Welch PSD — Control Systems/Spectral Measurements

Welch PSD

Control Systems / Spectral Measurements

The one-sided power spectral density of the signal, by Welch's method – the average of K windowed periodograms taken from overlapping segments of a running window:

P(f) = (1 ÷ (K·f_s·Σw²)) · Σ_k |FFT(w·segment_k)|²

This is MATLAB's pwelch. Averaging is what separates it from a single periodogram: one periodogram's variance does not fall however long the record, and K of them averaged reduce it by K.

Ports

• u – the sampled signal. Scalar: one channel and its own window.
• p – the density, [L/2+1, 1], one-sided with DC first and Nyquist last. This is the vector every other block in this family takes on its own p port, so it wires straight into Band Power, Occupied Bandwidth, Spectral Entropy and the rest.

Parameters

• Segment Length – L, the transform length. An even whole number from 4 to 32; even because a one-sided spectrum needs a Nyquist bin. Bounded above because the transform is unrolled at export and its cost grows as L².
• Segments – K, how many are averaged, from 1 to 8. More segments cut the variance and lengthen the window the block looks back over. K = 1 is allowed and is an ordinary windowed periodogram – it is not refused here, unlike on Magnitude-Squared Coherence, because a single periodogram is still a meaningful density.
• Segment Overlap – how far each segment is advanced:
  • None (0%) – segments do not share samples.
  • Half (50%) – MATLAB's default, and the one that recovers the most information from a given window.
  • Three quarters (75%) – more segments from the same samples, at the cost of correlating them.
  The window the block looks back over is L + (K−1)·hop samples.
• Window – Hamming (MATLAB's default), Hann or Rectangular. These are the symmetric windows, which is what MATLAB's hamming() and hann() return.
• Bin Spacing – Δf, the frequency step between output bins, in hertz. The sampling rate follows as f_s = L·Δf and is what makes the answer a density rather than a power. It is the same parameter, under the same name, that Band Power and the rest of this family read.
• 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.

No sine, no cosine and no division survives into a generated core. The window, the transform's basis and the whole 1÷(K·f_s·Σw²) scale are fixed by the configuration, so they are computed once at export time and a core carries the resulting numbers: two dot products per bin per segment, a square each, and one scale. All five settings are structural – they decide how many multiplies and how many outputs the core contains – so re-export after changing any of them.

The three HDL targets are genuine synthesizable Q16.16. ⚠ What to watch there is range rather than precision: the block squares, so an input that reaches ±180 already reaches the format's ceiling before any scaling, and a small Bin Spacing divides the answer down into the quantum from the other end. Scale the signal into roughly ±1 first.

Simulink bridge

None (Support::None). pwelch is a Signal Processing Toolbox function and that toolbox ships no Simulink library, so there is no path a diagram could name. The bridge reports this block rather than dropping it silently, and it therefore has no parity testbench; code export verification still covers it across all ten languages.

Notes

• Stateful, and discrete by nature (setDiscreteOnlyBlock(true)): one shift register of L + (K−1)·hop samples.
• ⚠ It is the producer this family was missing. Band Power, Occupied Bandwidth, Mean Frequency, Spectral Entropy, SNR, THD and the rest all take a spectrum on a p port and answer a measurement about it; this block makes that vector.
• ⚠ It is not FFT Magnitude, and the difference is not cosmetic. FFT Magnitude emits a single periodogram: full length rather than one-sided, unwindowed, and with no scaling at all, so it is a spectrum but not a density – and its variance does not fall however long the record is. Averaging K windowed segments is exactly what fixes that, and it is the whole point of Welch's method.
• ⚠ Three conventions are MATLAB's and each is a silent wrong answer if guessed. The divisor is K·f_s·Σw² and not the segment length; the interior bins are doubled while DC and Nyquist are not; and the windows are the symmetric ones, so hamming(8) starts at 0.08. All three were measured against R2026a rather than remembered – see this block's source banner for the five-bin comparison, which agrees to 6.9e−18.
• The window is zero-prefilled, so the first L + (K−1)·hop − 1 spectra of a run are a startup transient.
• Scalar only. One channel and its own history; wire one block per channel.
• No state space. |X|² is not linear in the window, so model reduction correctly declines to merge it.