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### EE513Audio Signals and Systems

LPC Analysis and Speech

Kevin D. DonohueElectrical and Computer EngineeringUniversity of Kentucky

Speech Generation

Speech can be divided into fundamental building blocks of sounds referred to as phonemes. All sounds result from turbulence through obstructed air flow

The vocal cords create quasi-periodic obstructions of air flow as a sound source at the base of the vocal tract. Phonemes associated with the vocal cord are referred to as voiced speech.

Single shot turbulence from obstructed air flow through the vocal tract is primarily generated by the teeth, tongue and lips. Phonemes associated with non-periodic obstructed air flow are referred to as unvoiced speech.

Taken from http://www.kt.tu-cottbus.de/speech-analysis/

Quasi-Periodic

Pulsed Air

Vocal Tract

Filter

Vocal Radiator

Air Burst or Continuous flow

Unvoiced Speech

Speech Production ModelsThe general speech model:

Sources can be modeled as quasi periodic impulse trains or random sequences of impulses.

Vocal tract filter can be modeled as an all-pole filter related to the tract resonances.

The radiator can be modeled as a simple gain with spatial direction (possibly some filtering)

Vocal Tract Resonances

First 3 resonances of tube with 1 closed end

Vocal tract length corresponds to signal wavelength (). It can be obtained from resonant frequencies (f ) estimated from recorded speech soundsand the speed of sound (c), using equation:

1/4 Wavelength

3/4 Wavelength

5/4 Wavelength

Image adapted from:hyperphysics.phy-astr.gsu.edu

Vocal Tract Resonances

The resonances of the vocal tract are called formants and

can be estimated from peaks of the spectrum where the effects

of pitch have been smoothed out (i.e. spectral envelope).

Low Order AR Modeling

If the voiced speech is characterized by an all pole model with low order (i.e. about 10 for sampling rate of 8kHz), then the pole frequencies correspond to the resonances of the vocal tract:

The above transfer function can represent a filter that computes the error between the current sample and the sample predicted from previous samples. Therefore, it is call a prediction error filter.

Example

Create an “auh” sound (as the “a” in about or “u” in hum) and use the (linear prediction coefficient) LPC command to model this sound being generated from a quasi-periodic sequence of impulses exciting an all pole filter.

The LPC command finds a vector of filter coefficients such that prediction error is minimized.

Predict x(n) from previous samples:

Compute prediction error sequence with:

Use Z-transforms to find transfer function of filter that recovers x(n) from the LPCs and error sequence e(n).

LPC Derivation

Derive an algorithm to compute LPC coefficients from a stream of data that minimizes the mean squared prediction error.

Let be the sequence of data points and

be the Mth order LPC coefficients, and be the prediction estimate.

The mean squared error for the prediction is given by:

LPC Computation

Put prediction equations in matrix form:

Each row of is a prediction of the corresponding sample in

LPC Computation

The mean squared error can be expressed as:

If derivative is taken with respect to a and set equal to 0, the result is:

LPC Computation

Transpose of the data matrix times itself results in the autocorrelation matrix:

The data matrix transpose times the future (p-vector) values become a sequence of autocorrelation values starting with the first lag:

Autocorrelation and LPC

Define the autocorrelation of a sequence as:

Note that the LPC coefficients are computed from the autocorrelation coefficients:

Autocorrelation Matrix

Script for Analysis

winlens = 50; %PSD window length in milliseconds

[y,fs] = wavread(\'../data/aaa3.wav\'); % Read in wavefile

winlen = winlens*fs/1000;

[cb,ca] = butter(5,2*100/fs,\'high\'); % Filter to remove LF recording noise

yf = filtfilt(cb,ca,y);

[a,er] = lpc(yf,10); % Compute LPC coefficient with model order 10

predy = filter(a,1,yf); % Compute prediction error with all zero filter

kd=1; % Starting figure number

figure(kd) ; plot(predy); hold on; plot(yf,\'g\'); hold off; title(\'Prediction error\'); xlabel(\'Samples\'); ylabel(\'Amplitude\')

recon = filter(1,a,predy); % Compute reconstructed signal from error and all-pole filter

figure(kd+1) % Plot reconstructed signal

plot(recon,\'b\')

hold on

% Plot with original delayed by a unit so it does not entirely overlap the perfectly reconstructed signal

plot(yf(2:end),\'r\')

hold off

xlabel(\'Samples\'); ylabel(\'Amplitude\')

title(\'Reconstructed Signal (blue) and Original (red)\')

% By examining a the error sequence, generate a simple impulse sequence to simulate its period (about 103 sample period)

g = [];

for k=1:150

g = [g, 1, zeros(1,55)];

end

Script for Analysis

% Run simulated error sequence through all pole filter

sim = filter(1,a,g);

soundsc([(sim\')/std(sim); zeros(fix(fs)*1,1); yf/std(yf)],fs)

% Plot pole zero diagram

figure(kd+2)

r = (roots(a))

w = [0:.001:2*pi];

plot(real(r),imag(r),\'xr\',real(exp(j*w)),imag(exp(j*w)),\'b\')

title(\'Pole diagram of vocal tract filter\')

xlabel(\'Real\'); ylabel(\'Imaginary\')

% Find resonant frequencies corresponding to poles

froots = (fs/2)*angle(r)/pi;

nf = find(froots > 0 & froots < fs/2); % Find those corresponding to complex conjugate poles

figure(kd+3)

% Examine average specturm with formant frequencies

[pd,f] = pwelch(yf,hamming(winlen),fix(winlen/2),2*winlen,fs);

dbspec = 20*log10(pd);

mxp = max(dbspec); % Find max and min points for graphing verticle lines

mnp = min(dbspec);

plot(f,dbspec,\'b\') % Plot PSD

hold

Script for Analysis

% Over lines on plot where formant frequencies were estimated from LPCs

for k=1:length(nf)

plot([froots(nf(k)), froots(nf(k))], [mnp(1), mxp(1)], \'k--\')

end

hold off

title(\'PSD plot with formant frequencies (Black broken lines)\')

xlabel(\'Hertz\')

ylabel(\'dB\')

% Get spectrum from the AR (LPC) parameters

[hz,fz] = freqz(1, a, 1024, fs);

figure(kd+4)

plot(fz,abs(hz))

title(\'Spectrum Generated by LPCs\')

xlabel(\'Hertz\')

ylabel(\'Amplitude\')

LPC Analysis Result

Pole Frequencies of LPC model from vocal tract shape

Frequency periodicities from harmonics of Pitch frequency

+

+

+

z-1

z-1

Vocal Tract Filter ImplementationsLattice implementation are popular because of good numerical error and stability properties. The filter is implement in modular stages with coefficients directly related to stability criterion and tube resonances of the vocal tract (example of 2nd order system):

Example

- Record a neutral vowel sound, estimate the formant frequencies, and estimate the size of the vocal tract based on a 345 m/s speed of sound and assume an open-at-one-end tube model.
- Use LPCs estimated from the neutral vowel sound, to filter another sample of speech from the same speaker. Use it as an all zero filter and then as an all pole filter. Listen to the sound and describe what is happening.
- Convert the LPC coefficients for all-pole filter into a second order section and implement filter. Describe advantages of this approach.
- Modify the filter by maintaining the angle of the poles/zeros but move their magnitudes closer to the unit circle. Listen to the sound and explain what is happening.

Homework (1)

- Record a free vowel sound and estimate the size of your vocal tract based on the formant frequencies.
- Compute the LPCs from a free vowel sound and use the LPCs to filter another segment of speech with –10dB of white noise added. Use the LPCs as an all-zero filter and as an all-pole filter. Describe the sound of the filtered outputs and explain what is happening between the 2 filters.
- Move the poles and zeros further away from the unit circles and repeat part b). Describe the effect on the filtered sound when pole and zeros are moved away from the unit circle. Submit this description and the mfiles used to process the data.

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