## 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%.

```

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

```

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

```

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

```

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

```

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

```

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

```

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

```
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

```

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

```
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

```