Covariance Method — Control Systems/Spectral Measurements

Covariance Method

Control Systems / Spectral Measurements

Estimates the power spectral density of a signal frame parametrically: it fits an order-p autoregressive model by the covariance method – exactly as the Covariance AR Estimator block does – and evaluates the spectrum that model implies,

P_xx[k] = G·Ts / |A(e^j2πk/nfft)|², k = 0 … nfft−1,

where A is the prediction-error filter, G the model's error variance and Ts the time between samples of the series. This is MATLAB's pcov with 'twosided' and a sampling frequency of 1/Ts.

It fits the forward predictor x[n] ≈ −Σa_ix[n−i] by least squares over the N−p rows that fit wholly inside the frame (no windowing), and the error variance is the mean squared residual over those rows.

Ports

• u – the signal frame x, an [N,1] column; N must be at least 2p – MATLAB's arcov rule; below it there are fewer prediction rows than unknowns and the normal equations are singular.
• Pxx – the power spectral density, [nfft,1]: all nfft bins, two-sided, bin k at frequency k/(nfft·Ts). Units of power per hertz when Ts is in seconds.

Parameters

• Estimation Order – p, a whole number of 1 or more. Defaults to 6, as Simulink's does.
• FFT Length – nfft, the number of frequency bins, a whole number of 1 or more. Defaults to 256. It need not be a power of two. An nfft smaller than p+1 samples the spectrum of A wrapped modulo nfft rather than truncated – measured against the Simulink block, which does the same.
• Series Sample Time (s) – Ts, the time between samples of the series the frame was cut from, a positive scalar. Defaults to 1. It only scales the answer: halving Ts halves every bin.
• 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. The frame length and the order are baked into the core at export time, and so are the FFT length, Ts and the nfft-point cosine and sine tables.

The three HDL targets are simulation-only: they carry the arithmetic in real and quantize only at the port boundary. The solve divides by elimination pivots that are not bounded away from zero, and the spectrum divides by |A|², which a Q16.16 datapath does not survive. The cores simulate correctly and are not offered as synthesizable.

Simulink bridge

Import and export, mapped to dspspect3/Covariance Method – the DSP System Toolbox block. "Estimation Order" to ord, "FFT Length" to fftsize and "Series Sample Time (s)" to Ts.

inheritFFT = off and inheritTs = off are always emitted. With inheritFFT on the Simulink block picks its own length, and with inheritTs on it takes Ts from the frame period divided by N – measured: a 16-sample frame arriving every second scales every bin by exactly 1/16. This block runs one frame per step and has no frame period to divide, so Ts is always the dialog's value. Importing a model whose block inherits its Ts keeps the block's dialog Ts, so its spectrum can come out scaled differently from Simulink's, by exactly that ratio.

"Sampling Time (s)" does not cross. dspspect3/Covariance Method defines no SampleTime parameter at all – verified against the R2026a block dialog – and set_param on a parameter a block does not define is a hard error that aborts the whole generated script.

Notes

• Stateless and algebraic: the whole solve runs on this sample's frame. Nothing is carried between steps.
• An all-zero frame gives NaN, as the Simulink block does – measured on the estimator: A = [1 NaN …], G = NaN, so every bin of the spectrum is NaN. There is no guard to switch on; feed it a frame with energy in it.
• The frame rule is stricter than the Simulink block's. That block accepted p = 7 on a 12-sample frame, where only 5 prediction rows exist for 7 unknowns; this one refuses anything MATLAB's arcov refuses, because below that rule the answer is a number rather than an estimate.
• The normal equations are solved by Gaussian elimination with partial pivoting. The Simulink block uses a Cholesky factorization and arcov a QR; on any frame this block accepts, the three agree to rounding.
• Two-sided, every bin. Bins 1 … nfft−1 mirror about nfft/2 for a real frame; nothing is folded or doubled. For the one-sided density pcov gives by default, double bins 1 to nfft/2−1.
• The three HDL targets' output port is Q16.16, whose largest value is 32767. The spectrum is G·Ts/|A|² and nothing keeps A's zeros off the unit circle (the covariance methods least of all) – a sharp enough peak overflows that port on those three targets while the other seven carry it. Scale Ts down to keep every bin in range; it only scales the answer.
• Reach for the Covariance AR Estimator block for the model itself rather than its spectrum.