Linear prediction filter coefficients
lpc determines the coefficients of a forward
linear predictor by minimizing the prediction error in the least squares
sense. It has applications in filter design and speech coding.
[a,g] finds the coefficients of
pth-order linear predictor (FIR filter) that
predicts the current value of the real-valued time series
on past samples.
p is the order of the prediction filter polynomial,
[1 a(2) ... a(p+1)]. If
p is unspecified,
as a default
x is a matrix containing a separate signal in
lpc returns a model estimate for each
column in the rows of matrix
a and a column vector
of prediction error variances
g. The length of
be less than or equal to the length of
Estimate a data series using a third-order forward predictor. Compare the estimate to the original signal.
First, create the signal data as the output of an autoregressive process driven by normalized white Gaussian noise. Use the last 4096 samples of the AR process output to avoid startup transients.
noise = randn(50000,1); x = filter(1,[1 1/2 1/3 1/4],noise); x = x(45904:50000);
Compute the predictor coefficients, estimated signal, prediction error, and autocorrelation sequence of the prediction error.
a = lpc(x,3); est_x = filter([0 -a(2:end)],1,x); e = x-est_x; [acs,lags] = xcorr(e,'coeff');
Compare the predicted signal to the original signal.
plot(1:97,x(4001:4097),1:97,est_x(4001:4097),'--'), grid title 'Original Signal vs. LPC Estimate' xlabel 'Sample number', ylabel 'Amplitude' legend('Original signal','LPC estimate')
Plot the autocorrelation of the prediction error.
plot(lags,acs), grid title 'Autocorrelation of the Prediction Error' xlabel 'Lags', ylabel 'Normalized value'
The prediction error is approximately white Gaussian noise, as expected for a third-order AR input process.
The prediction error, e(n), can be viewed as the output of the prediction error filter A(z) shown below, where H(z) is the optimal linear predictor, x(n) is the input signal, and is the predicted signal.
lpc uses the autocorrelation method of autoregressive
(AR) modeling to find the filter coefficients. The generated filter
might not model the process exactly even if the data sequence is truly
an AR process of the correct order. This is because the autocorrelation
method implicitly windows the data, that is, it assumes that signal
samples beyond the length of
x are 0.
lpc computes the least squares solution to
m is the length of
Solving the least squares problem via the normal equations
leads to the Yule-Walker equations
= [r(1) r(2) ... r(p+1)
an autocorrelation estimate for
x computed using
The Yule-Walker equations are solved in O(p2)
flops by the Levinson-Durbin algorithm (see
 Jackson, L. B. Digital Filters and Signal Processing. 2nd Edition. Boston: Kluwer Academic Publishers, 1989, pp. 255–257.