signal-processing

name	signal-processing
description	Engineering signal processing — FFT, windowing, aliasing, filtering (FIR/IIR), PSD, coherence, cross-correlation, order tracking, envelope analysis, digital signal conditioning.
metadata	{"priority":7,"promptSignals":{"phrases":["signal processing","FFT","sampling","aliasing","filter","PSD","Nyquist","windowing","order tracking","spectrum"],"minScore":3}}

Engineering Signal Processing — Complete Skill

Sampling and Nyquist

Nyquist-Shannon Theorem

Sample rate f_s must satisfy: f_s > 2 × f_max (signal bandwidth) Minimum sample rate: f_s = 2 × f_max (Nyquist rate) Practical: use f_s ≥ 2.5× f_max (accounting for anti-alias filter rolloff) Anti-alias filter: low-pass at f_s/2 — MUST be analog, before ADC

Aliasing

If f_signal > f_s/2: signal "folds" and appears at f_alias = |f_signal - n×f_s| Example: f_s = 1000 Hz, f_signal = 800 Hz → f_alias = |800 - 1000| = 200 Hz (wrong peak location) Prevention: low-pass filter before ADC at f_s/2 (Nyquist frequency)

Resolution and Record Length

Frequency resolution: Δf = f_s / N = 1/T (T = record length [s]) N = number of samples in FFT block For Δf = 0.5 Hz at f_s = 5000 Hz: N = 5000/0.5 = 10,000 samples (T = 2 sec) Zoom FFT: narrow frequency range at fine resolution (ZOOM FFT or complex demodulation)

FFT and Spectral Analysis

Discrete Fourier Transform (DFT)

X[k] = Σ_{n=0}^{N-1} x[n] × e^(-j2πkn/N), k = 0, 1, ..., N-1 FFT: fast algorithm, O(N log N) vs. O(N²) for DFT N must be power of 2 for radix-2 FFT (512, 1024, 2048, 4096...)

Amplitude scaling: FFT amplitude scaled to match engineering units One-sided spectrum: A[k] = |X[k]| × 2/N for k > 0, × 1/N for k = 0 Phase: φ[k] = angle(X[k]) [rad]

Power Spectral Density (PSD)

G(f) = |X(f)|² / (T × Δf) [units: (unit)²/Hz] E.g., acceleration: [g²/Hz] or [(m/s²)²/Hz]

From FFT: S[k] = |X[k]|²/(N × f_s) [two-sided PSD, scaled] G[k] = 2 × S[k] [one-sided PSD, k > 0]

RMS from PSD: a_rms = √(∫G(f)df) = √(Σ G[k] × Δf)

Averaging: N_avg linear averages: PSD variance reduced by 1/N_avg Random: 50-100 averages needed for stable PSD estimate (±10% variance)

Windowing (Spectral Leakage Control)

Leakage Problem

FFT assumes signal is periodic within record length T If f_signal ≠ exact multiple of Δf: energy "leaks" to adjacent bins Effect: broadened peaks, reduced amplitude accuracy

Common Windows

Window	Main lobe width	Side lobe level	Use
Rectangular	2/N	-13 dB	Impact (transient fits in window)
Hanning	4/N	-31 dB	Continuous random, harmonic
Hamming	4/N	-41 dB	Narrowband signals
Flat-top	8/N	-93 dB	Accurate amplitude calibration
Blackman-Harris	8/N	-92 dB	Better sidelobe (weaker resolution)

Rule: use Hanning for continuous signals (vibration, operational modal) Use Rectangular (no window) + Force/Exponential windows for impact testing

Force window: rectangular but zeroed after impact ends (avoids leakage from force signal) Exponential window: decays to zero at end of record (applies to response signal in impact test)

Filtering

FIR (Finite Impulse Response) Filter

y[n] = Σ_{k=0}^{M} b_k × x[n-k] Linear phase (no phase distortion) — preferred for measurement Design: windowed sinc, remez algorithm, Parks-McClellan Downside: high order M needed for sharp rolloff (computationally expensive)

IIR (Infinite Impulse Response) Filter

y[n] = Σ b_k x[n-k] - Σ a_k y[n-k] Non-linear phase (introduces distortion at cutoff region) Lower order for same rolloff vs. FIR → faster, but phase distortion Design: Butterworth, Chebyshev, Elliptic

Butterworth: maximally flat passband; -3 dB at cutoff, 20N dB/decade rolloff (N = order) Chebyshev I: ripple in passband, sharper rolloff Elliptic: ripple in both pass and stop bands; sharpest rolloff for given order

Zero-phase filtering (MATLAB filtfilt): apply filter forward + backward → zero phase lag Use for post-processing recorded data (not real-time)

High-pass, Low-pass, Band-pass, Band-stop

HP: remove DC and drift (accelerometers: cutoff 1-5 Hz) LP: anti-alias, noise reduction BP: isolate frequency band of interest (engine order) BS (notch): remove single frequency (60 Hz mains noise)

Cross-Correlation and Coherence

Cross-Correlation

R_xy(τ) = ∫x(t) y(t+τ) dt [time domain; measures lag τ between signals] Cross-power spectrum: S_xy(f) = FFT{R_xy(τ)} Used for: time delay measurement (acoustic localization, flow velocity), causality

Coherence Function

γ²(f) = |S_xy(f)|² / (S_xx(f) × S_yy(f)) ∈ [0, 1] γ² = 1: perfect linear causal relationship γ² < 0.9 at resonance: noise, nonlinearity, or multiple inputs Plot coherence alongside FRF to identify valid frequency regions

Order Tracking (Rotating Machinery)

Fixed Sample Rate → Angle-Domain

Standard FFT: fixed time sampling → peaks at frequency [Hz] Order tracking: resample to fixed angle steps → peaks at order × RPM Order n: frequency = n × RPM/60 [Hz]; 1st order = RPM/60

Computed order tracking (COT): Measure tachometer pulse → compute instantaneous speed ω(t) Resample time-domain signal to equal-angle increments (interpolation) FFT of angle-domain signal → order spectrum

Phase-locked order tracking (hardware): dedicated hardware synchronizes sampling to shaft rotation

Application: gearbox diagnostics (mesh frequency = N_teeth × RPM/60), bearing fault (non-integer orders)

Envelope Analysis (Bearing Fault Detection)

Procedure

1. Band-pass filter around resonance frequency excited by bearing impacts (typically 5-30 kHz)
2. Rectify (absolute value)
3. Low-pass filter (envelope extraction)
4. FFT of envelope → bearing fault frequencies

Bearing fault frequencies (from geometry): BPFO (outer race): f_BPFO = N/2 × RPM/60 × (1 - d/D cos α) BPFI (inner race): f_BPFI = N/2 × RPM/60 × (1 + d/D cos α) FTF (cage): f_FTF = RPM/120 × (1 - d/D cos α) BSF (ball spin): f_BSF = D/(2d) × RPM/60 × (1 - (d/D)² cos²α) N = number of balls, d = ball diameter, D = pitch diameter, α = contact angle

Time-Frequency Analysis

Short-Time Fourier Transform (STFT)

Move window through signal, compute FFT at each position Resolution tradeoff: short window → good time res, poor freq res; long window → good freq res, poor time res Output: spectrogram (time vs. frequency, color = amplitude)

Wavelet Transform

Variable window: short for high-freq (good time res), long for low-freq (good freq res) Morlet wavelet: Gaussian-windowed cosine (continuous WT for analysis) Scalogram: equivalent to spectrogram for wavelets

Hilbert Transform

HT{x(t)} → analytic signal z(t) = x(t) + i×HT{x(t)} = A(t)e^{iφ(t)} A(t) = instantaneous amplitude (envelope) dφ/dt = instantaneous frequency Used for: EMD (empirical mode decomposition), amplitude/phase demodulation

Key Engineering Applications

Application	Method	Output
Vibration diagnosis	FFT, order tracking	Fault frequency peaks
Bearing fault	Envelope analysis	BPFO, BPFI peaks
Modal testing	FRF = S_xF/S_FF	H(ω), coherence
Structural health	PSD change over time	Frequency shift, damping change
Engine knock	STFT	Time of knock event
Noise path analysis	Coherence	Source ranking

Output

Provide: sampling rate f_s [Hz], FFT block size N, frequency resolution Δf [Hz], window type, number of averages, filter type + cutoff, PSD estimate G(f) [unit²/Hz], RMS level, coherence at frequencies of interest.