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Detection of continuoustime quaternion signals in additive noise
EURASIP Journal on Advances in Signal Processing volume 2012, Article number: 234 (2012)
Abstract
Different kinds of quaternion signal detection problems in continuoustime by using a widely linear processing are dealt with. The suggested solutions are based on an extension of the KarhunenLoève expansion to the quaternion domain which provides uncorrelated scalar realvalued random coefficients. This expansion presents the notable advantage of transforming the original fourdimensional eigen problem to a onedimensional problem. Firstly, we address the problem of detecting a quaternion deterministic signal in quaternion Gaussian noise and a version of Pitcher’s Theorem is given. Also the particular case of a general quaternion Wiener noise is studied and an extension of the CameronMartin formula is presented. Finally, the problem of detecting a quaternion random signal in quaternion white Gaussian noise is tackled. In such a case, it is shown that the detector depends on the quaternion widely linear estimator of the signal.
Introduction
Quaternion signals are of great relevance to applications in the area of statistical signal processing in which the received signal is composed of a certain number of random components since they account naturally for their correlated nature [1, 2]. These are useful, for example, in studying communication, electromagnetics, seismology, acoustics, etc., problems frequently encountered in this area [3]. One of these problems where the application of the mathematical theory of quaternions has recently attracted significant attention is vectorsensor signals [4]. A vectorsensor array model uses an array of sensors whose output is a vector corresponding to the different magnitudes of the problem analyzed, i.e., it is a device that measures a complete physical vector quantity [5, 6].
As is the case with complexvalued random signals, the suitable statistical processing for quaternions requires the augmented statistics to be considered, i.e., requires the operation on the quaternion and its involutions over the three pure unit quaternions in an orthogonal basis. This approach, called quaternion widely linear (QWL) processing, can lead to better performances than the traditional quaternion linear processing for multiple problems [1].
On the other hand, one classical approach to addressing the signal detection problem is via an appropriate series representation [7–10]. Series expansions enable us to bridge the gap between the continuoustime observation set and the discretetime one in a straightforward manner. In fact, they provide a countable set of random coefficients with the same information content up to sets of measure zero as the observation process. If such random coefficients are uncorrelated, then they become an excellent tool to derive optimal detection structures. The KarhunenLoève (KL) expansion is the most widely used because of its optimality properties in information compression [11]. This series representation has been recently extended to the quaternion domain by using augmented statistics [12]. The technique to derive the QKL expansion is based on the definition of a realvalued univariate stochastic signal whose secondorder statistics match that of quaternion. This strategy avoids addressing a fourdimensional vectorial problem which notably simplifies the obtaining of the representation. Another advantage of such a series expansion is that the random coefficients take the form of scalar realvalued random variables.
This article handles the problem of detecting a quaternion signal corrupted by additive noise by means of the QKL expansion and following a WL processing. More specifically, two classes of quaternion signal detection problems are tackled. First, we study the detection of a quaternion deterministic signal in quaternion Gaussian noise. The main result is a version of the Pitcher’s Theorem adapted to the quaternion domain. The particular case in which the noise is a general quaternion Wiener process is also analyzed and, as a consequence, a version of the wellknown CameronMartin formula to the quaternion field is presented. Finally, we address the detection of quaternion random signals in quaternion white Gaussian noise (QWGN). In this case, we demonstrate that the loglikelihood ratio depends on the QWL estimator of the signal provided in [12].
The quaternion detection problem has been studied in the discretetime setting previously. For example, the problem of detecting a polarized signal corrupted by unpolarized noise, in the Gaussian case, in terms of different types of properness was formulated in [13]. In [14] an efficient colorimpulse detector for switching vector median filters based on the quaternion representation of color difference is presented. Our approach is different in that we formulate the problem in continuoustime and use the QKL expansion to extract the random coefficients.
The article is organized as follows. In the following section, we summarize some basic concepts about quaternion and outline the QKL expansion. In Section 3, we are concerned with the detection of a completely known quaternion signal in quaternion Gaussian noise. We obtain the expression of the general loglikelihood ratio and then, some particular cases are studied. The detection of quaternion random signals in QWGN is addressed in Section 4. The results obtained are first stated and then proved rigorously in an Appendix 1. Finally, a section of Conclusions ends this article.
Preliminaries
We use boldfaced uppercase letters to denote matrices, boldfaced lowercase letters for column vector, and lightfaced lowercase letters for scalar quantities. Superscripts (·)^{∗}, (·)^{T}, and (·)^{H}represent quaternion (or complex) conjugate, transpose, and Hermitian (i.e., transpose and quaternion conjugate), respectively. All the random variables considered are assumed with zeromean. Consider a quaternion q=q_{1} + q_{2}i + q_{3}j + q_{4}k, where q_{1},q_{2},q_{3},q_{4} are real random variables and i,j,k are the imaginary units. The conjugate of a quaternion is defined as q^{∗}=q_{1}−q_{2}i−q_{3}j−q_{4}k and the norm of a quaternion is \parallel q\parallel =\sqrt{q{q}^{\ast}}=\sqrt{{q}^{\ast}q}=\sqrt{{q}_{1}^{2}+{q}_{2}^{2}+{q}_{3}^{2}+{q}_{4}^{2}}. Denote q^{η}=−ηqη, η=i,j,k, the three perpendicular quaternion involutions, i.e.,
We define an augmented quaternion vector as \left(\right.separators="">\n \n q\n =\n \n \n [\n q\n ,\n \n \n q\n \n \n i\n \n \n ,\n \n \n q\n \n \n j\n \n \n ,\n \n \n q\n \n \n k\n \n \n ]\n \n \n T\n \n \n \n. The secondorder properties of q are fully specified by its augmented covariance matrix, E q q^{H}[1].
We now consider quaternions in a continuoustime setting. Given a quaternion random signal q(t)=q_{1}(t) + q_{2}(t)i + q_{3}(t)j + q_{4}(t)k, with t∈[0,T, a complete description of the secondorder characteristics of q(t) in the quaternion domain is attained by the augmented quaternion vector, q(t), or, equivalently, by the augmented correlation function, R_{ q }(t s)=E q(t)q^{H}(s)]. Also, if q_{ n }(t), n=1,…,4, are meansquare continuous signals, then an extension of the KL expansion to the quaternion field can be suggested [12]. This series representation presents two remarkable properties: the deterministic coefficients have the same structure as the augmented vector q(t) and the scalar random coefficients are realvalued and uncorrelated. Specifically, consider the realvalued random signal
and let λ_{ n } and a_{ n }(t) be the eigenvalues and eigenfunctions of its correlation function, respectively. Then, the augmented quaternion vector and its correlation function admit the following series representations [12]
where \left(\right.separators="">\n \n \n \n \phi \n \n \n n\n \n \n (\n t\n )\n =\n \n \n [\n \n \n \phi \n \n \n n\n \n \n (\n t\n )\n ,\n \n \n \phi \n \n \n n\n \n \n i\n \n \n (\n t\n )\n ,\n \n \n \phi \n \n \n n\n \n \n j\n \n \n (\n t\n )\n ,\n \n \n \phi \n \n \n n\n \n \n k\n \n \n (\n t\n )\n ]\n \n \n T\n \n \n \n with
and \left(\right.separators="">\n \n \n \n \u03f5\n \n \n n\n \n \n =\n \n \n \u222b\n \n \n 0\n \n \n T\n \n \n \n \n \phi \n \n \n n\n \n \n H\n \n \n (\n t\n )\n q\n (\n t\n )\n dt\n \n are real random variables such that E ϵ_{ n }ϵ_{ m }=β_{ n }δ_{ nm }, with β_{ n }=4λ_{ n }.
A potential application of the QKL expansion is found in the problem of estimating the quaternion signal q(t) in additive QWGN [12]. The solution provided is optimal in the minimum meansquared error sense and is derived following a QWL processing. For that, consider the observation quaternion process of the form
being w_{0}(t) a quaternion \mathbb{Q}proper^{a} Wiener process with parameter r_{0}and uncorrelated with q(s). Thus, the QWL estimator of q(t), {\widehat{q}}_{\text{QWL}}\left(t\right), is given by
with
Detection of quaternion deterministic signals in quaternion Gaussian noise
Our first objective is to study the problem of detection
with x(t) a quaternion continuous completely known signal and v(t) a quaternion meansquare continuous Gaussian noise. Denote {\mathcal{P}}_{0} and {\mathcal{P}}_{1} the probability measures corresponding to {\mathcal{H}}_{0} and {\mathcal{H}}_{1}, respectively. According to Grenander’s Theorem one way of computing likelihood ratios for continuoustime observation models is first to reduce the observation signal to an equivalent observation sequence, and then looking for the limit of the likelihood ratio for the truncated sequence. An alternative, somewhat more practical, representation of the likelihood ratio for problem (4) is provided by Pitcher’s Theorem. This result suggests a simpler and more efficient implementation of the corresponding signal detection system. In the particular case of Gaussian white noise the representation of the optimum detection statistic obtained is known as the CameronMartin formula. In the next result, we give an extension of Pitcher’s Theorem to the quaternion domain.
Theorem 3.1
Suppose that there exists a quaternion function g(t) with components of bounded variation such that
then the detection problem (4) is not singular ({\mathcal{P}}_{0}\equiv {\mathcal{P}}_{1}) and the loglikelihood ratio test is given by
with
Remark 1
From (5) and (7) we have the following alternative representation for Δ_{1}
Remark 2
If the quaternion function g(t) is differentiable with respect to p(t)=dg(t)/dt, then equation (5) becomes
and the first term of (6),
Particular case: the general quaternion Wiener process
Following the classical strategy, the detection problem of a deterministic signal x(t) in additive QWGN is formulated of the form [9]
with x(t) a known continuous quaternion signal and w_{0}(t) is the quaternion \mathbb{Q}proper Wiener process with parameter r_{0}defined in the previous section.
The fourdimensional structure of a quaternion allows us to give a more general definition of a quaternion Wiener process in a similar way to [15]. Next, we introduce this new process and afterwards, we tackle the detection problem for this type of process.
Definition 3.1
The general quaternion Wiener process is defined as a quaternion {w(t),t∈[0,T]} such that its augmented correlation function R_{ w }(t,s) is of the form
where A(t)=C(t)C^{H}(t), with the quaternion matrix C(t) having the particular form
and being b(t),c(t),d(t), and e(t) quaternion continuous functions.
Remark 3
If \mathit{C}\left(t\right)=\sqrt{{r}_{0}}{\mathit{I}}_{4\times 4}, then we get the quaternion \mathbb{Q}proper Wiener process w_{0}(t).
Using this new concept, we consider the detection problem with the hypotheses of the form
with x(t) a known continuous quaternion signal and w(t) the general quaternion Wiener process. Denoting \left(\right.separators="">\n \n y\n (\n t\n )\n =\n \n \n \u222b\n \n \n 0\n \n \n t\n \n \n x\n (\n s\n )\n ds\n \n and considering its augmented vector y(t) then, the generalized Pitcher’s equation (5) for this case is
In the following result we solve equation (9) and give an explicit expression for g(s).
Corollary 3.2
Suppose that the quaternion matrix A(t) given in (8) has inverse for t∈[0,T] then
where \mathit{g}\left(T\right)={\left[g,{g}^{\mathrm{i}},{g}^{\mathrm{j}},{g}^{\mathrm{k}}\right]}^{\mathtt{\text{T}}} with g arbitrary.
Remark 4
In the particular case that we have the quaternion \mathbb{Q}proper Wiener process w_{0}(t) with parameter r_{0}, then we obtain the extension of the wellknown CameronMartin formula to the quaternion domain, which is given by
Simulation example
In order to illustrate the performance of the proposed detector we consider the model (4) with the quaternion signal x(t) of the form
and the quaternion noise v(t)=v_{1}(t) + v_{2}(t)i + v_{3}(t)j + v_{4}(t)k the one given in the example of [12], i.e., {v_{ n }(t),t∈[0,1]}, n=1,…,4, are Gaussian processes with v_{3}(t)=v_{1}(t) + v_{2}(t) + w_{1}(t) and v_{4}(t)=v_{3}(t) + w_{2}(t), w_{1}(t) and w_{2}(t) realvalued independent Gaussian processes and also independent of v_{1}(t) and v_{2}(t). Moreover, E v_{1}(t)v_{1}(s)]=f(t)f(s), E v_{2}(t)v_{2}(s)]=g(t)g(s), E v_{1}(t)v_{2}(s)]=f(t)g(s), E w_{1}(t)w_{1}(s)]=d(t)d(s) and E w_{2}(t)w_{2}(s)]=h(t)h(s) with f(t)=1−6t, g(t)=6t^{2}, d(t)=2t−1 and h(t)=20t^{3}−30t^{2} + 12t−1.
In Figure 1, we show the detection probability versus the falsealarm probability by using the NeymanPearson criterion.
Detection of quaternion random signals in QWGN
So far we have considered quaternion deterministic signals. However, there are other situations in which the quaternion signals have a stochastic nature. In this framework, we study the detection problem of a quaternion random signal in additive QWGN, i.e, we consider the hypotheses pair
with x(t) a meansquare continuous quaternion random signal and w_{0}(t) the quaternion \mathbb{Q}proper Wiener process with parameter r_{0}. Suppose also that x(t) is independent of w_{0}(t).
Theorem 4.1
{\mathcal{P}}_{0}\equiv {\mathcal{P}}_{1} and the loglikelihood ratio is
where {\widehat{x}}_{\text{QWL}}\left(t\right) is the QWL estimator of x(t) given in (3) and β_{ n } and φ_{ n }(t) are the eigenvalues and eigenfunctions of R_{ x }(t,s), respectively.
Conclusions
Different quaternion detectors obtained from augmented statistics have been presented. Although we have avoided dealing with a fourdimensional eigen problem by introducing the signal x(t), we have to solve a unidimensional eigen problem which can be very involved in practice. In those cases where a closedform solution of the eigen problem is not available, a numerical method of solution can be used, as for example, the RayleighRitz method [16]. This numerical procedure allows us to solve operator equations approximately and thus, to obtain suboptimum detectors for the Gaussian detection problems addressed which converge to the optimum ones. To this end, we can use an approximate QKL expansion for quaternion signals based on the approximate eigenvalues and eigenfunctions obtained from the application of the RayleighRitz method.
Finally, we would like to give an outlook to the possible extensions of the results provided in this work. For instance, in the problem of detecting a random signal in white Gaussian noise it is wellknown the estimatorcorrelator representation of the loglikelihood ratio, which depends on the causal estimator of the signal. Our future goal will be the extension of this closed form for the detector to the quaternion domain. On the other hand, the application of the methodology proposed in the field of Reproducing Kernel Hilbert Spaces could allows us to find an interesting solution for the discrimination problem between two quaternion random signals.
Appendix 1
Proof of Theorem 3.1
From (1) and (2), v(t) and R_{ v }(t,s) admit the series representations
where \left(\right.separators="">\n \n \n \n \u03f5\n \n \n n\n \n \n =\n \n \n \u222b\n \n \n 0\n \n \n T\n \n \n \n \n \phi \n \n \n n\n \n \n H\n \n \n (\n t\n )\n v\n (\n t\n )\n dt\n \n. Thus, taking (5) and (14) into account, we get
with \left(\right.separators="">\n \n \n \n \chi \n \n \n n\n \n \n =\n \n \n \beta \n \n \n n\n \n \n \n \n \u222b\n \n \n 0\n \n \n T\n \n \n \n \n \phi \n \n \n n\n \n \n H\n \n \n (\n s\n )\n d\n g\n (\n s\n )\n \n. Then, to study the continuoustime problem (4) we can consider the equivalent discrete problem^{b}
On the other hand, since R_{ v }(t,s) is a continuous function we have that 2Δ_{1}<∞. Likewise, from (14)
Hence, applying Grenander’s Theorem [9] to (16) we obtain that {\mathcal{P}}_{0}\equiv {\mathcal{P}}_{1} and
Now, from (13) and (15) we have that z\left(t\right)=\sum _{n=1}^{\infty}{\mathit{\phi}}_{n}\left(t\right){\varsigma}_{n} and thus,
Finally, from (18), (19), and (17) we demonstrate (6).
Proof of Corollary 3.2
Consider the Hermitian matrix \left(\right.separators="">\n \n M\n (\n t\n )\n =\n \n \n \u222b\n \n \n 0\n \n \n t\n \n \n A\n (\n \tau \n )\n d\tau \n \n then, taking (8) into account, it follows that (9) is equivalent to
Thus, integrating by parts (20), we get
Now, since \left(\right.separators="">\n \n M\n (\n t\n )\n =\n \n \n \u222b\n \n \n 0\n \n \n t\n \n \n A\n (\n s\n )\n ds\n \n we have that (21) is equal to
Then the solution of (9) is given by (10).
Proof of Theorem 4.1
Consider the random variables \left(\right.separators="">\n \n \n \n \u03f5\n \n \n n\n \n \n =\n \n \n \u222b\n \n \n 0\n \n \n T\n \n \n \n \n \phi \n \n \n n\n \n \n H\n \n \n (\n t\n )\n x\n (\n t\n )\n dt\n \n and \left(\right.separators="">\n \n \n \n w\n \n \n n\n \n \n =\n \n \n \u222b\n \n \n 0\n \n \n T\n \n \n \n \n \phi \n \n \n n\n \n \n H\n \n \n (\n t\n )\n d\n \n \n \n w\n \n \n 0\n \n \n \n (\n t\n )\n \n. Then E[ϵ_{ n }ϵ_{ m }]=β_{ n }δ_{ nm }, E[w_{ n }w_{ m }]=r_{0}δ_{ nm } and E[ϵ_{ n }w_{ m }]=0, for all n and m.
The problem (11) is equivalent to the following problem
Unlike (16), ϵ_{ n } and w_{ n } are now both random variables. Thus, under {\mathcal{H}}_{0}, ς_{ n }∼N(0,r_{0}) and under {\mathcal{H}}_{1}, ς_{ n }∼N(0,β_{ n } + r_{0}). From these conditions, it is shown [9] that {\mathcal{P}}_{0}\equiv {\mathcal{P}}_{1} and
On the other hand, the random variables ς_{ n }take the form \left(\right.separators="">\n \n \n \n \u03c2\n \n \n n\n \n \n =\n \n \n \u222b\n \n \n 0\n \n \n T\n \n \n \n \n \phi \n \n \n n\n \n \n H\n \n \n (\n t\n )\n d\n z\n (\n t\n )\n \n. This fact is immediate under {\mathcal{H}}_{0} and, under {\mathcal{H}}_{1}, we have
Thus, the first term of (22) can be expressed in the following way
Finally, from (22), (23), and (3) we get (12).
Endnotes
^{a}That is, the augmented correlation function of w_{0}(t) is \left(\right.separators="">\n \n \n \n R\n \n \n \n \n \n w\n \n \n 0\n \n \n \n \n \n (\n t\n ,\n s\n )\n =\n \n \n r\n \n \n 0\n \n \n min\n (\n t\n ,\n s\n )\n \n \n I\n \n \n 4\n \xd7\n 4\n \n \n \n, where I_{4×4}is the fourdimensional identity matrix [17]. ^{b} Due to the random coefficients ϵ_{ n } having the same information up to sets of measure zero as that of v(t).
Abbreviations
 QWL:

Quaternion widely linear
 KL:

KarhunenLoève
 QKL:

Quaternion KarhunenLoève
 QWGN:

Quaternion white Gaussian noise.
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NavarroMoreno, J., RuizMolina, J.C., Oya, A. et al. Detection of continuoustime quaternion signals in additive noise. EURASIP J. Adv. Signal Process. 2012, 234 (2012). https://doi.org/10.1186/168761802012234
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DOI: https://doi.org/10.1186/168761802012234