LOGARITHMIC UNCERTAINTY PRINCIPLE FOR QUATERNION LINEAR CANONICAL TRANSFORM

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1 Proceedings of the 06 International Conference on Wavelet Analysis and Pattern Recognition Jeju South Korea 0-3 July LOGARITHMIC UNCERTAINTY PRINCIPLE FOR QUATERNION LINEAR CANONICAL TRANSFORM MAWARDI BAHRI RYUICHI ASHINO Department of Mathematics Hasanuddin University Makassar 9045 Indonesia Division of Mathematical Sciences Osaka Kyoiku University Osaka Japan Abstract: The quaternion linear canonical transform (QLCT) can be thought as a generalization of the linear canonical transform (LCT) to quaternion algebra. The relationship between the QLCT and the quaternion Fourier transform (QFT) is derived. Based on this fact and properties of the QLCT a logarithmic uncertainty principle associated with the quaternion linear canonical transform is established. Keywords: Linear canonical transform; Logarithmic uncertainty principle. Introduction The linear canonical transform (LCT) plays an important role in many field of optics and signal processing. It is a generalized form of many mathematical transforms such as the Fourier Laplace fractional Fourier Fresnel and the other transforms. The relationship between the linear canonical transform and the Fourier transform was discussed in [0]. Uncertainty principles for the linear canonical transform were discussed in [7 ]. A higher-dimensional extension of the LCT within quaternion algebra setting is called quaternion linear canonical transform which was first studied in [9]. They also established its Heisenberg type uncertainty principle which is one of the most fundamental properties of the QLCT. The classical uncertainty relation can be found in [4]. The logarithmic uncertainty principle is a more general form of Heisenberg type uncertainty principle which describes the relationship between a quaternion function and its QLCT. Therefore the purpose of this paper is to establish logarithmic uncertainty principle associated with the quaternion linear canonical transform. Our first step is in the proof to use the relationship between the QLCT and QFT (see []). In this case we find that the QLCT can be reduced to the QFT. The QFT has been applied to various areas for example two-dimensional linear time-invariant partial differential systems [5] and color images [6]. We also need to introduce the logarithmic uncertainty principle related to the quaternion Fourier transform. The quaternion algebra over R denoted by H = {q = q 0 + iq + jq + kq 3 ; q 0 q q q 3 R} is an associative non-commutative four-dimensional algebra with the following multiplication rules: ij = ji = k jk = kj = i ki = ik = j i = j = k = ijk =. For a quaternion q = q 0 + iq + jq + kq 3 H q 0 is called the scalar part of q denoted by Sc(q) and iq + jq + kq 3 is called the vector (or pure) part of q. The vector part of q is conventionally denoted by q. Let p q H and p q be their vector parts respectively. The multiplication rules imply the quaternionic multiplication qp as qp = q 0 p 0 q p + q 0 p + p 0 q + q p q p = (q p + q p + q 3 p 3 ) q p = i(q p 3 q 3 p ) + j(q 3 p q p 3 ) + k(q p q p ). The quaternion conjugate of q given by q = q 0 iq jq kq 3 q 0 q q q 3 R () is an anti-involution namely qp = p q /6/$ IEEE 40

2 Proceedings of the 06 International Conference on Wavelet Analysis and Pattern Recognition Jeju South Korea 0-3 July The norm or modulus of q H is defined by q = q q = q0 + q + q + q 3. Then we have qp = q p p q H. Using the conjugate () and the modulus of q we can define the inverse of q H \ {0} as q = q q which implies that H is a normed division algebra. Any quaternion q can be split up into which implies q = q + + q q ± = (q ± i q j) q ± = {(q 0 ± q 3 ) + i(q q )} ± k = ± k {(q 0 ± q 3 ) + j(q q )}. () Then we have the following lemma. Lemma (Modulus identity) Proof. By () we have q = q + q +. q = {(q 0 q 3 ) + i(q + q )} k q + = {(q 0 + q 3 ) + i(q q )} + k. Basic properties of imaginary unit quaternion give q = (q 0 q 3 ) + i(q + q ) + j(q + q ) k(q 3 q 0 ) q + = (q 0 + q 3 ) + i(q q ) j(q q ) + k(q 3 + q 0 ). Using () we obtain q = (q 0 q 3 ) + (q + q ) q + = (q 0 + q 3 ) + (q q ). Hence q + q + = (q 0 q 3 ) + (q + q ) + (q 0 + q 3 ) + (q q ) = q 0 + q + q + q 3 = q which completes the proof. Since p + q = (p 0 + p 3 ) + i(p p ) j(p p ) + k(p 3 + p 0 ) (q 0 q 3 ) i(q + q ) j(q + q ) + k(q 3 + q 0 ) it implies that Sc(p + q ) = (p 0 + p 3 )(q 0 + q 3 ) + (p p )(q + q ) (p p )(q + q ) + (p 3 + p 0 )(q 3 + q 0 ) = 0. Let us introduce the canonical inner product for quaternionvalued functions f g : R H as follows: (f g) = f(x)g(x) dx dx = dx dx. R Then the natural norm is given by f = ( ) / f f = f(x) d x R and the quaternion module L (R ; H) is defined by L (R ; H) = {f f : R H f < }. The symmetric real scalar part is defined by f g = [(f g) + (g f)] = Sc(f(x)ḡ(x)) dx. R 4

3 Proceedings of the 06 International Conference on Wavelet Analysis and Pattern Recognition Jeju South Korea 0-3 July Besides the quaternion units i j k and the vector part q of a quaternion q H we will use the following real vector notation: and so on. x = (x x ) R x = x + x x y = x y + x y f(x) = f(x x ). Quaternion Linear Canonical Transform Let us define the two-sided QFT and provide some properties used to prove the uncertainty principle. Hereinafter we use the abbreviation QFT stands for the two-sided QFT. Definition The QFT of f L (R ; H) is the transform F q {f} : R H given by the integral F q {f}(ω) = e iωx f(x)e jωx dx (3) (π) R x = x e + x e ω = ω e + ω e and F q is called the quaternion Fourier transform operator. The quaternion exponential product e iωx e jωx is called the quaternion Fourier kernel. Definition Let f L (R ; H) and F q {f} L (R ; H). Then the inverse transform of the QFT is given by F q f(x) = Fq [F q {f}](x) = e iωx F q {f}(ω) e jωx dω (π) R is called the inverse QFT operator. An important property of the QFT are stated in the following lemma which is needed to prove the QLCT-Parseval. For details of the QFT see [ 3 8]. Lemma (QFT Parseval) Let f g L L (R ; H). Then f g L (R ;H) = F q {f} F q {g} L (R ;H). In particular when f = g we have the quaternion version of the Plancherel formula that is f L (R ;H) = F q{f} L (R ;H). Let us consider the two-sided QLCT defined as follows: Definition 3 (Two-Sided QLCT) Let A = (a b c d ) A = (a b c d ) be two matrix parameters satisfying det(a s ) = a s d s b s c s = s =. The QLCT of f L (R ; H) is defined by L H A A {f}(ω) K A (x ω )f(x)k A (x ω ) dx R for b b 0; d e i( c d )ω f(d ω x )K A (x ω ) = for b = 0 b 0; d e j( c d )ω f(x d ω )K A (x ω ) for b 0 b = 0; d d e i( c d )ω f(d ω d ω )e j( c d )ω for b = b = 0. Here the kernel functions of the QLCT is given by K A (x ω ) = ( ) e i a b x b x ω + d b ω π πb for b 0; K A (x ω ) = ( ) e j a b x b x ω + d b ω π πb respectively. for b 0 Note that the QLCT the case when b b = 0 or b = b = 0 is not interesting because it is essentially a multiplication by a quaternion chirp. Therefore we deal with only the case when b b 0 in this paper. As a special case when A i = (a i b i c i d i ) = (0 0) i = the LCT reduces to the QFT. Namely L H A A {f}(ω) = R e i π 4 π f(x)e iωx e j π 4 π e jωx dx = e i π 4 Fq {f}(ω) e j π 4 F q {f} is the QFT of f given by (3). (4) 4

4 Proceedings of the 06 International Conference on Wavelet Analysis and Pattern Recognition Jeju South Korea 0-3 July Lemma 3 (QLCT Parseval) Let f g L L (R ; H). Then we have f g L (R ;H) = L H A A {f} L H A A {g} L (R ;H). For f = g we have the QLCT version of the Plancherel formula f L (R ;H) = LH A A {f} L (R ;H). (5) The following theorem given in [ Theorem 5] will be used to prove our main result. Theorem The QLCT of f L (R ; H) with matrix parameters A = (a b c d ) and A = (a b c d ) can be reduced to the QFT F q {g f }(ω) = e ixω g f (x)e jxω dx (6) (π) R with F(ω) = F q {g f }(ω) = F(bω) g f (x) = e i π 4 b f(x) e j π 4 b f(x) = e i a b x f(x)e j a b x e ix ω b g f (x)e jx ω b dx (π) R F(ω) = e i d b ω L H A A {f}(ω)e j d b ω. 3. Logarithmic Uncertainty Principle for QLCT The classical uncertainty principle of harmonic analysis s- tates that a non-trivial function and its Fourier transform cannot both be simultaneously sharply localized. In quantum mechanics an uncertainty principle asserts that one cannot be certain of the position and of the velocity of an electron (or any particle) at the same time. In other words increasing the knowledge of the position decreases the knowledge of the velocity or momentum of an electron. In this section we extend the logarithmic uncertainty principle for the QFT [3] to that for the QLCT. Based on the definition of the QFT (3) we construct the logarithmic uncertainty principle related to the QFT as follows. Denote by S(R ; H) the Schwartz class of quaternion functions. Theorem (QFT logarithmic uncertainty principle) Let f S(R ; H). Then ln x + x f(x) dx R + ln ω + ω F q{f}(ω) dω R D f(x) dx (7) R ( D = ψ ln π ) and Γ(t) is the gamma function. ψ(t) = d dt ln[γ(t)] Next we state our logarithmic uncertainty principle associated with the QLCT. Theorem 3 (QLCT logarithmic uncertainty principle) Let f L (R ; H) and L H A A {f} L (R ; H) be the QLCT of f. Then ln x + x f(x) dx R + b R ln ω + ω LH A A {f}(ω) dω ( ) D + ln b + b f(x) dx. (8) R Proof. Replacing f by g f on both sides of (7) we obtain R Set ω = ω b. Then we have ln x + x g f (x) dx + ln ω F q {g f }(ω) dω R D g f (x) dx. R ln x + x e i π 4 e f(x) i π 4 dx R b b + ln ω R b F q{g f }( ω b ) dω D e i π 4 e f(x) i π 4 dx. b b R 43

5 Proceedings of the 06 International Conference on Wavelet Analysis and Pattern Recognition Jeju South Korea 0-3 July Hence b b ln x + x f(x) dx ( ) + ln ω + ω ln b + b R R F q {g f }( ω b ) d ω b D f(x) dx. b b R Applying (6) we have R b b ln x + x f(x) dx ( ) + ln ω R b b + ω ln b + b L H A A {f}(ω) dω D f(x) dx. b b R Hence ln x + x f(x) dx R ( ) + ln ω + ω ln b + b R L H A A {f}(ω) dω D f(x) dx. R Applying the Plancherel theorem for the QLCT (5) we obtain ln x + x f(x) dx R + ln ω + ω LH A A {f}(ω) dω R ( ) D + ln b + b f(x) dx R which completes the proof. Remark If f is normalized namely f L (R ) = then (8) becomes ln x + x f(x) dx R + ln ω + ω LH A A {f}(ω) dω R ( ) D + ln b + b. 4. Conclusion The quaternion linear canonical transform (QLCT) is a nontrivial generalization of the classical linear canonical transform (LCT) to quaternion algebra. It is constructed by replacing the complex kernel function with the quaternion kernel function in the LCT definition. We introduce the fundamental relationship between the QLCT and the quaternion Fourier transform (QFT). Based on this relation we establish a logarithmic uncertainty principle associated with the QLCT. Our logarithmic uncertainty principle could play an important role in the timefrequency analysis in the QLCT space. Acknowledgments The first author is partially supported by Hibah Penelitian Kompetisi Internal Tahun 04 (No. 454/UN4.0/PL.9/04) from the Hasanuddin University Indonesia. The second author is partially supported by JSPS. KAKENHI (C) of Japan. References [] M. Bahri R. Ashino and R. Vaillancourt Convolution Theorems for quaternion Fourier transform: properties and applications Abstract and Applied Analysis vol. 03 Article ID pages. [] M. Bahri and R. Ashino A simplified proof of uncertainty principle for quaternion linear canonical transforms Abstract and Applied Analysis vol. 06 Article ID pages. [3] M. Bahri E. Hitzer A. Hayashi and R. Ashino An uncertainty principle for quaternion Fourier transform Comput. Math. Appl. 56(9) [4] R. Bracewell The Fourier Transform and its Applications Boston McGraw Hill [5] T. A. Ell Quaternion-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems Proceedings of the 3nd IEEE Conference on Decision and Control vol [6] T. A. Ell and S. J. Sangwine Hypercomplex Fourier transform of color images IEEE Trans. Signal Process. 6()

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