sftrans  R Documentation 
Transform band edges of a generic lowpass filter to a filter with different band edges and to other filter types (high pass, band pass, or band stop).
## Default S3 method:
sftrans(Sz, Sp, Sg, W, stop = FALSE, ...)
## S3 method for class 'Arma'
sftrans(Sz, W, stop = FALSE, ...)
## S3 method for class 'Zpg'
sftrans(Sz, W, stop = FALSE, ...)
Sz 
In the generic case, a model to be transformed. In the default case, a vector containing the zeros in a polezerogain model. 
Sp 
a vector containing the poles in a polezerogain model. 
Sg 
a vector containing the gain in a polezerogain model. 
W 
critical frequencies of the target filter specified in
radians. 
stop 

... 
additional arguments (ignored). 
Given a low pass filter represented by poles and zeros in the splane, you can convert it to a low pass, high pass, band pass or band stop by transforming each of the poles and zeros individually. The following summarizes the transformations:
LowPass Transform
S > C S/Fc
Zero at x  Pole at x 
zero: F_c x/C  F_c x/C 
gain: C/F_c  F_c/C 
HighPass Transform
S > C F_c/S
Zero at x  Pole at x 
zero: F_c C/x  F_c C/x 
pole: 0  0 
gain: x  1/x 
BandPass Transform
S > C \frac{S^2+F_hF_l}{S(F_hF_l)}
Zero at x  Pole at x 
zero: b \pm \sqrt(b^2F_hF_l)  b \pm \sqrt(b^2F_hF_l) 
pole: 0  0 
gain: C/(F_hF_l)  (F_hF_l)/C 
b = x/C (F_hF_l)/2  b=x/C (F_hF_l)/2 
BandStop Transform
S > C \frac{S(F_hF_l)}{S^2+F_hF_l}
Zero at x  Pole at x 
zero: b \pm \sqrt(b^2F_hF_l)  b \pm \sqrt(b^2F_hF_l) 
pole: \pm \sqrt(F_hF_l)  \pm \sqrt(F_hF_l) 
gain: x  1/x 
b = C/x (F_hF_l)/2  b=C/x (F_hF_l)/2 
Bilinear Transform
S > \frac{2}{T} \frac{z1}{z+1}
Zero at x  Pole at x 
zero: (2+xT)/(2xT)  (2+xT)/(2xT) 
pole: 1  1 
gain: (2xT)/T  (2xT)/T 
where C
is the cutoff frequency of the initial lowpass filter, F_c
is
the edge of the target low/high pass filter and [F_l,F_h]
are the edges
of the target band pass/stop filter. With abundant tedious algebra,
you can derive the above formulae yourself by substituting the
transform for S
into H(S)=Sx
for a zero at x
or H(S)=1/(Sx)
for a
pole at x
, and converting the result into the form:
H(S) = g \mbox{prod}(SXi) / \mbox{prod}(SXj)
Please note that a pole and a zero at the same place exactly cancel. This is significant for High Pass, Band Pass and Band Stop filters which create numerous extra poles and zeros, most of which cancel. Those which do not cancel have a ‘fillin’ effect, extending the shorter of the sets to have the same number of as the longer of the sets of poles and zeros (or at least split the difference in the case of the band pass filter). There may be other opportunistic cancellations, but it does not check for them.
Also note that any pole on the unit circle or beyond will result in an unstable filter. Because of cancellation, this will only happen if the number of poles is smaller than the number of zeros and the filter is high pass or band pass. The analytic design methods all yield more poles than zeros, so this will not be a problem.
For the default case or for sftrans.Zpg
, an object of class
“Zpg”, containing the list elements:
zero 
complex vector of the zeros of the transformed model 
pole 
complex vector of the poles of the transformed model 
gain 
gain of the transformed model 
For sftrans.Arma
, an object of class
“Arma”, containing the list elements:
b 
moving average (MA) polynomial coefficients 
a 
autoregressive (AR) polynomial coefficients 
Original Octave version by Paul Kienzle pkienzle@users.sf.net. Conversion to R by Tom Short.
Proakis & Manolakis (1992). Digital Signal Processing. New York: Macmillan Publishing Company.
Octave Forge https://octave.sourceforge.io/
Zpg
, bilinear
,
Arma
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