How a PSD means its area

For a power spectral density, the area under the curve is the measurement. The RMS of the signal is the square root of the area under its PSD, a specification's overall level is the area under its breakpoints, and octave banding is the same area divided up differently. So Visual Dynamics stores how a value is read between the points as one property of each object, interpolation, and both the integral and the plotted curve come from it. A spectrum is never drawn one way and integrated another.

There are two readings, 'bin' and 'log_log'. Every figure below is drawn by the same call the application uses (tools/make_concept_figures.py regenerates them).

'bin': constant across its own bin

A computed PSD is a density per analysis bin: each line's value is constant across its bin, the area is the sum of value times width, and the spectrum is drawn as steps. For a narrowband spectrum the bin edges are the midpoints between neighboring lines, which is exact. In this example the values of ten coarse lines sum to 10 over 10 Hz bins, so the area is exactly 100 (m/s²)² and the RMS is exactly 10 m/s². The steps show that sum directly.

The same power on octave bands

Banding onto proportional bands (the Octave Bands reading on the plot bar and its Apply Octave Bands button, or project.compute_octave in a script) conserves the area: the same power, arranged by band. A specification is banded the same way, with its warning and abort limits. A curve written at breakpoints contributes the exact power-law area over each band, a controller's target on lines is integrated as a density, and the result is a specification on the bands, to be compared band for band against a measurement banded the same way. An octave-band spectrum sets bandwidth, because its bins are geometric and its edges come from the standard, not from its neighbors. Edges inferred from the band centers would be slightly off at every band and wrong at the two ends. The steps land on the standard's own edges, and the RMS is still exactly 10 m/s².

'log_log': a power law between breakpoints

A specification written at a few breakpoints is a continuous requirement. Between two breakpoints it is the straight line the two points make on log-log axes, which is a power law, and its area has a closed form. It is drawn as that curve (on the linear frequency axis below, the power-law segments show their true curvature), and its area() integrates the same power law: 6.06 m/s² RMS for this one.

The reading belongs to each object, not to its class. A specification written at a dozen breakpoints is a curve, but a specification computed from a record, such as a transient target's PSD, is a density like any other and carries 'bin'.

Checking it

psd.area()                  # the one integral, read as drawn
psd.area(low=20.0, high=2000.0)      # over a band
rms = psd.area() ** 0.5

banded = psd.to_octave(6)   # sixth octaves, conserving the area
assert abs(banded.area() - psd.area()) < 1e-9 * psd.area()

area() is the only integral, and it reads a spectrum the same way it is drawn; nothing outside the object makes that choice. It integrates one record, the first unless record= says otherwise.

In the window, a specification selected on its own opens as its spectra. RMS on the plot bar is the other reading: one bar per channel showing the level the specification asks for (the square root of that same area), with a table of the numbers beneath. Nothing is colored, since a specification on its own has nothing to be compared against. Records picked in the tree restrict both readings. A specification selected together with a measurement offers the comparison's four readings instead: Spectra, RMS Error, Lines Out (How a comparison is judged) and Chance (Lines outside a band by chance).