LING124 Cepstrum, LPC, Autocorrelation

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LING124 Cepstrum, LPC, Autocorrelation

• September 18, 2008

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Class outline

• Source-filter model of speech production
• Cepstrum
• Linear Predictive Coding
• Autocorrelation

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Source-filter model

• Source glottal pulse
• Filter the vocal tract, characterized as a tube
• Harmonics whole number multiples of the
  fundamental frequency
• Different shapes of the filter (vocal tract)
  amplify different harmonics
• Different vowel ? different shape of the vocal
  tract ? amplification of different harmonics ?
  different spectral shape

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Source-filter model (2)

• x(t) excitation signal source (glottal pulse)
• h(t) transfer function filter shape of the
  vocal tract
• y(t) x(t) h(t)
• y(t) output signal speech sound
• The output signal is the convolution of the
  excitation signal and the transfer function

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Convolution

• The amount of overlap between two functions as
  one function is shifted over another function

From Wolfram MathWorld
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Cepstrum

• y(t) x(t) h(t)
• We want to find out the shape of the vocal tract
  that led to the speech sound
• We want to separate h(t) from y(t)
• A homomorphic transformation Dy(t) converts
  y(t) x(t) h(t) into y(t) x(t)h(t)
• One such transformation is the Cepstral analysis,
  which is, inverse Fourier transform of the log
  of Fourier transform of the signal

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Cepstrum (2)

• Spectrum of spectrum
• (Inverse) Fourier transform of the log of Fourier
  transform
• For xNn, a discrete-time signal with N samples
  in a cycle
• Magnitude and (phase) of the kth frequency
  component from DFT is
• Cepstral value for the nth sample is

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Cepstrum (3)

• Spectrum

• Cepstrum

Figures from JM (p.300), originally from Taylor
(2008)
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Linear Predictive Coding (LPC)
Figure 11.1 in Ladefoged (p.182)
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LPC (2)

• We assume that each sample (xn) of a speech
  sound can be predicted by a linear combination of
  a number of previous samples (e.g. xn-1,
  xn-2, ... , xn-p)
• xna1xn-1a2xn-2 ... apxn-p
• Of course there will be errors there will be
  difference between predicted value and the actual
  value
• en(xn-xn)2

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LPC (3)

• We want to find the coefficients that minimize
  the error, more specifically the sum of squared
  error within a window
• Assuming pth order and a window has w samples
• Define error as
• Solve for 1i p
• We will consider the coefficients as
  characterizing the filter

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LPC (4)

• The prediction error will be the greatest when
  there is an abrupt change in the sound wave
• This is most likely to happen whenever there is
  an impulse from the vocal fold vibration
• So if we plot the size of the error over time,
  the interval between two nearest peaks in error
  will equal the period of the glottal pulse

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Autocorrelation

• f(tT) f(t) for a periodic wave with period T
• If there are N samples in a cycle, xnNxn
• Take a copy of the wave
• Shift the wave by small number of samples
• Identify after how many samples the copy and the
  original completely overlaps
• e.g. Plot and compute the
  distance between two nearest minima

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Autocorrelation (2)