Determine coefficients of Nth-order forward linear predictors

Estimation / Linear Prediction

`dsplp`

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 *a _{2}*

$${\widehat{u}}_{M+1}=-\left({a}_{2}{u}_{M}\right)-\left({a}_{3}{u}_{M-1}\right)-\mathrm{...}-\left({a}_{N+1}{u}_{M-N+1}\right)$$

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

$$\underset{i\in {\Re}^{n}}{\mathrm{min}}\Vert U\tilde{a}-b\Vert $$

where $$\Vert \cdot \Vert $$ indicates the 2-norm and

$$U=\left[\begin{array}{cccc}{u}_{1}& 0& \cdots & 0\\ {u}_{2}& {u}_{1}& \ddots & \vdots \\ \vdots & {u}_{2}& \ddots & 0\\ \vdots & \vdots & \ddots & {u}_{1}\\ \vdots & \vdots & \vdots & {u}_{2}\\ \vdots & \vdots & \vdots & \vdots \\ {u}_{M}& \vdots & \vdots & \vdots \\ 0& \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & \vdots \\ 0& \cdots & 0& {u}_{M}\end{array}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\text{}\tilde{a}=\left[\begin{array}{c}{a}_{2}\\ \vdots \\ {a}_{n}+1\end{array}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}b=\left[\begin{array}{c}{u}_{2}\\ {u}_{3}\\ \vdots \\ {u}_{M}\\ 0\\ \vdots \\ 0\end{array}\right]$$

Solving the least squares problem via the normal equations

$${U}^{\ast}U\tilde{a}={U}^{\ast}b$$

leads to the system of equations

$$\left[\begin{array}{cccc}{r}_{1}& {r}_{2}^{\ast}& \cdots & {r}_{n}^{\ast}\\ {r}_{2}& {r}_{1}& \ddots & \vdots \\ \vdots & \ddots & \ddots & {r}_{2}^{\ast}\\ {r}_{n}& \cdots & {r}_{2}& {r}_{1}\end{array}\right]\text{\hspace{0.17em}}\left[\begin{array}{c}{a}_{2}\\ {a}_{3}\\ \vdots \\ {a}_{n+1}\end{array}\right]\text{\hspace{0.17em}}=\left[\begin{array}{c}-{r}_{2}\\ -{r}_{3}\\ \vdots \\ -{r}_{n+1}\end{array}\right]$$

where *r* = [*r _{1}*

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.

**Output(s)**The type of prediction coefficients output by the block. The block can output polynomial coefficients (

`A`

), reflection coefficients (`K`

), or both (`A and K`

).**Output prediction error power (P)**When selected, enables port

`P`

, which outputs the output prediction error power.**Inherit prediction order from input dimensions**When selected, the block inherits the prediction order from the input dimensions.

**Prediction order (N)**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.

Double-precision floating point

Single-precision floating point

Autocorrelation | DSP System Toolbox |

Levinson-Durbin | DSP System Toolbox |

Yule-Walker Method | DSP System Toolbox |

`lpc` | Signal Processing Toolbox |

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