R/bilinear.R

In signal: Signal Processing

## Copyright (C) 1999 Paul Kienzle
##
## This program is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this program; if not, write to the Free Software
## Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA

## usage: [Zz, Zp, Zg] = bilinear(Sz, Sp, Sg, T)
##        [Zb, Za] = bilinear(Sb, Sa, T)
##
## Transform a s-plane filter specification into a z-plane
## specification. Filters can be specified in either zero-pole-gain or
## transfer function form. The input form does not have to match the
## output form. T is the sampling frequency represented in the z plane.
##
## Theory: Given a piecewise flat filter design, you can transform it
## from the s-plane to the z-plane while maintaining the band edges by
## means of the bilinear transform.  This maps the left hand side of the
## s-plane into the interior of the unit circle.  The mapping is highly
## non-linear, so you must design your filter with band edges in the
## s-plane positioned at 2/T tan(w*T/2) so that they will be positioned
## at w after the bilinear transform is complete.
##
## The following table summarizes the transformation:
##
## Transform         Zero at x                  Pole at x
## ----------------  -------------------------  ------------------------
## Bilinear          zero: (2+xT)/(2-xT)        pole: (2+xT)/(2-xT)
##      2 z-1        pole: -1                   zero: -1
## S -> - ---        gain: (2-xT)/T             gain: (2-xT)/T
##      T z+1
## ----------------  -------------------------  ------------------------
##
## With tedious algebra, you can derive the above formulae yourself by
## substituting the transform for S into H(S)=S-x for a zero at x or
## H(S)=1/(S-x) for a pole at x, and converting the result into the
## form:
##
##    H(Z)=g prod(Z-Xi)/prod(Z-Xj)
##
## Please note that a pole and a zero at the same place exactly cancel.
## This is significant since the bilinear transform creates numerous
## extra poles and zeros, most of which cancel. Those which do not
## cancel have a "fill-in" effect, ext} #ing 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 I will
## 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.  The
## analytic design methods all yield more poles than zeros, so this will
## not be a problem.
##
## References:
##
## Proakis & Manolakis (1992). Digital Signal Processing. New York:
## Macmillan Publishing Company.

## Author: Paul Kienzle <pkienzle@users.sf.net>

bilinear <- function(Sz, ...) UseMethod("bilinear")

bilinear.Zpg <- function(Sz, T, ...)  
  bilinear(Sz$zero, Sz$pole, Sz$gain, T)

bilinear.Arma <- function(Sz, T, ...)  
  as.Arma(bilinear(as.Zpg(Sz), T))

bilinear.default <- function(Sz, Sp, Sg, T, ...)  { 
  p <- length(Sp)
  z <- length(Sz)
  if (z > p || p == 0)
    stop("must have at least as many poles as zeros in s-plane")

## ----------------  -------------------------  ------------------------
## Bilinear          zero: (2+xT)/(2-xT)        pole: (2+xT)/(2-xT)
##      2 z-1        pole: -1                   zero: -1
## S -> - ---        gain: (2-xT)/T             gain: (2-xT)/T
##      T z+1
## ----------------  -------------------------  ------------------------
  Zg <- Re(Sg * prod((2-Sz*T)/T) / prod((2-Sp*T)/T))
  Zp <- (2+Sp*T) / (2-Sp*T)
  if (is.null(Sz))
    Zz <- rep.int(-1, length(Zp))
  else {
    Zz <- (2+Sz*T) / (2-Sz*T)
    Zz <- c(Zz, rep.int(-1, p - z))
  } 
  Zpg(zero = Zz, pole = Zp, gain = Zg)
}