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Use scipy.signal.freqz to calculate a digital filter’s complex frequency response, then plot its magnitude with np.abs(h). For a magnitude-in-decibels plot, convert that magnitude with 20 * np.log10(...). Set fs to the filter’s sampling frequency to put the x-axis in hertz.
Calculate and plot magnitude
freqz returns two arrays: w, the frequencies at which the response was evaluated, and h, the complex-valued response. Because h is complex, plot its absolute value for magnitude rather than plotting h directly.
import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
# b and a are the filter numerator and denominator coefficients.
fs = 48_000 # Example sampling frequency in Hz
w, h = signal.freqz(b, a, fs=fs)
fig, ax = plt.subplots()
ax.plot(w, 20 * np.log10(np.maximum(np.abs(h), 1e-12)))
ax.set(
xlabel="Frequency (Hz)",
ylabel="Magnitude (dB)",
title="Digital filter frequency response",
)
ax.grid(True)
plt.show()The 1e-12 floor prevents taking the logarithm of zero. It is a display choice, not a measurement of the filter; values below the floor appear at the floor on this plot. For a linear magnitude plot instead, use ax.plot(w, np.abs(h)) and label the vertical axis “Magnitude” or “Gain.” The SciPy tutorial also shows plotting np.abs(H) against frequencies returned by freqz. SciPy signal-processing tutorial.
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Choose frequency units and range
The current documented signature is freqz(b, a=1, worN=512, whole=False, plot=None, fs=2*pi, include_nyquist=False). By default, worN=512 requests 512 frequency samples. With the default fs, frequencies are in radians per sample, and the default one-sided range runs from zero toward π, the Nyquist frequency.
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- Use hertz: pass the actual sample rate, such as
fs=48_000. The returnedwvalues and x-axis are then in hertz, and the default range runs from zero toward half that sample rate. - Use the full digital frequency range: set
whole=Trueto evaluate from zero tofs, rather than only through the Nyquist region. - Include the Nyquist endpoint: set
include_nyquist=True. It includes the final Nyquist sample only whenwhole=FalseandworNis an integer; in other cases the option is ignored. - Evaluate selected frequencies: pass an explicit frequency array as
worN. Its values must use the same units asfs.
For dense response curves, increase the integer worN to request more samples. SciPy documents cases where a fast FFT length can improve performance, but grid density and endpoint choices are usually the key plotting decisions. Keep the sampling-frequency setting consistent with the one used to specify filter design edges, since design functions can have different sampling-frequency defaults. SciPy freqz reference and signal-processing tutorial.
Add a phase plot
Phase comes from the angle of the complex response. Use np.unwrap to remove jumps by multiples of 2π, and plot phase separately so its scale does not compete with magnitude.
phase = np.unwrap(np.angle(h))
fig, (ax_mag, ax_phase) = plt.subplots(2, 1, sharex=True)
ax_mag.plot(w, np.abs(h))
ax_mag.set_ylabel("Magnitude")
ax_mag.grid(True)
ax_phase.plot(w, phase)
ax_phase.set(xlabel="Frequency (Hz)", ylabel="Phase (radians)")
ax_phase.grid(True)
plt.show()Here fs was supplied in the earlier freqz call, so the shared frequency axis is in hertz. If you use decibels for magnitude, replace the first subplot’s np.abs(h) with the decibel expression from the previous example.
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Do not pass matplotlib.pyplot.plot as freqz’s plot callback when you intend to show amplitude. SciPy warns that this plots the real part of the complex response, not its magnitude. Calculate w, h and plot np.abs(h) explicitly, or provide a callback that plots the absolute value. SciPy freqz reference.
Use the SOS function for second-order sections
If your filter is represented as second-order sections (SOS), use scipy.signal.freqz_sos rather than converting the sections to a single numerator and denominator polynomial just to calculate the response. SciPy documents a high-order example in which numerical error affects the response computed from a single transfer-function coefficient representation, and compares it with the SOS-based calculation. This is a representation and numerical-stability consideration, not evidence that every freqz result is inaccurate. SciPy freqz_sos reference. The SciPy signal function index distinguishes freqz for coefficient form from freqz_sos for SOS form.
Check plots before comparing them
Two response curves are meaningfully comparable only when their setup and presentation choices are clear. Check that the sampling rate and frequency-axis units match, that grid density and Nyquist endpoint handling are appropriate, and that both magnitude curves use the same scale (linear or decibels). For phase, check whether it is unwrapped and displayed on a legible separate axis. Also account for whether the filters use polynomial coefficients or SOS, particularly for high-order filters.
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