Signal axes, cadence, and units

Use this page when uniformly sampled measurements must be mapped from time indices to physically interpretable Fourier frequencies.

The scientific question¶

A sampled signal is evidence recorded on an axis. Before computing a spectrum, one must know the sample cadence, duration, axis units, missing-data policy, and whether sampling is actually uniform. Those choices determine which frequencies are represented and which cannot be distinguished.

For $N$ uniformly spaced samples with cadence $\Delta t$ , the sampled duration is $T=N\Delta t$ under the periodic-record convention. The Fourier-bin spacing is $\Delta f=1/T$ . For sample frequency $f_s=1/\Delta t$ , $f_s/2$ is the Nyquist limit, but content above it can alias into the represented band. The limit is not always a sampled Fourier bin: even $N$ includes $k=N/2$ at $f_s/2$ , whereas odd $N$ ends its nonnegative branch at $k=(N-1)/2$ , at $(N-1)f_s/(2N)$ , and has no Nyquist bin.

Mathematical objects¶

Let $x_n=x(t_n)$ with $t_n=t_0+n\Delta t$ for $n=0,\ldots,N-1$ . The sample axis carries units of time, while $x_n$ carries the observable’s units. Frequency has inverse-time units. A Fourier coefficient, amplitude spectrum, power spectrum, and power spectral density have different normalizations and units and must not be called simply “the spectrum.”

Negative-frequency ordering and the treatment of the Nyquist bin depend on whether $N$ is even and whether a full complex or real-input transform is used.

Core derivation¶

Choose the forward discrete Fourier transform (DFT) sign and leave its inverse normalization explicit:

X_k=\sum_{n=0}^{N-1}x_n \exp\left(-2\pi i\frac{kn}{N}\right), \qquad x_n=\frac{1}{N}\sum_{k=0}^{N-1}X_k \exp\left(+2\pi i\frac{kn}{N}\right).

(1)

The axis paired with (1) is:

f_k=\frac{k}{N\Delta t}, \qquad \Delta f=\frac{1}{N\Delta t}, \qquad f_{\mathrm{Nyq}}=\frac{1}{2\Delta t}.

(2)

For the full DFT, bins above the nonnegative half are interpreted as negative frequencies after subtracting the sampling frequency $f_s=1/\Delta t$ . The frequency equation does not make irregular samples uniform or recover aliased content.

This parity distinction controls one-sided endpoint handling. DC is always an unpaired endpoint, but $f_s/2$ is an unpaired endpoint only for even $N$ . For odd $N$ , every strictly positive real-FFT bin has a distinct negative-frequency partner. The corresponding power-doubling rule is stated in (2).

What the ecosystem already owns¶

JAX FFT owns FFT array mechanics, and JAX owns convolution mechanics. Those functions compute transforms; they do not select physical cadence, frequency units, normalization, missing-data policy, or a scientific claim.

What Jaxstro may add¶

The proposed value of jaxstro.signal is an explicit signal-axis record with cadence, duration convention, units, real or complex layout, frequency ordering, normalization metadata, and provenance. It may validate uniformity and construct frequency coordinates that agree with the declared DFT convention.

The module is planned only. It would not replace JAX FFT or silently repair irregular, gapped, or aliased data.

Evidence required before implementation¶

Readiness would require:

exact bin comparisons with JAX frequency helpers for odd and even lengths;
sinusoid fixtures at DC, interior bins, and the Nyquist boundary;
unit checks connecting cadence, frequency, and transformed quantities;
round-trip tests for the declared DFT and inverse normalization;
explicit rejection or handling tests for irregular and missing samples;
alias demonstrations with known above-Nyquist inputs; and
provenance round trips for cadence, origin, layout, and convention.

Claim boundary¶

This page defines conventions, not a detector, estimator, or implemented Jaxstro signal runtime.

Connected foundations and methods¶

Review Functions, units, and scales for axes and units and From mathematical relations to differentiable programs for the mathematical-map versus executed-program distinction. Continue to Windows and spectral leakage, Power and cross-spectral estimation, and Phase and delay. Current finite-grid ideas also appear in Grids and conservative bin transfer.