## SPpcXcep

#### Routine

void SPpcXcep (const float pc[], int Np, float cep[], int Ncep)

#### Purpose

Convert predictor coefficients to cepstral coefficients

#### Description

This routine calculates cepstral coefficients corresponding to a set of predictor coefficients. Consider the prediction error filter A(z),

```              Np       -k
A(z) = 1 - SUM p(k) z  .
k=1
```

The power spectrum corresponding to the all-pole LPC filter is

```  S(w) = g^2 / |A(w)|^2 ,
```

where A(w) is short-hand notation for A(exp(jw)). The power spectrum is the Fourier transform of the (infinite) set of autocorrelation coefficients The cepstrum for the autocorrelation coefficients is given by the inverse Fourier transform of the log power spectrum. Equivalently the cepstral coefficients are the Fourier series coefficients of the (periodic) log power spectrum,

```  ln[S(w)] = 2 ln[g] - ln[|A(w)|^2]
```

```              inf
=  SUM  c(k) exp(-jkw) .
k=-inf
```

The cepstral coefficients are symmetric, c(-k) = c(k). For a minimum-phase A(z), the integral of ln[|A(w)|^2] is zero. The coefficient c(0) is then the average of the log power spectrum, c(0) = 2 ln[g]. In this routine, g is assumed to be unity, giving c(0) = 0. An expression for the other cepstral coefficients is obtained by expanding the ln[A(w)] in a Laurent series and taking the derivatives. This gives the recursion

```                1  n-1
c(n) = p(n) + -  SUM (n-k) c(n-k) p(k),  n=1,2,... .
n  k=1
```

This routine returns the first Ncep cepstral coefficients.

Reference:
J. D. Markel and A. H. Gray, Jr., "Linear Prediction of Speech", Springer-Verlag, 1976.

Predictor coefficients are usually expressed algebraically as vectors with 1-offset indexing. The correspondence to the 0-offset C-arrays is as follows.

```  p(1) <==> pc       predictor coefficient corresponding to lag 1
p(i) <==> pc[i-1]     1 <= i < Np
```

#### Parameters

-> const float pc[]
Vector of predictor coefficients (Np values). These are the coefficients of the predictor filter, with pc being the predictor coefficient corresponding to lag 1, and pc[Np-1] corresponding to lag Np.
-> int Np
Number of predictor coefficients
<- float cep[]
Cepstral coefficients (Ncep values). The first cepstral coefficient cep corresponds to the zero quefrency term and is always returned with the value zero.
-> int Ncep
Number of cepstral coefficients (including cep)

#### Author / revision

P. Kabal / Revision 1.9 2003/05/09