/*! ****************************************************************************
\ingroup IIR_FILTER_DESIGN_GROUP
\fn int bilinear(double* bs, double* as, int ord, double* bz, double* az)

\brief
Transform a s-plane analog filter transfer function \f$H(s)\f$ to the
digital filter transfer function \f$H(z)\f$.

Bilinear transform is rational composition:

\f[
s \leftarrow \frac{1 - z^{-1}}{1 - z^{-1}}.
\f]

Digital filter order, passband magnitude ripple and stopband suppression 
still the same after bilinear transform as analog filter. 

Frequency \f$\Omega\f$ of analog filter and frequency  
\f$\omega\f$ of digital filter relations:

\f[
\Omega = \tan(\omega / 2).
\f]


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

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

\param[in]  ord
Analog and digital filters order. \n 
\n

\param[out]  bz
Pointer to the vector of digital filter \f$H(z)\f$ 
numerator coefficients after bilinear transform.
Vector size is `[ord+1 x 1]`. \n
Memory must be allocated. \n 
\n

\param[out]  az
Pointer to the vector of digital filter \f$H(z)\f$ 
denominator coefficients after bilinear transform.
Vector size is `[ord+1 x 1]`. \n
Memory must be allocated. \n 
\n

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


Example:

\include bilinear_test.c

This program calculates the transfer function \f$H(s)\f$ of analog 
Chebyshev filter of the first kind, with a cutoff frequency of 1 rad/s, 
and produces bilinear trandform to digital filter, 
with a normilized cutoff frequency equals 0.5.

Result:

\verbatim
bz[0] =   0.246    az[0] =   4.425
bz[1] =   0.983    az[1] =  -3.318
bz[2] =   1.474    az[2] =   4.746
bz[3] =   0.983    az[3] =  -2.477
bz[4] =   0.246    az[4] =   1.034
err = 0
\endverbatim

In addition, the frequency response of the resulting digital filter 
is calculated and plotted by GNUPLOT package.

\image html bilinear.png

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







/*! ****************************************************************************
\ingroup IIR_FILTER_DESIGN_GROUP
\fn int iir(double rp, double rs, int ord,  double  w0, double  w1, 
            int type, double* b,  double* a)
\brief
Digital IIR filter design.

The function calculates the coefficients of the digital IIR filter 
transfer fucntion \f$ H(z) \f$. 
Filter coeffitients can be used in \ref filter_iir function

\param[in]  rp
Magnitude ripple in passband (dB). \n
\n


\param[in]  rs
Suppression level in stopband (dB). \n
\n

\param[in]  ord
Filter order. \n
Number of \f$H(z)\f$ numerator and denominator coefficients is `ord+1`. \n
For bandpass and bandstop filters `ord` must be even. \n 
\n

\param[in]  w0
Normalized cutoff frequency (from 0 to 1) for lowpass or highpass filter. \n
Or left normalized cutoff frequency (from 0 to 1) for 
bandpass and bandstop filter. \n 
\n


\param[in]  w1
Right normalized cutoff frequency (from 0 to 1) for 
bandpass and bandstop filter. \n 
This parameter is ingnored for lowpass and highpass filters.
\n

\param[in]  type
Filter type. \n
This patameter sets combination of filter type (one of follow): \n
\verbatim
DSPL_FILTER_LPF   - lowpass  filter;
DSPL_FILTER_HPF   - highpass filter;
DSPL_FILTER_BPASS - bandpass filter;
DSPL_FILTER_BSTOP - bandstop filter,
\endverbatim
and of filter approximation type (one of follow):
\verbatim
DSPL_FILTER_BUTTER - Butterworth filter;
DSPL_FILTER_CHEBY1 - Chebyshev of the first kind filter;
DSPL_FILTER_CHEBY2 - Chebyshev of the second kind filter;
DSPL_FILTER_ELLIP  - Elliptic filter.
\endverbatim
\n 
\n

\param[out]  b
Pointer to the transfer function \f$H(z)\f$ 
numerator coefficients vector. \n 
Vector size is  `ord+1`. \n
Memory must be allocated. \n 
\n

\param[out]  a
Pointer to the transfer function \f$H(z)\f$ 
denominator coefficients vector. \n 
Vector size is  `ord+1`. \n
\n

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

Example:

\include iir_test.c

This program calcultes filter coefficients for different flags `type`.

In addition, the filters magnitudes 
is calculated and plotted by GNUPLOT package.

\image html iir_test.png

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