GLRT-based array receivers for the detection of a known signal with unknown parameters corrupted by noncircular interferences
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Abstract
The detection of a known signal with unknown parameters in the presence of noise plus interferences (called total noise) whose covariance matrix is unknown is an important problem which has received much attention these last decades for applications such as radar, satellite localization or time acquisition in radio communications. However, most of the available receivers assume a second order (SO) circular (or proper) total noise and become suboptimal in the presence of SO noncircular (or improper) interferences, potentially present in the previous applications. The scarce available receivers which take the potential SO noncircularity of the total noise into account have been developed under the restrictive condition of a known signal with known parameters or under the assumption of a random signal. For this reason, following a generalized likelihood ratio test (GLRT) approach, the purpose of this paper is to introduce and to analyze the performance of different array receivers for the detection of a known signal, with different sets of unknown parameters, corrupted by an unknown noncircular total noise. To simplify the study, we limit the analysis to rectilinear known useful signals for which the baseband signal is real, which concerns many applications.
Keywords
Detection GLRT Known signal Unknown parameters Noncircular Rectilinear Interferences Widely linear Arrays Radar GPS Time acquisition DS-CDMAI. Introduction
The detection of a known signal with unknown parameters in the presence of noise plus interferences (called total noise in the following), whose covariance matrix is unknown, is a problem that has received much attention these last decades for applications such as time or code acquisition in radio communications networks, time of arrival estimation in satellite location systems or target detection in radar and sonar.
Among the detectors currently available, a spatio-temporal adaptive detector which uses the sample covariance matrix estimate from secondary (signal free) data vectors is proposed in [1] and [2] by Brennan, Reed and Mallett. This detector is modified in [3] by Robey et al to derive a constant false-alarm rate test called the adaptive matched filter (AMF) detector, well suited for radar applications. In [4] the previous problem is reconsidered by Kelly as a binary hypothesis test: total noise only versus signal plus total noise. The Kelly's detector uses the maximum likelihood (ML) approach to estimate the unknown parameters of the likelihood ratio test, namely the total noise covariance matrix and the complex amplitude of the useful signal. This detection scheme is commonly referred to as the GLRT [5]. Extensions of the Kelly's GLRT approach assuming that no signal free data vectors are available are presented in [6] and [7] for radar and GPS applications respectively. In [8], Brennan and Reed propose a minimum mean square error detector for time acquisition purposes in the context of multiusers DS-CDMA radio communications networks. This problem is then reconsidered in [9] by Duglos and Scholtz from a GLRT approach under a Gaussian noise assumption and assuming the total noise covariance matrix and the useful propagation channel are two unknown parameters. The advantages of this detector are presented in [6] in a radar context, with regard to structured detectors that exploit an a priori information about the spatial signature of the targets.
Nevertheless, all the previous detectors assume implicitly or explicitly a second order (SO) circular [10] (or proper [11]) total noise and become suboptimal in the presence of SO noncircular (or improper [12]) interferences, which may be potentially present in radio communications, localization and radar contexts. Indeed, many modulated interferences share this feature, for example, Amplitude Modulated (AM), Amplitude Phase Shift Keying (ASK), Binary Phase Shift Keying (BPSK), Rectangular Quadrature Amplitude Modulated, offset QAM, Minimum Shift Keying (MSK) or Gaussian MSK (GMSK) [13] interferences. For this reason, the problem of optimal detection of a signal corrupted by SO noncircular total noise has received an increasing attention this last decade. In particular, a matched filtering approach in SO noncircular total noise is presented in [14] and [12] for radar and radio communications respectively, but under the restrictive assumption of a completely known signal. Alternative approaches, developed under the same restrictive assumptions, are presented in [16] and [15] using a deflection criterion and the LRT respectively. In [17] the problem of optimal detection in SO noncircular total noise is investigated but under the assumption of a noncircular random signal. In [18] a GLRT approach is also proposed to detect the noncircular character of the observations and its performance is studied in [19].
However, despite these works, the major issue of practical use consisting in detecting a known signal with unknown parameters in the presence of an arbitrary unknown SO noncircular total noise has been scarcely investigated up to now. To the best of our knowledge, it has only been analyzed recently in [20] and [21] for synchronization and time acquisition purposes in radio communications networks, assuming a BPSK, MSK or GMSK useful signal and both unknown total noise and unknown useful propagation channel. For this reason, to fill the gap previously mentioned and following a GLRT approach, the purpose of this paper is to introduce and to analyze the performance of different array receivers, associated with different sets of unknown signal parameters, for the detection of a known signal corrupted by an unknown SO noncircular total noise. To simplify the analysis, only rectilinear known useful signals are considered, i.e. useful signals whose complex envelope is real such as AM, PPM, ASK or BPSK signals, also called one dimensional signals. This assumption is not so restrictive since rectilinear signals, and BPSK signals in particular, are currently used in a large number of practical applications such as DS-CDMA radio communications networks, GNSS system [22], some IFF systems or some specific radar systems which use binary coding signal [23]. For such known waveforms, the new detectors introduced in this paper implement optimal widely linear (WL) [24] filters contrary to the detectors proposed in [1, 3, 4, 6, 7, 8, 9] and [25] which are deduced from optimal linear filters.
Section II introduces some hypotheses, data statistics and the problem formulation. In section III, the optimal receiver for the detection of a known rectilinear signal with known parameters corrupted by a SO noncircular total noise is presented as a reference receiver, jointly with some of its performance. Various extensions of this optimal receiver, assuming different sets of unknown signal's parameters, are presented in sections IV and V from a GLRT approach for known and unknown signal steering vector, respectively. Performance of all the developed receivers are compared to each other in section VI through computer simulations, displaying, in the detection process, the great interest to take the potential noncircular feature of the total noise into account. Finally section VII concludes the paper. Note that most of the results of the paper have been patented in [20] and [26], whereas some results of the paper have been partially presented in [27] and theoretical statistical performances of some receivers have been studied in [28].
II. Hypotheses and problem formulation
A. Hypotheses
where the known transmitted symbols, a_{ n } (0 ≤ n ≤ K - 1) are real and deterministic, T is the symbol duration and v(t) is a real-valued pulse shaped filter verifying the Nyquist condition, i.e., such that r(nT) = v(t) ⊗ v(-t)*/_{ t = nT } = 0 for n ≠ 0, where ⊗ is the convolution operation. The signal s(t) may correspond to the synchronization preamble of a radio communications link. For example, each burst of the military 4285 HF standard is composed of a synchronization sequence containing K = 80 known BPSK symbols, 3 × 16 known BPSK symbols for Doppler tracking and 4 × 32 QPSK information symbols. The filter v(t) corresponds to a raise cosine pulse shape filter with a roll off equal to 0.25 or 0.3. The signal s(t) may also correspond to the PN code transmitted by one satellite of a GNSS system where, in this case and as shown in Appendix A, a_{ n } and T correspond to the transmitted chips and chip duration respectively whereas v(t) is a rectangular pulse of duration T. Finally, although model (1) is generally not valid for conventional radar applications, it holds for some specific radar applications such as secondary surveillance radar (SSR), currently used for air traffic control surveillance and called Identification Friend and Foes (IFF) systems in the military domain. For example for the standardized S-mode of such systems, the signal transmitted by a target for its identification is a PPM signal which has the form (1) where v(t) is a rectangular pulse of duration T and where a_{ n } = 0 or 1. Other specific active radars transmit a series of N pulses such that each pulse is a known binary sequence (a_{ n } = ±1) of 13 chips (K = 13) corresponding to a Barker code, whereas v(t) is a rectangular pulse of duration T.
where b_{ Tv }(nT) is the zero mean sampled total noise vector at the output of v(-t)*, which is assumed to be uncorrelated with a_{ n }.
B. Second order statistics of the data
where ${\pi}_{s}\left(nT\right)\triangleq {\mu}_{s}^{2}{a}_{n}^{2}$ is the instantaneous power of the useful signal which should be received by an omnidirectional sensor of gain unity; R(nT) ≜ E[b_{ Tv }(nT)b_{ Tv }(nT)^{ H }] and C(nT) ≜ E[b_{ Tv }(nT)b_{ Tv }(nT)^{ T }] are the first and second correlation matrices of b_{ Tv }(nT) respectively. Note that C(nT) = 0 ∀n for a SO circular total noise vector and that the previous statistics depend on the time parameter since both the known signal (rectilinear) and the interferences (potentially digitally modulated) are not stationary.
C. Problem formulation
The problem addressed in this paper then consists in detecting, from a GLRT approach, the known symbols or chips a_{ n } (0 ≤ n ≤ K - 1), from the observation vectors x_{ v }(nT) (0 ≤ n ≤ K - 1), for different sets of unknown parameters, assuming the total noise b_{ Tv }(nT) is potentially SO noncircular. More precisely, we assume that each of the parameters μ_{ s }, ϕ_{ s }, s, R(nT) and C(nT) may be either known or unknown, depending on the application. We first address the unrealistic case of completely known parameters in section III, while the cases of practical interest corresponding to some unknown parameters are addressed in sections IV and V from a GLRT approach. To compute all these receivers, some theoretical assumptions, which are not necessary verified and which are not required in practical situations, are made. These assumptions are not so restrictive in the sense that GLRT-based receivers derived under these assumptions still provide good detection performance even if most of the latter are not verified in practice. These theoretical assumptions correspond to
A.1: the samples b_{ Tv }(nT), 0 ≤ n ≤ K - 1, are zero mean, statistically independent, noncircular and jointly Gaussian
A.2: the matrices R(nT) and C(nT) do not depend on the symbol indice n
A.3: the samples b_{ Tv }(nT) and a_{ m } are uncorrelated ∀n, m.
The statistical independence of the samples b_{ Tv }(nT) requires in particular propagation channels with no delay spread and may be verified for temporally white interferences. The Gaussian assumption is a theoretical assumption allowing to only exploit the SO statistics of the observations from a LRT or a GLRT approach whatever the statistics of interference, Gaussian or not. The noncircular assumption is true in the presence of SO noncircular interferences but is generally not exploited in detection problems up to now. Assumption A.2 is true for cyclostationary interferences with symbol period T. Finally A.3 is verified in particular for a useful propagation channel with no delay spread. It is also verified for a propagation channel with delay spread for which the main path is the useful signal whereas the others, sufficiently delayed, are included in b_{ Tv }(nT).
III. Optimal receiver for known parameters
A. Optimal receiver
where ${\mathbf{w}}_{1,c}\triangleq {e}^{j{\varphi}_{s}}{\mathbf{R}}^{-1}\mathbf{s}$ is the conventional SMF, ${y}_{1,c}\left(nT\right)\triangleq {\mathbf{w}}_{1,c}^{H}{\mathbf{x}}_{v}\left(nT\right)$, ${z}_{1,c}\left(nT\right)\triangleq \mathsf{\text{Re}}\left[{y}_{1,c}\left(nT\right)\right]$, ${\widehat{\mathbf{r}}}_{x,a}$, ${\widehat{\mathbf{r}}}_{{y}_{1,c},a}$ and ${\widehat{\mathbf{r}}}_{{z}_{1,c},a}$ are defined by (13) where ${\stackrel{\u0303}{\mathbf{x}}}_{v}\left(nT\right)$ has been replaced by x_{ v }(nT), y_{1,c}(nT) and z_{1,c}(nT) respectively. Expression (12) then corresponds to the correlation of the real part, z_{1,c}(nT), of the SMF's output, y_{1,c}(nT), with the known real symbols, a_{ n }, over the known signal duration.
B. Performance
Computation and comparison of SINR_{ o } and SINR_{ c } are done in [32] in the presence of one rectilinear interference plus background noise and is not reported here. This comparison displays in particular the great interest of taking the SO noncircularity of the total noise into account in the receiver's computation as well as the capability of the optimal receiver to perform, in this case, single antenna interference cancellation (SAIC) of a rectilinear interference by exploiting the phase diversity between the sources. Illustrations of CONV_{1} and OPT_{1} receiver performance are presented in section VI.
IV. GLRT receivers for a known signal steering vector
In most of situations of practical interest, the parameters μ_{ s }, ϕ_{ s }, R(nT) and C(nT) are unknown while, for some applications, the steering vector s is known. This is in particular the case for radar applications for which a Doppler and a range processing currently take place at the output of a beam, which is mechanically or electronically steered in a given direction and scanned to monitor all the directions of space. In this case, the steering vector s is associated with the current direction of the beam. Another example corresponds to satellite localization for which the satellite positions are known and the vector s may be associated, in this case, with the direction of one of the satellites. Moreover, in some cases, some signal free observation vectors (called secondary observation vectors) sharing the same total noise SO statistics are available in addition to the observation vectors containing the signal to be detected plus the total noise (called primary observation vectors). For example the secondary observation vectors may correspond to samples of data associated with another range than the range of the detected target in radar or to observations in the absence of useful signal. In such situations, we will say that a total noise alone reference (TNAR) is available. In other applications, a TNAR is difficult to built, due for example to the total noise potential nonstationarity or to the presence of multipaths. For all the reasons previously described, following a GLRT approach, we introduce in sections IV-A, IV-B and IV-C several new receivers for the detection of a known real-valued signal, with different sets of unknown parameters, corrupted by a SO noncircular total noise. More precisely, these receivers assume that the parameters μ_{ s } and ϕ_{ s } are unknown, the vector s is known and the matrices R(nT ) and C(nT) are either known (section IV-A) or unknown, assuming (section IV-B) or not (section IV-C) that a TNAR is available in this latter case.
A. Unknown parameters (μ_{s}, ϕ_{s}) and known total noise (R, C)
which is proportional to the square modulus of the correlation between the SMF's output, y_{1,c}(nT), and the known real-valued symbols, a_{ n }, over the known signal duration.
B. Unknown parameters (μ_{s}, ϕ_{s}) and total noise (R, C) with a TNAR
We assume in this section that s is known, parameters μ_{ s }, ϕ_{ s }, R and C are unknown and that a TNAR is available. We denote by b_{ Tv }(nT)' (0 ≤ n ≤ K' - 1) the K' samples of the secondary data, which contain the total noise only such that R(nT)' ≜ E[b_{ Tv }(nT)'b_{ Tv }(nT)'^{ H }] = R(nT) and C(nT)' ≜ E[b_{ Tv }(nT)'b_{ Tv }(nT)'^{ T }] = C(nT). Under both this assumption and A.1, A.2, matrices R and C may be estimated either from the secondary data only or from both the primary and the secondary data, which gives rise to two different receivers.
1) Total noise estimation from secondary data only
where $\widehat{\mathbf{R}}$ is defined by (24) but with b_{ Tv }(nT)' instead of ${\stackrel{\u0303}{\mathbf{b}}}_{Tv}{\left(nT\right)}^{\prime}$.
2) Total noise estimation from both primary and secondary data
where ${\widehat{\mathbf{R}}}_{b}$ is defined by (24) with b_{ Tv }(nT)' instead of ${\stackrel{\u0303}{\mathbf{b}}}_{Tv}{\left(nT\right)}^{\prime}$. Expression (34) is nothing else than the Kelly's detector [4], whose extensions to an arbitrary number of primary samples are given by (32) for a SO circular total noise and by (31) for both a SO noncircular total noise and a real-valued signal to be detected. Note finally that for a very large number of secondary snapshots (K' → ∞), (28) becomes equivalent to (24) and receiver (31) reduces to (26).
C. Unknown parameters (μ_{s}, ϕ_{s}) and total noise (R, C) without a TNAR
where ${\widehat{\mathbf{R}}}_{x}$ is defined by (35) with x_{ v }(nT) instead of ${\stackrel{\u0303}{\mathbf{x}}}_{v}\left(nT\right)$. Note that when K becomes very large (K → ∞), (38) and (39) also correspond to (31) and (32) respectively. Moreover, for a very weak desired signal and (SINR_{ o } ≪ 1), ${\widehat{\mathbf{R}}}_{\stackrel{\u0303}{x}}\approx {\widehat{\mathbf{R}}}_{\stackrel{\u0303}{b}}$ defined by (24) with K and ${\stackrel{\u0303}{\mathbf{b}}}_{Tv}\left(nT\right)$ instead of K' and ${\stackrel{\u0303}{\mathbf{b}}}_{Tv}{\left(nT\right)}^{\prime}$,${\widehat{\mathbf{R}}}_{x}\approx {\widehat{\mathbf{R}}}_{b}$ defined by (24) with K and b_{ Tv }(nT) instead of K' and ${\stackrel{\u0303}{\mathbf{b}}}_{Tv}{\left(nT\right)}^{\prime}$, ${\widehat{\mathbf{r}}}_{\stackrel{\u0303}{x},a}^{H}{\widehat{\mathbf{R}}}_{\stackrel{\u0303}{x}}^{-1}{\widehat{\mathbf{r}}}_{\stackrel{\u0303}{x},a}\ll 1$ and ${\widehat{\mathbf{r}}}_{x,a}^{H}{\widehat{\mathbf{R}}}_{x}^{-1}{\widehat{\mathbf{r}}}_{x,a}\ll 1$. We then deduce that (38) and (39) reduce to (26) and (27) respectively.
V. GLRT receiver for an unknown signal steering vector
In some situations of practical interest such as in radio communications, the steering vector s is often unknown jointly with the parameters μ_{ s }, ϕ_{ s }, R(nT) and C(nT). Moreover, in some cases, some signal free observation vectors (secondary observation vectors) sharing the same total noise SO statistics are still available in addition to the primary observation vectors and may correspond to samples of data associated with adjacent channels, adjacent time slots or guard intervals. For these reasons, we introduce in sections V-A, V-B and V-C several new receivers for the detection of a known real-valued signal, with different sets of unknown parameters and whose steering vector is unknown, corrupted by a SO noncircular total noise.
A. Unknown parameters (μ_{s}, ϕ_{s}, s) and known total noise (R, C)
B. Unknown parameters (μ_{s}, ϕ_{s}, s) and total noise (R, C) with a TNAR
We assume in this section that parameters μ_{ s }, ϕ_{ s }, R, C and s are unknown but that a TNAR is available. We note b_{ Tv }(nT)' (0 ≤ n ≤ K' - 1) the K' samples of the secondary data, which only contain the total noise such that R(nT)' ≜ E[b_{ Tv }(nT)'b_{ Tv }(nT)'^{ H }] = R(nT) and C(nT)' ≜ E[b_{ Tv }(nT)'b_{ Tv }(nT)'^{ T }] = C(nT).
1) Total noise estimation from secondary data only
where $\widehat{\mathbf{R}}$ is defined by (24) but with b_{ Tv }(nT)' instead of ${\stackrel{\u0303}{\mathbf{b}}}_{Tv}{\left(nT\right)}^{\prime}$.
2) Total noise estimation from both primary and secondary data
where ${\widehat{\mathbf{R}}}_{b,0}$ is defined by (33). Note finally that for a very large number of secondary snapshots (K' → ∞), (45) becomes equivalent to (43) and receiver (46) reduces to (44).
C. Unknown parameters (μ_{s}, ϕ_{s}, s) and total noise (R, C) without a TNAR
which is nothing else than the detector introduced in [8] and [9] for synchronization purposes in SO circular contexts. Note finally that for very large values of K (K → ∞), (47) becomes equivalent to (45) and receiver (48) reduces to (46).
VI. Performances of receivers in the presence of so noncircular interferences
A. Total noise model
where η_{2} is the mean power of the background noise per sensor; I is the (N × N) identity matrix; π_{ p }(kT) ≜ E[|j_{p,v}(kT)|^{2}] is the instantaneous power of the interference p at the output of the filter v(-t)* received by an omnidirectional sensor for a free space propagation; c_{ p }(kT) ≜ E[j_{p,v}(kT)^{2}] characterizes the SO noncircularity of the interference p. In particular, c_{ p }(kT) = π_{ p }(kT) for a BPSK interference p, whereas c_{ p }(kT) = 0 for a QPSK interference p.
B. Computer simulations
1) Hypotheses
Synthesis of the different receivers and associated unknown parameters and hypotheses
Known parameters | Unknown parameters | Hypotheses | Receivers |
---|---|---|---|
μ_{ s }, ϕ_{ s }, s, R(nT), C(nT) | No | No | OPT(CONV)_{1}(x_{ v }, K) |
s, R(nT), C(nT ) | μ _{ s } , ϕ _{ s } | No | OPT(CONV)_{2}(x_{ v }, K) |
s | μ_{ s }, ϕ_{ s }, R(nT), C(nT) | TNAR available, R, C on sec. data | OPT(CONV)_{3}(x_{ v }, K, K') |
s | μ_{ s }, ϕ_{ s }, R(nT), C(nT) | TNAR available, R, C on sec.+prim. Data | OPT(CONV)_{4}(x_{ v }, K, K') |
s | μ_{ s }, ϕ_{ s }, R(nT), C(nT) | No TNAR | OPT(CONV)_{5}(x_{ v }, K) |
R(nT), C(nT) | μ_{ s }, ϕ_{ s }, s | No | OPT(CONV)_{6}(x_{ v }, K) |
No | μ_{ s }, ϕ_{ s }, s, R(nT), C(nT) | TNAR available, R, C on sec. data | OPT(CONV)_{7}(x_{ v }, K, K') |
No | μ_{ s }, ϕ_{ s }, s, R(nT), C(nT) | TNAR available, R, C on sec.+prim. data | OPT(CONV)_{8}(x_{ v }, K, K') |
No | μ_{ s }, ϕ_{ s }, s, R(nT), C(nT) | No TNAR | OPT(CONV)_{9}(x_{ v }, K) |
2) Scenarios with P= 1 interference
Figures 2a, 3a, 4a and 5a show, for N = 1 sensor, the poor detection of the desired signal from all the conventional detectors due to their incapability to reject the strong interference. On the contrary, the optimal detectors, which exploit the SO noncircularity of both the desired signal and the interference, perform SAIC due to the exploitation of the phase diversity between the sources. Note that SAIC is possible since the SO noncircularity of both the desired signal and interference are exploited by the receiver, which is not the case for the WL MVDR beamformer introduced in [33] which does not exploit the SO noncircularity of the desired signal. Comparison of Figures 2a and 3a or 4a and 5a shows increasing performance of the optimal detectors as the phase diversity between the sources increases. In both cases, the O_{1} detector, which assumes that all the parameters of the sources are known, gives the best performance. In a same way, the O_{9} detector, which assumes that all the parameters of the sources are unknown, has the lowest performance. Moreover, for a given set of unknown desired signal parameters, the a priori knowledge of the noise statistics (O_{2} and O_{6}) increases the performance with respect to the absence of knowledge of the latter. In a same way, the knowledge of a TNAR (O_{3}, O_{4}, O_{7}, O_{8}) allows to roughly increase the performance with respect to an absence of TNAR (O_{5}, O_{9}). Finally, counterintuitively, the use of both primary and secondary data for the estimation of the noise correlation matrix (O_{4}, O_{8}) degrades the performance with respect to the use of secondary data only (O_{3}, O_{7}) for this estimation. This is due to the fact that contrary to the LRT receiver which is optimal for detection, GLRT receivers are sub-optimal receivers which generate estimates of the noise covariance matrix with more variance when primary data are used. More precisely, the variance of the noise covariance matrix estimate and then the associated performance degradation increases with an increasing relative weight given to the primary data with respect to secondary data in the linear combination of the two estimates, which explains the result. On the contrary, in such situations, an optimal receiver would necessarily decide to discard the primary data and to keep only the secondary data not to increase the variance of the noise covariance matrix estimate and then not to decrease the performance. However, this optimal process does not correspond to a GLRT receiver and is perhaps to invent. The same reasoning holds for OPT_{7}, OPT_{8} and OPT_{9} receivers.
Figures 2b, 3b, 4b and 5b show that, for N = 2 sensors, all the conventional detectors have an increased detection probability with respect to the case N = 1 due to their capability to reject the interference thanks to the spatial discrimination between the sources. Moreover, we note, for a given set of estimated parameters, much better performance of the optimal detectors due to the joint spatial and phase discriminations between the sources. Comparison of Figures 2b and 3b or 4b and 5b shows again increasing performance of the optimal detectors as the phase diversity between the sources increases. We still note the best performance of the completely informed detectors (C_{1} and O_{1}) and the lowest performance of the less informed detectors (C_{9} and O_{9}). We note again, for a given set of unknown desired signal parameters, that better performance are obtained when the total noise is either known or estimated from secondary data only. In a same way, the knowledge of a TNAR allows to increase the performance in comparison with an absence of TNAR. Finally, for a given set of total noise parameters, the a priori knowledge of the signal steering vector s increases the performance.
3) Scenarios with P= 2 interferences
We note the poor detection of the desired signal from all the conventional detectors compared to the optimal ones, due to their difficulty to reject the two strong interferences since the array is overconstrained (P = N = 2). On the contrary, the optimal detectors, which discriminate the sources by both the direction of arrival and the phase, succeed in rejecting these two interferences since one is rectilinear, which generates a good detection of the desired signal in most cases. More precisely, it has been shown in [33] and [32] that a BPSK source generates only one source in the extended observation vector, while a QPSK source generates two sources. The protection of the desired signal and the rejection of the two interferences then require 1 + 1 + 2 = 4 degrees of freedom, which in fact corresponds to the number of degrees of freedom, 2N = 4, effectively available, hence the result. Comparison of Figures 6a and 6b or 7a and 7b shows again increasing performance of the optimal detectors as the phase diversity between the desired signal and the BPSK interference increases. Again, the O_{1} detector gives the best performance while the O_{9} detector gives the lowest ones. Again, the a priori knowledge of the noise statistics or of a TNAR or of the desired signal steering vector allows an increase in performances.
VII. Conclusion
Several new receivers for the detection of a known rectilinear signal, with different sets of unknown parameters, corrupted by SO noncircular interferences have been presented in this paper. It has been shown that taking the potential noncircularity property of the interferences into account may dramatically improve the performance of both mono and multi-sensors receivers, due to the joint exploitation of phase and spatial discrimination between the sources. In particular, the capability of the new detectors to do SAIC of rectilinear interferences, by exploiting the phase diversity between the sources has been verified for all the new detectors. It also puts forward that the more a priori information on the signal, the better the performance.
Appendix A
which has the same form as (1) where real-valued symbols a_{ n } are replaced by real-valued chips d_{ l }(±1), where T is replaced by T_{ c } and where K is replaced by K SF - 1. We easily verify that $w\left(t\right)\otimes w{\left(-t\right)}^{*}{|}_{t=n{T}_{c}}=0$ for n ≠ 0.
Appendix B
Using (20) into (B.4), it is straightforward to verify that a sufficient statistics of (B.4) is given by (22).
Appendix C
which proves that LR(x_{ v }, K) defined by (C.11) is an increasing function of the sufficient statistic $\frac{{\mathbf{z}}^{H}{\mathbf{A}}^{-1}\mathbf{z}}{\alpha -{\mathbf{u}}^{H}\mathbf{u}}$ which is finally proportional to (31).
Appendix D
Using (40) into (B.4), it is straightforward to verify that a sufficient statistics of (D.3) is given by (41).
Appendix E
which proves that LR(x_{ v }, K) defined by (C.11) is an increasing function of the sufficient statistic $\frac{{\mathbf{u}}^{H}\mathbf{u}}{\alpha}$