FREQZ

This function computes the complex frequency response of a digital filter defined by numerator and denominator coefficients.

For filter transfer function

H(z) = \frac{\sum_{m=0}^{M} b_m z^{-m}}{\sum_{n=0}^{N} a_n z^{-n}}

it evaluates H(e^{j\omega}) on a frequency grid and returns frequency, magnitude, and phase so response behavior can be analyzed directly in Excel.

Excel Usage

=FREQZ(b, a, wor_n, whole, fs, include_nyquist)

b (list[list], required): Numerator (feed-forward) filter coefficients.
a (list[list], optional, default: [[1]]): Denominator (feedback) filter coefficients.
wor_n (int, optional, default: 512): Number of frequency points to evaluate.
whole (bool, optional, default: false): If True, compute response over the whole unit circle.
fs (float, optional, default: 6.283185307179586): Sampling frequency in frequency units (radians per sample by default).
include_nyquist (bool, optional, default: false): If True and wor_n is an integer, include the Nyquist frequency endpoint.

Returns (list[list]): A 2D array with rows for frequency, magnitude response, and phase response.

Example 1: Frequency response for a simple low-pass filter on default grid

Inputs:

b		a		wor_n
1	1	1	-0.5	16

Excel formula:

=FREQZ({1,1}, {1,-0.5}, 16)

Expected output:

Result
0	0.19635	0.392699	0.589049	0.785398	0.981748	1.1781	1.37445	1.5708	1.76715	1.9635	2.15984	2.35619	2.55254	2.74889	2.94524
4	3.83605	3.43491	2.95836	2.50777	2.11663	1.78561	1.50525	1.26491	1.05546	0.869597	0.701632	0.547095	0.40241	0.264635	0.131251
0	-0.2873	-0.538017	-0.738334	-0.893173	-1.01319	-1.10805	-1.18498	-1.24905	-1.30377	-1.35165	-1.39448	-1.43359	-1.47002	-1.50459	-1.53798

Example 2: Frequency response with custom number of points

Inputs:

b			a	wor_n
1	0	-1	1	16

Excel formula:

=FREQZ({1,0,-1}, {1}, 16)

Expected output:


0.19635	0.392699	0.589049	0.785398	0.981748	1.1781	1.37445	1.5708	1.76715	1.9635	2.15984	2.35619	2.55254	2.74889	2.94524
0.390181	0.765367	1.11114	1.41421	1.66294	1.84776	1.96157	2	1.96157	1.84776	1.66294	1.41421	1.11114	0.765367	0.390181
1.37445	1.1781	0.981748	0.785398	0.589049	0.392699	0.19635	0	-0.19635	-0.392699	-0.589049	-0.785398	-0.981748	-1.1781	-1.37445

Example 3: Frequency response over full unit circle

Inputs:

b					a	whole	wor_n
0.2	0.2	0.2	0.2	0.2	1	true	16

Excel formula:

=FREQZ({0.2,0.2,0.2,0.2,0.2}, {1}, TRUE, 16)

Expected output:

Result
0	0.392699	0.785398	1.1781	1.5708	1.9635	2.35619	2.74889	3.14159	3.53429	3.92699	4.31969	4.71239	5.10509	5.49779	5.89049
1	0.852395	0.482843	0.0702307	0.2	0.235916	0.0828427	0.113291	0.2	0.113291	0.0828427	0.235916	0.2	0.0702307	0.482843	0.852395
0	-0.785398	-1.5708	-2.35619	0	-0.785398	-1.5708	0.785398	0	-0.785398	1.5708	0.785398	0	2.35619	1.5708	0.785398

Example 4: Frequency response including Nyquist endpoint

Inputs:

b		a	wor_n	include_nyquist
1	-1	1	16	true

Excel formula:

=FREQZ({1,-1}, {1}, 16, TRUE)

Expected output:


0.20944	0.418879	0.628319	0.837758	1.0472	1.25664	1.46608	1.67552	1.88496	2.0944	2.30383	2.51327	2.72271	2.93215	3.14159
0.209057	0.415823	0.618034	0.813473	1	1.17557	1.33826	1.48629	1.61803	1.73205	1.82709	1.90211	1.9563	1.98904	2
1.46608	1.36136	1.25664	1.15192	1.0472	0.942478	0.837758	0.733038	0.628319	0.523599	0.418879	0.314159	0.20944	0.10472	0

Python Code

Show Code

import numpy as np
from scipy.signal import freqz as scipy_freqz

def freqz(b, a=[[1]], wor_n=512, whole=False, fs=6.283185307179586, include_nyquist=False):
    """
    Compute the frequency response of a digital filter.

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

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

    Args:
        b (list[list]): Numerator (feed-forward) filter coefficients.
        a (list[list], optional): Denominator (feedback) filter coefficients. Default is [[1]].
        wor_n (int, optional): Number of frequency points to evaluate. Default is 512.
        whole (bool, optional): If True, compute response over the whole unit circle. Default is False.
        fs (float, optional): Sampling frequency in frequency units (radians per sample by default). Default is 6.283185307179586.
        include_nyquist (bool, optional): If True and wor_n is an integer, include the Nyquist frequency endpoint. Default is False.

    Returns:
        list[list]: A 2D array with rows for frequency, magnitude response, and phase response.
    """
    try:
        def flatten_numeric(x):
            if not isinstance(x, list):
                return [float(x)]
            values = []
            for row in x:
                if isinstance(row, list):
                    for v in row:
                        values.append(float(v))
                else:
                    values.append(float(row))
            return values

        b_vals = flatten_numeric(b)
        a_vals = flatten_numeric(a)

        if len(b_vals) == 0:
            return "Error: b must contain at least one numeric coefficient"
        if len(a_vals) == 0:
            return "Error: a must contain at least one numeric coefficient"

        w, h = scipy_freqz(
            b_vals,
            a_vals,
            worN=int(wor_n),
            whole=bool(whole),
            fs=float(fs),
            include_nyquist=bool(include_nyquist)
        )

        return [
            w.tolist(),
            np.abs(h).tolist(),
            np.angle(h).tolist()
        ]
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

b *

Numerator (feed-forward) filter coefficients.

Denominator (feedback) filter coefficients.

wor_n

Number of frequency points to evaluate.

whole

If True, compute response over the whole unit circle.

Sampling frequency in frequency units (radians per sample by default).

include_nyquist

If True and wor_n is an integer, include the Nyquist frequency endpoint.