circular_autocorrelation

std.seq.circular_autocorrelation · Level L4

The circular autocorrelation of a real signal, rₖ = Σⱼ xⱼ·x₍ⱼ₊ₖ₎ mod n, by the Wiener–Khinchin theorem: the inverse transform of the power spectrum.

r = ifft(|fft(x)|²)

Signature

circular_autocorrelation(x: f64[n]) → f64[n]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Agrees with the reference Σⱼ x[j]·x[(j + k) mod n], summed directly to 80 digits (100-digit arithmetic), on all 40 test cases.
All 143 float64 results inside the running error bound; the closest uses 36% of it.
Interpreter and NumPy backend return bit-identical results.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 46%
bit-equal to the NumPy formula in float64: 45%
largest error, in units in the last place: 1.9e+8

Large ulp counts appear where a result is tiny next to the numbers it is computed from (after cancellation, for example), so one unit in the last place is tiny too; the absolute error is still inside the bound. Results within their own error of zero are not counted.

Note

The reference is the correlation sum itself, with no transform at all.

Identity

Calls

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Called by

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sha256:f81611b049ad6ee54bd4388f28621ace99896072a3d2fd7ac71b50e03b2ea744

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