Interfering Channel Estimation in Radar-Cellular Coexistence: How Much Information Do We Need?
Fan Liu, Adrian Garcia-Rodriguez, Christos Masouros, Giovanni Geraci
aa r X i v : . [ ee ss . SP ] J u l Interfering Channel Estimation in Radar-CellularCoexistence: How Much Information Do We Need?
Fan Liu,
Student Member, IEEE,
Adrian Garcia-Rodriguez,
Member, IEEE,
Christos Masouros,
Senior Member, IEEE, and Giovanni Geraci,
Member, IEEE
Abstract —In this paper, we focus on the coexistence between aMIMO radar and cellular base stations. We study the interferingchannel estimation, where the radar is operated in the “searchand track” mode, and the BS receives the interference from theradar. Unlike the conventional methods where the radar and thecellular systems fully cooperate with each other, in this work weconsider that they are uncoordinated and the BS needs to acquirethe interfering channel state information (ICSI) by exploiting theradar probing waveforms. For completeness, both the line-of-sight (LoS) and Non-LoS (NLoS) channels are considered in thecoexistence scenario. By further assuming that the BS has limiteda priori knowledge about the radar waveforms, we proposeseveral hypothesis testing methods to identify the working modeof the radar, and then obtain the ICSI through a variety ofchannel estimation schemes. Based on the statistical theory,we analyze the theoretical performance of both the hypothesistesting and the channel estimation methods. Finally, simulationresults verify the effectiveness of our theoretical analysis anddemonstrate that the BS can effectively estimate the interferingchannel even with limited information from the radar.
Index Terms —Radar interfering channel estimation, Radar-communication coexistence, spectrum sharing, hypothesis testing,search and track.
I. I
NTRODUCTION R ECENT years have witnessed an explosive growth ofwireless services and devices. As a consequence, thefrequency spectrum has become one of the most valuableresources. Since 2015, mobile network operators in the UKhave been required to pay a combined annual total of £ £ A. Existing Approaches
Aiming for realizing the spectral coexistence of radar andcommunication, existing contributions mainly focus on miti-gating the mutual interference between the two systems by use
This work has been submitted to the IEEE for possible publication.Copyright may be transferred without notice, after which this version mayno longer be accessible.F. Liu and C. Masouros are with the Department of Electronic and ElectricalEngineering, University College London, London, WC1E 7JE, UK. (e-mail:[email protected], [email protected]).A. Garcia-Rodriguez and G. Geraci are with Nokia BellLabs, Dublin 15, Ireland. (e-mail: [email protected],[email protected]). of precoding/beamforming techniques [6]–[10]. Such effortscan be found in the pioneering work of [7], in which the radarsignals are precoded by a so-called null-space projector (NSP),and thus the interference generated to the communicationsystems is zero-forced. To achieve a favorable performancetrade-off, the NSP method is further improved in [9], [10] viaSingular Value Decomposition (SVD), where the interferencelevel can be adjusted considering the singular values of thechannel matrix.As a step further, more recent works have exploited convexoptimization techniques for jointly designing transmit wave-forms/precoders of radar and communication systems, suchthat certain performance metrics can be optimized [11]–[20].For instance, in [13], the receive signal-to-interference-plus-noise ratio (SINR) of the radar is maximized in the presenceof both the clutters and the communication interference, whilethe capacity of the communication system is guaranteed. Theinverse problem has been tackled in [14], where the communi-cation rate has been maximized subject to the radar SINR con-straint, as well as the power budgets for both systems. Whilethe aforementioned works are well-designed via sophisticatedtechniques, it is in general difficult for them to be applied tocurrent radar applications, given the fact that the governmentaland military agencies are unwilling to make major changes intheir radar deployments, which may impose huge costs ontheir financial budgets [21]. Hence, a more practical approachis to develop transmission schemes at the communicationside only, where the radar is agnostic to the interference oreven the operation of the communication system. In this line,[16] considers the coexistence between a MIMO radar anda BS performing multi-user MIMO (MU-MIMO) downlinktransmissions, in which the precoder of the BS is the onlyoptimization variable. In [18], [19], the BS precoder has beenfurther developed by exploiting the constructive multi-userinterference, which demonstrates orders-of-magnitude power-savings.It is worth highlighting that precoding based techniquesrequire the knowledge of the interfering channel either atthe radar or the communication BS. In fact, perfect/imperfectchannel state information (CSI) assumptions are quite typicalin the above works. To obtain such information, the radar andthe BS are supposed to fully cooperate with each other andtransmit training symbols, in line with conventional channelestimation methods. In [10], the MIMO radar needs to estimatethe channel based on the received pilot signals sent by the BS,which inevitably occupies extra computational and signalingresources. Other works such as [13] require an all-in-one control center to be connected to both systems via a dedi-cated side information link, which conducts the informationexchange and the waveform optimization. In practical scenar-ios, however, the control center brings forward considerablecomplexity in the system design, and is thus difficult toimplement. Moreover, since it is the cellular operator whoexploits the spectrum of the radar, it is the performance ofthe latter that should be primarily guaranteed, i.e., the radarresources should be allocated to target detection rather thanobtaining the CSI. Unfortunately, many existing contributionsfailed to address this issue, and, to the best of our knowledge,the channel estimation approaches tailored for the radar-cellular coexistence scenarios remain widely unexplored. Inlight of the above drawbacks regarding the CSI acquisition,the natural question is, 1) is it possible to estimate the channelwhen there is limited cooperation between the radar and thecommunication systems?
And if so, 2) how much informationdo we need for the estimation?B. The Contribution of Our Work
This paper aims at answering the above issues, where wefocus on interfering channel estimation between a MIMO radarand a MIMO BS:1) To cope with the first issue above, we hereby proposeto exploit the radar probing waveforms for estimatingthe interfering channel. In this case the radar does notneed to send training symbols or estimate the channelby itself, and thus the need for cooperation is fullyeliminated. Following the classic MIMO radar literature[22], [23], we assume that the radar has two workingmodes, i.e., searching and tracking. In the search mode,the radar transmits a spatially orthogonal waveform,which formulates an omni-directional beampattern forsearching potential targets over the whole angular do-main. In the track mode, the radar transmits directionalwaveforms to track the target located at the angle ofinterest, and thus to obtain a more accurate observation.In the meantime, the BS is trying to estimate the channelbased on the periodically received radar interference,which is tied to the radar’s duty cycle. As the searchingand tracking waveforms are randomly transmitted, wepropose to identify the operation mode of the radar byuse of the hypothesis testing approach, and then estimatethe channel at the BS.2) To answer the second question raised above, we furtherinvestigate different cases under both LoS and NLoSchannels, where different levels of priori knowledgeabout the radar waveforms are assumed to be knownat the BS, i.e., from full knowledge of searching andtracking waveforms by the BS, to knowledge of search-ing waveform only, to a fully agnostic BS to the radarwaveforms. From a realistic perspective, the second andthe third cases are more likely to appear in practicewhile the first case serves as a performance benchmark.Accordingly, the theoretical performance analysis of theproposed approaches are provided. (cid:258) (cid:258)
Radar TX
InterferingChannel
Comms RX (a) (cid:258) (cid:258)
Radar TX
InterferingChannel
Comms RX (b)Fig. 1. MIMO Radar and BS coexistence. (a) Radar search mode; (b) Radartrack mode.
For the sake of clarity, we summarize below the contribu-tions of this paper:1) We consider the interfering channel estimation for thecoexistence of radar and cellular, where the radar prob-ing waveforms are exploited to obtain ICSI at the BS.2) We propose hypothesis testing approaches for the BS toidentify the operation mode of the radar, based on thelimited priori information available at the BS.3) We analyze the theoretical performance of the proposeddetectors and estimators, whose effectiveness is furtherverified via numerical results.The remainder of this paper is arranged as follows. SectionII introduces the system model, Section III and Section IVpropose interfering channel estimation approaches for NLoSand LoS scenarios, respectively. Subsequently, Section V an-alyzes the theoretical performance of the proposed schemes,Section VI provides the corresponding numerical results, andfinally Section VII concludes the paper.
Notations : Unless otherwise specified, matrices are denotedby bold uppercase letters (i.e., X ), vectors are representedby bold lowercase letters (i.e., z ), and scalars are denoted bynormal font (i.e., ρ ). tr ( · ) and vec ( · ) denote the trace and thevectorization operations. ⊗ denotes the Kronecker product. k·k and k·k F denote the l norm and the Frobenius norm. ( · ) T , ( · ) H , and ( · ) ∗ stand for transpose, Hermitian transpose andcomplex conjugate, respectively.II. S YSTEM M ODEL
As shown in Fig. 1, we consider a MIMO radar with M t transmit antennas and M r receive antennas that is detectingtargets located in the far field. For simplicity, we assume that Fig. 2. Radar working mode - “search and track” and operations performedby the communications BS. the MIMO radar employs the same antenna array for bothtransmission and reception, and denote M t = M r = M .Meanwhile, an N -antenna BS operating in the same frequencyband is receiving interference from the radar and trying toacquire the ICSI between them. Below we provide the systemmodels for both the radar and the BS. A. Radar Signal Transmission - Search and Track
It is widely known that by employing incoherent waveforms,the MIMO radar achieves higher Degrees of Freedom (DoFs)and better performance than the conventional phased-arrayradar that transmits correlated waveforms [22]. By denotingthe MIMO radar probing waveform as X ∈ C M × L , its spatialcovariance matrix can be given as [22]–[26] R X = 1 L XX H , (1)where L is the length of the radar pulse. Throughout the paperwe consider L ≥ N ≥ M > , and assume uniform lineararrays (ULA) at both the radar and the BS. The correspondingbeampattern can be thus given in the form [22]–[26] P d ( θ ) = a H ( θ ) R X a ( θ ) , (2)where θ denotes the azimuth angle, and a ( θ ) = (cid:2) , e j π ∆ sin( θ ) , ..., e j π ( M − θ ) (cid:3) T ∈ C M × is the steer-ing vector of the transmit antenna array with ∆ being thespacing between adjacent antennas normalized by the wave-length.When the orthogonal waveform is transmitted by the MIMOradar, it follows that [24], [27] R X = P R M I M , (3)where P R is the total transmit power of the radar, and I M isthe M -dimensional identity matrix. It is easy to see from (2)that the covariance matrix (3) generates an omni-directionalbeampattern, which is typically used for searching when thereis limited information about the target locations [22]. Oncea target is detected, the radar switches to the tracking mode,where it will no longer transmit orthogonal waveforms andwill generate a directional beampattern that points to thespecific location, thus obtaining a more accurate observation.This, however, results in a non-orthogonal transmission, i.e., R X = P R M I M . In this paper, we assume that the radar adoptsboth the searching and tracking modes subject to a probability transition model. This model is illustrated in Fig. 2 and canbe summarized as follows [28]: Assumption 1 : At the i -th pulse repetition interval (PRI)of the radar, the probability that the radar is operating at thetracking mode is P ( i − D , where P ( i − D is the target detectionprobability of the ( i − -th PRI.The above assumption entails that the MIMO radar changesits probing waveform X randomly within each PRI, whichmakes it challenging for the BS to estimate the interferingchannel between them. B. Interfering Channel Model
The interfering channel between the BS and the radar couldbe characterized through different models, depending on theirspecific positions. For instance, the military and weather radarsare typically located at high-altitude places such as top of thehills, in which case the channel between the BS and radaris likely to be a Line-of-Sight (LoS) channel. On the otherhand, if the radar is used for monitoring the low-altitudeflying objects (such as drones) or the urban traffic, it isusually deployed in urban areas at similar heights than theBS, thus resulting in a Non-Line-of-Sight (NLoS) channel.For completeness, we will discuss both cases in this paper.Since both the radar and the BS are located in fixed positions,we also adopt the following assumption:
Assumption 2 : For the LoS coexistence scenario, we assumethat the interfering channel from the radar to the BS is fixed.For the NLoS coexistence scenario, we assume the interferingchannel is flat Rayleigh fading, and remains unchanged duringseveral radar PRIs.
C. BS Signal Reception Model
Denoting the interfering channel as G ∈ C N × M , thereceived signal matrix at the BS can be given as Y = GX + W , (4)where W = [ w , w , ..., w L ] ∈ C N × L is the noise matrix,with w l ∼ CN ( , N I N ) , ∀ l . In the proposed hypothesistesting framework, the noise power N plays an important rolefor normalizing the test statistic. Note that when radar keepssilent, the BS will receive nothing but the noise, and N canbe measured at this stage. Since the radar antenna number andits transmit power are fixed parameters, they can also be easilyknown to the BS operators. Therefore, it is reasonable to adoptthe following assumption: Assumption 3 : The BS knows the value of N , M and P R .In order to estimate the channel and the noise power N ,the BS needs to know when is radar transmitting, i.e., it mustsynchronize its clock with the radar pulses. As shown in Fig.2, during one PRI, the radar only transmits for a portion of thetime, typically below 10%, and employs the remaining 90%for receiving, during which the radar remains silent. Such aratio is called duty cycle [14]. By exploiting this property,the BS is able to blindly estimate the beginning and the endof a radar pulse by some simple methods, such as energydetection. Note that for the NLoS channel scenario, there willbe random time-spread delays within each pulse, which makes the synchronization inaccurate. However, since we assume aflat fading channel in the NLoS case, the time-spread delaywill be contained within one snapshot of the radar, whichresults in negligible errors [29]. We summarize the abovethrough the following assumption: Assumption 4 : The BS can perfectly synchronize its clockwith the radar pulses, i.e., it is able to know the beginning andthe end of each radar pulse.
D. Channel Estimation Procedure
In light of the above discussion, we summarize below thechannel estimation procedure at the BS:1) Synchronize the system clock with the radar transmittedpulses.2) Identify the working mode of the radar based on thereceived radar interference, i.e., whether the radar issearching or tracking.3) Estimate the interfering channel by exploiting the limitedknowledge about the radar waveforms.In the following, we will develop several approaches for theBS to acquire the ICSI when radar is randomly changing itsprobing waveform. We will first consider the NLoS channelcase, and then the LoS channel case.III. NL O S C
HANNEL S CENARIO
Consider the ideal case where the BS knows exactly thewaveform sent by the radar in each PRI. Recalling (4), thewell-known maximum likelihood estimation (MLE) of thechannel G is given as [30] ˆG = YX H (cid:0) XX H (cid:1) − , (5)which is also known as the Least-Squares estimation (LSE)for G . Unfortunately, the BS is not able to identify whichwaveform is transmitted, since the radar changes its waveformrandomly at each PRI. Hence, (5) can not be directly appliedand it is difficult to estimate the channel directly. In thissection, we discuss several cases where different levels ofknowledge about the radar waveforms are available at the BS.At each level, we propose specifically tailored approaches toacquire the ICSI. A. BS Knows the Searching and Tracking Waveforms - Gen-eralized Likelihood Ratio Test (GLRT)
In this reference case, we assume that the BS knows boththe searching and the tracking waveforms that the radar maytransmit at the i -th PRI, which we denote as X and X ,respectively. Since X is orthogonal, we have L X X H = P R M I M ⇒ X X H = LP R M I M . (6)Before estimating the channel, the BS needs to identify whichwaveform is transmitted based on the received noisy data Y ∈ C N × L . This leads to the following hypothesis testing (HT)problem [31] Y = ( H : GX + W , H : GX + W . (7) As per Assumption 1, the priori probabilities of the above twohypotheses can be given as P ( H ) = 1 − P ( i − D , P ( H ) = P ( i − D . (8)The HT problem (7) can be solved via the generalized likeli-hood ratio test (GLRT), which is given by [31] L G ( Y ) = p (cid:16) Y ; ˆG , H (cid:17) P ( H ) p (cid:16) Y ; ˆG , H (cid:17) P ( H )= p (cid:16) Y ; ˆG , H (cid:17) P ( i − D p (cid:16) Y ; ˆG , H (cid:17) (cid:16) − P ( i − D (cid:17) H ≷ H γ, (9)where γ is the detection threshold, p (cid:16) Y ; ˆG , H (cid:17) and p (cid:16) Y ; ˆG , H (cid:17) are the likelihood functions (LFs), for the twohypotheses respectively, and can be given in the form p (cid:16) Y ; ˆG , H (cid:17) = ( πN ) − NL exp (cid:18) − N tr (cid:18)(cid:16) Y − ˆGX (cid:17) H (cid:16) Y − ˆGX (cid:17)(cid:19)(cid:19) , (10) p (cid:16) Y ; ˆG , H (cid:17) = ( πN ) − NL exp (cid:18) − N tr (cid:18)(cid:16) Y − ˆGX (cid:17) H (cid:16) Y − ˆGX (cid:17)(cid:19)(cid:19) . (11)In the above expressions, ˆG and ˆG are the MLEs under H and H , which are obtained as ˆG = YX H (cid:0) X X H (cid:1) − , (12) ˆG = YX H (cid:0) X X H (cid:1) − = MLP R YX H . (13)Overall, once the BS determines which hypothesis to choosebased on Y , it can successfully estimate the channel byuse of (12) or (13). However, it can be observed that theGLRT detector in (9) requires the detection probability P ( i − D to be known to the BS, which is impossible in practice.Therefore, the detector (9) can only serve as the optimalperformance bound . Since the actual P ( i − D is unknown tothe BS, the reasonable choice for the priori probabilities is P ( H ) = P ( H ) = 0 . , namely P ( i − D = 0 . . We can thenapply the similar GLRT procedure for solving the HT problem.The test statistic in (9) is thus simplified as L G ( Y ) = p (cid:16) Y ; ˆG , H (cid:17) p (cid:16) Y ; ˆG , H (cid:17) H ≷ H γ. (14) B. BS Knows Only the Searching Waveform - Rao Test
In a realistic scenario, the tracking waveform X may varyfrom pulse to pulse. This is because the target to be detectedmay move very fast, which results in rapid changes in itsparameters such as the distance, velocity and the azimuthangle. Hence, it is far from realistic to assume the BS knows X , not to mention P D (in fact, both quantities are onlydetermined after a target is detected). Nevertheless, as anomni-directional searching waveform, there is no reason for X to be changed rapidly. Indeed, in some cases, the radarmay only use one waveform for omni-searching. Based on theabove, to assume that the BS only knows X seems to be amore practical choice . In this case, the HT problem (7) canbe recast as H : X = X , G , H : X = X , G . (15)In (15), the channel to be estimated is called the nuisanceparameter [31]. Remark 1:
At first glance, the GLRT procedure seems tobe applicable to (15) as well. However, note that to obtain theMLE of G under H is equivalent to solving the followingoptimization problem min G , X k Y − GX k F s.t. k X k F = LP R , (16)where the constraint is to ensure the power budget of theradar-transmitted waveform. While the above problem is non-convex, it yields trivial solutions that achieve zero with ahigh probability. This is because the problem (16) is likelyto have more than enough DoFs to ensure that Y = GX ,since G is unconstrained, and X can be always scaled tosatisfy the norm constraint, where the scaling factor can beincorporated in G . Therefore, the likelihood function under H will always be greater than that of H , which makes theHT design meaningless.Realizing the fact above, we propose to use the Rao test(RT) to solve the HT problem (15), which does not need theMLE under H . Based on [32]–[34], let us define Θ = (cid:2) vec T ( X ) , vec T ( G ) (cid:3) T , (cid:2) θ Tr , θ Ts (cid:3) T . (17)Then, the RT statistic for the complex-valued parameters canbe given in the form T R ( Y )= 2 ∂ ln p ( Y ; Θ ) ∂ vec ( X ) (cid:12)(cid:12)(cid:12)(cid:12) T Θ = ˜Θ h J − (cid:16) ˜Θ (cid:17)i θ r θ r ∂ ln p ( Y ; Θ ) ∂ vec * ( X ) (cid:12)(cid:12)(cid:12)(cid:12) Θ = ˜Θ H ≷ H γ, (18)where ˜Θ = h θ Tr , ˆ θ Ts i T = h vec T ( X ) , vec T (cid:16) ˆG (cid:17)i T is theMLE under H , and h J − (cid:16) ˜Θ (cid:17)i θ r θ r is the upper-left partitionof J − (cid:16) ˜Θ (cid:17) , with J ( Θ ) being the Fisher Information Matrix(FIM).Unlike the GLRT, the Rao test only lets the BS determineif the radar is using the searching mode, i.e., whether theorthogonal waveform matrix X is transmitted in the currentradar PRI. In that case, the BS could obtain the MLE of thechannel by use of (13). Otherwise, the BS is required to waituntil an orthogonal waveform is transmitted by the radar. At this stage we note the fact that such information exchange can be easilyperformed once prior to transmission, since the searching waveform of theradar remains unchanged. In contrast, conventional training based techniquesrequire the radar or the BS to frequently send pilot symbols, which entails amuch tighter cooperation between both systems.
C. Agnostic BS
We now consider the hardest case that the BS does not knowany of the waveforms transmitted by the radar. In this case, theBS still knows that XX H = LP R M I M for an omni-directionalradar transmission. Therefore, the HT problem in (15) can berecast as H : XX H = LP R M I M , G , H : XX H = LP R M I M , G . (19) Remark 2:
At first glance, we might be able to apply ageneralized RT to solve the HT problem, where both the truevalues of G and X are replaced by their MLEs. This isbecause X is also unknown to the BS. Note that to obtainthe MLEs of these two parameters is equivalent to solving thefollowing optimization problem min G , X k Y − GX k F s.t. XX H = LP R M I M . (20)Again, the above problem will unfortunately yield trivial so-lutions and make the HT design meaningless. This is because X can be viewed as a group of orthogonal basis, and theunconstrained G spans the whole space, which makes anygiven Y achievable with a high probability.The above remark involves that it is challenging to blindlyestimate the ICSI for an agnostic BS under the NLoS channelscenario. Instead, we will show in the next section that blindchannel estimation is feasible for the LoS channel scenario.IV. L O S C
HANNEL S CENARIO
In this section, we consider the scenario that the interferingchannel between radar and BS is a LoS channel, where thereceived signal matrix at the BS is given by Y = α b ( θ ) a H ( θ ) X + W , (21)where α represents the large-scale fading factor, θ is theangle of arrival (AoA) from the radar to the BS, b ( θ ) = (cid:2) , e j π Ω sin( θ ) , ..., e j π ( N − θ ) (cid:3) T ∈ C N × is the steeringvector of the BS antenna array, with Ω being the normalizedspacing, and a ( θ ) is radar’s steering vector defined in Sec.II-A. Since the ULA geometry of the radar is fixed, weassume that the BS knows the spacing between the adjacentantennas of radar. Hence, the channel parameters that need tobe estimated at the BS are α and θ .Adopting the ideal assumption that the BS has instantaneousknowledge of the radar-transmitted waveform X in each PRI,the MLEs of the two parameters could be obtained by solvingthe optimization problem min α,θ (cid:13)(cid:13) Y − α b ( θ ) a H ( θ ) X (cid:13)(cid:13) F . (22)Note that if θ is fixed, the MLE of α can be given as ˆ α = b H ( θ ) YX H a ( θ ) L k b ( θ ) k a H ( θ ) R X a ( θ ) = b H ( θ ) YX H a ( θ ) N L a H ( θ ) R X a ( θ ) , (23) which suggests that the MLE of α depends on that of θ .Substituting (23) into the objective function of (22), the MLEof θ can be thus given by ˆ θ = arg min θ f ( Y ; θ, X ) , (24)where f ( Y ; θ, X ) = (cid:13)(cid:13)(cid:13)(cid:13) Y − b H ( θ ) YX H a ( θ ) b ( θ ) a H ( θ ) X N L a H ( θ ) R X a ( θ ) (cid:13)(cid:13)(cid:13)(cid:13) F . (25)While (25) is non-convex, the optimum can be easily obtainedthrough a 1-dimensional search over θ . A. BS Knows the Searching and Tracking Waveforms - GLRT
By assuming that the BS knows both X and X , the HTproblem (7) can be reformulated as Y = ( H : α b ( θ ) a H ( θ ) X + W , H : α b ( θ ) a H ( θ ) X + W . (26)The GLRT detector can be again applied to the LoS channel, inwhich case the likelihood functions under the two hypothesesare given as p (cid:16) Y ; ˆ θ , H (cid:17) = ( πN ) − NL exp (cid:18) − N f (cid:16) Y ; ˆ θ , X (cid:17)(cid:19) ,p (cid:16) Y ; ˆ θ , H (cid:17) = ( πN ) − NL exp (cid:18) − N f (cid:16) Y ; ˆ θ , X (cid:17)(cid:19) , (27)where f is defined in (25), and ˆ θ and ˆ θ are the MLEs of θ under the two hypotheses, respectively. By recalling (9), theGLRT detector can be expressed as L LoSG ( Y ) = 1 N (cid:16) f (cid:16) Y ; ˆ θ , X (cid:17) − f (cid:16) Y ; ˆ θ , X (cid:17)(cid:17) H ≷ H γ. (28)The analytic distribution for (28) is not obtainable, since thereis no closed-form solution of ˆ θ under both hypotheses. B. BS Knows Only the Searching Waveform - Energy Detec-tion
Similar to the NLoS channel case, a more practical as-sumption is to consider that the BS knows only the searchingwaveform X . In this case, the GLRT detector is no longerapplicable and the HT is given by H : X = X , α, θ, H : X = X , α, θ. (29)At first glance, it seems that the Rao detector (18) can betrivially extended from the NLoS channel scenario to the LoScase. Nevertheless, the following proposition puts an end tosuch a possibility. Proposition 1.
The Rao test does not exist for the scenarioof the LoS channel.Proof.
See Appendix A. (cid:4)
The algebraic explanation behind Proposition 1 is intuitive.As shown in (21), by multiplying the rank-1 LoS channel
Fig. 3. Searching and tracking beampatterns of the MIMO radar. to the radar waveform, the latter is mapped to a rank-1subspace, which leads to serious information losses and yieldsa non-invertible FIM. Recalling (18), the Rao test requires tocompute the inverse of the FIM. Hence, it simply does notwork in this specific case.To resolve the aforementioned issue, we consider an energydetection (ED) approach for the LoS channel. According to(21), the average power of the received radar signal is givenas P LoS = E (cid:0) tr (cid:0) YY H (cid:1)(cid:1) = E tr (cid:16) | α | b ( θ ) a H ( θ ) XX H a ( θ ) b H ( θ ) + WW H (cid:17) + (cid:0) tr (cid:0) α b ( θ ) a H ( θ ) XW H (cid:1)(cid:1) = E (cid:16) tr (cid:16) | α | b ( θ ) a H ( θ ) XX H a ( θ ) b H ( θ ) + WW H (cid:17)(cid:17) ≈ L tr (cid:16) | α | b ( θ ) a H ( θ ) XX H a ( θ ) b H ( θ ) (cid:17) + N N = | α | P d ( θ ) tr (cid:0) b ( θ ) b H ( θ ) (cid:1) + N N = N | α | P d ( θ ) + N N , (30)where P d ( θ ) is the radar transmit beampattern defined in (2),and the approximation in the fourth line of (30) is based onthe Law of Large Numbers. From (30), it is obvious thatthe received power at the BS is proportional to the radar’stransmit power at the angle θ . If the searching waveform X is transmitted, we have P d ( θ ) = P R M a H ( θ ) I M a ( θ ) = P R , (31)which suggests that the BS will receive equal power at eachangle θ . On the other hand, if the tracking waveform X istransmitted, most of the power will focus at the mainlobe,while less power will be distributed among the sidelobes, inwhich case the BS receives high power when it is located at themainlobe of the radar, and much lower power at other angles.According to the aforementioned observations, in this paperwe let the BS define two power measurement thresholds to determine whether the radar is in searching or tracking mode.As shown in Fig. 3 , the BS chooses H if the received powerfalls between the two proposed thresholds, and it chooses H otherwise. Accordingly, the ED detector can be given as T E ( Y ) = 1 L tr (cid:0) YY H (cid:1) ∈ [ γ, η ] → H ,T E ( Y ) = 1 L tr (cid:0) YY H (cid:1) ∈ (0 , γ ] ∪ [ η, + ∞ ) → H , (32)where γ and η are the two power thresholds. Remark 3:
Note that the performance of the detector in (32)depends on the size of the ambiguity regions shown in Fig.3. By narrowing the distance between γ and η , the detectortrades-off the tolerance of the noise with the ambiguity area.By using the ED detector, the BS could choose from thetwo hypotheses without knowing both waveforms. Once H ischosen, the BS can estimate the AoA by finding the minimumof f ( Y ; θ, X ) . C. Agnostic BS
Finally, we consider the hardest case where the BS does notknow either the searching or tracking waveform. Note thatthe energy detector (32) still works in this case, as it doesnot require any information about X or X . The remainingquestion is how to estimate the channel. In order to do so, wefirst note that for the case of omni-directional transmission wehave P d ( θ ) = P R M a H ( θ ) I M a ( θ ) = P R , (33)in which case (31) can be rewritten as P LoS = E (cid:0) tr (cid:0) YY H (cid:1)(cid:1) ≈ L tr (cid:0) YY H (cid:1) ≈ N P R | α | + N N . (34)From (34), it follows that | α | ≈ tr (cid:0) YY H (cid:1) LN P R − N P R , (35)which can be used for estimating the absolute value of α . It canbe further observed that for the omni-directional transmission,we also have L YY H = | α | P R M b ( θ ) a H ( θ ) I M a ( θ ) b H ( θ ) + ˜W = | α | P R b ( θ ) b H ( θ ) + ˜W , (36)where ˜W is the noise matrix. The LSE of θ can be thus givenby ˆ θ = arg min θ (cid:13)(cid:13)(cid:13)(cid:13) YY H LP R − | α | b ( θ ) b H ( θ ) (cid:13)(cid:13)(cid:13)(cid:13) F . (37)Overall, once H is chosen by the energy detector (32),one can estimate | α | and θ by (35) and (37) respectively,even without any knowledge about the radar waveforms. Weremark that, since the noise matrix ˜W is no longer Gaussiandistributed, (35) and (37) are not MLEs of the parameters.For clarity, we summarize the proposed approaches fordifferent scenarios in Table. I. The tracking beampattern in Fig. 3 is generated based on the convexoptimization method in [22], which we show in (73) in Sec. VI. TABLE IP
ROPOSED A PPROACHES FOR D IFFERENT S CENARIOS
NLoS Channel LoS Channel
BS Knows Both Waveforms GLRT GLRTBS Knows Searching Waveform Rao Test Energy DetectionAgnostic BS None Energy Detection
V. T
HEORETICAL P ERFORMANCE A NALYSIS
In this section, we provide the theoretical performanceanalysis for the proposed hypothesis testing and channel esti-mation approaches. With this purpose, we use decision errorprobability and the mean squared error (MSE) as performancemetrics.
A. GLRT for NLoS Channels
To analyze the performance of the GLRT detector, the MLEsof the unknown parameters under different hypotheses mustbe derived in closed-forms. While we consider GLRT for bothNLoS and LoS channels in the previous discussion, the closed-form MLE of the AoA is not obtainable for the LoS channel.Therefore, we will only analyze the GLRT performance forthe NLoS channel in this subsection. Firstly, let us substitute(12) and (13) into (10) and (11), which yield p (cid:16) Y ; ˆG , H (cid:17) = ( πN ) − NL exp (cid:18) − N tr (cid:18)(cid:16) Y − ˆG X (cid:17) H (cid:16) Y − ˆG X (cid:17)(cid:19)(cid:19) = ( πN ) − NL exp (cid:18) − N tr (cid:18) Y (cid:18) I − MLP R X H X (cid:19) Y H (cid:19)(cid:19) , (38)and p (cid:16) Y ; ˆG , H (cid:17) = ( πN ) − NL exp (cid:18) − N tr (cid:18)(cid:16) Y − ˆG X (cid:17) H (cid:16) Y − ˆG X (cid:17)(cid:19)(cid:19) = ( πN ) − NL exp (cid:18) − N tr (cid:18) Y (cid:18) I − X H (cid:16) X X H (cid:17) − X (cid:19) Y H (cid:19)(cid:19) . (39)Taking the logarithm of (9) we obtain ln p (cid:16) Y ; ˆG , H (cid:17) P ( i − D p (cid:16) Y ; ˆG , H (cid:17) (cid:16) − P ( i − D (cid:17) = 1 N tr (cid:18) Y (cid:18) X H (cid:0) X X H (cid:1) − X − MLP R X H X (cid:19) Y H (cid:19) − ln − P ( i − D P ( i − D ! H ≷ H γ . (40)Finally, the GLRT detector can be given as L G ( Y ) = 1 N tr (cid:18) Y (cid:18) X H (cid:16) X X H (cid:17) − X − MLP R X H X (cid:19) Y H (cid:19) H ≷ H γ = γ + ln − P ( i − D P ( i − D ! . (41) T R ( Y ) = 2 N tr (cid:18)(cid:18) I L − MLP R X H X (cid:19) Y H YX H (cid:16) X Y H YX H (cid:17) − X Y H Y (cid:19) H ≷ H γ. (51)Note that both X H (cid:0) X X H (cid:1) − X and MLP R X H X are projec-tion matrices [30]. The physical meaning of (41) is intuitive,i.e., to project the received signal onto the row spaces of X and X respectively, and to compute the difference betweenthe lengths of the projections to decide which hypothesis tochoose. Letting P ( i − D = 0 . , we have ln (cid:18) − P ( i − D P ( i − D (cid:19) = 0 and γ = γ , which represents the case that P D is unknown.We now derive the Cumulative Distribution Function (CDF)of L G . Defining A = X H (cid:0) X X H (cid:1) − X , B = MLP R X H X , ˜y = vec (cid:0) Y H (cid:1) √ N , D = I N ⊗ ( A − B ) , (42)it follows that L G ( Y ) = ˜y H ( I N ⊗ ( A − B )) ˜y = ˜y H D˜y . (43)If D is an idempotent matrix, then the test statistic subjectsto the non-central chi-squared distribution [30]. While both A and B are idempotent, it is not clear if their difference is stillidempotent. Moreover, their is no guarantee that D is semidef-inite. Hence, D is an indefinite matrix in general, which makes L G an indefinite quadratic form (IQF) in Gaussian variables.Given the non-zero mean value of ˜y , L G becomes a non-central Gaussian IQF, which is known to have no closed-formexpression for its CDF [35], [36]. Based on [37], here weconsider a so-called saddle-point method to approximate theCDF of the test statistic. It is clear that ˜y ∼ CN ( b , I NL ) ,where b = H : vec (cid:0) X H G H (cid:1).p N , H : vec (cid:0) X H G H (cid:1).p N , (44)which are the mean values for ˜y under H and H respec-tively. Let us denote the eigenvalue decomposition of D as D = QΛQ H , where Λ = diag ( λ , λ , ..., λ NL ) contains theeigenvalues. Based on the saddle-point approximation [37], theCDF of L G is given as P ( L G ≤ γ ) ≈ π exp ( s ( ω )) s π | s ′′ ( ω ) | , (45)where s ( ω ) = ln (cid:18) e γ ( jω + β ) e − c ( ω ) ( jω + β ) det ( I + ( jω + β ) Λ ) (cid:19) , (46) c ( ω ) = NL X i =1 (cid:12)(cid:12) ¯ b i (cid:12)(cid:12) − NL X i =1 (cid:12)(cid:12) ¯ b i (cid:12)(cid:12) − ( jω + β ) λ i , (47) ¯b = Q H b = (cid:2) ¯ b , ¯ b , ..., ¯ b NL (cid:3) T . (48) The above results hold for any β > . ω is the so-calledsaddle point, which is the solution of the following equation s ′ ( jω ) = − − ω + β ) − NL X i =1 λ i λ i ( − ω + β )+ γ − NL X i =1 (cid:12)(cid:12) ¯ b i (cid:12)(cid:12) λ i (1 + λ i ( − ω + β )) = 0 , (49)where ω = j ( β + p ) . It has been proved that (49) has a singlereal solution on p ∈ ( −∞ , [37], which can be numericallyfound through a 1-dimensional searching.At the i -th PRI, it is natural to measure the performance ofGLRT by use of the decision error probability given the CDFof L G , which is obtained as P ( i ) G = P ( L G ≥ γ ; H ) P ( H ) + P ( L G ≤ γ ; H ) P ( H )= (1 − P ( L G ≤ γ ; H )) (cid:16) − P ( i − D (cid:17) + P ( L G ≤ γ ; H ) P ( i − D , (50)where the CDF of L G under each hypothesis can be computedusing the above equations (45)-(49), by accordingly substitut-ing the values of b under the two hypotheses, which are givenin (44). B. Rao Test for NLoS Channels
We start from the following proposition.
Proposition 2.
The Rao detector for solving (15) is given by(51), shown at the top of this page.Proof.
See Appendix B. (cid:4)
It is clear from (51) that we do not need any informationabout X for solving the HT problem (15), which makes it asuitable detector for the practical scenario where the BS onlyknows X . While Y is Gaussian distributed, it is very difficultto analytically derive the CDF of (51) due to the highly non-linear operations involved. By realizing this, here we onlyfocus our attention on a special case, where the distributionbecomes tractable. Note that if L ≥ M = N holds true, YX H ∈ C N × N and X Y H ∈ C N × N become the invertiblesquare matrices with a high probability, in which case we have YX H (cid:16) X Y H YX H (cid:17) − X Y H = (cid:18)(cid:0) X Y H (cid:1) − X Y H YX H (cid:16) YX H (cid:17) − (cid:19) − = I N . (52) It follows that T Rs ( Y ) = 2 N tr (cid:18)(cid:18) I L − MLP R X H X (cid:19) Y H Y (cid:19) = 2 N tr (cid:18) Y (cid:18) I L − MLP R X H X (cid:19) Y H (cid:19) , N tr (cid:0) YPY H (cid:1) H ≷ H γ (53)is the Rao detector under this special case. It can be seenthat (53) is also a quadratic form in Gaussian variables.Interestingly, the matrix P = I L − MLP R X H X is a projectionmatrix, which projects any vector to the null-space of X H .Therefore, we have tr (cid:16) GX PX H G H (cid:17) = 0 , (54)which leads to tr (cid:16) GX PX H G H (cid:17) ≥ (cid:16) GX PX H G H (cid:17) . (55)The above equations (54) and (55) can be viewed as thehypothesis testing for the noise-free scenario, where we seethat the Rao detector (53) is effective in differentiating thetwo hypotheses. By adding the Gaussian noise to GX and GX , it can be inferred that T Rs ( Y ; H ) ≥ T Rs ( Y ; H ) with a high probability in the high SNR regime, which makesthe detector (53) valid. Proposition 3. T Rs subjects to central and non-central chi-squared distributions under H and H , respectively, whichare given as T Rs ∼ ( H : X K , H : X K ( µ ) , (56) where µ = N tr (cid:16) GX (cid:16) I L − MLP R X H X (cid:17) X H G H (cid:17) is thenon-centrality parameter, and K = 2 N ( L − M ) representsthe DoFs of the distributions.Proof. See Appendix C. (cid:4)
Similar to (50), the decision error probability at the i -th PRIfor the special Rao detector (53) is given by P ( i ) Rs = (cid:16) − F X K ( γ ) (cid:17) (cid:16) − P ( i − D (cid:17) + F X K ( µ ) ( γ ) P ( i − D , (57)where F X K and F X K ( µ ) are the CDFs of central and non-central chi-squared distributions, respectively. C. Channel Estimation Performance for NLoS Channels
As discussed in Sec. IV, there are no closed-form solutionsfor the estimations of the AoA under the LoS channel.Hence, we only consider the channel estimation performancefor the NLoS channel case, where the MSE is used as theperformance metric. By denoting the estimated channel as ˆG = YX H (cid:0) XX H (cid:1) − , the squared error can be given in theform φ = (cid:13)(cid:13)(cid:13) ˆG − G (cid:13)(cid:13)(cid:13) F = (cid:13)(cid:13)(cid:13) YX H (cid:0) XX H (cid:1) − − G (cid:13)(cid:13)(cid:13) F = (cid:13)(cid:13)(cid:13)(cid:0) XX H (cid:1) − XY H − G H (cid:13)(cid:13)(cid:13) F . (58) Let us define ¯y = vec (cid:0) Y H (cid:1) ∼ CN (cid:0) vec (cid:0) X H G H (cid:1) , N I NL (cid:1) , T = I N ⊗ (cid:0) XX H (cid:1) − X , ¯g = vec (cid:0) G H (cid:1) . (59)Then, (58) can be simplified as φ = k T¯y − ¯g k . (60)Based on basic statistics and linear algebra, we also have y eq , T¯y − ¯g ∼ CN (cid:0) , N TT H (cid:1) , (61)where TT H = I N ⊗ (cid:0) XX H (cid:1) − X · I N ⊗ X H (cid:0) XX H (cid:1) − = I N ⊗ (cid:0) XX H (cid:1) − . (62)Based on the above, the MSE of the channel estimation canbe obtained as E ( φ ) = E (cid:16) k y eq k (cid:17) = E (cid:0) tr (cid:0) y eq y Heq (cid:1)(cid:1) = tr (cid:0) E (cid:0) y eq y Heq (cid:1)(cid:1) = N tr (cid:16) I N ⊗ (cid:0) XX H (cid:1) − (cid:17) = N NL tr (cid:0) R − X (cid:1) . (63)In the case that the directional waveform is transmitted, wehave R X = L X X H . For the orthogonal transmission, theMSE can be given as E ( φ ) = N NL tr (cid:18) P R M I M (cid:19) − ! = N M NLP R . (64)It is clear from (63) that the MSE is determined by thecovariance matrix and the length of the radar waveforms, aswell as the antenna number at the BS. D. Energy Detection for LoS Channels
In this subsection, we analyze the performance of the energydetector (32). First of all, let us rewrite (32) as N tr (cid:0) YY H (cid:1) = 2 ˜y H ˜y , (65)where ˜y ∼ CN ( d , I NL ) is given in (42), with d being definedas d = H : vec (cid:0) α ∗ X H a ( θ ) b H ( θ ) (cid:1).p N , H : vec (cid:0) α ∗ X H a ( θ ) b H ( θ ) (cid:1).p N . (66)Eq. (65) is the sum of the squared Gaussian variables, whichsubjects to the non-central chi-squared distribution [30]. Recallthe proof of Proposition 3. By replacing the matrix P in (89) asthe identity matrix I L , we obtain the non-centrality parametersunder two hypotheses as ε = 2 | α | N tr (cid:0) b ( θ ) a H ( θ ) X X H a ( θ ) b H ( θ ) (cid:1) = 2 | α | N LP R N , (67) ε = 2 | α | N tr (cid:0) b ( θ ) a H ( θ ) X X H a ( θ ) b H ( θ ) (cid:1) . (68) The DoFs of both distributions are obtained as κ = 2 rank ( I NL ) = 2 N L. (69)Given any ˜ η ≥ ˜ γ ≥ as the thresholds for the energy detector(32), it follows that L tr (cid:0) YY H (cid:1) ∈ [˜ γ, ˜ η ] ⇔ N tr (cid:0) YY H (cid:1) ∈ (cid:20) L ˜ γN , L ˜ ηN (cid:21) . (70)Let γ , L ˜ γN , η , L ˜ ηN . Under the two hypotheses, theprobability that the test statistic does not fall into the decisionregion can be accordingly given by P ( T E ( Y ) / ∈ [˜ γ, ˜ η ] ; H ) = P (cid:18) N tr (cid:0) YY H (cid:1) / ∈ [ γ, η ] ; H (cid:19) = 1 − (cid:0) − F X κ ( ε ) ( γ ) (cid:1) F X κ ( ε ) ( η ) ,P ( T E ( Y ) ∈ [˜ γ, ˜ η ] ; H ) = P (cid:18) N tr (cid:0) YY H (cid:1) ∈ [ γ, η ] ; H (cid:19) = (cid:0) − F X κ ( ε ) ( γ ) (cid:1) F X κ ( ε ) ( η ) . (71)Finally, at the i -th PRI, the decision error probability for theenergy detector is P ( i ) E = (cid:2) − (cid:0) − F X κ ( ε ) ( γ ) (cid:1) F X κ ( ε ) ( η ) (cid:3) (cid:16) − P ( i − D (cid:17) + (cid:0) − F X κ ( ε ) ( γ ) (cid:1) F X κ ( ε ) ( η ) P ( i − D . (72) E. Discussion on the Hypothesis Testing Thresholds
It is worth highlighting that the performance of all thedetectors above relies on the given thresholds. Typically, thethreshold is chosen to optimize certain performance metrics,i.e., the decision error probability in our case. Note that theGLRT detector is equivalent to the maximum likelihood ratio.Hence the optimal threshold can be straightforwardly givenas γ = ln − P ( i − D P ( i − D ! . Nevertheless, as the true value of P ( i − D is unknown to the BS, only the suboptimal threshold γ = 0 can be adopted.For the Rao and energy detectors, the BS is unable todetermine the optimal hypothesis testing thresholds, sinceit does not know the tracking waveform X under suchscenarios. Therefore, the hypothesis testing thresholds canonly be obtained by numerical simulations. We address thisissue in the next section.VI. N UMERICAL R ESULTS
In this section, numerical results are provided to verify theeffectiveness of the proposed approaches. Below we introducethe parameters used in our simulations.1)
Radar Waveforms:
We use X = q LP R M U as the radarsearching waveform, where U ∈ C M × L is an arbitrarilygiven unitary matrix. For the tracking waveform X , wefirstly solve the classic 3dB beampattern design problem to obtain the waveform covariance matrix R ∈ C M × M ,which is [22] min t, R − ts.t. a H ( θ ) Ra ( θ ) − a H ( θ m ) Ra ( θ m ) ≥ t, ∀ θ m ∈ Ψ , a H ( θ ) Ra ( θ ) = a H ( θ ) Ra ( θ ) / , a H ( θ ) Ra ( θ ) = a H ( θ ) Ra ( θ ) / , R (cid:23) , R = R H , diag ( R ) = P R M , (73)where θ denotes the azimuth angle of the target, i.e., thelocation of the radar’s mainlobe, whose 3dB beamwidthis determined by ( θ − θ ) , and Ψ stands for the sideloberegion. According to [22], problem (73) is convex, andcan be easily solved via numerical tools. The trackingbeampattern generated by (73) can accurately achievethe desired 3dB beamwidth, while maintaining the min-imum sidelobe level. We then obtain the tracking wave-form X by the Cholesky decomposition of R . Withoutloss of generality, we assume that the mainlobe focuseson the angle of ◦ , and the desired 3dB beamwidth is ◦ .2) Threshold Setting:
For the GLRT detectors, we con-sider both the optimal threshold γ = ln (cid:18) − P ( i − D P ( i − D (cid:19) and its suboptimal counterpart γ = 0 . Since the optimalthreshold for Rao test is difficult to obtain, we providethe ergodic empirical thresholds, which are computed byMonte Carlo simulations with a large number of channelrealizations, and can guarantee that the average errorprobability is minimized. Meanwhile, we also computethe optimal threshold that corresponds to one singlechannel realization for M = N , where the theoreticalerror probability is given in (57). Note that such athreshold is not obtainable in practical scenarios, as itrequires the BS to know the channel a priori. In oursimulations, it serves as the performance benchmark forthe Rao test. For the energy detector under the LoSchannel, the empirical thresholds are simply given as γ = N (cid:0) P R + N (cid:1) , η = N (2 P R + N ) , while theperformance of the optimal thresholds for one singlechannel realization is also presented for comparison.3) Other Parameters:
For simplicity, we assume that thedetection probability of radar is the same at each PRI,namely P iD = P D , ∀ i . Without loss of generality, weset P R = 1 , and define the transmit SNR of radaras SNR = P R /N . Unless otherwise specified, we fix L = 20 , and assume half-wavelength separation betweenadjacent antennas. A. NLoS Channel Scenario
In this subsection, we assume a Rayleigh fading channel G ,i.e., the entries of G are independent and identically distributed(i.i.d.) and subject to the standard complex Gaussian distribu-tion. We firstly consider the case that M = N = 16 , L =20 , P D = 0 . . To understand the impact of the ergodic HT thresholds on the performance of the Rao test, Fig. 4 showsthe decision error probability computed through Monte Carlosimulations for increasing values of the HT thresholds. It canbe observed that, for each SNR value, the error probabilitycurve has a unique minimum point, which determines theergodic threshold for the detector. We then use these resultsfor the following Rao test simulations. Hypothesis Testing Thresholds D e c i s i on E rr o r P r obab ili t y -2 -1 -4dB-6dB-10dB-8dB-12dB-14dB -2dB Solid Lines: Error Probability forDifferent SNR Values;Dashed Line: Empirical ErgodicThresholds
Fig. 4. Decision error probability of the Rao test for varying HT thresholds γ . M = N = 16 , L = 20 , P D = 0 . . In Fig. 5, the performances of the GLRT and the Raotest are compared under the same parameter configurationof Fig. 4, where the theoretical and simulated curves aredenoted by solid and dashed lines, respectively. For GLRT,we employ both the optimal and suboptimal thresholds men-tioned above. For the Rao test, we investigate not only theempirical thresholds shown in Fig. 4, but also the optimalthresholds for the specific instantaneous channel realization. Itcan be noted that the theoretical curves match well with theirsimulated counterparts for both detectors, which validates ourperformance analysis of (50) and (57) in Sec. V. Moreover, theRao detector outperforms the GLRT in the low SNR regime,where the associated error probability is close to 0.1. Thereason for this is explained as follows. In light of Fig. 4, theoptimal threshold for Rao test is close to 0 when the SNR islow. Due to the non-negativity of the Rao test statistic (53),hypothesis H will always be chosen by the detector, whichhas the prior probability of P ( H ) = P D = 0 . , leading toan error probability of 0.1. It can be further noted that theGLRT statistic (41) can be either positive or negative. Whenthe SNR is low, the GLRT detector choose randomly from thetwo hypotheses, resulting in an error probability of 0.5. Atthe high SNR regime, however, GLRT outperforms the Raodetector, as it employs the information of both X and X .We further show in Fig. 6 the detection performance for P D = 0 . , where we fix N = 16 , and set M = 10 and M = 16 respectively. Note that the optimal and the suboptimalthresholds for GLRT are exactly the same, given the priorprobability of . for each hypothesis. For the Rao test, since SNR (dB) -16 -14 -12 -10 -8 -6 -4 -2 D e c i s i on E rr o r P r obab ili t y -3 -2 -1 Rao Test, Optimal ThresholdRao Test, Empirical ThresholdGLRT, Optimal ThresholdGLRT, Suboptimal Threshold
Solid Lines: Theoretical Values;Dashed Lines:Simulated Values
Fig. 5. Decision error probability vs. SNR for the GLRT and Rao tests. M = N = 16 , L = 20 , P D = 0 . . the analytical performance for the nonequal-antenna case isintractable, we only show the performance with empiricalthreshold for M = 16 . It can be observed that, when M = 10 ,the performance for both detectors are superior to that ofthe case of M = 16 , which is sensible given that the BSexploits more DoFs for hypothesis testing in the former case.In addition, the GLRT outperforms the Rao test for both lowand high SNR regimes. This is because the priori probabilityfor H is now 0.5, leading to an error probability of 0.5 forRao test for the low SNR regime, which further verifies thecorrectness of our observations in the analysis of Fig. 5. SNR (dB) -16 -12 -8 -4 D e c i s i on E rr o r P r obab ili t y -3 -2 -1 Rao Test, Empirical Threshold, M = 10GLRT, M = 10Rao Test, Optimal Threshold, M = 16Rao Test, Empirical Threshold, M = 16GLRT, M = 16
Solid Lines: Theoretical ValuesDashed Lines:Simulated ValuesM = 10 M = 16
Fig. 6. Decision error probability vs. SNR for the GLRT and Rao tests. M = { , } , N = 16 , L = 20 , P D = 0 . . We investigate the channel estimation performance in Fig.7, where we fix the radar antenna number as M = 5 , andincrease the BS antennas from N = 4 to N = 20 . Note that the hypothesis testing exploits the power of all the entries inthe received signal matrix to make the binary decision, whichdoes not require a high SNR per entry to guarantee a successfuloutcome. This is very similar to the concept of diversitygain. Nevertheless, for the NLoS channel estimation, we needto estimate each entry individually, where the diversity gaindoes not exist. For this reason, we fix the SNR at 15dB toachieve the normal estimation performance. It can be seenfrom Fig. 7 that the theoretical curves match well with thesimulated ones, which proves the correctness of (63) and(64). Secondly, the MSE increases with the rise of the BSantenna number, owing to the increasing number of the matrixentries to be estimated. Finally, it is worth highlighting thatbetter estimation performance can be achieved by use of thesearching waveform X rather than the tracking waveform X . This is because the optimal pilot signals are orthogonalwaveforms such as X according to the channel estimationtheory [38]. Antenna Number at BS (N) M SE ( d B ) -10-8-6-4-202 Simulated, X0, L = 20Theoretical, X0, L = 20Simulated, X1, L = 20Theoretical, X1, L = 20Simulated, X0, L = 30Theoretical, X0, L = 30Simulated, X1, L = 30Theoretical, X1, L = 30
Fig. 7. Channel estimation MSE vs. number of antennas at the BS. M =5 , SNR = 15 dB.
B. LoS Channel Scenario
In this subsection, we show the numerical results for theLoS channel scenario. Unless otherwise specified, we assumethat the BS is located at θ = 20 ◦ relative to the radar. In eachMonte Carlo simulation, a unit-modulus path-loss factor α israndomly generated.We first look at the detection performance of GLRT andED in Fig. 8 with M = N = 16 , L = 20 , P D = 0 . . Forsimplicity, we use “ED” to refer to the energy detection inFig. 8. Again, we observe that the theoretical curves matchwell with their simulated counterparts. It is interesting to seethat the energy detector outperforms the GLRT detector underhigh SNR regime. This is a counter-intuitive behavior, as theGLRT exploits both X and X while the energy detectorrequires nothing from the radar. However, this result can beexplained by realizing that the performance of GLRT is highly dependent on the information contained in the received signals.Specifically, since the LoS channel projects the received signalmatrix onto a rank-1 subspace, this breaks down the structureof the transmitted waveforms. In contrast, the energy detectionexploits the difference between the two beampatterns, whichis equivalent to utilizing the intrinsic structure of X and X ,and hence leads to better performance. SNR (dB) -16 -12 -8 -4 D e c i s i on E rr o r P r obab ili t y -4 -3 -2 -1 GLRT, Optimal ThresholdGLRT, Suboptimal ThresholdED, Empirical Threshold, SimulatedED, Empirical Threshold, TheoreticalED, Optimal Threshold, SimulatedED, Optimal Threshold, Theoretical
Fig. 8. Decision error probability vs. SNR for the GLRT and energy detectionHT under a LoS channel. M = N = 16 , L = 20 , P D = 0 . . Azimuth Angle (deg) -90 -60 -30 0 30 60 90 D e c i s i on E rr o r P r obab ili t y -4 -3 -2 -1 Empirical Threshold, SimulatedOptimal Threshold, SimulatedEmpirical Threshold, TheoreticalOptimal Threshold. Theoretical
Fig. 9. Decision error probability vs. azimuth angle for the energy detectionHT under a LoS channel. M = N = 16 , L = 20 , P D = 0 . , SNR = − dB. As discussed above, the ED exploits the difference betweenthe omnidirectional and directive beampatterns, in which casethe performance of the energy detector relies on the angle ofthe BS relative to the radar. We therefore show in Fig. 9 thedecision error probability at the BS by varying its azimuthangle θ , where the SNR is set as − dB, and the detectorswith both optimal and empirical thresholds are considered. Interestingly, all of the curves in the figure show a shapesimilar to that of the tracking beampattern in Fig. 3. This isbecause the detection performance of ED is mainly determinedby the power gap between the two beampatterns. In themainlobe area, we see that the error performance is better thanthat of the other areas, owing to the largest power gap withinomnidirectional and directive antenna patterns in this region,as illustrated in Fig. 3. Finally, as predicted in Sec. IV-B, thedetection performance becomes worse if the BS is locatedat an angle that falls into the ambiguity region, where thetwo beampatterns are unable to be effectively differentiated.Fortunately, the BS is unlikely to be located in such area sincethe it only occupies a small portion of the whole space.
Antenna Number of BS (N) M SE ( d B ) -20-15-10-505 X0, ML Estimator (Eq. 24)X1, ML Estimator (Eq. 24)ED, LS Estimator (Eq. 37)X0, ML Estimator (Eq. 24)X1, ML Estimator (Eq. 24)
Solid Lines: MSE of θ Dashed Lines: MSE of α Fig. 10. Channel estimation MSE vs. number of antennas at the BS for LoSscenario, M = 4 , L = 20 , SNR = − dB. Fig. 10 shows the channel estimation performance for theLoS scenario with an increasing number of BS antennas,where M = 4 , L = 20 , SNR = − dB. In this figure,the maximum likelihood (ML) and the least-squares (LS)estimators (24) and (37) are employed for the cases of knownand unknown waveforms, respectively. In contrast to the NLoSchannel shown in Fig. 7, Fig. 10 illustrates that the MSE ofboth the estimated θ and α decreases with the increase of theBS antennas under the LoS channel. This is because θ and α are the only two parameters to be estimated, which can bemore accurately obtained by increasing the DoFs at the BS.It can be again observed that the accuracy of X is superiorto that of X when the ML estimator is used, thanks to theorthogonal nature of the searching waveform. Nevertheless, westill need to identify the working mode of the radar before wecan estimate the channel parameters. Moreover, there existsa 3dB performance gap between the LS estimator and theML estimator using X . This is because the LS estimator(24) is solely based on the searching waveform X , whichis definitely worse than the associated ML estimator, as thelatter is typically the optimal estimator in a statistical sense.Even so, the performance of the LS estimator is satisfactoryenough, as it does not require any information of the radar waveforms. VII. C ONCLUSIONS
This paper deals with the issue of interfering channelestimation for radar and cellular coexistence, where we assumethat the radar switches randomly between the searching andtracking modes, and the BS is attempting to estimate theradar-cellular interfering channel by use of the radar probingwaveforms. To acquire the channel state information, the BSfirstly identifies the working mode of the radar by use ofhypothesis testing approaches, and then estimates the channelparameters. For completeness, both the LoS and NLoS chan-nels are considered, where different detectors are proposed asper the available priori knowledge at the BS, namely GLRT,Rao test and energy detection. As a step further, the theoreticalperformance of the proposed approaches are analyzed in detailusing statistical techniques. Our simulations show that thetheoretical curves match well with the numerical results, andthat the BS can effectively estimate the interfering channel,even with limited information from the radar.A
PPENDIX AP ROOF OF P ROPOSITION ln p ( Y ) = − NL ln πN − N tr (cid:18)(cid:16) Y − α b ( θ ) a H ( θ ) X (cid:17) H (cid:16) Y − α b ( θ ) a H ( θ ) X (cid:17)(cid:19) . (74)According to [30], the FIM can be partitioned as J ( Θ ) = (cid:20) J rr J rs J sr J ss (cid:21) , (75)where J rr = E (cid:18) ∂ ln p∂ vec * ( X ) ∂ ln p∂ vec T ( X ) (cid:19) = 4 N | α | N I L ⊗ a ∗ ( θ ) a T ( θ ) ∈ C ML × ML . (76)Let θ s = [ α, θ ] T ∈ C × be the nuisance parameters, then J rs = E ∂ ln p∂ vec * ( X ) (cid:18) ∂ ln p∂ θ s (cid:19) T ! ∈ C ML × , J sr = J Hrs ∈ C × ML , J ss = E ∂ ln p∂ θ ∗ s (cid:18) ∂ ln p∂ θ s (cid:19) T ! ∈ C × . (77)From (77) and (78), it can be observed that rank ( J rr ) = L, rank ( J rs ) ≤ , rank ( J sr ) ≤ , rank ( J ss ) ≤ . (78)To compute the upper-left partition of the inverse FIM, let usdefine ¯J = J rr (cid:16) ˜Θ (cid:17) − J rs (cid:16) ˜Θ (cid:17) J − ss (cid:16) ˜Θ (cid:17) J sr (cid:16) ˜Θ (cid:17) . (79) By using the property of the rank operator, and recalling that L ≥ M > , we have rank (cid:0) ¯J (cid:1) ≤ L + 2 < M L, (80)which indicates that ¯J ∈ C ML × ML is a singular matrix andis thus non-invertible. Hence, the Rao test statistic does notexist. This completes the proof.A PPENDIX BP ROOF OF P ROPOSITION ln p = − N L ln πN − N tr (cid:16) ( Y − GX ) H ( Y − GX ) (cid:17) . (81)To compute the Fisher Information, we calculate the deriva-tives as ∂ ln p∂ vec ( X ) = 2 N (cid:16) I L ⊗ G H (cid:17) z , ∂ ln p∂ vec * ( X ) = 2 N (cid:16) I L ⊗ G T (cid:17) z ∗ ,∂ ln p∂ vec ( G ) = 2 N ( X ∗ ⊗ I N ) z , ∂ ln p∂ vec ∗ ( G ) = 2 N ( X ⊗ I N ) z ∗ , (82)where z = vec ( Y − GX ) . Recalling (75)-(77), and by usingthe fact that E (cid:0) z ∗ z T (cid:1) = N I NL , we have J rr = E (cid:18) ∂ ln p∂ vec * ( X ) ∂ ln p∂ vec T ( X ) (cid:19) = 4 N (cid:0) I L ⊗ G T (cid:1) E (cid:0) z ∗ z T (cid:1) ( I L ⊗ G ∗ )= 4 N I L ⊗ G T G ∗ , (83) J rs = E (cid:18) ∂ ln p∂ vec * ( X ) ∂ ln p∂ vec T ( G ) (cid:19) = 4 N X H ⊗ G T , (84) J sr = J Hrs = 4 N X ⊗ G ∗ , (85) J ss = E (cid:18) ∂ ln p∂ vec * ( G ) ∂ ln p∂ vec T ( G ) (cid:19) = 4 N XX H ⊗ I N . (86)The FIM can be therefore expressed as J ( Θ ) = 4 N (cid:20) I L ⊗ G T G ∗ X H ⊗ G T X ⊗ G ∗ XX H ⊗ I N (cid:21) . (87)By recalling the definition of ˜Θ , and noting that X X H = LP R M I M , ρ I M , we have h J − (cid:16) ˜Θ (cid:17)i θ r θ r = (cid:16) J rr (cid:16) ˜Θ (cid:17) − J rs (cid:16) ˜Θ (cid:17) J − ss (cid:16) ˜Θ (cid:17) J sr (cid:16) ˜Θ (cid:17)(cid:17) − = N (cid:18) I L ⊗ ˆG T ˆG ∗ − ρ (cid:16) X H ⊗ ˆG T (cid:17) I MN (cid:16) X ⊗ ˆG ∗ (cid:17)(cid:19) − = N (cid:18)(cid:18) I L − ρ X H X (cid:19) ⊗ (cid:16) ˆG T ˆG ∗ (cid:17)(cid:19) − , (88)where ρ = LP R M , and ˆG is given by (13). By using (13), (18),(82), and (88), the Rao test statistic can be expressed as (51),which completes the proof. A PPENDIX CP ROOF OF P ROPOSITION T Rs ( Y ) = 2 N tr (cid:0) YPY H (cid:1) = 2 ˜y H ( I N ⊗ P ) ˜y , (89)where ˜y is defined in (42). In this expression, both the realand imaginary parts of √ ˜y subject to the standard normaldistribution. Since I N ⊗ P is also an idempotent matrix, (89)subjects to non-central chi-squared distribution under bothhypotheses [30]. Under H , the non-centrality parameter isgiven by µ = 2 N tr (cid:18) GX (cid:18) I L − MLP R X H X (cid:19) X H G H (cid:19) = 0 , (90)which indicates that T Rs ( Y ; H ) is in fact central chi-squareddistributed. Under H , the non-centrality parameter is givenas µ = 2 N tr (cid:18) GX (cid:18) I L − MLP R X H X (cid:19) X H G H (cid:19) . (91)The DoFs of the two distributions are given by K = 2 rank ( I N ⊗ P )= 2 N rank ( P ) = 2 N tr ( P ) = 2 N ( L − M ) , (92)where we use the property of the idempotent matrix that rank ( P ) = tr ( P ) . This completes the proof.R , April 2014, pp. 7–14.[7] S. Sodagari, A. Khawar, T. C. Clancy, and R. McGwier, “A projectionbased approach for radar and telecommunication systems coexistence,”in , Dec2012, pp. 5010–5014.[8] A. Babaei, W. H. Tranter, and T. Bose, “A nullspace-based precoderwith subspace expansion for radar/communications coexistence,” in , Dec 2013,pp. 3487–3492.[9] A. Khawar, A. Abdelhadi, and C. Clancy, “Target detection performanceof spectrum sharing MIMO radars,” IEEE Sensors Journal , vol. 15,no. 9, pp. 4928–4940, Sept 2015.[10] J. A. Mahal, A. Khawar, A. Abdelhadi, and T. C. Clancy, “Spectral coex-istence of MIMO radar and MIMO cellular system,”
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