functions in filter.i -

             butter(np, w)  
          or butter(np, w, wc, db)  
 
     return frequency response (amplitude) for Butterworth filter;  
     the parameters are the same as for fil_butter.  

             cauer(np, ripple, atten, w)  
          or cauer(np, ripple, atten, w, wc, db)  
 
     return frequency response (amplitude) for Cauer filter;  
     the parameters are the same as for fil_cauer.  

             cheby1(np, ripple, w)  
          or cheby1(np, ripple, w, wc, db)  
 
     return frequency response (amplitude) for Chebyshev filter;  
     the parameters are the same as for fil_cheby1.  

             cheby2(np, atten, w)  
          or cheby2(np, atten, w, wc, db)  
 
     return frequency response (amplitude) for inverse Chebyshev filter;  
     the parameters are the same as for fil_cheby2.  

             fft_good(n)  
 
     returns the smallest number of the form 2^x*3^y*5^z greater  
     than or equal to n.  An fft of this length will be much faster  
     than a number with larger prime factors; the speed difference  
     can be an order of magnitude or more.  
     For n>100, the worst cases result in a little over a 11% increase  
     in n; for n>1000, the worst are a bit over 6%; still larger n are  
     better yet.  The median increase for n<=10000 is about 1.5%.  

             fil_analyze, filt, poles, zeroes  
 
     given a FILT, return the complex POLES and ZEROES, sorted in  
     order of increasing imaginary part.  The real parts of POLES will  
     all be negative if the FILT is stable.  

             filt= fil_bessel(np, wc, db)  
 
     returns the lowpass Bessel filter with NP poles, normalized  
     such that at angular frequency WC, the attenuation is DB decibels.  
     (That is, the attenuation factor is 10^(.05*DB) at WC,  
      so that to_db(response(filt,WC))==DB.)  
     A Bessel filter has the most nearly constant group delay time  
     d(phase)/dw of any filter of the same order.  It minimizes pulse  
     distortion, but does not cut off very rapidly in frequency.  
     If WC is nil or zero, it defaults to 1.0.  
     If DB is nil, the filter is normalized such that both the s^0  
     and s^NP terms are 1, unless the natural= keyword is non-zero,  
     in which case the filter is normalized such that the group delay  
     d(phase)/dw is -1 at w=0.  

             filt= fil_butter(np, wc, db)  
 
     returns the lowpass Butterworth filter with NP poles, normalized  
     such that at angular frequency WC, the attenuation is DB decibels.  
     (That is, the attenuation factor is 10^(.05*DB) at WC,  
      so that to_db(response(filt,WC))==DB.)  
     A Butterworth filter is the best Taylor series approximation to  
     the ideal lowpass filter (a step in frequency) response at both  
     w=0 and w=infinity.  
     For wc=1 and db=10*log10(2), the square of the Butterworth frequency  
     response is 1/(1+w^(2*np)).  
     If WC is nil or zero, it defaults to 1.0.  
     If DB is nil, the filter is normalized "naturally", which is the  
     same as DB=10*log10(2).  

             filt= fil_cauer(np, ripple, atten, wc, db)  
          or filt= fil_cauer(np, ripple, -skirt, wc, db)  
 
     returns the lowpass Cauer (elliptic) filter with NP poles, passband  
     ripple RIPPLE and stopband attenuation ATTEN decibels, normalized  
     such that at angular frequency WC, the attenuation is DB decibels.  
     (That is, the attenuation factor is 10^(.05*DB) at WC,  
      so that to_db(response(filter,WC))==DB.)  
     If the third parameter is negative, its absolute value is SKIRT,  
     the ratio of the frequency at which the stopband attenuation is  
     first reached to the frequency at which the passband ends (where  
     the attenuation is RIPPLE).  The closer to 1.0 SKIRT is, the  
     smaller the equivalent ATTEN would be.  The external variable  
     cauer_other is set to ATTEN if you provide SKIRT, and to SKIRT  
     if you provide ATTEN.  
     The Cauer filter has NP zeroes as well as NP poles.  
     Consider the four parameters: (1) filter order, (2) transition  
     ("skirt") bandwidth, (3) passband ripple, and (4) stopband ripple.  
     Given any three of these, the Cauer filter minimizes the fourth.  
     If WC is nil or zero, it defaults to 1.0.  
     If DB is nil, the filter is normalized "naturally", which is the  
     same as DB=RIPPLE.  

             filt= fil_cheby1(np, ripple, wc, db)  
 
     returns the lowpass Chebyshev type I filter with NP poles, and  
     passband ripple RIPPLE decibels, normalized such that at  
     angular frequency WC, the attenuation is DB decibels.  
     (That is, the attenuation factor is 10^(.05*DB) at WC,  
      so that to_db(response(filter,WC))==DB.)  
     A Chebyshev type I filter gives the smallest maximum error over the  
     passband for any filter that is a Taylor series approximation to  
     the ideal lowpass filter (a step in frequency) response at  
     w=infinity.  It has NP/2 ripples of amplitude RIPPLE in its passband,  
     and a smooth stopband.  
     For wc=1 and db=ripple, the square of the Chebyshev frequency  
     response is 1/(1+eps2*Tnp(w)), where eps2 = 10^(ripple/10)-1,  
     and Tnp is the np-th Chebyshev polynomial, cosh(np*acosh(x)) or  
     cos(np*acos(x)).  
     If WC is nil or zero, it defaults to 1.0.  
     If DB is nil, the filter is normalized "naturally", which is the  
     same as DB=RIPPLE.  

             filt= fil_cheby2(np, atten, wc, db)  
 
     returns the lowpass Chebyshev type II filter with NP poles, and  
     stopband attenuation ATTEN decibels, normalized such that at  
     angular frequency WC, the attenuation is DB decibels.  
     (That is, the attenuation factor is 10^(.05*DB) at WC,  
      so that to_db(response(filter,WC))==DB.)  
     This is also called an inverse Chebyshev filter, since its poles  
     are the reciprocals of a Chebyshev type I filter.  It has NP zeroes  
     as well as NP poles.  
     A Chebyshev type II filter gives the smallest maximum leakage over  
     the stopband for any filter that is a Taylor series approximation to  
     the ideal lowpass filter (a step in frequency) response at  
     w=0.  It has NP/2 ripples of amplitude ATTEN in its stopband,  
     and a smooth passband.  
     For wc=1 and db=ripple, the square of the inverse Chebyshev frequency  
     response is 1 - 1/(1+eps2*Tnp(1/w)), where eps2 = 10^(ripple/10)-1 =  
     1/(10^(atten/10)-1) and Tnp is the np-th Chebyshev polynomial,  
     cosh(np*acosh(x)) or cos(np*acos(x)).  
     If WC is nil or zero, it defaults to 1.0.  
     If DB is nil, the filter is normalized "naturally", which is the  
     same as DB=ATTEN.  

             fil_delay(filt)  
          or fil_delay(filt, 1)  
 
     return the group delay d(phase)/dw at w=0 (zero frequency) for  
     filter FILT.  By default, FILT is assumed to be normalized   
     to an angular frequency (e.g.- radians per second), but if  
     the 2nd parameter is non-nil and non-0 FILT is assumed to be  
     normalized to a circular frequency (e.g.- Hz or GHz).  

             filt= fil_make(poles, zeroes)  
 
     given the complex POLES and ZEROES, return a FILT.  The real  
     parts of POLES must all be negative to make a stable FILT.  
     Both POLES and ZEROES must occur in conjugate pairs in order to  
     make a real filter (the returned filter is always real).  
     The returned filter always has a0=1 (its DC gain is 1).  

             fil_poly(c, x)  
 
     return c(1) + c(2)*x + c(3)*x^2 + c(4)*x^3 + ...  

             fil_response(filt, w)  
 
     return the complex response of FILT at the frequencies W.  
     The frequency scale for W depends on how FILT has been scaled;  
     filters are rational functions in W.  
     The to_db and to_phase functions may be useful for extracting  
     the attenuation and phase parts of the complex response.  

             filter(filt, dt, signal)  
 
     apply the filter FILT to the input SIGNAL, which is sampled  
     at times spaced by DT.  The filter is assumed to be normalized  
     to an angular frequency (e.g.- radians per second), unless  
     DT<0, in which case FILT is assumed to be normalized to a  
     circular frequency (e.g.- Hz or GHz).  
     The result will have the same length as SIGNAL; be sure to pad  
     SIGNAL if you need the response to go beyond that time, or  
     you can use the pad=n keyword to force the returned result to  
     have N samples more than SIGNAL.  
     If the shift= keyword is non-nil and non-0, then the result  
     is shifted backward in time by the filter group delay at  
     zero frequency.  
     The impulse response of the FILT is also assumed to be shorter  
     than the duration of signal, and SIGNAL is assumed to be sampled  
     finely enough to resolve the FILT impulse response.  
     FILT is an array of double, which represents a filter with  
     a particular finite list of zeroes and poles.  See the specific  
     functions to construct filters from poles and zeroes (fil_make),  
     or classic Bessel, Butterworth, Chebyshev, inverse Chebyshev, or  
     Cauer (elliptic) designs.  With fil_analyze, you can find the  
     poles and zeroes of a FILT.  The format for FILT is:  
     FILT is an array of double with the following meanings:  
       FILT(1) = np = number of poles  (integer >= 0)  
       FILT(2) = nz = number of zeroes (integer >= 0)  
       FILT(3) = reserved  
       FILT(4:4+nz) = coefficients for numerator  
                 = [a0, a1, a2, a3, ..., anz]  
       FILT(5+nz:4+nz+np) = coefficents for denominator (if np>0)  
                 = [b1, b2, b3, ..., bnp]  
     The Laplace transform (s-transform) of the filter response is  
       L[FILT] = (a0 + a1*s + a2*s^2 + a3*s^3 + ...) /  
                 ( 1 + b1*s + b2*s^2 + b3*s^3 + ...)  

             to_db(signal, ref)  
          or to_db(signal)  
 
     return 20.*log10(abs(SIGNAL)/REF), the number of decibels  
     corresponding to the input SIGNAL.  REF defaults to 1.0.  

             to_phase(signal)  
          or to_phase(signal, 1)  
 
     return atan(SIGNAL.im,SIGNAL.re), the phase of the input SIGNAL.  
     If the second argument is present and non-0, the phase will be in  
     degrees; by default the phase is in radians.  
     To_phase attempts to unroll any jumps from -180 to +180 degrees  
     or vice-versa; zero phase will be taken somewhere near the middle  
     of the signal.  The external variable to_phase_eps controls the  
     details of this unrolling; you can turn off unrolling by setting  
     to_phase_eps=0.0 (initially it is 0.3).