Power and cross-spectral estimation

Use this page when Fourier coefficients must become an amplitude, power, or power-density estimate with explicit units and sidedness.

The scientific question¶

The squared magnitude of a DFT coefficient is not yet a fully specified scientific observable. Its meaning depends on the DFT normalization, cadence, window, record length, sidedness, detrending, averaging, and whether the output is per bin or per unit frequency.

For a real signal, negative-frequency coefficients mirror positive-frequency coefficients. A one-sided representation may combine their power, but DC and an even-length Nyquist bin have no distinct negative partner and must not be doubled.

Mathematical objects¶

Let $X_k$ be the DFT defined on the signal-axes page and let $f_k$ identify its frequency bin. An amplitude spectrum has the observable’s units. A power per bin has squared observable units. A power spectral density has squared observable units per frequency.

A periodogram is a finite-record estimator. Segment averaging can reduce variance while changing frequency resolution and introducing choices about overlap and segment windows. Those choices are estimator metadata, not incidental array details.

Core derivation¶

For the unwindowed DFT convention on a record with cadence $\Delta t$ , define a two-sided density whose sum over bins satisfies a Parseval-compatible account:

P_k^{(2)}=\frac{\Delta t}{N}|X_k|^2, \qquad \sum_{k=0}^{N-1}P_k^{(2)}\Delta f =\frac{1}{N}\sum_{n=0}^{N-1}|x_n|^2, \qquad \Delta f=\frac{1}{N\Delta t}.

(1)

For a real signal, a one-sided density follows from (1) by retaining nonnegative frequencies and doubling only bins with a distinct negative-frequency partner:

P_k^{(1)}= \begin{cases} P_k^{(2)}, & k=0, \\ P_k^{(2)}, & k=N/2\ \text{for even }N, \\ 2P_k^{(2)}, & \text{otherwise on the positive-frequency branch}. \end{cases}

(2)

Therefore an odd-length one-sided spectrum doubles every strictly positive bin: there is no Nyquist endpoint. An even-length spectrum instead leaves both DC and its $k=N/2$ Nyquist endpoint undoubled. This is the endpoint consequence of the even- and odd- $N$ axes on (2).

A window replaces the unwindowed normalization with its declared coherent or power gain. That correction must be stated rather than inferred from the output name.

What the ecosystem already owns¶

JAX FFT owns real-input FFT mechanics, and JAX owns convolution mechanics. It does not decide whether a returned array represents amplitude, energy, power per bin, or power density.

What Jaxstro may add¶

The proposed jaxstro.signal may define records that carry sidedness, cadence, frequency units, DFT normalization, window correction, detrending, segment plan, and density units. It may provide validation helpers for Parseval accounts and one-sided endpoint handling.

The module would delegate FFT execution to JAX and would not treat estimator variance reduction as new physical information.

Evidence required before implementation¶

Readiness would require:

analytic sinusoid tests for amplitude and power under each supported convention;
Parseval checks for real and complex signals at odd and even lengths;
one-sided versus two-sided integral parity, including DC and Nyquist bins;
white-noise tests for density units and window ENBW corrections;
segment and overlap fixtures with deterministic averaging metadata;
dtype, precision, jit, and batch-shape tests; and
provenance round trips for every normalization choice.

Claim boundary¶

No planned estimator is implemented, and this page does not report detection or uncertainty performance.

Connected foundations and methods¶

Use Signal axes, cadence, and units for DFT and bin conventions, Windows and spectral leakage for coherent gain and equivalent noise bandwidth, and Phase and delay for cross spectrum and delay. Probability language is developed in Probability and distributions, while fixed accumulation appears in Sampled Newton-Cotes integration.