/*! ****************************************************************************
\ingroup IIR_FILTER_DESIGN_GROUP
\fn int  low2high (double* b, double* a, int ord, double w0, double w1,
                   double* beta, double* alpha)
\brief Lowpass to highpass filter frequency transform

Function transforms lowpass filter transfer function \f$ H(s) \f$ 
to the highpass filter transfer function \f$ F(s) \f$.

Filter order, magnitude ripple in passband and stopband 
supression still the same.

\param[in]  b
Pointer to the lowpass filter transfer function \f$H(s)\f$ numerator 
coefficients vector. \n
Vector size is `[ord+1 x 1]`. \n 
\n

\param[in]  a
Pointer to the lowpass filter transfer function \f$H(s)\f$ denominator 
coefficients vector. \n
Vector size is `[ord+1 x 1]`. \n 
\n

\param[in]  ord
Filter order. \n 
\n

\param[in]  w0
Lowpass filter cutoff frequency. \n 
\n

\param[in]  w1
Highpass filter cutoff frequency after transformation. \n 
\n

\param[in,out]  beta
Pointer to the highwpass filter transfer function \f$F(s)\f$ numerator 
coefficients vector after transformation. \n
Vector size is `[ord+1 x 1]`. \n 
Memory must be allocated. \n
\n

\param[in,out]  alpha
Pointer to the highwpass filter transfer function \f$F(s)\f$ denominator 
coefficients vector after transformation. \n
Vector size is `[ord+1 x 1]`. \n 
Memory must be allocated. \n
\n

\return
`RES_OK` if filter coefficients is calculated successfully. \n 
Else \ref ERROR_CODE_GROUP "code error".

\author Sergey Bakhurin www.dsplib.org
***************************************************************************** */







/*! ****************************************************************************
\ingroup IIR_FILTER_DESIGN_GROUP
\fn int  low2low(double* b, double* a, int ord, double w0, double w1,
                 double* beta, double* alpha)

Lowpass to lowpass filter frequency transform

Function transforms lowpass filter transfer function \f$ H(s) \f$ 
to the lowpass filter transfer function \f$ F(s) \f$ 
with other cutoff frequency.

Filter order, magnitude ripple in passband and stopband 
supression still the same.

\param[in]  b
Pointer to the input lowpass filter transfer function \f$H(s)\f$ numerator 
coefficients vector. \n
Vector size is `[ord+1 x 1]`. \n 
\n

\param[in]  a
Pointer to the input lowpass filter transfer function \f$H(s)\f$ denominator 
coefficients vector. \n
Vector size is `[ord+1 x 1]`. \n 
\n

\param[in]  ord
Filter order. \n 
\n

\param[in]  w0
Input lowpass filter cutoff frequency. \n 
\n

\param[in]  w1
Lowpass filter cutoff frequency after transformation. \n 
\n

\param[in,out]  beta
Pointer to the lowpass filter transfer function \f$F(s)\f$ numerator 
coefficients vector after transformation. \n
Vector size is `[ord+1 x 1]`. \n 
Memory must be allocated. \n
\n

\param[in,out]  alpha
Pointer to the lowpass filter transfer function \f$F(s)\f$ denominator 
coefficients vector after transformation. \n
Vector size is `[ord+1 x 1]`. \n 
Memory must be allocated. \n
\n

\return
`RES_OK` if filter coefficients is calculated successfully. \n 
Else \ref ERROR_CODE_GROUP "code error".

\author Sergey Bakhurin www.dsplib.org
***************************************************************************** */









/*! ****************************************************************************
\ingroup IIR_FILTER_DESIGN_GROUP
\fn int ratcompos(  double* b, double* a, int n, 
                    double* c, double* d, int p, 
                    double* beta, double* alpha)  
\brief  Rational composition

Function calcultes composition \f$Y(s) = (H \circ F)(s) = H(F(s))\f$, here

\f[
H(s) = \frac{\sum\limits_{m = 0}^{n} b_m s^m}
{\sum\limits_{k = 0}^{n} a_k s^k}, \quad 
F(s) = \frac{\sum\limits_{m = 0}^{p} d_m s^m}
{\sum\limits_{k = 0}^{p} c_k s^k}, \quad 
Y(s) = \frac{\sum\limits_{m = 0}^{n p} \beta_m s^m}
{\sum\limits_{k = 0}^{n  p} \alpha_k s^k}
\f]

This function is using for filter frequency transform.

\param[in]  b
Pointer to the \f$H(s)\f$ polynomial function 
numerator coefficients vector. \n
Vector size is `[n+1 x 1]`. \n
\n

\param[in]  a
Pointer to the \f$H(s)\f$ polynomial function 
denominator coefficients vector. \n
Vector size is `[n+1 x 1]`. \n
\n

\param[in]  n
Order of \f$H(s)\f$ numerator and denominator polynomials. \n 
\n

\param[in]  c
Pointer to the \f$F(s)\f$ polynomial function 
numerator coefficients vector. \n
Vector size is `[p+1 x 1]`. \n
\n

\param[in]  d
Pointer to the \f$F(s)\f$ polynomial function 
denominator coefficients vector. \n
Vector size is `[p+1 x 1]`. \n
\n

\param[in]  p
Order of \f$F(s)\f$ numerator and denominator polynomials. \n 
\n

\param[in,out]  beta
Pointer to the numerator coefficients vector of 
\f$Y(s) = (H \circ F)(s)\f$. \n
Vector size is `[n*p+1 x 1]`. \n
Memory must be allocated. \n 
\n

\param[in,out]  alpha
Pointer to the denominator coefficients vector of 
\f$Y(s) = (H \circ F)(s)\f$. \n
Vector size is `[n*p+1 x 1]`. \n
Memory must be allocated. \n 
\n


\return
`RES_OK` if rational composition is calculated successfully. \n 
Else \ref ERROR_CODE_GROUP "code error".

\author Sergey Bakhurin www.dsplib.org
***************************************************************************** */