Time Fourier Transform
• Where
• M is the window length
• s is the hop size
• l is an integer (0,1,2,.L)
• h is the impulse response of the window
• The STFT is for Xm,l for all l (and m)
• Note element of the STFT represents a frequency

Were fs is the sample rate
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In Matlab
clear all close all x,FS,NBITSwavread('bassoon.
wav') M4092 Window length sM/8 Hop
size sectionslength(x)/s - M/s Xzeros(M,sectio
ns) xwindowedzeros(M,1) for l0sections
xwindowedx((ls1)(lsM)) . hamming(M)
specabs(fft(xwindowed)) X(,l1)spec end
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Choosing Hop Size

• If hop size equals window size then half the
  signal will be attenuated
• If the overlap point corresponds to -3dB then the
  subjective importance of any attenuation will be
  very small.
• Typically some fraction of window size, e.g. M/8
• Decimation Factor e.g. a decimation factor of 2
  will have two overlaps in the window.

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Re-synthesis

• Put the musical signal back together again from
  the STFTs

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Re-synthesis

• Inverse STFTs
• where is now the part of the new time
  signal valid for frame l
• And knit them all together with the overlap-add
  method
• Where is the final output signal

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Time expansion or compression without pitch
distortion

• Vary the value of s on playback

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Pitch shifting without time distortion

• Spectrum scaling - Multiply all fm by scale
  factor (or by a function)
• Pitch shifting important tool in recording studio
  for correcting poor vocal performance

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Tricks with a phase vocoder Vary of recorded
sound

• Can modify the amplitude and frequency content of
  the STFTs in pretty much anyway you like
• E.g. the phase vocoder can act as a very flexible
  and accurate filter by varying to amplitude of
  the values in the frequency bins of the STFTs.

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Other creative applications

• Timbral interpolation from on instrument to
  another by interpolating between STFTs
• Cross-synthesis using a characteristic from one
  sound (amp, freq or phase) to modulate a
  characteristic from another.

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Analogy of Subtractive Synthesiser Vocoders

• Included with popular synths (Moogs, Korgs, etc)
• Array of analog band pass filters that take say a
  voice signal and split it up into a range of
  amplitudes corresponding to different frequencies
  (a crude STFT).
• These time varying amplitudes can then be used to
  control another signal such as a synthesiser
  timbre or an electric guitar sound.
• ELOs Mr Blue Sky

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Parameters and choices

• Window size M larger means good freq resolution,
  poor time resolution, long calculation time
• Window type anything non-rectangular (Hamming,
  Hanning, Blackman, etc)
• FFT size Optimisation restricts to 2n e.g. 1024
• Hop size, s fraction of M e.g. sM/8 overlap
  at -3dB points of window

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Improvements

• Window closing
• Repeat analysis of signal xk with smaller and
  smaller window sizes
• Tracking phase vocoder
• Find peaks in magnitude spectrum and attempt to
  track these through successive spectra (producing
  an envelop for each peak) discard other
  magnitude information
• Different Transforms
• Reduce effects of time/frequency trade off by
  using another transform e.g. wavelet, Wigner

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Summary

• A phase vocoder can shift or scale a musical
  spectrum, compress or expand signal duration and
  interpolate timbres all independently of other
  (unwanted) effects
• The phase vocoder implements the STFT (and its
  inverse), developed from the DFT
• Understanding of the SFTF is essential for proper
  control of a phase vocoder
• The concept of the phase vocoder can be extended
  by window closing, tracking or novel base
  transforms