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# functions in filter.i - f

 fft_good ``` 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%. ``` SEE ALSO: fft,   fft_setup,   convol

 fil_analyze ``` 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. interpreted function, defined at i/filter.i line 160 ``` SEE ALSO: filter,   fil_make

 fil_bessel ``` 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. interpreted function, defined at i/filter.i line 178 ``` SEE ALSO: filter,   fil_analyze

 fil_butter ``` 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). interpreted function, defined at i/filter.i line 226 ``` SEE ALSO: filter,   fil_analyze,   butter

 fil_cauer ``` 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. interpreted function, defined at i/filter.i line 357 ``` SEE ALSO: filter,   fil_analyze,   cauer

 fil_cheby1 ``` 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. interpreted function, defined at i/filter.i line 263 ``` SEE ALSO: filter,   fil_analyze,   cheby1

 fil_cheby2 ``` 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. interpreted function, defined at i/filter.i line 305 ``` SEE ALSO: filter,   fil_analyze,   cheby2

 fil_delay ``` 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). interpreted function, defined at i/filter.i line 93 ``` SEE ALSO: filter,   fil_butter,   fil_bessel,   fil_cheby1,   fil_cheby2,   fil_response,   to_db,   to_phase

 fil_make ``` 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). interpreted function, defined at i/filter.i line 126 ``` SEE ALSO: filter,   fil_analyze

 fil_normalize ``` fil_normalize interpreted function, defined at i/filter.i line 473 ```

 fil_poly ``` fil_poly(c, x) return c(1) + c(2)*x + c(3)*x^2 + c(4)*x^3 + ... interpreted function, defined at i/filter.i line 115 ```

 fil_response ``` 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. interpreted function, defined at i/filter.i line 72 ``` SEE ALSO: filter,   fil_butter,   fil_bessel,   fil_cheby1,   fil_cheby2,   fil_delay,   to_db,   to_phase

 filter ``` 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 + ...) interpreted function, defined at i/filter.i line 8 ``` SEE ALSO: filter,   fil_bessel,   fil_butter,   fil_cheby1,   fil_cheby2,   fil_cauer,   fil_response,   fil_make,   fil_analyze,   to_db,   to_phase