Instantaneous Bandwidth — Control Systems/Spectral Measurements

Instantaneous Bandwidth

Control Systems / Spectral Measurements

The second central moment of a power spectral density – how widely its mass is spread about its own centre – scaled by a constant:

b = s·√(Σ_k(f_k − f̄)²·P_k ÷ Σ_kP_k), with f̄ = Σ_kf_k·P_k ÷ Σ_kP_k

This is MATLAB's instbw given a time-frequency distribution, and Mean Frequency is the first moment of the same spectrum – the two are the centre and the width of one distribution. The block is three weighted sums, two divisions and a square root: no branch and nothing carried between samples.

Ports

• p – the power spectral density, one nonnegative value per bin. A vector, either an [N,1] column or a [1,N] row, with N between 2 and 256. Bin k is the frequency k·Δ.
• b – the bandwidth, in the same units as Bin Spacing times Scale Factor. Always scalar, whatever the input's length.

Parameters

• Bin Spacing – Δ, the frequency step between neighbouring bins, a positive number. Left at 1 the answer comes out in bins; set it to f_s÷(2·(N−1)) for a one-sided spectrum reaching the Nyquist frequency and the answer comes out in Hz.
• Integration Range – which bins the moments are taken over:
  • Full spectrum – every bin. This is instbw(P, F, T), and it is the default.
  • Frequency band – only the bins inside the band. This is instbw(..., 'FrequencyLimits', [flo fhi]).
• Frequency Band – [f_lo f_hi], a two element vector in the same units as Bin Spacing, read only in Frequency band mode. The band is narrowed to the bins it contains: a bin counts when f_lo ≤ f_k ≤ f_hi, which is instbw's own rule and Mean Frequency's. Band Power snaps the other way – see Notes.
• Scale Factor – s, a positive number the moment is multiplied by. The default 3.5449077018110318 is 2·√π, which is instbw's own default, measured rather than read off a page. Set it to 1 for the plain standard deviation of the spectrum.
• Epsilon – ε, added to the total power before each division. It is what makes an all-zero band return 0 instead of a NaN – a degeneracy guard, not an accuracy knob, so the default is 1e−30: far below any total power a real spectrum carries, and small enough not to move the answer.
• 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.

All three weight vectors are structural and are inlined into the arithmetic at export time rather than exposed as tunable parameters – they follow from the bin spacing, the bin count and the band, and changing any of them changes how many multiplies the core contains, which no runtime parameter can do. Scale Factor and Epsilon are inlined too. Re-export after changing any of them.

The three HDL targets are simulation-only: the block divides twice and takes a square root, and neither belongs in this tree's Q16.16 fixed-point base. They compute in real and quantize only at the port boundary, so they simulate correctly and are not offered as synthesizable.

Simulink bridge

No equivalent (Support::None). Signal Processing Toolbox ships no Simulink library at all, and instbw is one of its MATLAB functions. The bridge reports this block rather than dropping it silently, and it therefore has no parity testbench. Code export verification covers it across all ten languages.

Notes

• Stateless. The answer depends on this sample's spectrum and nothing else, so there is no startup transient and no history to seed.
• ⚠ This is not the Spread that Spectral Features reports. That one weights by the magnitude spectrum and centres on its own magnitude centroid; this one weights by power. On the same data the two differ in the first decimal – 2.0637 against 1.5146, from centroids 4.1603 and 4.0643 – so they are different statistics of one signal rather than two spellings of one.
• ⚠ The emitted form is the raw-moment identity, and it costs a cancellation. f̄ is not known until the spectrum arrives, so a centred sum would need it inside every term; the block emits (A − (B÷D)·B) ÷ D with A = Σf²P, B = ΣfP and D = ΣP + ε instead, which is three sums rather than one sum per bin. Measured on an eight-bin spectrum the two forms differ by 5.3e−15 on a moment of 2.294. The error grows when the spread is small against the centre frequency: a spectrum concentrated in one bin of a high-frequency axis is where to be careful.
• The moment is taken through an absolute value before the square root. The identity above can land a hair below zero when the spread is at the edge of what the arithmetic can resolve, and a negative under a square root is an error in several of the ten targets rather than a NaN. Taking the magnitude of a quantity that cannot truly be negative removes the question without a branch.
• The band is narrowed, not widened – and Band Power does the opposite. This block keeps the bins inside the band, as instbw and Mean Frequency do; Band Power keeps the bins that bracket it.
• An all-zero band answers 0, where MATLAB answers NaN. That is the deliberate divergence Epsilon exists for, and it is the same one Mean Frequency makes for the same reason.
• No state space. A ratio of weighted sums is not linear in the input, so there is no A/B/C/D pair and model reduction correctly declines to merge the block.