## SPpcXlsf

#### Routine

void SPpcXlsf (const float pc[], float lsf[], int Np)

#### Purpose

Convert predictor coefficients to line spectral frequencies

#### Description

The transfer function of the prediction error filter is formed from the predictor coefficients. This polynomial is transformed into two reciprocal polynomials having roots on the unit circle. The roots of these polynomials interlace. The line spectral frequencies are the angles corresponding to these roots. The two reciprocal polynomials are expressed as series expansions in Chebyshev polynomials with real roots in the range -1 to +1. The inverse cosine of the roots of the Chebyshev polynomial expansion gives the line spectral frequencies.

If Np line spectral frequencies are not found, this routine stops with an error message. This error occurs if the input coefficients do not give a prediction error filter with minimum phase or the line spectral frequencies are too close together.

The LSF search algorithm uses an initial coarse search with a resolution of 0.01 pi (corresponding to 60 Hz for a 8000 Hz sampling rate). This search alternates between the two Chebyshev polynomials. Roots may be missed if two roots of the same polynomial are spaced closer than 0.01 pi apart. The LSF's correspond to roots taken alternately from the two polynomials. Then LSF's may be missed if two non-adjacent LSF's are separated by less than 0.01 pi. The present resolution is adequate for predictor coefficients derived from speech signals.

Line spectral frequencies and predictor coefficients are usually expressed algebraically as vectors with one-offset indexing. The correspondence to the zero-offset C-arrays is as follows.

      l(1) <==> lsf[0]       first (lowest frequency) line spectral frequency
      l(i) <==> lsf[i-1]     1 <= i < Np
      p(1) <==> pc[0]        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[0] being the predictor coefficient corresponding to lag 1, and pc[Np-1] corresponding to lag Np. The predictor coefficients must correspond to a minimum phase prediction error filter.
<- float lsf[]
  Array of Np line spectral frequencies (in ascending order). Each line spectral frequency lies in the range 0 to pi.
-> int Np
  Number of coefficients (at most 50)

#### Author / revision

P. Kabal / Revision 1.33 2003/05/09

#### See Also

SPpcXcep, SPpcXcor, SPpcXec, SPpcXrc

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