Bilangan: Jurnal Ilmiah Matematika.
Kebumian dan Angkasa Volume.
3 Nomor.
Desember 2025 e-ISSN : 3032-7113.
p-ISSN : 3032-6389.
Hal.
DOI: https://doi.
org/10.
62383/bilangan.
Tersedia: https://journal.
id/index.
php/Bilangan Uncertainty Principle for the Hartley Transform:
Direct and FourierAeBased Approaches Andi Tenri Ajeng Nur1*.
Husnul Khotimah2.
Putri Nilam Cayo3 Department of Mathematics.
Universitas Sriwijaya.
Indonesia *Correspondence Authorr: tentri@unsri.
Abstract.
The Hartley transform provides a real-valued alternative to the classical Fourier transform, offering structural advantages for the analysis of real-valued signals.
This paper presents a systematic study of the continuous Hartley transform, including its definition, inversion formula.
Plancherel identity, and core operational properties such as shifting, modulation, and convolution.
The analytical framework is developed in parallel with the classical Fourier theory to highlight structural similarities and distinctions between the two transforms.
Furthermore, we establish a Hartley-type Heisenberg uncertainty principle using two complementary approaches:
a direct method based on intrinsic properties of the Hartley kernel, and a Fourier-based method that exploits the algebraic relationship between the Hartley and Fourier transforms.
These results provide a unified and rigorous foundation for understanding uncertainty relations within real-valued transform frameworks, and they demonstrate the continued relevance of the Hartley transform in harmonic analysis, integral transforms, and modern signal processing.
Keywords: Fourier transform.
Hartley transform.
Heisenberg inequality.
Signal Analysis.
Uncertainty principle.
INTRODUCTION
The Fourier transform is one of the most fundamental tools in mathematical analysis, signal processing, and communication theory.
Its ability to decompose signals into frequency components has led to powerful analytic methods used across science and engineering (Stein & Shakarchi, 2003.
Folland, 2.
Despite its widespread utility, the Fourier transform is inherently complex-valued, which may be unnecessary or computationally inefficient in applications where the underlying data are entirely real.
To address this issue.
Hartley introduced in 1942 a fully real-valued analogue of the Fourier transform (Hartley, 1.
The Hartley transform employs the real kernel ycaycayc.
= cos ycu sin ycu, producing a transform that is self-inverse and avoids complex arithmetic.
Its theoretical foundations and computational significance were later solidified through BracewellAos modern treatment (Bracewell, 1.
Subsequent work extended HartleyAos framework to multidimensional, generalized, and fast computational settings (Lohmann et al.
Bracewell, 1984.
Hargreaves, 1.
The discrete Hartley transform (DHT), first introduced in Bracewell .
, broadened the transformAos impact in digital signal processing, enabling efficient real-valued convolution, filtering, and fast algorithmic implementations (Feldman, 1999.
McLaren & Smith, 1998.
Martucci, 2.
Additional studies have highlighted its advantages in numerical integration, image processing, and real-valued filter design (Vlsek & Novyk, 1999.
Zadeh & Reibman.
Bose & Boo, 2.
Naskah Masuk: 29 Agustus 2025.
Revisi: 30 September 2025.
Diterima: 28 Oktober 2025.
Terbit: 03 Desember Uncertainty Principle for the Hartley Transform: Direct and FourierAeBased Approaches Beyond computational considerations, several authors have emphasized the structural and functional-analytic relationships between the Hartley and Fourier transforms, including equivalence of energy identities, symmetry properties, and harmonic-analytic behavior (Oppenheim & Willsky, 1.
More advanced works have connected Hartley-type transforms to generalized uncertainty principles and real-valued harmonic analysis (Cowling & Price.
Goh & Pfander, 1.
, providing improved understanding of localization and transformdomain constraints.
The purpose of this paper is to provide a rigorous and coherent presentation of the continuous Hartley transform and its key analytic properties.
Section 2 introduces the necessary functional-analytic preliminaries, including Lebesgue spaces and the Fourier transform.
Section 3 develops the Hartley transform, its inversion formula.
Plancherel identity, and core operational properties.
Section 4 establishes a Hartley version of the Heisenberg uncertainty principle using two approaches: a direct analytic proof and a Fourier-based method, emphasizing the structural parallels and distinctions between the two transform frameworks.
PRELIMINARIES
In this section we recall several basic definitions and notations used throughout the paper.
For 1 O yc O O, the Lebesgue space yayc (E.
consists of all measurable functions on Ey whose yayc -norm is finite.
Definition 2.
1 (The ycyee (E.
Spac.
The space yayc (E.
is defined as yc AnyceAnyayc (E.
= (O .
c yccy.
, 1 O yc < O, .
Ey and for yc = O.
AnyceAnyaO(E.
= ess supA.
ycOOEy .
The space ya2 (E.
is a Hilbert space with inner product yce, yciya2(E.
= O yce.
i yci.
Ey We recall the definition of the Fourier transform, which will be used extensively in later BILANGAN Ae VOLUME.
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Definition 2.
2 (Fourier Transfor.
For a function yce OO ya1 (E.
, the Fourier transform is defined by O E.
= ya.
= O yce.
yce Oeycycyc yccyc.
OeO Lemma 2.
3 (Inverse Fourier Transfor.
If ya = E.
OO ya1 (E.
O ya2 (E.
, then yce.
= O ya.
yce ycycyc yccyc.
2yuU Ey Using the identity yce Oeycycyc = cosA.
Oe ycsinA.
, the Fourier transform decomposes as .
= yaycI .
yaycI .
= O yce.
Ey yaya .
= Oe O yce.
Ey Lemma 2.
4 (Parseval's Identit.
For all yce, yci OO ya2 (E.
, the following identity holds:
O yce.
i yci.
yccyc = Ey O E.
i E.
2yuU Ey In particular.
AnE.
An2ya2(E.
2yuU AnyceAn2ya2(E.
= Lemma 2.
5 (CauchyAeSchwarz Inequalit.
For yc, yc OO ya2 (E.
|O yc.
O (O .
(O .
OeO OeO OeO A standard reference for this inequality is (Stein & Shakarchi, 2.
HARTLEY TRANSFORM AND ITS PROPERTIES
The Hartley transform serves as a real-valued analogue of the classical Fourier transform and provides a convenient framework for the analysis of real signals.
This section establishes the basic definition of the transform and develops several fundamental properties that form the analytical foundation for later results, including the inversion formula.
Plancherel identity, and operational rules.
Uncertainty Principle for the Hartley Transform: Direct and FourierAeBased Approaches Definition 3.
1 (Hartley Transfor.
Let yce OO ya1 (E.
O ya2 (E.
The Hartley transform of yce is defined by O .
= EU.
= O yce.
OeO where the ycaycayc is given by ycaycaycA.
= cos.
Example 1.
Consider the Gaussian function yce.
= yce ycayc ,ayca > 0, compute its Hartley transform Solution.
The Hartley transform of yce.
is defined by O = EU.
= O yce.
yccyc OeO = O yce ycayc .
)yccycA .
OeO By separating the integral into cosine and sine components, equation .
= O yce aOeO yccyc O yce ycayc sin.
aOeO ya1 .
ya2 To evaluate the cosine part ya1 , one can complete the square in the exponent, leading to O yuU ya1 = O yce ycayc cos.
yccyc = Oo Ayce Oeyc /.
, yca OeO .
where this uses the standard Gaussian integral O yuU O yce yca.
cOeycnyc/.
) yccyc = Oo .
OeO .
The sine part ya2 of equation .
vanishes due to symmetry, because yce ycayc is even and sin.
is odd:
ya2 = O yce Oeycayc sin.
yccyc = 0.
OeO By substituting equations .
, the Hartley transform of the Gaussian function becomes yuU ya.
= Oo yce Oeyc /.
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Thus, the Hartley transform preserves the Gaussian shape, analogous to the Fourier transform, which illustrates one of the convenient properties of the Hartley transform in signal analysis.
Figure 1.
Gaussian function.
Figure 2.
Hartley transform of the Guassian function.
Figure 1 illustrates the graph of the Gaussian function which exhibits a bellAeshaped curve symmetric about the vertical axis.
The function attains its maximum at yc = 0 and decays exponentially as yc moves away from the center.
This plot highlights the strong time-domain localization characteristic of the Gaussian.
Figure 2 shows the graph of the Hartley transform of the Gaussian function.
The resulting curve remains smooth, symmetric, and well-localized, reflecting the fact that the Gaussian is preserved .
p to scaling factor.
under various integral transforms, including the Uncertainty Principle for the Hartley Transform: Direct and FourierAeBased Approaches Hartley transform.
This frequency-domain graph illustrates how the energy of the original signal is distributed with respect to the variable yc.
Combining equation .
with the expression for ya.
yields the fundamental identity ya.
= yaycI .
Oe yaya .
, .
which expresses the Hartley transform as a real linear combination of the real and imaginary parts of the Fourier transform.
Theorem 3.
1 (Invers Formul.
Let yce OO ya1 O ya2 .
If ya = EU.
, then yce.
= 1 O O ya.
2yuU OeO .
A full proof may be found in (Bracewell, 1.
Theorem 3.
2 (Plancherel Identit.
For yce OO ya2 (E.
, the Hartley transform satisfies O O .
yccyc = OeO O .
2yuU OeO .
The proof follows from the self-inverse property of the Hartley transform.
see (Bracewell.
For convenience, we introduce the notation ya.
= yaycI .
, so that by .
= yaycI .
Oe yaya AA.
, ya.
= yaycI .
Theorem 3.
3 (Time-shift Identit.
For yce OO ya1 O ya2 and yca OO Ey.
EU.
c Oe yc.
= cos.
Proof.
Using the Fourier shift rule.
c Oe yc.
= yce Oeycayc ya.
, and writing yce Oeycayc = cos.
Oe yc sin.
with the decomposition .
, we obtain Eu.
ce Oeycnycayc y.
= cos.
yaycI sin.
Ec.
ce Oeycnycayc y.
= cos.
yaya Oe sin.
yaycI .
Since EU.
= Eu(Eyc.
Oe Ec(Eyc.
, the identity follows.
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Theorem 3.
4 (Modulation Identit.
For yce OO ya1 O ya2 and yca OO Ey.
EU.
ce ycnycayc yce.
= Eu.
c Oe yc.
} Oe Ec.
c Oe yc.
Proof.
Using the modulation rule for Fourier transform.
ce ycnycayc yce.
= ya.
c Oe yc.
, and applying EU = EuE Oe EcE, the identity follows.
Theorem 3.
5 (Convolution Identit.
For yce, yci OO ya1 O ya2 , .
EU.
ce O yc.
= Eu.
} Oe Ec.
Proof.
The Fourier convolution rule gives E.
ce O yc.
= ya UI ya.
Applying EU = EuE Oe EcE yields the result.
HEISENBERG
UNCERTAINTY
PRINCIPLE
FOR
THE
HARTLEY
TRANSFORM
This section establishes an analogue of the classical Heisenberg uncertainty principle in the setting of the Hartley transform.
The discussion develops a frequencyAetime inequality consistent with the Fourier case, but expressed entirely in terms of the real-valued Hartley Central to the analysis are the time and frequency variances associated with a function and its Hartley transform.
Let ya.
= EU.
, denote the Hartley transform of yce OO ya2 (E.
Define the time and frequency variances yuayc2 = O yc 2 .
yccyc, yuayc2 = O yc 2 .
Theorem 4.
1 (Heisenberg Uncertainty Principle: Direct Proo.
For every yce OO ya2 (E.
,A yuayc2 yuayc2 Ou (O.
| yccy.
Proof.
Using the Plancherel identity .
, we obtain O OeO yccyc = O .
2yuU OeO .
Uncertainty Principle for the Hartley Transform: Direct and FourierAeBased Approaches Next, consider the integral O A .
i ya Oi O ycyce.
OeO By integration by parts .
nd using the fact that the boundary terms vanish for yce OO ya2 (E.
), .
yields 1 O ya = Oe O .
2 OeO .
The derivative of the Hartley kernel is given by ycc yuU ycaycayc.
= yc.
Oe sin.
) = ycAycaycayc .
cyc Oe ).
To estimate the moments, apply Lemma 2.
3 with yc.
= ycyce.
, yc.
= ycaycaycA.
, which leadsAiafter transforming one factor into the frequency domain using .
to the O O yc OeO yccyc Ou (O .
4 OeO .
Finally, multiplying .
24 ) by yuayc2 = O yc 2 .
yccyc, we obtain the desired uncertainty inequality yuayc2 yuayc2 Ou (O.
| yccy.
, which completed the proof.
Theorem 4.
2 (Heisenberg Uncertainty Principle: Fourier Relatio.
For every yce OO ya2 (E.
,A yuayc2 yuayc2 Ou (O.
| yccy.
Proof.
We begin by recalling that the Fourier transform of yce can be written as .
= .
aycI .
, .
where yaycI and yaya denote the real and imaginary parts of ya, respectively.
Moreover, the Hartley transform satisfies ya.
= yaycI .
Oe yaya .
Substituting .
gives BILANGAN Ae VOLUME.
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= .
aycI .
Oe 2yaycI .
Integrating .
over Ey, and observing that the mixed term integrates to zero by symmetry, yields O .
yccyc = O .
Ey .
Ey Using .
, the Plancherel identity for the Hartley transform follows:
O .
yccyc = Ey O .
yccyc = O .
2yuU Ey 2yuU Ey .
From .
, it follows that the Hartley and Fourier transforms preserve energy in the same way.
Using this equivalence, the classical Fourier Heisenberg inequality .
Bracewell,1.
) becomes applicable.
Thus, (O yc 2 .
Ey yccy.
(O yc 2 .
Ey yccy.
Ou (O .
4 Ey .
Next, combining .
29 )with the definition of yuayc2 gives O yc 2 .
yccyc = O yc 2 .
Ey .
Ey Substituting .
yields yuayc2 yuayc2 Ou (O.
| yccy.
, which completes the derivation of the Hartley Heisenberg uncertainty principle.
CONCLUTION
This paper has presented the fundamental properties of the Hartley transform, including its inversion formula.
Plancherel identity, and core operational rules.
The relation between the Hartley and Fourier transforms has been clarified, enabling a unified analytical framework.
Two versions of the Heisenberg uncertainty principle for the Hartley transform were establishedAione derived directly from its kernel structure and the other obtained via its connection to the Fourier transform.
Uncertainty Principle for the Hartley Transform: Direct and FourierAeBased Approaches REFERENCES