Frequency response function (FRF) and coherence from CSV

The magnitude peaks at the mast’s resonance, about ten times its response at rest (0.05 mm/N), and the phase falls through −90° there, as a 2nd-order system’s does. Near the peak the sway is large and the coherence is close to 1. At 20 Hz the mast barely moves, the 0.5 mm of measurement noise on the sway is a large part of what is measured, and the coherence drops to 0.692: H1 still reads the mast’s own value, while H2 reads too high by a factor of 1 ÷ 0.692.

The same in Python

With the sample file downloaded, these lines give SciPy’s answer for the same settings (skiprows=[1] skips the units row):

import numpy as np, pandas as pd
from scipy import signal

d = pd.read_csv("megarows-sample-frf.csv", skiprows=[1])
x, y = d["Wind_Force"].to_numpy(), d["Mast_Sway"].to_numpy()
kw = dict(fs=100, window="hann", nperseg=2048, noverlap=1024)
f, pxy = signal.csd(x, y, **kw)
_, pxx = signal.welch(x, **kw)
_, pyy = signal.welch(y, **kw)
_, coh = signal.coherence(x, y, **kw)
h1, h2 = pxy / pxx, pyy / np.conj(pxy)
k = np.argmax(np.abs(h1[1:])) + 1
print(f"peak {f[k]:.2f} Hz  |H1| {abs(h1[k]):.3f} mm/N  coherence {coh[k]:.3f}")
i = np.argmin(np.abs(f - 20))
print(f"{f[i]:.2f} Hz  coherence {coh[i]:.3f}  |H1| {abs(h1[i]):.5f}  |H2| {abs(h2[i]):.5f}")

They print:

peak 8.06 Hz  |H1| 0.499 mm/N  coherence 0.995
20.02 Hz  coherence 0.692  |H1| 0.00705  |H2| 0.01018

We ran them on 8 October 2026 with NumPy 2.1.3, SciPy 1.14.1, pandas 2.2.3 and Python 3.12. A test runs the same calculation through MegaRows on every change to the site and checks that its numbers are the ones on this page, and that they match SciPy’s to about 12 significant digits. Download the sample file (CSV, 427 KB).

H1 or H2

Both estimate the same frequency response from averaged spectra: Gxx and Gyy, the power spectral densities of the input and the output, and Gxy, their cross spectral density.

• H1 = Gxy / Gxx is not biased by noise on the output, the usual case: a response measured with sensor noise, as in the sample. It is the default.
• H2 = Gyy / Gyx is not biased by noise on the input. It is often the better estimate right at a sharp resonance, where a shaker’s force drops and the input is the noisier signal.
• H1 and H2 on the same plot show where the two agree: H1 is never above H2, and H1 ÷ H2 is the coherence, so they agree where the coherence is close to 1 and part where it drops.

What the coherence tells you

The magnitude-squared coherence of the two signals, |Gxy|² / (Gxx Gyy), runs from 0 to 1 at each frequency: the share of the output’s power that follows the input in a straight line. High coherence supports a consistent linear relation between input and output at these settings. Check calibration, alignment, noise and leakage before relying on the FRF. Low coherence means something else is in the output:

• noise on the output or the input, as at 20 Hz in the sample;
• another input you did not measure, such as a second load or a vibration from nearby;
• a nonlinear system, such as a joint that rattles or a spring that stiffens;
• segments too short for a sharp resonance, or a delay between the signals of more than a fraction of a segment: try a longer segment.

Units and the phase

The magnitude is in the output’s unit per the input’s: mm/N for the sample, g/N for an accelerometer against a force, (m/s²)/N in SI. Give each column its unit in a units row under the header. The phase is in degrees, from −180° to 180°: negative when the output lags the input. The coherence has no unit.

Checked against SciPy

Private by design: nothing is uploaded

Test data is often confidential. This tool reads your file and calculates the FRF in your browser, so the file and its name never leave your device. It works on files with millions of rows: the spectra use every sample of both channels.