On uncertainty principle for the two-sided quaternion linear canonical transform

In this paper, the relation between linear canonical transform (LCT) and quaternion linear canonical transform (QLCT) by using orthogonal plan split (OPS) method is studied. Further, several inequalities like sharp Young–Hausdorff inequality and Pitt's inequality are proved. Moreover, entropy uncertainty principle, sharp local uncertainty principle, and Heisenberg's uncertainty principle for the two‐sided QLCT are also discussed.

“…Remark The two‐sided QLCT reduces to two‐sided QFT, 35 if we take

A_{1} = false(0, - 1; 1,0 false), A_{1} = false(0, - 1; 1,0 false)

.…”

Section: Two‐sided Qlctmentioning

confidence: 99%

“…Let

f \in L^{1} false(ℝ^{2}, ℍ false)

and

{scriptL}_{A_{1}, A_{2}}^{ℍ} f, {scriptF}^{q} f

be the two‐sided QLCT and QFT, 35 respectively. Then by simple calculation, we get

\begin{matrix} false({scriptL}_{A_{1}, A_{2}}^{ℍ} f false) false(ξ false) = \frac{1}{\sqrt{i b_{1}}} e^{\frac{i}{2 b_{1}} false(d_{1} ξ_{1}^{2} false)} {scriptF}^{q} ((), e^{\frac{i}{b_{1}} a_{1} t_{1}^{2}} f false(t false) e^{\frac{j}{b_{2}} a_{2} t_{2}^{2}}) ((), \frac{ξ}{b}) \frac{1}{\sqrt{j b_{2}}} e^{\frac{j}{2 b_{2}} false(d_{2} ξ_{2}^{2} false)} . \end{matrix}

…”

Section: Two‐sided Qlctmentioning

confidence: 99%

“…Proposition Let p ∈ [1, 2] and q be such that

\frac{1}{p} + \frac{1}{q} = 1

, then

false‖ {scriptL}_{A_{1}, A_{2}} f {false‖}_{L_{q} false(ℝ^{2} false)} \leq scriptK false‖ f {false‖}_{L^{p} false(ℝ^{2} false)},

where

scriptK = false| b_{1} b_{2} {false|}^{\frac{1}{q} - \frac{1}{2}} ((), \frac{p^{\frac{1}{p}}}{q^{\frac{1}{q}}}) {false(2 π false)}^{\frac{1}{q} - \frac{1}{p}}

. Proof The sharp Hausdorff–Young inequality of a function and its FT in

ℝ^{2}

is given by 35

{((), \int_{ℝ^{2}} {(||, false(scriptF f false) false(ξ false))}^{q} d ξ)}^{\frac{1}{q}} \leq ((), \frac{p^{\frac{1}{p}}}{q^{\frac{1}{q}}}) {false(2 π false)}^{\frac{1}{q} - \frac{1}{p}} {((), \int_{ℝ^{2}} false| f false(t false) {false|}^{p} d t)}^{\frac{1}{p}} .

Now, replacing ( ξ 1 , ξ 2 ) by

<...>

…”

Section: Two‐sided Qlctmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Uncertainty principles associated with quaternion linear canonical transform and their estimates

Kundu

Prasad

2022

“…We [13] exploited Heisenberg's uncertainty principle associated with the quaternion windowed linear canonical transform. Some scholars have studied other types of uncertainty principles as well, such as logarithmic uncertainty principle [14], Benedicks-Amrein-Berthier uncertainty principle [15], sharp local uncertainty principle [16], Donoho-Stark's uncertainty principle [17], and entropic uncertainty principle [18]. As applications for the QLCT, the generalized filters and color images are introduced [19,20].…”

Section: Introductionmentioning

confidence: 99%

Theories and applications associated with biquaternion linear canonical transform

Gao

2023

The quaternion linear canonical transform (QLCT) has been widely used in color image processing. Biquaternion is a more generalized algebra of quaternion, which has attracted scholars' research interest in recent years. In this paper, a new transform is proposed called the biquaternion linear canonical transforms (BiQLCTs). Due to the noncommutativity of biquaternion algebra multiplication, there are three different types of the BiQLCTs: Left‐sided BiQLCT, right‐sided BiQLCT, and two‐side BiQLCT. The transforms are the extension of the complex linear canonical transforms. Then, the relationships between the three kinds of transforms are obtained. Next, based on the right‐side biquaternion linear canonical transform (RBiQLCT), some general properties of this transform are proved. Moreover, the convolution and correlation theorems of the RBiQLCT are studied. As an application, according to the convolution operator and convolution theorem, the biquaternion linear time‐invariant system is analyzed. Finally, the Heisenberg uncertainty principle for the RBiQLCT is established.

Uncertainty principles of the fractional Clifford–Fourier transform

Shi

Gao

Xie

et al. 2023

In this paper, we define the two‐sided fractional Clifford–Fourier transform (FrCFT). Using its properties, we get some uncertainty principles of the FrCFT. Two parts are obtained. One part is a modified uncertainty principle. The uncertainty principle states a lower bound on the spreads of two specific transform domains. It is shown that only a Gaussian‐type signal minimizes the uncertainty. We also give a Heisenberg‐type uncertainty principle. The other part is a logarithmic uncertainty principle, which may be obtained from a sharp of Pitt's inequality.