# Butterworth filter order and cutoff Compute butterworth filter order and cutoff for the desired response characteristics. ```r buttord(Wp, Ws, Rp, Rs) ``` ## Arguments - `Wp, Ws`: pass-band and stop-band edges. For a low-pass or high-pass filter, `Wp` and `Ws` are scalars. For a band-pass or band-rejection filter, both are vectors of length 2. For a low-pass filter, `Wp < Ws`. For a high-pass filter, `Ws > Wp`. For a band-pass (`Ws[1] < Wp[1] < Wp[2] < Ws[2]`) or band-reject (`Wp[1] < Ws[1] < Ws[2] < Wp[2]`) filter design, `Wp` gives the edges of the pass band, and `Ws` gives the edges of the stop band. Frequencies are normalized to [0,1], corresponding to the range [0, Fs/2]. - `Rp`: allowable decibels of ripple in the pass band. - `Rs`: minimum attenuation in the stop band in dB. ## Details Deriving the order and cutoff is based on: $$ |H(W)|^2 = \frac{1}{1+(W/Wc)^{2n}} = 10^{-R/10}%|H(W)|^2 = 1 / [1+(W/Wc)^(2n)] = 10^(-R/10) $$ With some algebra, you can solve simultaneously for `Wc` and `n` given `Ws`, `Rs` and `Wp`, `Rp`. For high-pass filters, subtracting the band edges from Fs/2, performing the test, and swapping the resulting `Wc` back works beautifully. For bandpass- and bandstop-filters, this process significantly overdesigns. Artificially dividing `n` by 2 in this case helps a lot, but it still overdesigns. ## Returns An object of class `FilterOfOrder` with the following list elements: - **n**: filter order - **Wc**: cutoff frequency - **type**: filter type, one of low , high , stop , or pass This object can be passed directly to `butter` to compute filter coefficients. ## References Octave Forge [https://octave.sourceforge.io/](https://octave.sourceforge.io/) ## Author(s) Original Octave version by Paul Kienzle, pkienzle@user.sf.net . Conversion to R by Tom Short. ## See Also `butter`, `FilterOfOrder`, `cheb1ord` ## Examples ```r Fs <- 10000 btord <- buttord(1000/(Fs/2), 1200/(Fs/2), 0.5, 29) plot(c(0, 1000, 1000, 0, 0), c(0, 0, -0.5, -0.5, 0), type = "l", xlab = "Frequency (Hz)", ylab = "Attenuation (dB)") bt <- butter(btord) plot(c(0, 1000, 1000, 0, 0), c(0, 0, -0.5, -0.5, 0), type = "l", xlab = "Frequency (Hz)", ylab = "Attenuation (dB)", col = "red", ylim = c(-10,0), xlim = c(0,2000)) hf <- freqz(bt, Fs = Fs) lines(hf$f, 20*log10(abs(hf$h))) ```

buttord function