Magnitude Squared Coherence — Control Systems/Spectral Measurements

Magnitude-Squared Coherence

Control Systems / Spectral Measurements

Welch's magnitude-squared coherence between two scalar streams, over a sliding window: C_xy(f) = |P_xy(f)|² / (P_xx(f)·P_yy(f)), one value per frequency bin and always between 0 and 1. This is MATLAB's mscohere(x, y, window, noverlap, nfft), transcribed from its source.

It answers how much of y at each frequency is linearly explained by x: near 1 where the two are related by a fixed gain and phase, near 0 where they are independent or the relation is nonlinear. The block keeps the last W samples of each input, splits them into K overlapping segments, windows and transforms every one, and averages the three spectra before dividing.

Ports

• x – the first stream, a scalar. The reference, in the sense that the answer says how much of y it explains.
• y – the second stream, a scalar.
• Cxy – the coherence, a column of L/2 + 1 entries: one per frequency bin from DC to Nyquist, where L is the segment length. Entry m is the bin at m/L cycles per sample – m·f_s/L hertz at a sampling rate of f_s.

Parameters

• Segment Length – L, the transform length, an even whole number from 4 to 32. It sets the frequency resolution and the number of output bins.
• Segments – K, how many segments are averaged, from 2 to 8. One is refused: with a single segment the ratio above is an identity and the answer is a flat 1 on every signal. More segments mean a steadier estimate and a longer window.
• Segment Overlap – how far the segments overlap.
  • None (0%) – the hop is L.
  • Half (50%) – the hop is L/2. MATLAB's default, and this block's.
  • Three quarters (75%) – the hop is L/4; needs L divisible by four.
  The window the block holds is W = L + (K−1)·hop samples long.
• Window – the taper applied to each segment: Hamming (MATLAB's default), Hann or Rectangular. Both tapers are the symmetric form, which is what hamming and hann return unless asked otherwise.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. Every transform is a fixed weighted sum of window taps, and the weights are computed once at export time, so no emitted core contains a sine, a cosine or a loop over data. The two windows become shift registers. Nothing is exposed as a tunable parameter: changing the window or the segmentation changes the whole coefficient table, which is structural.

The three HDL targets are simulation-only: they carry the arithmetic in real and quantize only at the port boundary. The answer is a ratio of two run-time quantities, each a sum of squares spanning several decades, which does not belong in a Q16.16 datapath. VHDL additionally carries each bin in an emitted function, because a clocked process here has no place to keep the eight running sums a bin needs.

Simulink bridge

No equivalent (Support::None), so nothing crosses in either direction, and no parity testbench is owed. Measured rather than assumed: mscohere is a Signal Processing Toolbox function, that toolbox ships no Simulink library at all, and a find_system sweep with LookUnderMasks over the DSP System Toolbox and Simulink library roots matched no coherence block – that library has a cross-spectrum estimator and nothing that divides by the two auto-spectra. No configuration of it crosses either, including "Sampling Time (s)", which has no counterpart to be written to. Code export verification still covers the block across all ten languages.

Notes

• Stateful and inherently discrete: two shift registers of W samples, advanced once per sample. There is no derivative to integrate, so the block runs on the discrete path whatever the model's solver is (setDiscreteOnlyBlock(true)), and its period comes from its own "Sampling Time (s)" – or, when that is zero or less, from the model's global sampling time. The first W−1 samples of a run are read against a zero-prefilled window, as every sliding-window block here is, so the estimate is only complete once the window has filled – and until the second segment holds a sample, it necessarily reads 1.
• The averaging IS the measurement. A single segment gives |X*Y|² = |X|²|Y|² bin by bin – Cauchy-Schwarz with equality – so K = 1 would report perfect coherence for any two signals whatever. That is why one segment is refused rather than allowed.
• Welch's normalizing constants are not computed, because they cancel: the segment count and the window power each appear once above and once below. The answer matches MATLAB's; the intermediate sums do not.
• It measures a LINEAR relation. A perfectly repeatable nonlinear link reads low, and that is the finding, not a fault.
• A silent window answers zero, not a NaN: the denominator is floored, and below the floor the bin reports 0.
• Related blocks. Correlation And Convolution / Cross-Correlation answers the same question in the time domain and over all lags at once; Spectral Measurements / Band Power reports how much power a band carries, which this block deliberately says nothing about.