Introduction

A standard Fourier transform assumes samples are evenly spaced and present at every expected time step. Missing values violate that assumption, so there is no universally correct “FFT with gaps” trick. The right answer depends on why samples are missing, how many are missing, and whether you need a quick approximation or a defensible scientific estimate.

Why Missing Values Break a Plain FFT

The discrete Fourier transform is defined on a complete sequence. When your array contains gaps, you no longer have the signal the FFT expects. Replacing missing values blindly changes the signal before the transform even begins.

Two common mistakes are:

• filling missing points with zero
• dropping missing points and pretending the time grid stayed uniform

Zero filling inserts artificial discontinuities. Dropping points changes the sampling pattern. Both can create false spectral peaks or distort amplitudes.

Small Gaps: Interpolation Can Be Good Enough

If the data is mostly complete and the gaps are short relative to the dominant frequencies, interpolation is often a practical approximation. It does not recover the true missing samples, but it may produce a reasonable working signal for exploratory analysis.

1import numpy as np
2from scipy.fft import rfft, rfftfreq
3
4x = np.array([0.0, 1.0, np.nan, 1.0, 0.0, -1.0, np.nan, -1.0])
5idx = np.arange(len(x))
6mask = ~np.isnan(x)
7
8x_interp = np.interp(idx, idx[mask], x[mask])
9spectrum = rfft(x_interp)
10freqs = rfftfreq(len(x_interp), d=1.0)
11
12print(x_interp)
13print(np.abs(spectrum))
14print(freqs)

This approach is acceptable only when the interpolation assumption is justified by the signal and the size of the gaps.

Irregular Sampling: Use a Method Designed for It

If missing values mean the sampling times are no longer uniformly spaced, forcing the data back onto a regular grid may be the wrong move. In that case, a method for uneven sampling is usually more defensible.

A common example is Lomb-Scargle period analysis:

1import numpy as np
2from scipy.signal import lombscargle
3
4sample_times = np.array([0.0, 1.0, 3.0, 4.0, 7.0, 8.0])
5values = np.sin(2 * np.pi * 0.25 * sample_times)
6angular_freqs = np.linspace(0.01, 5.0, 500)
7
8power = lombscargle(sample_times, values, angular_freqs)
9print(power[:10])

This does not produce the same object as a classic FFT, but it is often a better way to estimate periodic structure when the data is uneven or gappy.

Imputation Is a Modeling Step, Not Just a Cleanup Step

Interpolation, spline fitting, Kalman smoothing, or model-based imputation all make the transform possible by inventing values in the missing regions. That can be perfectly reasonable, but it is still a modeling choice.

The more missing data you have, the more the spectrum reflects the assumptions of the fill method rather than the original observations. This is why the phrase “just fill missing values and run FFT” is too casual for serious analysis.

A practical rule is:

• small isolated gaps: interpolation may be acceptable
• long gaps: imputation becomes much more opinionated
• heavily irregular sampling: use an uneven-sampling method instead of hiding the irregularity

Windowing and Detrending Still Matter

Missing-data handling does not replace normal spectral hygiene. If the signal has a trend, offset, or edge discontinuities, you may still need detrending or windowing after the preprocessing step.

For example:

1import numpy as np
2from scipy.signal import detrend, windows
3from scipy.fft import rfft
4
5signal = detrend(x_interp)
6windowed = signal * windows.hann(len(signal))
7spectrum = rfft(windowed)
8print(np.abs(spectrum))

Without that step, you can misread low-frequency energy caused by drift or abrupt edges as meaningful structure.

Choose the Method Based on the Question

If you are doing rough visualization, a simple interpolation-plus-FFT pipeline may be enough. If you are trying to estimate physically meaningful frequencies for a publication or a diagnostic system, you need to justify the handling of missingness much more carefully.

That distinction matters more than the code. Many disputes about “the right transform” are actually disputes about what level of approximation is acceptable for the use case.

Common Pitfalls

The biggest mistake is replacing missing values with zeros and interpreting the resulting spectrum as if the original signal had those zeros. That changes the signal substantially.

Another mistake is forgetting that missing samples may destroy uniform sampling altogether. In that case, a standard FFT is no longer the right mathematical tool.

People also focus on the transform before quantifying the missing-data pattern. The fraction, distribution, and length of gaps are often more important than the FFT code itself.

Summary

• A plain FFT assumes complete, evenly spaced samples.
• Missing values require either careful preprocessing or a method for uneven sampling.
• Interpolation can work for small gaps but it changes the signal.
• For irregular timing, methods such as Lomb-Scargle are often a better fit.
• Treat missing-data handling as a modeling decision, not as a trivial preprocessing step.