BUTTER

The Butterworth filter is a type of signal processing filter designed to have a frequency response as flat as possible in the passband. It is also referred to as a maximally flat magnitude filter.

For an Nth-order Butterworth low-pass filter, the magnitude squared of the frequency response is:

|H(j\omega)|^2 = \frac{1}{1 + (\frac{\omega}{\omega_c})^{2N}}

where N is the filter order and \omega_c is the cutoff frequency.

Excel Usage

=BUTTER(N, Wn, btype, filter_output_form, analog, fs)

N (int, required): The order of the filter.
Wn (list[list], required): Critical frequency or frequencies.
btype (str, optional, default: “lowpass”): The type of filter.
filter_output_form (str, optional, default: “ba”): Type of output representation.
analog (bool, optional, default: false): When True, return an analog filter.
fs (float, optional, default: null): The sampling frequency of the digital system.

Returns (list[list]): Filter coefficients or representation.

Example 1: Low-pass (b, a)

Inputs:

N	Wn	btype
2	0.1	lowpass

Excel formula:

=BUTTER(2, 0.1, "lowpass")

Expected output:

Result
0.0200834	0.0401667	0.0200834
1	-1.56102	0.641352

Example 2: High-pass (SOS)

Inputs:

N	Wn	btype	fs	filter_output_form
4	100	highpass	1000	sos

Excel formula:

=BUTTER(4, 100, "highpass", 1000, "sos")

Expected output:

Result
0.432847	-0.865693	0.432847	1	-1.0486	0.29614
1	-2	1	1	-1.32091	0.632739

Python Code

Show Code

import numpy as np
from scipy.signal import butter as scipy_butter

def butter(N, Wn, btype='lowpass', filter_output_form='ba', analog=False, fs=None):
    """
    Butterworth digital and analog filter design.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.butter.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        N (int): The order of the filter.
        Wn (list[list]): Critical frequency or frequencies.
        btype (str, optional): The type of filter. Valid options: Lowpass, Highpass, Bandpass, Bandstop. Default is 'lowpass'.
        filter_output_form (str, optional): Type of output representation. Valid options: Numerator/Denominator, Pole-Zero, Second-Order Sections. Default is 'ba'.
        analog (bool, optional): When True, return an analog filter. Default is False.
        fs (float, optional): The sampling frequency of the digital system. Default is None.

    Returns:
        list[list]: Filter coefficients or representation.
    """
    try:
        if isinstance(Wn, list):
            wn_flat = [float(v) for row in Wn for v in row]
            wn_val = wn_flat[0] if len(wn_flat) == 1 else wn_flat
        else:
            wn_val = float(Wn)

        result = scipy_butter(
            int(N), 
            wn_val, 
            btype=btype, 
            analog=bool(analog), 
            output=filter_output_form, 
            fs=float(fs) if fs is not None else None
        )

        if filter_output_form == 'ba':
            b, a = result
            max_len = max(len(b), len(a))
            return [b.tolist() + [""] * (max_len - len(b)), a.tolist() + [""] * (max_len - len(a))]
        elif filter_output_form == 'zpk':
            z, p, k = result
            max_len = max(len(z), len(p), 1)
            return [z.tolist() + [""] * (max_len - len(z)), p.tolist() + [""] * (max_len - len(p)), [float(k)] + [""] * (max_len - 1)]
        else: # sos
            return result.tolist()
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

N *

The order of the filter.

Wn *

Critical frequency or frequencies.

btype

The type of filter.

filter_output_form

Type of output representation.

analog

When True, return an analog filter.

The sampling frequency of the digital system.