Uniqueness of the discrete Fourier transform

doi:10.1016/j.sigpro.2023.109041

Signal Processing

Volume 209, August 2023, 109041

https://doi.org/10.1016/j.sigpro.2023.109041 Get rights and content

Highlights

•
The discrete Fourier transform has the fundamental property of carrying convolution into componentwise product.
•
Is it the only linear and invertible function with this property?
•
The answer is negative: in $C^{M}$ , there are $M!$ such functions including the DFT.
•
However, in another point of view, the answer is positive up to a permutation.

Abstract

The discrete Fourier transform has the fundamental property of carrying convolution into componentwise product. A natural question is whether it is the only one with this ability. The answer is positive in some sense. This uniqueness is little known and its proof [1] might be difficult to access for users of mathematics, engineers and physicists. The ambition of this communication is to rewrite the proof of this fundamental property of uniqueness in a detailed way for a varied scientific audience.

Graphical abstract

Introduction

The Discrete Fourier Transform (DFT) is a fundamental tool for engineers thanks to its numerous properties [2]: linearity, inversion, periodicity, translation,... It brings a spectral point of view which is complementary to the temporal vision. There is a very powerful algorithm to compute it (FFT). Its ability to condense energy at low frequencies makes it an essential compression tool in the digital world.

The proof of its properties is done from its analytical formulation (1) and it uses classical mathematical techniques for engineers/physicists such as substitution rules or manipulation of indices. In this article, we propose a property whose proof is not straightforward: the DFT is unique in the sense that it is the only linear and invertible map, modulo a permutation of indices, which carries convolution into the componentwise product.

We found three references on the subject. The first one is an unpublished article [3] which approaches the problem from the perspective of computational complexity. However, it is easy to find a counterexample which invalidates the core of this work (section 3 of [3]) and therefore its conclusion.

The second paper (section II of [4]) presents an algebraic proof in the general framework of rings. However, to state the equation (9) of [4], the authors made two arbitrary choices (called “without loss of generality” in the paper) that hide the exact meaning of the uniqueness (i.e. up to a permutation). The choices are not easy to overcome to find all the invertible and linear maps with the convolution property.

The only other reference to our knowledge is a book from the Springer series “Computational Music Science” [1]. The proof given there on page 8 is original. The presentation of the proof is concise and might discourage a reader who is not a mathematician by training. It overlooks certain points and we noted some easily rectifiable clumsiness of notation. On the other hand, the converse of the theorem is not mentioned.

Our ambition is to rewrite and complete the proof of this fundamental property of uniqueness from Amiot [1] in a detailed and relevant way for a larger scientific audience.

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Section snippets

Prerequisites

It is enough to know that the DFT is a linear and invertible map which transforms the convolution into a term-by-term product. To make this communication self-containing and to specify notation, we recall the definitions in this section. The mathematical tools used in this communication are not beyond the reach of a user of mathematics, physicist or engineer: the notion of linear and invertible map (i.e. invertible matrices), the canonical basis for a complex vector of dimension

M

, and index

Uniqueness of the discrete Fourier transform

Theorem 1

Let

F

be the DFT map. The only linear and invertible maps from

C^{M}

C^{M}

which carry convolution

*

into componentwise product

\times

are written as

G = H \circ F

for some permutation matrix

H

. Moreover, for any permutation

H

, the map

H \circ F

satisfies (8).

H \circ F (f * g) = H \circ F (f) \times H \circ F (g)

Since the number of permutations of

M

elements equals

M!

, Theorem 1 shows:

Corollary 2

There exists exactly

M!

linear and invertible maps with the same fundamental property as the DFT with respect to convolution.

For instance, for a discrete signal of

Illustration

The formula (1) is generalised by (24) which is the expression of the map

F^{σ} = H \circ F

from

f_{m}

F_{u}^{σ}

with

σ

the permutation associated to

H

F_{u}^{σ} = \sum_{m = 1}^{M} f_{m} e^{- 2 i π \frac{(m - 1) (σ^{- 1} (u) - 1)}{M}} with 1 \leq u \leq M

These matrices, expressed in the canonical basis, can be obtained thanks to Eq. (24) or by multiplying the permutation matrices by the Fourier matrix

A

(3). For example, in

C^{4}

, there are 24 maps

F^{σ} = H \circ F

. In particular, the Fourier matrix in

C^{4}

corresponds to the identity permutation.

Conclusion

We present here in a detailed manner the proof of a fundamental and non-trivial property of the discrete Fourier transform: this is the only linear and invertible map, up to a permutation, which carries the convolution into the componentwise product.

The proof of the implication is composed of two steps. First, we note that, for any linear and invertible map

G

with the same property, the map

H = G \circ F^{- 1}

is a linear and invertible map which preserves the componentwise product. Then, we determine all

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (4)

E. Amiot
Music Through Fourier Space
(2016)
I. Amidror
Mastering the Discrete Fourier Transform in One, Two or Several Dimensions
(2013)

There are more references available in the full text version of this article.

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¹: orcid:0000-0002-4365-8270

²: orcid:0000-0003-2447-5118

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Signal Processing

Highlights

Abstract

Graphical abstract

Introduction

Access through your organization

Section snippets

Prerequisites

Uniqueness of the discrete Fourier transform

Illustration

Conclusion

Declaration of Competing Interest

References (4)

Music Through Fourier Space

Mastering the Discrete Fourier Transform in One, Two or Several Dimensions

Cited by (8)

TFLM: A time-frequency domain learning model for underwater acoustic signal reconstruction

A denoising method for ECG signals based on CEEMDAN-TSO and stacked sparse autoencoders

Research on non-stationary characteristic test and decomposition for dynamic response of floating structures

Next Generation Human Action Recognition: A Comprehensive Review of State-of-the-Art Signal Processing Techniques

Weakly Supervised Object Detection for Diabetes Retinopathy Combining Spatial and Frequency Domains

En-NBOR: An Enhanced Newton-Raphson-Based Optimizer for Underwater Acoustic Signal Reconstruction

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Short communicationUniqueness of the discrete Fourier transform

Highlights

Abstract

Graphical abstract

Introduction

Access through your organization

Section snippets

Prerequisites

Uniqueness of the discrete Fourier transform

Illustration

Conclusion

Declaration of Competing Interest

Music Through Fourier Space

Mastering the Discrete Fourier Transform in One, Two or Several Dimensions

Short communication
Uniqueness of the discrete Fourier transform