Sampling, FFT, Windowing, and Filtering

Visual: sampling and frequency-domain diagram showing aliasing, FFT bins, spectral leakage, windowing, filter response, and implementation diagnostics.

Sampling turns continuous sensor signals into discrete data. FFTs expose frequency structure. Windowing controls spectral leakage. Filtering shapes signal bandwidth, delay, and noise. The first-principles question is always: what information exists in the signal, what information did sampling preserve, and what did processing distort?

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Related docs

• Radar Ambiguity, Chirp Design, and Doppler Limits
• CFAR Detection and Thresholding
• Sensor Filtering: Alpha-Beta, Kalman, and Complementary
• Sensor Likelihoods, Noise, and Error Budgets
• Time Sync, PTP, Timestamping, and Latency Models

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Why it matters for AV, perception, SLAM, and mapping

Vehicle sensors are sampled systems. Cameras expose over finite intervals, LiDAR scans accumulate over rotating or solid-state patterns, radars sample ADC chirps, IMUs sample acceleration and angular rate, and wheel encoders quantize motion. Signal-processing choices affect detection latency, velocity estimates, map alignment, and filter consistency.

Bad sampling or filtering can create aliasing, phase lag, ringing, false peaks, or over-smoothed signals that look plausible but are late. In a moving vehicle, latency is a spatial error.

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Core math and algorithm steps

Sampling and Nyquist

For sampling period T_s:

f_s = 1 / T_s

A band-limited signal with maximum frequency f_max can be sampled without aliasing only if:

f_s > 2 * f_max

Frequencies above f_s / 2 fold into lower frequencies. This is aliasing, not noise. Anti-alias filtering must happen before sampling or before downsampling.

Discrete Fourier transform

For N samples x[n]:

X[k] = sum_{n=0}^{N-1} x[n] exp(-j 2 pi k n / N)

Frequency bin spacing:

delta_f = f_s / N

For real-valued input, use a real FFT (rfft) and corresponding real frequency bins. Zero padding interpolates the displayed spectrum; it does not improve the true resolution set by observation time.

Windowing

The FFT assumes the finite block is periodic. If the block does not end at an integer number of cycles, discontinuities create spectral leakage. Windowing multiplies the data:

x_w[n] = w[n] * x[n]

Tradeoffs:

Window	Strength	Cost
Rectangular	narrowest main lobe	high sidelobes
Hann/Hamming	lower sidelobes	wider main lobe
Blackman/Blackman-Harris	strong sidelobe suppression	more resolution loss
Flat top	better amplitude estimate	wide main lobe

Use FFT-periodic windows for spectral analysis and symmetric windows for FIR filter design.

Filtering

Linear time-invariant filtering:

y[n] = sum_k b[k] x[n-k] - sum_k a[k] y[n-k]

FIR filters use only feed-forward terms. IIR filters use feedback and can reach sharp frequency responses with fewer coefficients, but they introduce nonlinear phase unless designed or applied carefully.

Common choices:

Filter	Use	Risk
Moving average	quick smoothing	delay and poor frequency selectivity
Butterworth IIR	smooth passband	phase delay and stability at high order
Bessel IIR	phase behavior	slower rolloff
FIR linear phase	predictable delay	more coefficients
Median	impulse rejection	nonlinear, can distort dynamics
Savitzky-Golay	smooth derivatives	window assumptions can fail at edges

Delay and phase

For real-time systems, filter delay is part of the measurement model. A linear phase FIR of length M has group delay:

delay = (M - 1) / 2 samples

Forward-backward filtering can remove phase delay offline, but it uses future samples and is not causal. Do not use offline zero-phase filtering to estimate the latency of an online stack.

Basic processing checklist

choose sample rate from signal bandwidth and latency budget
anti-alias before ADC or decimation
remove invalid samples and saturations explicitly
window blocks before spectral estimation
scale FFT amplitudes consistently
filter with known passband, stopband, delay, and edge behavior
validate on synthetic signals with known frequency, phase, and amplitude

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Implementation notes

• Use scipy.fft for FFTs and scipy.signal for windows and filters.
• Use second-order sections (output="sos") for higher-order IIR filters to improve numerical stability.
• Record sample time from hardware timestamps when available, not loop arrival time.
• Do not infer physical frequency from FFT index without f_s and N.
• Check units after every transform: ADC counts, volts, power, dB, meters, or radians.
• Treat missing samples and dropped frames as timing events, not ordinary zero-valued samples.
• Validate filter response with impulse, step, sine sweep, and known-noise tests before using it in a safety-relevant estimator.

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Failure modes and diagnostics

Failure mode	Symptom	Diagnostic
Aliasing	Low-frequency artifact from high-frequency source.	Repeat test at higher sample rate or with stronger anti-alias filter.
Spectral leakage	Energy smeared across bins.	Compare rectangular and tapered windows.
Wrong FFT scaling	Amplitude changes with N.	Test a sine wave with known amplitude.
Filter lag	Perception or control output arrives late.	Step response and group delay measurement.
Edge artifacts	Startup transient or replay boundary spike.	Inspect first and last window lengths.
IIR instability	Output grows or rings unexpectedly.	Check pole locations and use SOS implementation.
Offline/online mismatch	Lab results better than vehicle behavior.	Remove noncausal filtering from online-equivalent tests.

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Sources

• SciPy FFT documentation: https://docs.scipy.org/doc/scipy/reference/fft.html
• SciPy FFT tutorial: https://docs.scipy.org/doc/scipy/tutorial/fft.html
• SciPy signal processing API: https://docs.scipy.org/doc/scipy/reference/signal.html
• SciPy window generation documentation: https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.windows.get_window.html
• SciPy Butterworth filter documentation: https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.butter.html
• SciPy filtering with second-order sections: https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.sosfilt.html
• Smith, "The Scientist and Engineer's Guide to Digital Signal Processing": https://www.dspguide.com/