kaiserord: Kaiser filter order and cutoff frequency

Description

Return the parameters needed to produce a FIR filter of the desired specification from a Kaiser window.

Usage

kaiserord(f, m, dev, fs = 2)

Arguments

frequency bands, given as pairs, with the first half of the first pair assumed to start at 0 and the last half of the last pair assumed to end at 1. It is important to separate the band edges, since narrow transition regions require large order filters.

magnitude within each band. Should be non-zero for pass band and zero for stop band. All passbands must have the same magnitude, or you will get the error that pass and stop bands must be strictly alternating.

dev

deviation within each band. Since all bands in the resulting filter have the same deviation, only the minimum deviation is used. In this version, a single scalar will work just as well.

sampling rate. Used to convert the frequency specification into the c(0, 1) range, where 1 corresponds to the Nyquist frequency, fs / 2.

Value

A list of class FilterSpecs with the following list elements:

n: filter order
Wc: cutoff frequency
type: filter type, one of "low", "high", "stop", "pass", "DC-0", or "DC-1".
beta: shape parameter

Details

Given a set of specifications in the frequency domain, kaiserord estimates the minimum FIR filter order that will approximately meet the specifications. kaiserord converts the given filter specifications into passband and stopband ripples and converts cutoff frequencies into the form needed for windowed FIR filter design.

kaiserord uses empirically derived formulas for estimating the orders of lowpass filters, as well as differentiators and Hilbert transformers. Estimates for multiband filters (such as band-pass filters) are derived from the low-pass design formulas.

The design formulas that underlie the Kaiser window and its application to FIR filter design are $$\beta =$$ $$0.1102(\alpha - 8.7), \alpha > 50$$ $$0.5842(\alpha -21)^{0.4} + 0.07886(\alpha - 21), 21 \le \alpha \le 50$$ $$0, \alpha < 21$$

where $\alpha = -20log_{10}(\delta)$ is the stopband attenuation expressed in decibels, $n=(\alpha - 8) / 2.285(\Delta\omega)$, where $n$ is the filter order and $\Delta\omega$ is the width of the smallest transition region.

Examples

Run this code

# NOT RUN {
fs <- 11025
op <- par(mfrow = c(2, 2), mar = c(3, 3, 1, 1))
for (i in 1:4) {
  if (i == 1) {
    bands <- c(1200, 1500)
    mag <- c(1, 0)
    dev <- c(0.1, 0.1)
  }
  if (i == 2) {
    bands <- c(1000, 1500)
    mag <- c(0, 1)
    dev <- c(0.1, 0.1)
  }
  if (i == 3) {
    bands <- c(1000, 1200, 3000, 3500)
    mag <- c(0, 1, 0)
    dev <- 0.1
  }
  if (i == 4) {
    bands <- 100 * c(10, 13, 15, 20, 30, 33, 35, 40)
    mag <- c(1, 0, 1, 0, 1)
    dev <- 0.05
  }
  kaisprm <- kaiserord(bands, mag, dev, fs)
  d <- max(1, trunc(kaisprm$n / 10))
  if (mag[length(mag)] == 1 && (d %% 2) == 1) {
     d <- d + 1
  }
  f1 <- freqz(fir1(kaisprm$n, kaisprm$Wc, kaisprm$type,
                   kaiser(kaisprm$n + 1, kaisprm$beta),
                   scale = FALSE),
              fs = fs)
  f2 <- freqz(fir1(kaisprm$n - d, kaisprm$Wc, kaisprm$type,
                   kaiser(kaisprm$n - d + 1, kaisprm$beta),
                   scale = FALSE),
              fs = fs)
  plot(f1$w, abs(f1$h), col = "blue", type = "l",  xlab = "", ylab = "")
  lines(f2$w, abs(f2$h), col = "red")
  legend("right", paste("order", c(kaisprm$n-d, kaisprm$n)),
         col = c("red", "blue"), lty = 1, bty = "n")
  b <- c(0, bands, fs/2)
  for (i in seq(2, length(b), by=2)) {
    hi <- mag[i/2] + dev[1]
    lo <- max(mag[i/2] - dev[1], 0)
    lines(c(b[i-1], b[i], b[i], b[i-1], b[i-1]), c(hi, hi, lo, lo, hi))
  }
}
par(op)

# }