Determine the coefficients of an FIR filter that predicts the next value in a sequence from the present and past inputs.
Library
Signal Operations, in General DSP
Description
The LPC block performs linear predictive coding, which determines the coefficients of an FIR filter that predicts the next value in a sequence from the present and past inputs. This type of filter is also known as a one-step forward linear predictor. This technique has applications in filter design, speech coding, spectral analysis, and system identification.
At the top port, the LPC block outputs the coefficients of an nth-order moving average linear process that models the input vector (time series x) as
where n is specified by the Prediction order parameter. The filter coefficients are output in vector a = [1 a(2) ... a(n+1)]. At the bottom port the block outputs the square root of the minimum error energy, g:
If a value of -1 is specified for the Prediction order parameter, the block uses length(x)-1 for n.
A matrix input, x, is treated as a vector, x(:).
Algorithm
The LPC block computes the least-squares solution to
where 
indicates the 2-norm and
m is the length of x. Solving the least-squares problem via the normal equations
leads to the system of equations
where r = [r(1) r(2) ... r(n+1)] is an autocorrelation estimate for x computed using the Autocorrelation block. The normal equations are solved in O(n2) flops by the Levinson-Durbin block. The square root of the minimum error energy is given by
Note that the solution to the LPC problem is very closely related to the Yule-Walker AR method of spectral estimation. In that context, the normal equations above are referred to as the Yule-Walker AR equations.
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Real
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Complex
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Scalar
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Vector
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Matrix
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Sample Time
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Scalar Expansion
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Input
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inherited from input
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Output
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Dialog Box
References
Haykin, S. Adaptive Filter Theory. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1996.
Ljung, L. System Identification: Theory for the User. Englewood Cliffs, NJ: Prentice Hall, 1987. Pgs. 278-280.
Proakis, J. and D. Manolakis. Digital Signal Processing. 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996.
See Also
Complex LPC
Levinson-Durbin
Yule-Walker AR
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