R/butter.R

In gjmvanboxtel/gsignal: Signal Processing

# butter.R
# Copyright (C) 2019 Geert van Boxtel <gjmvanboxtel@gmail.com>
# Octave signal package:
# Copyright (C) 1999 Paul Kienzle <pkienzle@users.sf.net>
# Copyright (C) 2003 Doug Stewart <dastew@sympatico.ca>
# Copyright (C) 2011 Alexander Klein <alexander.klein@math.uni-giessen.de>
# Copyright (C) 2018 John W. Eaton
#
# 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 3 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; see the file COPYING. If not, see
# <https://www.gnu.org/licenses/>.
#
# 20200513 Geert van Boxtel           First version for v0.1.0
# 20200519 Geert van Boxtel           Added plane parameter to butter.IIRfspec
# 20200708 GvB                        renamed IIRfspec to FilterSpecs
# 20210308 GvB                        bug in passing w to sftrans;
#                                     added output parameter
#------------------------------------------------------------------------------

#' Butterworth filter design
#'
#' Compute the transfer function coefficients of a Butterworth filter.
#'
#' Butterworth filters have a magnitude response that is maximally flat in the
#' passband and monotonic overall. This smoothness comes at the price of
#' decreased rolloff steepness. Elliptic and Chebyshev filters generally provide
#' steeper rolloff for a given filter order.
#'
#' Because butter is generic, it can be extended to accept other inputs, using
#' \code{buttord} to generate filter criteria for example.
#'
#' @param n filter order.
#' @param w critical frequencies of the filter. \code{w} must be a scalar for
#'   low-pass and high-pass filters, and \code{w} must be a two-element vector
#'   c(low, high) specifying the lower and upper bands in radians/second. For
#'   digital filters, w must be between 0 and 1 where 1 is the Nyquist
#'   frequency.
#' @param type filter type, one of \code{"low"}, (default) \code{"high"},
#'   \code{"stop"}, or \code{"pass"}.
#' @param plane "z" for a digital filter or "s" for an analog filter.
#' @param output Type of output, one of:
#' \describe{
#'   \item{"Arma"}{Autoregressive-Moving average (aka numerator/denominator, aka
#'   b/a)}
#'   \item{"Zpg"}{Zero-pole-gain format}
#'   \item{"Sos"}{Second-order sections}
#' }
#' Default is \code{"Arma"} for compatibility with the 'signal' package and the
#' 'Matlab' and 'Octave' equivalents, but \code{"Sos"} should be preferred for
#' general-purpose filtering because of numeric stability.
#' @param ... additional arguments passed to butter, overriding those given by
#'   \code{n} of class \code{\link{FilterSpecs}}.
#'
#' @return Depending on the value of the \code{output} parameter, a list of
#'   class \code{\link{Arma}}, \code{\link{Zpg}}, or \code{\link{Sos}}
#'   containing the filter coefficients
#'
#' @examples
#' ## 50 Hz notch filter
#' fs <- 256
#' bf <- butter(4, c(48, 52) / (fs / 2), "stop")
#' freqz(bf, fs = fs)
#'
#' ## EEG alpha rhythm (8 - 12 Hz) bandpass filter
#' fs <- 128
#' fpass <- c(8, 12)
#' wpass <- fpass / (fs / 2)
#' but <- butter(5, wpass, "pass")
#' freqz(but, fs = fs)
#'
#' ## filter to remove vocals from songs, 25 dB attenuation in stop band
#' ## (not optimal with a Butterworth filter)
#' fs <- 44100
#' specs <- buttord(230/(fs/2), 450/(fs/2), 1, 25)
#' bf <- butter(specs)
#' freqz(bf, fs = fs)
#' zplane(bf)
#'
#' @references \url{https://en.wikipedia.org/wiki/Butterworth_filter}
#'
#' @seealso \code{\link{Arma}}, \code{\link{Zpg}}, \code{\link{Sos}},
#'   \code{\link{filter}}, \code{\link{cheby1}}, \code{\link{ellip}},
#'   \code{\link{buttord}}.
#'
#' @author Paul Kienzle, \email{pkienzle@@users.sf.net},\cr
#'   Doug Stewart, \email{dastew@@sympatico.ca},\cr
#'   Alexander Klein, \email{alexander.klein@@math.uni-giessen.de},\cr
#'   John W. Eaton.\cr
#'   Conversion to R by Tom Short,\cr
#'   adapted by Geert van Boxtel \email{G.J.M.vanBoxtel@@gmail.com}.
#'
#' @rdname butter
#' @export

butter <- function(n, ...) UseMethod("butter")

#' @rdname butter
#' @export

butter.FilterSpecs <- function(n, ...)
  butter(n$n, n$Wc, n$type, n$plane, ...)

#' @rdname butter
#' @export

butter.default <- function(n, w,
                           type = c("low", "high", "stop", "pass"),
                           plane = c("z", "s"),
                           output = c("Arma", "Zpg", "Sos"), ...) {

  # check input arguments
  type <- match.arg(type)
  plane <- match.arg(plane)
  output <- match.arg(output)
  if (!isPosscal(n) || !isWhole(n)) {
    stop("filter order n must be a positive integer")
  }
  stop <- type == "stop" || type == "high"
  digital <- plane == "z"
  if (!is.vector(w) || (length(w) != 1 && length(w) != 2)) {
    stop(paste("frequency w must be specified as a vector of length 1 or 2",
               "(either w0 or c(w0, w1))"))
  }
  if ((type == "stop" || type == "pass") &&  length(w) != 2) {
    stop("w must be two elements for stop and bandpass filters")
  }
  if (digital && !all(w >= 0 & w <= 1)) {
    stop("critical frequencies w must be in the range [0 1]")
  } else if (!digital && !all(w >= 0)) {
    stop("critical frequencies w must be in the range [0 Inf]")
  }

  ## Prewarp to the band edges to s plane
  if (digital) {
    T <- 2                    # sampling frequency of 2 Hz
    w <- 2 / T * tan(pi * w / T)
  }

  ## Generate splane poles for the prototype Butterworth filter
  ## source: Kuc
  C <- 1                      # default cutoff frequency
  pole <- C * exp(1i * pi * (2 * 1:n + n - 1) / (2 * n))
  if (n %% 2 == 1) {
    pole[(n + 1) / 2] <- -1   # pure real value at exp(i*pi)
  }
  zero <- numeric(0)
  gain <- C^n

  zpg <- Zpg(z = zero, p = pole, g = gain)

  ## splane frequency transform
  zpg <- sftrans(zpg, w = w, stop = stop)

  ## Use bilinear transform to convert poles to the z plane
  if (digital) {
    zpg <- bilinear(zpg, T = T)
  }

  if (output == "Arma") {
    retval <- as.Arma(zpg)
  } else if (output == "Sos") {
    retval <- as.Sos(zpg)
  } else {
    retval <- zpg
  }
  retval
}