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Normalized Chebyshev Type 1 Lowpass Filter
Syntax fncheb1(Norder, Ap, b, a)
Include: include spt\fncheb1.oms
See Also fnbut , fncheb2 , fnbes

ARGUMENTS:
   INPUTS:
      Norder = SCALAR. Requested order of transfer function. Coerced to
               INTEGER before local processing. Norder >= 2.
      Ap     = SCALAR. Passband ripple in dB, Ap > 0.0 dB. Coerced to
               double for local processing.
      b      = VECTOR, COLUMN, for return (type disregarded on input).
               Numerator polynomial coefficients. Type DOUBLE.
      a      = VECTOR, COLUMN, for return (type disregarded on input).
               Denominator polynomial coefficients. Type DOUBLE.
   RETURN: novalue. Filter functions are returned in 'b' and 'a'.

Description

This function creates a normalized TYPE 1 Chebyshev s-domain lowpass transfer function of the form: H(s) = b(s)/a(s). The 3-dB cutoff frequency is set to 1 radian/sec. The requested filter order 'Norder' must be >= 2. The numerator polynomial of the transfer function is returned through argument 'b', and is a column vector where the elements form a polynomial as follows:

b => b(1) + b(2)*s + b(3)*s^2 + ...

Denominator polynomial is returned in argument 'a', and is of the same form. The function calls SPT function 'fnc1pole()'.

A Chebyshev TYPE 1 filter has equi-ripple behavior in the passband and monotonically increasing attenuation in the stopband. The passband peak-to-peak ripple is set via parameter 'Ap' in dB, where Ap > 0.0dB (strictly positive). Due to the nature of Chebyshev functions, even order filters cannot have a gain of unity at 0 Hz as can odd order filters. Even order filters start at Ap dB down at 0 Hz and begin passband ripple characteristics from there. All filters are passive, i.e., they can a gain of 1.0 at most at any given frequency.

Example

include spt\fncheb1.oms

# Design CHEBYSHEV Type 1 lowpass prototype filter, fc = 1 [radian/sec]
Norder  = 5;                 # Filter Order
Ap      = 2d0;               # Passband Ripple
b       = novalue;           # Declare numerator   polynomial
a       = novalue;           # Declare denominator polynomial
fncheb1(Norder, Ap, b, a);   # Make prototype filter

# Evaluate this filter around its cutoff.
fmin    =  1d-2; # Plotting Limits
fmax    =  1d0;
ymax    =  10d0;
ymin    = -60d0;

N       = 201; # Plotting information
n       = seq(N)'-1d0;
f       = logspace(log10(fmin),log10(fmax),N)';
H       = gains(b,a,f);
HdB     = db20(H);
fc      = 1d0/2/PI; # 3dB down at this cutoff

A plot of the normalized CHEBYSHEV Type 1 filter apperas as:



Reference
Blinchikoff