ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Translation and reflection ambiguities for a finite Fourier magnitude map

Example

Let $C_{3} = Z / 3 Z$ , put $ω = (- 1 - i \sqrt{3}) / 2$ , and for a real signal $f : C_{3} \to R$ define its unnormalised discrete Fourier transform by $\hat{f} (k) = \sum_{n = 0}^{2} f (n) ω^{k n}$ for the representatives $k = 0, 1, 2$ . Its Fourier magnitude data are $(∣ \hat{f} (k) ∣)_{k = 0}^{2}$ . Define translation by $(τ_{1} f) (n) = f (n - 1)$ and reflection by $(R f) (n) = f (- n)$ , with indices in $C_{3}$ . For $f (0) = 1$ , $f (1) = 2$ , and $f (2) = 0$ , both $τ_{1} f$ and $R f$ have the same Fourier magnitude data as $f$ , while neither equals $f$ or $- f$ .

Example references

Philipp Grohs, Sarah Koppensteiner, and Martin Rathmair, finite Fourier translation and conjugate-reflection ambiguities, Proposition 3.3

Facts & Assumptions

Given: The cyclic group $C_{3}$ , $ω = (- 1 - i \sqrt{3}) / 2$ , the stated DFT convention, and the real signal $f$ in the example.

[F1]

Complex arithmetic is a field operation system; conjugation and modulus are given by $\overline{a + b i} = a - b i$ and $∣ a + b i ∣ = \sqrt{a^{2} + b^{2}}$ , and conjugation respects sums, products and modulus ( $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ , Real and imaginary parts, complex conjugation, and modulus, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F2]

The nonnegative square root of $3$ squares to $3$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

Verification

1.1givenF1F2algebra

Write $s = \sqrt{3}$ . By [F1] and [F2], $ω^{2} = (- 1 + i s) / 2$ , $ω^{3} = 1$ , and $∣ ω ∣^{2} = 1$ , hence $∣ ω ∣ = 1$ . Reindexing the finite sum with $j = n - 1$ gives $\hat{τ_{1} f} (k) = \sum_{j} f (j) ω^{k (j + 1)} = ω^{k} \hat{f} (k)$ . Multiplicativity of the modulus [F1] gives $∣ ω^{k} ∣ = 1$ , so every Fourier magnitude is unchanged.

2.1step 1.1F1algebra

Since $∣ ω ∣ = 1$ , [F1] gives $\overline{ω} = ω^{- 1}$ . Reindexing with $j = - n$ gives $\hat{R f} (k) = \sum_{j} f (j) ω^{- k j}$ . Because every $f (j)$ is real, this equals $\overline{\sum_{j} f (j) ω^{k j}} = \overline{\hat{f} (k)}$ by the conjugation laws in [F1], so reflection preserves every Fourier magnitude.

3.1step 1.1step 2.1givenconstructalgebra∎

The support of $f$ and $- f$ is ${0, 1}$ , while the supports of $τ_{1} f$ and $R f$ are ${1, 2}$ and ${0, 2}$ respectively. Thus neither transformed signal equals $f$ or $- f$ . Combined with the invariances in steps 1.1 and 2.1, both transformed signals are distinct from the original up to global sign but have identical Fourier magnitude data.

Depends on

Used by

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Sources

Philipp Grohs, Sarah Koppensteiner, and Martin Rathmair, Phase Retrieval: Uniqueness and Stability, arXiv:1901.07911v2, Proposition 3.3 (standard reference, not scraped)