Determine coefficients of Nth-order forward linear predictors
The Autocorrelation LPC block determines the coefficients of an N-step forward linear predictor for the time-series in each length-M input channel, u, by minimizing the prediction error in the least squares sense. A linear predictor is an FIR filter that predicts the next value in a sequence from the present and past inputs. This technique has applications in filter design, speech coding, spectral analysis, and system identification.
The Autocorrelation LPC block can output the prediction error for each channel as polynomial coefficients, reflection coefficients, or both. It can also output the prediction error power for each channel. The input u can be a scalar, unoriented vector, column vector, sample-based row vector, or a matrix. Frame-based row vectors are not valid inputs. The block treats all M-by-N matrix inputs as N channels of length M.
When you select Inherit prediction order from input dimensions, the prediction order, N, is inherited from the input dimensions. Otherwise, you can use the Prediction order parameter to specify the value of N. Note that N must be a scalar with a value less than the length of the input channels or the block produces an error.
When Output(s) is set to A, port A is enabled. For each channel, port A outputs an (N+1)-by-1 column vector, a = [1 a2 a3 ... aN+1]T, containing the coefficients of an Nth-order moving average (MA) linear process that predicts the next value, ûM+1, in the input time-series.
When Output(s) is set to K, port K is enabled. For each channel, port K outputs a length-N column vector whose elements are the prediction error reflection coefficients. When Output(s) is set to A and K, both port A and K are enabled, and each port outputs its respective set of prediction coefficients for each channel.
When you select Output prediction error power (P), port P is enabled. The prediction error power is output at port P as a vector whose length is the number of input channels.
The Autocorrelation LPC block computes the least squares solution to
where indicates the 2-norm and
Solving the least squares problem via the normal equations
leads to the system of equations
where r = [r1 r2 r3 ... rn+1]T is an autocorrelation estimate for u computed using the Autocorrelation block, and * indicates the complex conjugate transpose. The normal equations are solved in O(n2) operations by the Levinson-Durbin block.
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.
The type of prediction coefficients output by the block. The block can output polynomial coefficients (A), reflection coefficients (K), or both (A and K).
When selected, enables port P, which outputs the output prediction error power.
When selected, the block inherits the prediction order from the input dimensions.
Specify the prediction order, N, which must be a scalar. This parameter is disabled when you select the Inherit prediction order from input dimensions parameter.
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.