Convert linear prediction coefficients to reflection coefficients or reflection coefficients to linear prediction coefficients

Estimation / Linear Prediction

`dsplp`

The LPC to/from RC block either converts linear prediction coefficients
(LPCs) to reflection coefficients (RCs) or reflection coefficients
to linear prediction coefficients. Set the **Type of conversion** parameter
to `LPC to RC`

or `RC to LPC`

to
select the domain into which you want to convert your coefficients.
The A port corresponds to LPC coefficients, and the K port corresponds
to the RC coefficients. For more information, see Algorithm.

The block input can be an *N*-by-*M* matrix
or an unoriented vector. Each column of the matrix is treated as
a channel. When the input is an unoriented vector, the input is treated
as one channel.

Consider a signal *x* (*n*)
as the input to an FIR analysis filter represented by LPC coefficients.
The output of the this analysis filter, *e* (*n*),
is known as the prediction error signal. The power of this error signal
is denoted by *P*. When the zero lag autocorrelation
coefficient of *x* (*n*) is one,
the autocorrelation sequence and prediction error power are said to
be normalized.

Select the **Output normalized prediction error power** check
box to enable port P. The normalized prediction error power output
at `P`

is a vector with one element per input channel.
Each element varies between zero and one.

Select the **Output LPC filter stability** check
box to output the stability of the filter represented by the LPCs
or RCs. The synthesis filter represented by the LPCs is stable when
the absolute value of each of the roots of the LPC polynomial is less
than one. The lattice filter represented by the RCs is stable when
the absolute value of each reflection coefficient is less than 1.
When the filter is stable, the block outputs a Boolean value of `1`

for
each input channel at the S port. When the filter is unstable, the
block outputs a Boolean value of `0`

for each input
channel at the S port.

**If first input value is not 1** parameter
specifies the behavior of the block when the first coefficient of
the LPC coefficient vector in any channel is not 1. The following
options are available:

`Replace it with 1`

— Changes the first value of the coefficient channel to 1. The other coefficient values are unchanged.`Normalize`

— Divides the entire channel of coefficients by the first coefficient so that the first coefficient of the LPC coefficient vector is 1.`Normalize and Warn`

— Divides the entire channel of coefficients by the first coefficient so that the first coefficient of the LPC coefficient vector is 1. The block displays a warning message telling you that your vector of coefficients has been normalized.`Error`

— Displays an error telling you that the first coefficient of the LPC coefficient channel is not 1.

When in this mode, this block uses backward Levinson recursion to convert linear prediction coefficients (LPCs) to reflection coefficients (RCs). For a given Nth order LPC vector $$LP{C}_{N}=\left[\begin{array}{ccccc}1& {a}_{N1}& {a}_{N2}& \dots & {a}_{NN}\end{array}\right]$$, the block calculates the Nth reflection coefficient value using the formula $${\gamma}_{N}=-{a}_{NN}$$. The block then finds the lower order LPC vectors , $$LP{C}_{N-1},\text{}LP{C}_{N-2},\text{}\mathrm{...}\text{,}LP{C}_{1}$$, using the following recursion.

for *p* = *N*, *N* –
1, ..., 2,

$$\begin{array}{l}{\gamma}_{p}={a}_{pp}\hfill \\ F=1-{\gamma}_{p}{}^{2}\hfill \\ {a}_{p-1,m}=\frac{{a}_{p,m}}{F}-\frac{{\gamma}_{p}{a}_{p,p-m}}{F},\text{}1\le mp\hfill \end{array}$$

end

Finally, $${\gamma}_{1}=-{a}_{11}$$. The reflection coefficient vector is $$\left[\begin{array}{cccc}{\gamma}_{1},& {\gamma}_{2},& \dots ,& {\gamma}_{N}\end{array}\right]$$.

When in this mode, this block uses Levinson recursion to convert reflection coefficients (RCs) to linear prediction coefficients (LPCs). In this case, the input to the block is $$RC=\left[\begin{array}{cccc}{\gamma}_{1}& {\gamma}_{2}& \mathrm{...}& {\gamma}_{N}\end{array}\right]$$. The zeroth order LPC vector term is 1. Starting with this term, the block uses recursion to calculate the higher order LPC vectors, $$LP{C}_{2},LP{C}_{3},\mathrm{...}LP{C}_{N}$$, until it has calculated the entire LPC matrix.

$$LP{C}_{matrix}=\left[\begin{array}{c}LP{C}_{0}\\ LP{C}_{1}\\ LP{C}_{2}\\ \cdots \\ LP{C}_{N}\end{array}\right]=\left[\begin{array}{cccccc}1& 0& 0& 0& \cdots & 0\\ 1& {a}_{11}& 0& 0& \cdots & 0\\ 1& {a}_{21}& {a}_{22}& 0& \cdots & 0\\ 1& {a}_{31}& {a}_{32}& {a}_{33}& \cdots & 0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 1& {a}_{N1}& {a}_{N2}& {a}_{N3}& \cdots & {a}_{NN}\end{array}\right]$$

This LPC matrix consists of LPC vectors of order 0 through *N* found
by using the Levinson recursion. The following are the formulas for
the recursion steps, for *p* =
0, 1, ...,*N* – 1.

$$\begin{array}{l}{a}_{p+1,m}={a}_{p,m}+{\gamma}_{p+1}{a}_{p,p+1-m},\text{}1\le m\le p\hfill \\ {a}_{p+1,p+1}={\gamma}_{p+1}\hfill \end{array}$$

**Type of conversion**Select

`LPC to RC`

or`RC to LPC`

to select the domain into which you want to convert your coefficients.**Output normalized prediction error power**Select this check box to output the normalized prediction error power at port P.

**Output LPC filter stability**Select this check box to output the stability of the filter. When the filter represented by the LPCs or RCs is stable, the block outputs a Boolean value of

`1`

for each input channel at the S port. When the filter represented by the LPCs or RCs is unstable, the block outputs a Boolean value of`0`

for each input channel at the S port.**If first input value is not 1**Select what you would like the block to do when the first coefficient of the LPC coefficient vector is not 1. You can choose

`Replace it with 1`

,`Normalize`

,`Normalize and Warn`

, and`Error`

.

Makhoul, J *Linear Prediction: A tutorial review.* Proc.
IEEE. 63, 63, 56 (1975).

Markel, J.D. and A. H. Gray, Jr., *Linear Prediction
of Speech.* New York, Springer-Verlag, 1976.

Double-precision floating-point

Single-precision floating-point

Levinson-Durbin | DSP System Toolbox |

LPC to LSF/LSP Conversion | DSP System Toolbox |

LSF/LSP to LPC Conversion | DSP System Toolbox |

LPC/RC to Autocorrelation | DSP System Toolbox |

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