Signal Processing Workbench

Averaged modified periodograms, density scaling (unit²/Hz), mean removed per segment. Averaging reduces variance at the cost of resolution Δf = Fs/L. The confidence band assumes a Gaussian, stationary process and uses Welch's equivalent degrees of freedom \(\nu=2K/(1+2\sum_{j}(1-j/K)\rho^2(jD))\); the interval is \(\nu\hat P/\chi^2_{\nu,1-\alpha/2}\le S\le\nu\hat P/\chi^2_{\nu,\alpha/2}\). DC and Nyquist bins use ν/2.

Window properties

Window-method FIR identical in algorithm to scipy.signal.firwin (symmetric window, gain normalised at DC, Nyquist or the first passband centre), applied causally with zero initial state. Linear phase: output delayed by (taps−1)/2 samples. Cutoffs strictly in (0, Fs/2). Kaiser estimate uses kaiserord: taps = ⌈(A−7.95)/(2.285·π·Δω)+1⌉. Equiripple (Parks–McClellan) is not offered.

Auto mode applies the operation to the entered series with itself (autocorrelation). Full linear convolution of [1,2,3] and [1,1] is [1,3,5,3]; circular convolution with N = 3 is [4,3,5] (the tail wraps). same keeps La samples centred on the full result; valid keeps only complete-overlap samples. Correlation: \(r_{xy}[k]=\sum_n x[n+k]y[n]\); positive k means x is delayed relative to y. Normalized coefficient divides by \(\sqrt{E_xE_y}\) and is undefined for zero-energy input.

Vertical axis: frequency 0 (bottom) to Fs/2 (top). Horizontal axis: frame time. Pale = low, dark red = high.

Polyphase rational resampling with an anti-alias / anti-imaging FIR: output rate Fs·L/M. The processed sample rate is tracked separately from raw Fs; every downstream tool uses the processed Fs for processed data. Not Fourier resampling (no periodicity assumption). Output length must stay ≤ 32 768.

Analytic signal (z[n]=x[n]+j,mathcal{H}{x}[n]) via FFT. Envelope = |z|, phase = arg z (rad, unwrapped by the core), instantaneous frequency in Hz = (1/2π)·dφ/dt. Meaningful for single-component, narrow-band signals; near-zero envelope and multi-component signals give spurious IF. FFT-based: end samples are affected by the implicit periodic extension.

Welch-averaged: (gamma^2_{xy}=|P_{xy}|^2/(P_{xx}P_{yy})in[0,1]); H1 estimator (H=P_{xy}/P_{xx}), magnitude shown as (20log_{10}|H|) dB re 1 (output unit / input unit). Needs equal length and equal Fs. One segment gives γ² ≡ 1 (meaningless). Low coherence means noise or non-linearity; no causality or stability claim is made.

Complex STFT then weighted overlap-add inverse. Error is measured sample by sample against the actual input series, not assumed. Reconstruction requires COLA-compatible window/overlap (e.g. Hann with 50 % overlap); incompatible settings are rejected or show large error.

Power ratios in dB = (10log_{10}(P_1/P_2)). THD = harmonic power / fundamental power (harmonics 2…H); SNR excludes harmonics and DC; SINAD = fundamental / (noise + harmonics). Non-coherent sampling leaks power between bins — heed the core's warning. Ideal quantizer: mid-tread, step 2·FS/2^bits, values beyond ±FS clip; ideal SQNR for a full-scale sine ≈ 6.02·bits + 1.76 dB.

Components are measured by summing PSD × Δf over ±half-width bins around each peak, so tapered windows give correct power without coherent sampling. THD = \(\sqrt{P_h/P_1}\) and THD+N = \(\sqrt{(P_{tot}-P_1)/P_1}\) are RMS-amplitude ratios relative to the fundamental; SNR = \(10\log_{10}(P_1/P_{noise})\) excludes harmonics; SINAD = \(10\log_{10}(P_1/(P_{tot}-P_1))\). The DC band (bins 0…half-width) is excluded everywhere. Peak frequency uses parabolic interpolation of the log magnitude; amplitude interpolation is approximate (≈1–2 % for Hann).

Order tracking needs a measured speed reference. Tach mode detects rising threshold crossings (linear sub-sample timing) and assumes constant speed between pulses; angle mode unwraps a shaft-angle channel. The vibration channel is interpolated (4-point cubic) at uniform shaft angle, then transformed: the axis is order (cycles per revolution) with resolution 1/revolutions. Orders above (Fs/2)/(max rev/s) were never captured by the source sampling and are flagged. An ordinary FFT of a run-up smears orders and is not order tracking.

The processed series is treated as a high-rate stand-in for the analog input. "Without AA" keeps every M-th sample; "with AA" first applies a causal 8th-order elliptic low-pass (0.1 dB ripple, 80 dB stopband) at the chosen fraction of the new Nyquist. Both are then quantized by an ideal mid-tread converter (LSB = 2·FS/2^bits, clipping beyond ±FS). Ideal model only: no aperture jitter, DNL/INL or thermal noise. Alias of f at Fs: \(|f-F_s\,\mathrm{round}(f/F_s)|\).