Uncertainty relations of the complex signals from generalized Hilbert transform on LCT

Abstract

It is well known that as the generalized case for Fourier transform (FT) and fractional Fourier transform (FrFT), the linear canonical transform (LCT) has attracted more and more attention in signal processing and optics and so on due to more freedoms. On the other hand, as the tool of turning the real signals into the complex ones, the Hilbert transform (HT) is of much signification to information science. Since the complex signals through HT have the characteristic of the absence in negative frequency, HT plays an important role in communication. In the present paper, first, the generalized HT (GHT) in terms of LCT is given. Then, five types of generalized Uncertainty relations with novel uncertainty bounds of the complex signals in terms of generalized HT (GHT) in terms of LCT are demonstrated. These newly refined uncertainty bounds proved to be different from and much lower or sharper in many cases than that of the traditional complex and real signals. Finally, the examples are given to show the efficiency of the proposed uncertainty relations in this paper.

Appendices

Appendix 1

Proof of Theorem 1

Proof

Note the result in [23, 25], we have the generalized uncertainty relation on Shannon entropy in terms of LCT as follows.

$$ - \int_{ - \infty }^{\infty } {\left| {F_{A} (u)} \right|^{2} \ln \left| {F_{A} (u)} \right|^{2} } {\text{d}}u - \int_{ - \infty }^{\infty } {\left| {F_{B} (v)} \right|^{2} \ln } \left| {F_{B} (v)} \right|^{2} {\text{d}}v\; \ge \ln (\pi e\left| {a_{1} b_{2} - a_{2} b_{1} } \right|) $$

(19)

Firstly, based on the definition of GHT shown in (3)–(5) and (19), we will obtain

$$- \int_{{ - \infty }}^{\infty } {\left| {F_{A}^{H} (u)} \right|^{2} \ln \left| {F_{A}^{H} (u)} \right|^{2} } {\text{d}}u $$

$= - 4\int_{0}^{\infty } {\left| {F_{A} (u)} \right|^{2} \left( {2\ln 2 + \ln \left| {F_{A} (u)} \right|^{2} } \right)} {\text{d}}u$.

So we obtain the following relation

$$ - \int_{ - \infty }^{\infty } {\left| {F_{A}^{H} (u)} \right|^{2} \ln \left| {F_{A}^{H} (u)} \right|^{2} } {\text{d}}u = - 4\int_{0}^{\infty } {\left| {F_{A} (u)} \right|^{2} \ln 4} {\text{d}}u $$

$- 4\int_{0}^{\infty } {\left| {F_{A} (u)} \right|^{2} \ln \left| {F_{A} (u)} \right|^{2} } {\text{d}}u$.

Here if $F_{A} (u)$ is odd, then $F_{A} ( - u) = - F_{A} (u)$, so if we let $u^{\prime} = - u$, then taking into account the irrelevance of variable form we will have.

$\int_{0}^{\infty } {\left| {F_{A} (u)} \right|^{2} } {\text{d}}u = \int_{ - \infty }^{0} {\left| {F_{A} (u)} \right|^{2} } {\text{d}}u$.

At the same time, if $F_{A} (u)$ is even, then $F_{A} ( - u) = F_{A} (u)$, so if we let $u^{\prime} = - u$, thus taking into account of the irrelevance of variable form we will get the same result as follows.

$\int_{0}^{\infty } {\left| {F_{A} (u)} \right|^{2} } {\text{d}}u = \int_{ - \infty }^{0} {\left| {F_{A} (u)} \right|^{2} } {\text{d}}u$.

Based on the analysis in above, we always have.

$\int_{0}^{\infty } {\left| {F_{A} (u)} \right|^{2} } {\text{d}}u = \int_{ - \infty }^{0} {\left| {F_{A} (u)} \right|^{2} } {\text{d}}u$.

Meanwhile, note $\left\| {f(t)} \right\|_{2} = 1$, based on the Parseval’s Principle [3,4,5] we will have.

$\int_{ - \infty }^{\infty } {\left| {F_{A} (u)} \right|^{2} } {\text{d}}u = \int_{ - \infty }^{\infty } {\left| {F_{B} (v)} \right|^{2} } {\text{d}}v = 1$.

Thus, we will get the equation as follows.

$\int_{0}^{\infty } {\left| {F_{A} (u)} \right|^{2} } {\text{d}}u = \int_{ - \infty }^{0} {\left| {F_{A} (u)} \right|^{2} } {\text{d}}u = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}$.

Hence we obtain.

$- \int_{ - \infty }^{\infty } {\left| {F_{A}^{H} (u)} \right|^{2} \ln \left| {F_{A}^{H} (u)} \right|^{2} } {\text{d}}u = - \ln 16 - 4\int_{0}^{\infty } \left| {F_{A} (u)} \right|^{2} \ln \break \left| {F_{A} (u)} \right|^{2} {\text{d}}u$.

So, via the inequality in (7) we will get.

$- \ln 16 - 4\int_{ - \infty }^{\infty } {\left| {F_{A} (u)} \right|^{2} \ln \left| {F_{A} (u)} \right|^{2} } {\text{d}}u - \ln 16$$- 4\int_{ - \infty }^{\infty }\break \left| {F_{B} (v)} \right|^{2} \ln \left| {F_{B} (v)} \right|^{2} {\text{d}}v$$\ge - 8\ln 2 + 4\ln (\pi e\left| {a_{1} b_{2} - a_{2} b_{1} } \right|)$.

Appendix 2

Proof of Theorem 4

Proof

From [23, 25], we know the following inequality of logarithm Uncertainty principle associated with LCT.

$$ \int_{ - \infty }^{\infty } {\ln \left| u \right|^{2} \left| {F_{A} (u)} \right|^{2} } {\text{d}}u + \int_{ - \infty }^{\infty } {\ln \left| v \right|^{2} \left| {F_{B} (v)} \right|^{2} } {\text{d}}v $$

$$ \ge \ln \left( {\left| {a_{1} b_{2} - a_{2} b_{1} } \right|^{2} } \right) + \frac{{2\Gamma ^{\prime}\left( {1/4} \right)}}{{\Gamma \left( {1/4} \right)}} $$

(20)

On the other hand, we have.

$\int_{ - \infty }^{\infty } {\ln \left| u \right|^{2} \left| {F_{A}^{H} (u)} \right|^{2} } {\text{d}}u$$= 4\int_{0}^{\infty } {\ln \left| u \right|^{2} \left| {F_{A} (u)} \right|^{2} } {\text{d}}u$.

Here we note that if $F_{A} (u)$ is odd, then $F_{A} ( - u) = - F_{A} (u)$, so here we set $u^{\prime} = - u$, then by considering the irrelevance of variable form we will obtain

$$ \int_{0}^{\infty } {\ln \left| u \right|^{2} \left| {F_{A} (u)} \right|^{2} } {\text{d}}u = \int_{0}^{ - \infty } {\ln \left| { - u^{\prime}} \right|^{2} \left| {F_{A} ( - u^{\prime})} \right|^{2} } d( - u^{\prime}) $$

$= \int_{0}^{ - \infty } {\ln \left| { - u^{\prime}} \right|^{2} \left| { - F_{A} (u^{\prime})} \right|^{2} } d( - u^{\prime})$$= \int_{ - \infty }^{0} \ln \left| u \right|^{2} \left| {F_{A} (u)} \right|^{2} {\text{d}}u$.

Similarly, if $F_{A} (u)$ is even, then $F_{A} ( - u) = F_{A} (u)$, we have the same result as follows:

$\int_{0}^{\infty } {\ln \left| u \right|^{2} \left| {F_{A} (u)} \right|^{2} } {\text{d}}u$$= \int_{ - \infty }^{0} {\ln \left| u \right|^{2} \left| {F_{A} (u)} \right|^{2} } {\text{d}}u$.

Therefore, we can obtain.

$\int_{0}^{\infty } {\ln \left| u \right|^{2} \left| {F_{A} (u)} \right|^{2} } {\text{d}}u$$= \frac{1}{2}\int_{ - \infty }^{\infty } {\ln \left| u \right|^{2} \left| {F_{A} (u)} \right|^{2} } {\text{d}}u$.

So we have the relation as follows.

$\int_{ - \infty }^{\infty } {\ln \left| u \right|^{2} \left| {F_{A}^{H} (u)} \right|^{2} } {\text{d}}u = 2\int_{ - \infty }^{\infty } {\ln \left| u \right|^{2} \left| {F_{A} (u)} \right|^{2} } {\text{d}}u$.

By the same way, we have.

$\int_{ - \infty }^{\infty } {\ln \left| v \right|^{2} \left| {F_{B}^{H} (v)} \right|^{2} } {\text{d}}v = 2\int_{ - \infty }^{\infty } {\ln \left| v \right|^{2} \left| {F_{B} (v)} \right|^{2} } {\text{d}}v$.

Finally, we obtain the relation in Theorem 4.