Signal Detection in Distributed MIMO Radar with Non-Orthogonal Waveforms and Sync Errors
11 Signal Detection in Distributed MIMO Radar withNon-Orthogonal Waveforms and Sync Errors
Hongbin Li, Fangzhou Wang, Cengcang Zeng, and Mark A. Govoni
Abstract —Although routinely utilized in literature, orthogonalwaveforms may lose orthogonality in distributed multi-inputmulti-output (MIMO) radar with spatially separated transmit(TX) and receive (RX) antennas, as the waveforms may expe-rience distinct delays and Doppler frequency offsets unique todifferent TX-RX propagation paths. In such cases, the output ofeach waveform-specific matched filter (MF), employed to unravelthe waveforms at the RXs, contains both an auto term andmultiple cross terms , i.e., the filtered response of the desiredand, respectively, undesired waveforms. We consider the impactof non-orthogonal waveforms and their cross terms on targetdetection with or without timing, frequency, and phase errors. Tothis end, we present a general signal model for distributed MIMOradar, examine target detection using existing coherent/non-coherent detectors and two new detectors, including a hybriddetector that requires phase coherence locally but not acrossdistributed antennas, and provide a statistical analysis leading toclosed-form expressions of false alarm and detection probabilitiesfor all detectors. Our results show that cross terms can behavelike foes or allies , respectively, if they and the auto term add destructively or constructively , depending on the propagationdelay, frequency, and phase offsets. Regarding sync errors, weshow that phase errors affect only coherent detectors, frequencyerrors degrade all but the non-coherent detector, while all areimpacted by timing errors, which result in a loss in the signal-to-noise ratio (SNR). Index Terms —Distributed MIMO radar; non-orthogonal wave-forms; asynchronous propagation; timing, frequency, and phaseerrors; target detection
I. I
NTRODUCTION
A. Background
Multi-input multi-output (MIMO) radar, equipped with mul-tiple transmit/receive (TX/RX) antennas, has been of signif-icant interest for civilian and military applications in recentyears [1]–[27]. There are two broad categories, namely co-located MIMO radar (e.g., [1]), where the antennas in the TXand, respectively, RX array are closely spaced (to within afew wavelengths), and distributed MIMO radar [2], where theantennas are widely separated from each other. A distributedMIMO radar can be deployed with its sensors placed closeto the radar scene (e.g., via unmanned aerial vehicles) andprobe the scene from different aspect angles, allowing theradar to exploit the spatial and geometric diversity to enhancetarget detection and localization performance [2], [3], [8], [9].A large body of works have been devoted to the develop-ment of distributed MIMO radar related techniques, such as
H. Li, F. Wang, and C. Zeng are with the Department of Electrical andComputer Engineering, Stevens Institute of Technology, Hoboken, NJ 07030USA (e-mail: [email protected]; [email protected]; [email protected]).M. A. Govoni is with Army Research Laboratory, Aberdeen ProvingGround, MD 21005 USA. power allocation [10], antenna placement [11], neural networkbased optimization [12], detection in homogeneous [13] andnon-homogeneous [14] clutter environments, low-complexitymulti-target detection and localization [15], among others.This paper considers target detection in distributed MIMOradar with non-orthogonal waveforms. A MIMO radar trans-mits multiple waveforms from its TXs to probe the environ-ment. The RXs employ a set of matched filters (MFs), one foreach waveform, which are intended to unravel the radar echoesand separate the information carried by different waveforms.Under the condition that the waveforms are orthogonal withzero cross-correlation, a frequently used assumption in theliterature, the MF output would contain only the filtered echoof the desired waveform, i.e., the auto term , but no crossterms from the undesired waveforms, thus resulting in idealwaveform separation. However, it is impossible to maintainorthogonality with arbitrary delays and frequency shifts [16].The problem becomes more severe in distributed MIMO radar,since the received signals are inherently asynchronous , i.e.,waveforms sent from different TXs in general cannot simulta-neously arrive at an RX due to different propagation distances.Moreover, different TX-RX pairs observe distinct Dopplerfrequencies for the same moving target due to different aspectangles [3]. Such delay and frequency offsets would renderorthogonal waveform on transmit non-orthogonal at the RXs.Hence, the often neglected cross terms can be significant.The effects of cross terms were examined in [17], whichtreat them as deterministic unknowns, whereas in [18], [19],they were modeled as random quantities with an unknowncovariance matrix. In either case, the waveform correlation,which is known, was not utilized. We introduce herein anefficient and more general representation for the cross termsby taking into account the asynchronous signal propagation indistributed MIMO radar.Another objective of this paper is to consider the impactof synchronization errors, or sync errors for brevity, on targetdetection in distributed MIMO radar. Synchronization amongTXs and RXs in distributed MIMO radar is non-trivial, asthe sensors are spatially separated, driven by individual localclocks and oscillators. Phase synchronization, which is essen-tial in applications requiring coherent processing such as direc-tion finding, were considered in several studies. Specifically,the phase identifiability problem in self-calibrating MIMOradar was discussed in [20]. Various phase synchronizationschemes involving centralized or distributed processing wereproposed in [21]. A number of works examined signal detec-tion [22], direction finding [23], imaging [24], and beamform-ing [25] in the presence of phase errors when timing/frequency a r X i v : . [ ee ss . SP ] F e b errors are negligible. Meanwhile, [26] proposed a solution fortarget detection when the observations from different RXsare not correctly registered, i.e., aligned in the same spatialcoordinate system, due to possible timing mismatches amongthe sensors. While these studies underscore the importance ofsynchronization, joint investigations of the effects of timing,frequency, and phase errors, which are coupled with eachother, on distributed MIMO radar are lacking. Another limita-tion is that orthogonal waveforms are assumed in most cases. B. Main Contributions
A systematic framework is presented in this paper, whichcovers signal modeling, detection methods, and statistical anal-ysis, for target detection in distributed MIMO radar with non-orthogonal waveforms and sync errors. The main contributionsare summarized as follows.
1) Modeling:
We develop a general asynchronous signalmodel for distributed MIMO radar, which can incorporatetiming, frequency, and phase errors among RXs and TXs. Theauto and cross terms in the MF output are the auto- and cross-ambiguity functions of the waveforms sampled at distinctdelays and Doppler frequencies associated with individualpropagation paths. We show the model encompasses the co-located MIMO radar as a special case, which will be used tobenchmark the distributed MIMO radar and shed light on theimpact of cross terms and sync errors on target detection.
2) Detection Methods:
We consider coherent and non-coherent target detection methods for distributed MIMO radar.We first briefly review a classical non-coherent detector (NCD)[2], which is an energy detector, and a recently introducedapproximate coherent detector (ACD) [27], which phase-compensates the auto terms in the MF output but neglects crossterms. We then introduce an improved coherent detector (CD)that allows for cross terms and, moreover, exploits diversitiesin signal strength among different TX-RX paths. We alsopropose a new hybrid detector (HD) as a trade-off solution tobridge NCD and CD. HD coherently processes output samplesof each MF and non-coherently integrates across differentMFs. Since it requires phase coherence locally but not acrossspatially distributed antennas, HD bypasses the stringent phasesynchronization requirement of CD and, meanwhile, enjoysadditional coherent processing gain over NCD.
3) Analysis:
We provide a statistical analysis of the 4 detec-tors NCD, ACD, CD, and HD. Closed-form expressions of theprobability of false alarm and, respectively, the probability ofdetection are derived for either non-fluctuating or fluctuatingtargets. With simple tuning, these expressions are applicable tocases with or without sync errors and, furthermore, can be ex-tended to co-located MIMO radar. Therefore, they offer usefultools to investigate the impacts of asynchronous propagation,cross terms, and sync errors in distributed MIMO radar.
4) Key Observations:
To examine the impact of non-orthogonal waveforms, we consider different distributed set-ups with varying offsets in delay, frequency, and phase Such offsets, caused by spatially distributed and inherently asynchronoussensors, exist even if the RXs have perfect timing, frequency and phaseinformation of the TXs, and therefore shall not be confused as sync errors. among different TX-RX paths, which lead to cross terms withvarying magnitudes and phases compared with those of theauto terms. For benchmarking, we compare the distributedMIMO radar with a co-located MIMO radar which employsorthogonal waveforms and contains only auto terms in itsMF outputs. This is possible since co-located MIMO radarhas co-located sensors which can operate synchronously with-out delay, frequency, and phase offsets, thus obviating crossterms. Our comparative study reveals that, interestingly, thenon-orthogonal distributed MIMO radar may under- or out-performs the orthogonal co-located MIMO radar in target de-tection . The performance depends on if the cross terms behaveas interference , which occur when the auto and cross termsadd destructively , or as resources , which occurs when theyadd constructively . It should be noted that delay, frequency,and phase offsets, which are coupled with each other (e.g., adelay offset implies a phase offset, see Section II), all affecthow cross terms add with auto terms.Numerical simulations are also provided to illustrate theimpact of sync errors on the 4 detectors NCD, ACD, CD, andHD. Our results indicate that: (a) NCD and HD are immunefrom phase errors, which affect only coherent detectors ACDand CD; (b) frequency errors, caused by either carrier offsetsor Doppler estimation errors, affect all but NCD; and (c)all detectors are affected by timing errors, which cause theMF output to be sampled off the peak location of the autoambiguity function (see Section II), thus resulting in a lossin the signal-to-noise ratio (SNR). If the timing error issufficiently small (i.e., much smaller than the reciprocal ofthe waveform bandwidth but still significant relative to thecarrier period) so that the SNR loss is negligible, then it willonly impact coherent detectors such as ACD or CD as thetiming-error-induced phase error may not be negligible.The remainder is organized as follows. A general signalmodel for distributed MIMO radar is presented in Section II.Target detection methods are discussed in Section III, and theirstatistical analysis in Section IV. Section V contains numericalresults, followed by conclusions in Section VI.
Notations : We use boldface symbols for vectors (lowercase) and matrices (upper case). ( · ) T denotes the transposeand ( · ) H the conjugate transpose. (cid:107) · (cid:107) and | · | denote thevector 2-norm and absolute value, respectively. E {·} representsthe statistical expectation. CN ( u , Σ ) denotes the complexGaussian distribution with mean u and covariance matrix Σ . [ X ] mm indicates the ( m, m ) -th element of the diagonal matrix X while [ x ] m denotes the m -th element of the vector x .II. S IGNAL M ODEL
We first present a signal model for distributed MIMO radarwith or without sync errors, and then briefly discuss the co-located MIMO radar, which is a special case of the formerand will be employed as a benchmark for comparison.
A. Distributed MIMO Radar
We consider a distributed MIMO radar system with M TXs and N RXs as shown in Fig. 1. The TXs employ pulsed
TX
TX M RX
RX N target r, n R t, m R Figure 1. Transmit and receive configuration of a distributed MIMO radar. transmission to probe an area of interest by using M wave-forms. During a coherent processing interval, a succession of K periodic pulses are transmitted by each TX. Specifically, atthe m -th TX, the transmitted pulses are given by ˜ s m ( t ) = b m u m ( t ) e [2 π ( ˆ f c +∆ ct ,m ) t + φ t ,m ] , (1)where u m ( t ) = (cid:80) K − k =0 p m ( t − kT s ) is the baseband transmittedsignal, p m ( t ) is the complex envelope of a single pulse for TX m , T s is the pulse repetition interval (PRI), b m is the transmitamplitude, ˆ f c is the nominal carrier frequency, ∆ ct ,m denotesthe carrier frequency error introduced by the m -th TX, and φ t ,m is the carrier initial phase. The pulse waveform p m ( t ) has unit energy and is of the same duration T p for all TXs.Therefore, | b m | denotes the energy transmitted in a singlepulse.Suppose there is a moving target at a distance R t ,m to the m -th TX and a distance R r ,n to the n -th RX. The signal ˜ s n ( t ) observed at the n -th RX consists of echoes from the targetilluminated by M waveforms ˜ s n ( t ) = M (cid:88) m =1 αb m ξ mn u m ( t − τ mn ) × e π ( ˆ f c +∆ ct ,m + (cid:101) f mn )( t − τ mn ) e φ t ,m , (2)where α is the target amplitude, τ mn = ( R t ,m + R r ,n ) /c is the ( m, n ) -th TX-RX propagation delay, and (cid:101) f mn is the bistatic target Doppler frequency [3], [13] observed by the n -th RXin response to the radar waveform transmitted from the m -thTX. In addition, ξ mn is the channel coefficient associated withthe ( m, n ) -th TX-RX pair [28]: ξ mn = (cid:115) G r ,n G t ,m λ (4 π ) R t ,m R r ,n , (3)where λ is the wavelength of the signal and G t ,m and G r ,n are the m -th TX and, respectively, n -th RX antenna gain.A local carrier e [2 π ( ˆ f c +∆ cr ,n ) t + φ r ,n ] is generated at the n -th RX for down conversion, where ∆ cr ,n and φ r ,n denote thelocal carrier frequency error and initial phase, respectively. After down conversion, the baseband signal is s n ( t ) = M (cid:88) m =1 αb m ξ mn u m ( t − τ mn ) e π ( ˆ f c +∆ ct ,m + (cid:101) f mn )( t − τ mn ) × e φ t ,n e − [2 π ( ˆ f c +∆ cr ,n ) t + φ r ,n ] = M (cid:88) m =1 αb m ξ mn u m ( t − τ mn ) e ψ mn × e − π ( ˆ f c +∆ cr ,n ) τ mn e πf mn ( t − τ mn ) , (4)where ψ mn (cid:44) φ t ,m − φ r ,n denotes the initial phase offset and f mn (cid:44) (cid:101) f mn + ∆ ct ,m − ∆ cr ,n denotes the combined frequencyoffset between the m -th TX and n -th RX. A set of M matchedfilters (MFs), each matched to one of M waveforms, are usedat the n -th RX. Each MF requires estimates of the target delay τ mn and Doppler f mn for compensation. In the following, wefirst consider the general case with possible sync errors, andthen extend the result to the ideal case of no sync error, whichis included as a benchmark for comparative studies.
1) With Sync Errors:
At the n -th RX, s n ( t ) is con-volved with M MFs, g m ( t ) = p ∗ m ( − t ) e π ( f mn +∆ f mn ) t , m =1 , . . . , M , where ∆ f mn denotes the frequency error betweenthe effective Doppler frequency f mn and its estimate ˆ f mn . Let us define the cross ambiguity function (CAF) as χ m ¯ m ( ν, f ) = (cid:90) p m ( µ ) p ∗ ¯ m ( µ − ν ) e πfµ dµ. (5)Then, the output of the m -th MF at the n -th RX x mn ( t ) canbe written as x mn ( t ) = M (cid:88) ¯ m =1 αb ¯ m ξ ¯ mn e ψ ¯ mn e − π ( ˆ f c +∆ cr ,n ) τ ¯ mn e − πf ¯ mn τ ¯ mn × e π ( f mn +∆ f mn ) t K − (cid:88) k =0 (cid:90) p ¯ m ( µ − kT s − τ ¯ mn ) × p ∗ m ( µ − t ) e π ( f ¯ mn − f mn − ∆ f mn ) µ dµ = M (cid:88) ¯ m =1 αb ¯ m ξ ¯ mn e − π ( ˆ f c +∆ cr ,n ) τ ¯ mn e π ( f mn +∆ f mn )( t − τ ¯ mn ) × e ψ ¯ mn K − (cid:88) k =0 χ m ¯ m ( t − τ ¯ mn − kT s , f ¯ mn − f mn − ∆ f mn ) × e πkT s ( f ¯ mn − f mn − ∆ f mn ) . (6)The continuous-time signal x mn ( t ) is sampled at the pulserate, leading to K slow-time samples obtained at time instants t = τ mn +∆ t mn + kT s , k = 0 , · · · , K − , where ∆ t mn denotes Although the frequency error ∆ f mn includes both the carrier frequencyerror and Doppler mismatch, the former is usually much smaller as theTXs and RXs are cooperative, which enables accurate tracking of the carrierfrequency, e.g., via phase-locked loop. The Doppler mismatch can be moresignificant due to target motion uncertainty. Therefore, we may refer to ∆ f mn as the frequency error or the Doppler error interchangeably. the timing error between the true propagation delay τ mn andits estimate ˆ τ mn . Then, the output samples can be written as x mn ( k ) = x mn ( t ) (cid:12)(cid:12)(cid:12) t = τ mn +∆ t mn + kT s = αb m ξ mn e πkT s f mn × χ mm (∆ t mn , − ∆ f mn ) e − π ( ˆ f c +∆ cr ,n ) τ mn e π ( f mn +∆ f mn )∆ t mn × e ψ mn + (cid:88) ¯ m (cid:54) = m αb ¯ m ξ ¯ mn e ψ ¯ mn e − π ( ˆ f c +∆ cr ,n ) τ ¯ mn e πkT s f ¯ mn × χ m ¯ m ( τ mn + ∆ t mn − τ ¯ mn , f ¯ mn − f mn − ∆ f mn ) × e π ( f mn +∆ f mn )( τ mn +∆ t mn − τ ¯ mn ) , (7) m = 1 , . . . , M ; n = 1 , . . . , N ; k = 0 , . . . , K − . Remark 1:
It can be seen that the output sample x mn ( k ) consists of M components: the first term is the auto term between the m -th waveform and the m -th MF, and the othercomponents represent the cross terms between the other M − waveforms and the m -th MF. The cross terms vanish whenwaveforms p m ( t ) are orthogonal to each other, which isa routine assumption in the MIMO literature. In practice,maintaining strict orthogonality across time and frequencyin distributed MIMO radar with asynchronous propagation isinfeasible [16]. With non-orthogonal waveforms or waveformsthat are orthogonal only with zero delay/Doppler, cross termsare present as residuals, which may become non-negligibleand need to be accounted for. Remark 2:
The derivation of (6) and (7) appears to suggestthat the radar receiver requires prior estimates of the targetdelay and Doppler, which are unnecessary. The problem isaddressed by having the receiver scanning through the de-lay/Doppler uncertainty region, which is discretized into a setof range/Doppler bins. In our derivation, range/Doppler mea-surements are obtained by using a set of MFs, each matchedto a distinct Doppler frequency, and sampling the MF outputsat the Nyquist rate. In practice, the above process is oftenapproximated by a more efficient procedure, which involvesprocessing the target return using a fixed MF, sampling theMF output in fast- and slow-times, and then converting to thefrequency domain by the fast Fourier transform (FFT) [28].Note that (7) describes the observed signal only for the range-Doppler bin with the target. For non-target range-Dopplerbins, the measurements contain noise. These two types ofmeasurements are summarized by the hypothesis testing datamodel in (18).Next, we stack the K slow-time samples and form x mn =[ x mn (0) , · · · , x mn ( K − T , which can be expressed as x mn = α S n X mn h mn , (8)where the K × M Doppler steering matrix S n is S n = [ s ( f n ) , · · · , s ( f Mn )] , (9) s ( f ) = [1 , e πT s f , · · · , e π ( K − T s f ] T , the M × M ambiguity function matrix X mn is diagonal withdiagonal elements given by [ X mn ] ¯ m ¯ m = χ m ¯ m ( τ mn + ∆ t mn − τ ¯ mn , f ¯ mn − f mn − ∆ f mn ) , (10) and the ¯ m -th element of the M × channel vector h mn is [ h mn ] ¯ m = b ¯ m ξ ¯ mn e ψ ¯ mn e − π ( ˆ f c +∆ cr ,n ) τ ¯ mn × e π ( f mn +∆ f mn )( τ mn +∆ t mn − τ ¯ mn ) . (11)
2) Without Sync Errors:
Equations (7)–(11), which describethe general signal model for distributed MIMO radar, also holdfor the ideal case without sync errors, by setting ∆ f mn = 0 , ∆ cr ,n = 0 , and ∆ t mn = 0 , ∀ m, ∀ n . In other words, we needreplace (10) and (11) by [ X mn ] ¯ m ¯ m = χ m ¯ m ( τ mn − τ ¯ mn , f ¯ mn − f mn ) , (12) [ h mn ] ¯ m = b ¯ m ξ ¯ mn e ψ ¯ mn e − π ˆ f c τ ¯ mn e πf mn ( τ mn − τ ¯ mn ) . (13) Remark 3:
It is worth to note that matched filtering andsampling only require the knowledge of the delay τ mn andDoppler f mn , but not the carrier phase offset ψ mn . Therefore,phase errors are absent from the signal model Equations (7)–(13). However, for coherent detection, the observed signal x mn will be phase-compensated, and phase estimation errors willimpact such detectors (see Sections III-B and IV-B for details). B. Co-Located MIMO Radar
Co-located MIMO radar, which is a special case of dis-tributed MIMO radar, can be described by Equations (7)–(13) with some simplifications. Specifically, with co-locatedantennas, we have identical target delay τ mn , ∀ m, n , andidentical Doppler frequency f mn , ∀ m, n . In addition, if theTXs share the same oscillator, and so are the RXs, then thephase offset ψ mn is constant ∀ m, n .Assume the radar employs waveforms that are orthogonalwith zero delay and Doppler, i.e., χ m ¯ m (0 ,
0) = 0 , ∀ m (cid:54) = ¯ m. (14)Then, the cross terms in (7) disappear for co-located MIMOradar when there is no sync error. In this case, (7) can besimplified as x mn ( k ) = αb m ξ mn e − π ˆ f c τ mn χ mm (0 , e πkT s f mn e ψ mn . (15)In turn, (10) and (11) reduces to [ X mn ] ¯ m ¯ m = , ¯ m (cid:54) = m,χ mm (0 , , ¯ m = m, (16) [ h mn ] ¯ m = b ¯ m ξ ¯ mn e − π ˆ f c τ ¯ mn e ψ ¯ mn . (17)Equations (16) and (17) along with (9), describe the orthog-onal co-located MIMO radar , which will be employed tobenchmark the non-orthogonal distributed MIMO radar.III. T ARGET D ETECTION
Let y mn denote the noise contaminated observation of x mn .The target detection problem is described by the followinghypothesis testing: H : y mn = w mn , H : y mn = α S n X mn h mn + w mn , (18) m = 1 , , · · · , M, n = 1 , , · · · , N, where w mn is the noise, assumed to be Gaussian distributed, w mn ∼ CN ( , σ mn I ) . Note that the above hypothesis testingapplies to both distributed and co-located MIMO radars. In thefollowing, we first consider target detection approaches for thegeneral case, i.e., distributed MIMO radar with possible syncerrors, and then extend/simplify the solutions to the cases withno sync error and co-located MIMO radar. For target detection,we discuss several detectors, including a conventional non-coherent detector (NCD) [2], an approximate coherent detector(ACD) [27], a coherent detector (CD), and a hybrid detector(HD). The latter two are new. A. Non-Coherent Detector
A simple detector for the hypothesis testing (18) is basedon non-coherent integration of the MF outputs [2]: T NCD (cid:44) M (cid:88) m =1 N (cid:88) n =1 y Hmn y mn H ≷ H γ NCD , (19)where γ NCD is a threshold set for a given level of false alarm.It is clear that the above NCD is an energy detector.
B. Coherent Detectors
The above NCD does not require any phase synchronization.Improved detection performance can be achieved by exploitingphase information. One such detector, ACD, was introduced in[27], which performs phase compensation for the auto terms inthe MF output (7). Specifically, let ˆ ψ mn , ˆ τ mn , and ˆ f mn denoteestimates of the phase offset, delay, and Doppler frequency.The ACD is given by T ACD = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 N (cid:88) n =1 K − (cid:88) k =0 e − ˆ θ mnk y mn ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H ≷ H γ ACD , (20)where γ ACD is the threshold, y mn ( k ) denotes the k -th elementof y mn and ˆ θ mnk = ˆ ψ mn − π ˆ f c ˆ τ mn + 2 πkT s ˆ f mn , (21) ˆ ψ mn (cid:44) ψ mn + ∆ p mn , (22) ˆ τ mn (cid:44) τ mn + ∆ t mn , (23) ˆ f mn (cid:44) f mn + ∆ f mn , (24)where ∆ p mn , ∆ t mn , and ∆ f mn denote the phase, timing, andDoppler errors.Albeit simple, the ACD has two limitations. First, it per-forms phase compensation only for the auto-term, while ne-glecting the cross terms in (7), which is non-negligible whenthe waveforms are not orthogonal. Second, it applies equalweights in combining the outputs from different MFs, which issuboptimal since the TX-RX propagation paths associated withdifferent MFs are different with potentially different SNRs.To address these limitations, we propose an improved CDwith derivation presented in Appendix I-A. Specifically, let ˆ S n , ˆ X mn , and ˆ h mn be formed as in Equations (9)–(11), byusing the phase, delay, and Doppler frequency estimates: ˆ S n = [ s ( ˆ f n ) , · · · , s ( ˆ f Mn )] , (25) [ ˆ X mn ] ¯ m ¯ m = χ m ¯ m (ˆ τ mn − ˆ τ ¯ mn , ˆ f ¯ mn − ˆ f mn ) , (26) [ˆ h mn ] ¯ m = b ¯ m ξ ¯ mn e ˆ ψ ¯ mn e − π ˆ f c ˆ τ ¯ mn e π ˆ f mn (ˆ τ mn − ˆ τ ¯ mn ) . (27) Then, the new CD is given by T CD = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 N (cid:88) n =1 (ˆ S n ˆ X mn ˆ h mn ) H y mn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H ≷ H γ CD , (28)where γ CD denotes the test threshold. It can be seen that theCD sequentially performs Doppler filtering by ˆ S n , joint phasecompensation and amplitude weighting by ˆ X mn and ˆ h mn ,followed by coherent integration across antennas. C. Hybrid Detector
The above CD requires the knowledge of the phases,the CAFs of all waveforms, and the channel coefficients ξ mn . Although achieving the best performance with idealknowledge, CD is sensitive to knowledge/estimation errors. InAppendix I-B, we derive a new HD detector that bypasses thestringent requirement of CD and can still achieve considerableimprovement over the NCD, i.e., it offers a compromisebetween CD and NCD. Specifically, the HD is given by T HD = M (cid:88) m =1 N (cid:88) n =1 (cid:107) ˆ S n (ˆ S Hn ˆ S n ) − ˆ S Hn y mn (cid:107) H ≷ H γ HD , (29)where it is assumed K ≥ M so that the matrix inverse exists.Clearly, HD projects y mn onto the subspace spanned by theDoppler steering vectors ˆ S n , which is coherent processing ofthe signal observed at the ( m, n ) -th MF, followed by non-coherent integration across different RXs and TXs. Hence, itis a hybrid detector. The projection preserves the target signal y mn undistorted, while rejecting the noise component in theorthogonal subspace. This leads to an improved SNR, allowingHD to outperform the NCD. Note that HD requires phasecoherent only locally, i.e., within the output of each MF, butnot across spatially distributed antennas. D. Impact of Sync Errors
For NCD, sync errors in timing, Doppler, and phase onlyaffect the MF measurements in (18), but do not have anyimpact on the NCD implementation (19). This implies thatwithout sync errors, NCD is still given by (19), except thatthe MF outputs y mn are represented by (18), along with (9),(12), and (13).For ACD and CD, sync errors affect both measurementsand implementation, due to the additional phase/amplitudecompensation employed by the latter. In the case of no syncerrors, the MF measurements are given by (18), along with(9), (12), and (13). The phase compensations { ˆ θ mnk } in (21)should be replaced by their true values for ACD (20), while forCD (28), the amplitude/phase compensation quantities { ˆ S n } , { ˆ X mn } , and { ˆ h mn } should be replaced by their error-freecounterparts.For HD, sync errors also affect both the measurements andimplementation due to the Doppler projection in (29). In theabsence of sync errors, the measurements (18) have the samerepresentation as discussed above, whereas (29) should beimplemented with the true Doppler matrices { S n } . E. Extension to Co-Located MIMO Radar
For co-located MIMO radar with orthogonal waveforms, theMF measurements are represented by (18), along with (9),(16), and (17). The implementations of the NCD, ACD/CD,and HD remain the same as in distributed MIMO radar, whichare given by (19), (20), (28) and (29), respectively. Finally,the discussions on the impact of sync errors in Section III-Dfor distributed MIMO radar are also applicable to co-locatedMIMO radar. IV. S
TATISTICAL A NALYSIS
In this section, we provide a statistical analysis of the 4detectors, NCD, CD, ACD, and HD, introduced in SectionIII and derive expressions of their false alarm and detectionprobabilities. We only consider the general case with syncerrors, since the test statistic of each detector retains thesame form, although the statistical distributions are affectedby the presence/absence of sync errors, but can be easilydetermined by using the corresponding signal representationsas detailed in Section III-D. For each detector, we derive theprobability of detection for the case of non-fluctuating targetand, respectively, the case of fluctuating target assuming theSwerling I model [28]. Finally, we briefly discuss how toextend these results to co-located MIMO radar.
A. NCDTheorem 1:
Given the signal model (18) with or withoutsync errors, the probability of false alarm and the probabilityof detection with a non-fluctuating target for the NCD (19)are given by P f = Γ( KM N ) − ¯Γ( KM N, γ
NCD /σ )Γ( KM N ) , (30) P d = Q KMN (cid:16)(cid:112) λ NCD , (cid:114) γ NCD σ (cid:17) , (31)where ¯Γ( · , · ) denotes the upper incomplete Gamma function, Γ( · ) the Gamma function, Q m ( a, b ) the generalized Marcum-Q function, and the noncentrality parameter is given by λ NCD = M (cid:88) m =1 N (cid:88) n =1 | α | σ (cid:107) S n X mn h mn (cid:107) . (32) Proof:
See Appendix II-A.Next, we examine the average probability of detection in thecase of fluctuating target. For Swerling I target, the probabilitydensity function (pdf) of the radar cross section (RCS) ρ = | α | is [28] f ( ρ ) = 1¯ ρ e − ρ/ ¯ ρ , (33) where ¯ ρ = E { ρ } . Let λ (cid:48) NCD (cid:44) λ NCD /ρ . The average probabil-ity of detection is given by ¯ P d = (cid:90) ∞ f ( ρ ) P d dρ = (cid:90) ∞ ρ e − ρ/ ¯ ρ Q KMN (cid:16)(cid:113) λ (cid:48) NCD ρ, (cid:114) γ NCD σ (cid:17) dρ = ¯Γ( KM N, γ
NCD /σ )Γ( KM N )+ λ (cid:48) NCD (cid:0) γ NCD σ (cid:1) KMN F (1 , KM N + 1 , γ NCD λ (cid:48) NCD σ ( λ (cid:48) NCD +2 / ¯ ρ ) )( KM N )!( λ (cid:48) NCD + 2 / ¯ ρ ) e γ NCD σ , (34)where x ! denotes the factorial, F ( · , · , · ) is the Kummerconfluent hypergeometric function [29], and the third equalityis obtained by using [30, Theorem 1]. B. CDTheorem 2:
Given the signal model (18) with or withoutsync errors, the probability of false alarm and the probabilityof detection with a non-fluctuating target for the CD (28) aregiven by P f = e − γ CD ςσ , (35) P d = Q ( (cid:112) λ CD , (cid:114) γ CD ςσ ) , (36)where the noncentrality parameter is given by λ CD = 2 | α | (cid:12)(cid:12)(cid:12)(cid:80) Mm =1 (cid:80) Nn =1 (ˆ S n ˆ X mn ˆ h mn ) H S n X mn h mn (cid:12)(cid:12)(cid:12) σ (cid:80) Mm =1 (cid:80) Nn =1 (cid:107) ˆ S n ˆ X mn ˆ h mn (cid:107) , (37)and the scaling factor is ς = M (cid:88) m =1 N (cid:88) n =1 (cid:107) ˆ S n ˆ X mn ˆ h mn (cid:107) . (38) Proof:
See Appendix II-B.Similar to NCD, the average probability of detection for thecase of Swerling I fluctuating target is given by ¯ P d = (cid:90) ∞ ρ e − ρ/ ¯ ρ Q ( (cid:113) λ (cid:48) CD ρ, (cid:114) γ CD ςσ ) dρ = ¯Γ(1 , γ CD ςσ ) + λ (cid:48) CD γF (1 , , γ CD λ (cid:48) CD ςσ ( λ (cid:48) CD +2 / ¯ ρ ) ) ςσ ( λ (cid:48) CD + 2 / ¯ ρ ) e γ CD ςσ , (39)where λ (cid:48) CD (cid:44) λ CD /ρ . C. ACDTheorem 3:
Given the signal model (18) with or withoutsync errors, the probability of false alarm and the probabilityof detection with a non-fluctuating target for the ACD (20) aregiven by P f = e − γ ACD
KMNσ , (40) P d = Q ( (cid:112) λ ACD , (cid:114) γ ACD
KM N σ ) , (41) where λ ACD = 2 (cid:12)(cid:12)(cid:12)(cid:80) Mm =1 (cid:80) Nn =1 (cid:80) K − k =0 e − ˆ θ mnk x mn ( k ) (cid:12)(cid:12)(cid:12) KM N σ , (42)where ˆ θ mnk is defined in (21) and x mn ( k ) is defined in (7). Proof:
See Appendix II-C.Define λ (cid:48) ACD (cid:44) λ ACD /ρ . The average probability of detec-tion for the case of Swerling I fluctuating target is given by ¯ P d = ¯Γ(1 , γ ACD
KM N σ )+ λ (cid:48) ACD γ ACD F (1 , , γ ACD λ (cid:48) ACD
KMNσ ( λ (cid:48) ACD +2 / ¯ ρ ) ) KM N σ ( λ (cid:48) ACD + 2 / ¯ ρ ) e γ ACD
KMNσ . (43) D. HDTheorem 4:
Given the signal model (18) with or withoutsync errors, the probability of false alarm and the probabilityof detection with a non-fluctuating target for the HD (29) aregiven by P f = Γ( N M ) − ¯Γ( N M , γ HD /σ )Γ( N M ) , (44) P d = Q NM (cid:16)(cid:112) λ HD , (cid:114) γ HD σ (cid:17) , (45)where the noncentrality parameter is given by λ HD = | α | (cid:80) Mm =1 (cid:80) Nn =1 (cid:107) ˆ S n (ˆ S Hn ˆ S n ) − ˆ S Hn S n X mn h mn (cid:107) σ / . (46) Proof:
See Appendix II-D.Then, the average probability of detection for the case ofSwerling I fluctuating target is given by ¯ P d = ¯Γ( N M , γ HD /σ )Γ( N M )+ λ (cid:48) HD (cid:0) γ HD σ (cid:1) NM F (1 , N M + 1 , γ HD λ (cid:48) HD σ ( λ (cid:48) HD +2 / ¯ ρ ) )( N M )!( λ (cid:48) HD + 2 / ¯ ρ ) e γ HD σ , (47)where λ (cid:48) HD (cid:44) λ HD /ρ . E. Extension to Co-Located MIMO Radar
As discussed in Section III-E, NCD, CD, ACD, and HD canbe applied for target detection in co-located MIMO radar. Itis easy to see that Theorems 1 to 4, as well as the expressions(34), (39), (43), and (47) for the average probability of detec-tion, still hold for co-located MIMO radar. The only differenceis that the noncentrality parameter λ involved in each detector,as well as ς (38) for the CD, should be calculated by using thecorresponding signal representations, as discussed in SectionIII-E. V. S IMULATION R ESULTS
In this section, simulation results are presented to demon-strate the performance of the NCD [2], ACD [27], along withthe proposed CD and HD, for target detection in distributedMIMO radar. The performance of these detectors are assessedby using both computer simulation and the analytical resultsreported in Section IV. The SNR of the ( m, n ) -th propagationpath, which is measured at the n -th RX matched to the m -thTX waveform, is defined asSNR mn = | b m ξ mn | E {| α | } σ mn , (48)where the noise variance is chosen as σ mn = 1 . We considera Swerling I target mdel, where the target amplitude α ∼CN (0 , σ ) is randomly generated from trial to trial but remainsfixed within a coherent processing interval (CPI) in MonteCarlo simulations. We assume identical SNR for all paths,i.e., SNR mn = SNR, ∀ m, n , except in Section V-B where theeffect of different SNRs is examined. The simulation scenariosinvolve a distributed MIMO radar with M = 2 TXs and N = 1 RX. The propagation delays are τ = 0 . T p and τ =0 . T p unless otherwise stated, where T p = 10 − s is the pulseduration. The pulse repetition frequency (PRF) is 500 Hz, thecarrier frequency is 3 GHz, the target Doppler frequencies are f = 200 Hz and f = 190 Hz, unless otherwise stated, andthe number of pulses within a CPI is K = 12 . The phases are ψ = 0 . π and ψ = 0 . π unless otherwise stated and theprobability of false alarm is P f = 10 − .In the following, we first introduce two sets of linearfrequency modulation (LFM) based waveforms, also known aschirps, which are used by the MIMO radar for testing. Then,we examine the performance of these detectors in variousdistributed environments with non-identical propagation pathstrengths, different delays, phases, and Doppler frequencies,as well as in the presence of sync errors. A. Test Waveforms
LFM waveforms, which are frequently used in radar, areemployed as test waveforms. We consider two types of LFMwaveforms with different ambiguity characteristics. The firstare multi-band chirps: p m ( t ) = 1 (cid:112) T p e πβ ( t /T p + ηmt ) (49) ≤ t ≤ T p , m = 1 , . . . , M, where β is the bandwidth of the waveform and η is abandwidth gap parameter that is selected to keep the frequencybands of different waveforms non-overlapping. The ambiguityfunctions of the multi-band chirps can be obtained by using(49) in (5), which are shown in Fig. 2 (a) when M = 2 , η = 3 ,and β = 400 kHz. The multi-band chirps are orthogonal withzero cross ambiguity at zero delay and Doppler, i.e., whenthe waveforms arrive at the RX synchronously . However, theyare not strictly orthogonal in distributed MIMO radar due toasynchronous propagation, but can be considered as approxi-mately orthogonal since the cross ambiguity is relatively lowfor small delay/Doppler offsets. -1 -0.5 0 0.5 1 normalized delay -3 -2 -1 a m p li t ud e multi-band chirps m = 1 & m = 1 m = 1 & m = 2 m = 2 & m = 1 m = 2 & m = 2 (a) -1 -0.5 0 0.5 1 normalized delay -3 -2 -1 a m p li t ud e single-band chirps m = 1 & m = 1 m = 1 & m = 2 m = 2 & m = 1 m = 2 & m = 2 (b) Figure 2. Auto- and cross-ambiguity function versus the delay (normalizedby the pulse duration T p ) with zero Doppler. The second are single-band chirps with overlapping instan-taneous frequency. For M = 2 , we employ an up chirp givenby p u ( t ) = 1 (cid:112) T p e ( πβt /T p + κπβt ) , ≤ t < T p , (50)and a down chirp p d ( t ) = 1 (cid:112) T p e ( − πβt /T p +2 πβt + κπβt ) , ≤ t ≤ T p , (51)where κ is a constant that controls the center frequency of thechirps. The general expression of the single-band chirps can befound in [27, eq.(9)]. Fig. 2 (b) depicts the ambiguity functionsof the single-band chirps when M = 2 , κ = 3 , and β = 400 kHz, which shows the single-band chirps are non-orthogonal waveforms with high cross ambiguity. B. Effect of Unequal Channel Strength
We consider a scenario when the two propagation pathsfrom the TXs to the RX have different SNR. In particular, wefix SNR = 0 dB while varying SNR . Figs. 3 (a) and (b)depict the average probability of detection ¯ P d versus SNR ,where ¯ P d is determined by using the theoretical analysis inSection IV and simulation, respectively. It is seen that theanalysis perfectly matches the computer simulation for all 4detectors. With the multi-band chirps, Fig. 3 (a) shows that CDoutperforms ACD when SNR (cid:54) = SNR , where the benefitcomes from the amplitude weighting employed by CD. In -20 -10 0 10 20 SNR o , set (dB) a v e r ag e p r o b a b ili t y o f d e t ec t i o n multi-band chirps CDACDHDNCDsimulation (a) -20 -10 0 10 20
SNR o , set (dB) a v e r ag e p r o b a b ili t y o f d e t ec t i o n single-band chirps CDACDHDNCDsimulation (b)
Figure 3. ¯ P d of distributed MIMO radar versus SNR offset SNR − SNR ,where SNR = 0 dB. The solid lines are obtained from theoretical analysiswhile the markers (circles) are obtained by simulation. addition, HD is slightly worse than ACD but outperforms NCDsince it employs partial coherent combining within each CPIbut non-coherent combining across different antennas.With single-band chirps, the 4 detectors exhibit similar per-formance behaviors in Fig. 3 (b) except that the gap betweenCD and ACD is larger and, furthermore, CD outperformsACD even at SNR = SNR . This is because with single-band chirps, the cross terms in the MF outputs (7) are moresignificant, which are accounted for by CD in its phasecompensation but are neglected by ACD. C. Effect of Propagation-Induced Offsets and Cross Terms
As shown in Section II-A, asynchronous propagation isinherent in distributed MIMO radar, leading to inevitableoffsets in delays, Doppler frequencies, and phases, as wellas cross terms in the MF output (7), even when the RX isperfectly synchronized with the TXs. Next, we examine theeffect of such asynchronous propagation induced offsets ontarget detection. From now on, we no longer consider ACD,which is superseded by CD. In addition, we include the co-located MIMO radar as a benchmark, which assumes the TXsand RX are synchronous with zero delay/Doppler/phase offsetsand orthogonal waveforms are employed, and as a result, thereare no cross terms in the MF output (see Section II-B andIII-E).Fig. 4 shows the performance of the CD, HD, and NCDof the distributed MIMO radar under various timing offsets, -1 -0.5 0 0.5 1 timing o , set a v e r ag e p r o b a b ili t y o f d e t ec t i o n multi-band chirps CDCD(co-located)HDHD (co-located)NCDNCD(co-located) (a) -1 -0.5 0 0.5 1 timing o , set a v e r ag e p r o b a b ili t y o f d e t ec t i o n single-band chirps CDCD(co-located)HDHD (co-located)NCDNCD(co-located) (b)
Figure 4. ¯ P d of distributed MIMO radar versus delay offset τ − τ (normalized by the pulse duration T p ) in comparison with the synchronousco-located MIMO radar, when SNR = 0 dB, ψ = ψ = 0 , and f = f = 0 . in comparison with the co-located MIMO radar, where ¯ P d iscomputed analytically. It can be seen that the performanceof all 3 detectors for distributed MIMO radar fluctuates asthe delay offset varies. This is because the propagation delayaffects the phase of the auto and cross terms, as shown in(7). The auto and cross terms may add constructively whenthe difference of their phases is between − π/ and π/ , ordestructively when otherwise, which causes the fluctuation ofthe detection performance. A comparison between Figs. 4 (a)and (b) shows that the single-band chirps exhibit a largerfluctuation. This is because the single-band chirps have a largercross terms than the multi-band chirps.Figs. 5 and 6 show the detection performance of the CD,HD, and NCD for the distributed MIMO radar versus phaseoffset and, respectively, Doppler offset, in comparison withthe benchmark co-located MIMO radar. These detectors areseen to exhibit similar performance fluctuations as observedin Fig. 4 for similar reasons. Note that even though HD andNCD do not use phase information for detection, the observedsignal varies with the phase offset, which leads to performancevariation for these detectors.It is interesting to note from Figs. 4 to 6 that the distributedMIMO radar, whose MF outputs include both auto- and cross-terms, may out- or under-perform the co-located MIMO radar,which only has the auto-terms, due to the aforementionedconstructive or destructive addition. The challenge is that it is phase o , set a v e r ag e p r o b a b ili t y o f d e t ec t i o n multi-band chirps CDCD(co-located)HDHD (co-located)NCDNCD(co-located) (a) phase o , set a v e r ag e p r o b a b ili t y o f d e t ec t i o n single-band chirps CDCD(co-located)HDHD (co-located)NCDNCD(co-located) (b)
Figure 5. ¯ P d of distributed MIMO radar versus phase offset ψ − ψ (normalized by π ) in comparison with the synchronous co-located MIMOradar, when SNR = 0 dB, τ = 0 . T p , τ = 0 . T p , and f = f = 0 . non-trivial to control how the auto- and cross-terms are addedwith each other, which are affected by many factors includingthe delays, phases, and Doppler frequencies of the propagationpaths, as well as the ambiguities of the waveforms. D. Effect of Sync Errors
Finally, we evaluate the effects of sync errors, includingtiming, phase, and Doppler frequency errors, on detectionperformance. Fig. 7 depicts the simulated and analytical ¯ P d forCD, HD, and NCD under various timing conditions. Again,there is a perfect match between the simulated and analyticalresults. In addition, it is seen that in general, as the timing errorincreases, the performance of all 3 detectors degrades. This isbecause a larger timing error implies the sampling location isfurther away from the peak of the auto ambiguity function,which results in a higher loss of the energy of the desiredauto term and the associated SNR. It was observed in [27],if the timing error is much smaller than the reciprocal of thewaveform bandwidth but still significant relative to the carrierperiod so that the SNR loss is negligible, then it will onlyimpact coherent detectors such as ACD as the timing-error-induced phase error may not be negligible. The observationapplies to CD as well. For space limitation, we do not duplicatethe result here.The impact of phase error is shown in Fig. 8. It is ob-served that the phase error only affects CD, which is becausethe implementation of HD and NCD does not require any Doppler o , set a v e r ag e p r o b a b ili t y o f d e t ec t i o n multi-band chirps CDCD(co-located)HDHD (co-located)NCDNCD(co-located) (a)
Doppler o , set a v e r ag e p r o b a b ili t y o f d e t ec t i o n single-band chirps CDCD(co-located)HDHD (co-located)NCDNCD(co-located) (b)
Figure 6. ¯ P d of distributed MIMO radar versus Doppler offset f − f (normalized by the PRF) in comparison with the synchronous co-locatedMIMO radar, when SNR = 0 dB, τ = 0 . T p , τ = 0 . T p , and ψ = ψ = 0 . knowledge of the phase while the CD requires it for coherentintegration across antennas. On the other hand, Fig. 9 showsthe impact of Doppler frequency error. It is seen that Dopplerfrequency error degrades the performance of both CD and HDbut not that of NCD. This is because the Doppler knowledge isrequired for the implementation of CD and HD. Interestingly,HD outperforms NCD with or without the Doppler error, whileit exhibits much better performance than CD when Dopplererror is present. VI. C ONCLUSION
We examined the impact of non-orthogonal waveforms andsync errors on target detection in distributed MIMO radar. Ourmain contributions include the general asynchronous signalmodel for distributed MIMO radar, the new CD and HDdetectors, a complete statistical analysis of CD, HD, and thepreviously introduced NCD and ACD for distributed MIMOradar with sync errors. Our results indicate that cross termsstemmed from non-orthogonal waveforms can be beneficialor detrimental to target detection, while sync errors in timing,frequency, and phase have different impacts on different detec-tors. The fact that detection can benefit from cross terms opensup future research possibilities for TX-side encoding to reapsuch performance gain, if propagation related delay/frequencyoffsets in delay can be made available to the TXs. Another -10 0 10 20 30
SNR (dB) a v e r ag e p r o b a b ili t y o f d e t ec t i o n multi-band chirps CDHDNCDsimulation " t mn = [0 0] " t mn = [0 :
23 0 : T p " t mn = [0 :
43 0 : T p (a) -10 0 10 20 30 SNR (dB) a v e r ag e p r o b a b ili t y o f d e t ec t i o n single-band chirps CDHDNCDsimulation " t mn = [0 0] " t mn = [0 :
23 0 : T p " t mn = [0 :
43 0 : T p (b) Figure 7. ¯ P d of distributed MIMO radar versus SNR without timingerrors ( ∆ t mn = 0 ) or with two sets of timing errors. The solid lines areobtained from theoretical analysis while the markers (circles) are obtained bysimulation. future topic of interest is to extend the study to cases involvingclutter and extended target detection.A PPENDIX ID ERIVATIONS OF
CD (28)
AND
HD (29)
A. Derivation of CD (28)Coherent detection requires the knowledge of the Dopplersteering matrices S n , ambiguity function matrices X mn , andchannel vectors h mn for phase and amplitude compensation.Given these estimates, a CD can be obtained by using ageneralized likelihood ratio test (GLRT) framwork, which isdetailed next.Let Y (cid:44) [ y , · · · , y MN ] T ∈ C MN × K , which containsobservations from all RXs. According to (18), the likelihoodfunction under H and H can be expressed as p ( Y ; α ) = 1( πσ ) KMN × exp (cid:16) − σ M (cid:88) m =1 N (cid:88) n =1 (cid:107) y mn − α S n X mn h mn (cid:107) (cid:17) , (52) p ( Y ) = 1( πσ ) KMN exp (cid:16) − σ M (cid:88) m =1 N (cid:88) n =1 (cid:107) y mn (cid:107) (cid:17) . (53) -15 -10 -5 0 5 10 15 SNR (dB) a v e r ag e p r o b a b ili t y o f d e t ec t i o n multi-band chirps CD(w/o phase error)CD(w/ phase error)HD(w/o phase error)HD(w/ phase error)NCD(w/o phase error)NCD(w/ phase error) (a) -15 -10 -5 0 5 10 15
SNR (dB) a v e r ag e p r o b a b ili t y o f d e t ec t i o n single-band chirps CD(w/o phase error)CD(w/ phase error)HD(w/o phase error)HD(w/ phase error)NCD(w/o phase error)NCD(w/ phase error) (b)
Figure 8. ¯ P d of distributed MIMO radar versus SNR without phase error( ∆ p mn = 0 ) or with phase errors ( ∆ p = 0 . π and ∆ p = 0 . π ). It follows the log-likelihood ratio (LLR) is l ( Y ) = log p ( Y ; α ) p ( Y )= 1 σ M (cid:88) m =1 N (cid:88) n =1 (cid:16) (cid:107) y mn (cid:107) − (cid:107) y mn − α S n X mn h mn (cid:107) (cid:17) . (54)The GLRT requires the maximum likelihood estimate (MLE)of α under H . Taking the derivative of the log-likelihood ln p ( Y ; α ) w.r.t. α and setting it to zero yields the MLE ˆ α = (cid:80) Mm =1 (cid:80) Nn =1 ( S n X mn h mn ) H y mn (cid:80) Mm =1 (cid:80) Nn =1 ( S n X mn h mn ) H ( S n X mn h mn ) . (55)Substituting the MLE into the LLR: l ( Y ) = (cid:80) Mm =1 (cid:80) Nn =1 y Hmn S n ( S Hn S n ) − S Hn y mn σ (cid:80) Mm =1 (cid:80) Nn =1 ( S n X mn h mn ) H ( S n X mn h mn ) . (56)The denominator of l ( Y ) can be absolved into the testthreshold, which reduces the GLRT to (cid:101) T CD = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 N (cid:88) n =1 ( S n X mn h mn ) H y mn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (57)For practical implementation, S n , X mn , and h mn are con-structed from delay/phase/Doppler estimates. In the presenceof sync errors, they are formed by Equations (25)–(27), inwhich case the CD is given by (28). -15 -10 -5 0 5 10 15 SNR (dB) a v e r ag e p r o b a b ili t y o f d e t ec t i o n multi-band chirps CD(w/o Doppler error)CD(w/ Doppler error)HD(w/o Doppler error)HD(w/ Doppler error)NCD(w/o Doppler error)NCD(w/ Doppler error) (a) -15 -10 -5 0 5 10 15
SNR (dB) a v e r ag e p r o b a b ili t y o f d e t ec t i o n single-band chirps CD(w/o Doppler error)CD(w/ Doppler error)HD(w/o Doppler error)HD(w/ Doppler error)NCD(w/o Doppler error)NCD(w/ Doppler error) (b)
Figure 9. ¯ P d of distributed MIMO radar versus SNR without Doppler fre-quency errors ( ∆ f mn = 0 Hz) or with Doppler frequency errors ( ∆ f = − Hz and ∆ f = 10 Hz).
B. Derivation of HD (29)The HD can be obtained by using GLRT and treating β mn = α X mn h mn , which lumps the target amplitude α ,ambiguity function matrix X mn , and channel coefficient h mn , as an unstructured unknown vector . Specifically, let β (cid:44) [ β , · · · , β MN ] T ∈ C MN × K . According to (18), thelikelihood functions can be expressed as p ( Y ; β ) = 1( πσ ) KMN × exp (cid:16) − σ M (cid:88) m =1 N (cid:88) n =1 (cid:107) y mn − S n β mn (cid:107) (cid:17) , (58) p ( Y ) = 1( πσ ) KMN exp (cid:16) − σ M (cid:88) m =1 N (cid:88) n =1 (cid:107) y mn (cid:107) (cid:17) , (59)The MLE of β mn is obtained by taking the derivative of ln p ( Y ; β ) w.r.t. β mn and setting it to zero: ˆ β mn = ( S Hn S n ) − S Hn y mn . (60) Using the above MLE in the LLR along with some simplifi-cations leads to the following test statistic (cid:101) T HD = M (cid:88) m =1 N (cid:88) n =1 y Hmn S n ( S Hn S n ) − S Hn y mn = M (cid:88) m =1 N (cid:88) n =1 (cid:107) S n ( S Hn S n ) − S Hn y mn (cid:107) . (61)In practice, S n is replaced by ˆ S n formed from (25) along withDoppler estimates, leading to (29).A PPENDIX
IIP
ROOFS OF T HEOREMS TO A. Proof of Theorem 1
Consider the NCD (19), which is included below for easyreference (dropping the subscript “NCD” for simplicity) T = M (cid:88) m =1 N (cid:88) n =1 y Hmn y mn . (62)Clearly, T is a square sum involving KM N independent andidentically distributed (i.i.d.) real Gaussian random variableswith variance σ / and zero mean (under H ) and non-zeromean (under H ), respectively. Hence, T (cid:118) (cid:40) σ X KMN under H σ X (cid:48) KMN ( λ ) under H , (63)where X KMN and X (cid:48) KMN ( λ ) denote the central and, respec-tively, noncentral chi-square distribution with KM N degreesof freedom and the noncentrality parameter λ is given by (32).Based on the above distributions, the probability of false alarmand the probability of detection can be calculated as P f = (cid:90) ∞ γ f ( T |H ) dT = 2 σ (cid:90) ∞ γ KMN Γ( KM N ) (cid:16) xσ (cid:17) KMN − e − xσ dx = Γ( KM N ) − ¯Γ( KM N, γ/σ )Γ( KM N ) (64) P d = (cid:90) ∞ γ f ( T |H ) dT = 1 σ (cid:90) ∞ γ e − x + λσ / σ (cid:16) xλσ (cid:17) KMN − I ( KMN − (cid:16)(cid:114) λxσ (cid:17) dx = Q KMN (cid:16) √ λ, (cid:114) γσ (cid:17) . (65) B. Proof of Theorem 2
Let Y (cid:44) (cid:80) Mm =1 (cid:80) Nn =1 (ˆ S n ˆ X mn ˆ h mn ) H y mn . The teststatistic of CD (28) can be written as T = | Y | . (66) It is easy to show that Y is complex Gaussian with E { Y |H } = 0 , (67) E { Y |H } = α M (cid:88) m =1 N (cid:88) n =1 (ˆ S n ˆ X mn ˆ h mn ) H S n X mn h mn , (68)var { Y |H } = var { Y |H } = σ M (cid:88) m =1 N (cid:88) n =1 (cid:107) ˆ S n ˆ X mn ˆ h mn (cid:107) . (69)Let (cid:101) T (cid:44) Tσ (cid:80) Mm =1 (cid:80) Nn =1 (cid:107) ˆ S n ˆ X mn ˆ h mn (cid:107) . (70)It follows that (cid:101) T (cid:118) (cid:40) X under H X (cid:48) ( λ ) under H , (71)where the noncentrality parameter λ is given by (37). Hence,the probability of false alarm is given by P f = P ( T > γ |H ) = P (cid:16) (cid:101) T > γςσ (cid:12)(cid:12)(cid:12) H (cid:17) = e − γςσ , (72)where ς is defined in (38). Likewise, the probability ofdetection is given by P d = P ( T > γ |H ) = P ( (cid:101) T > γςσ |H )= 1 − F χ (cid:48) ( λ ) ( 2 γςσ ) = Q ( √ λ, (cid:114) γςσ ) , (73)where F χ (cid:48) ( λ ) ( x ) = 1 − Q ( √ λ, √ x ) is the cumulative distri-bution function (CDF) of the non-central chi-square randomvariable χ (cid:48) ( λ ) . C. Proof of Theorem 3
Define X = (cid:80) Mm =1 (cid:80) Nn =1 (cid:80) K − k =0 e − ˆ θ mnk y mn ( k ) . The teststatistic of ACD (20) is equivalent to T = | X | . (74)Based on the distribution of y mn , we can obtained the dis-tribution of y mn ( k ) as y mn ( k ) (cid:118) CN (0 , σ ) under H and y mn ( k ) (cid:118) CN ( x mn ( k ) , σ ) under H . Thus, we have X (cid:118) CN (0 , σ ) under H CN ( M (cid:80) m =1 N (cid:80) n =1 K − (cid:80) k =0 e − ˆ θ mnk x mn ( k ) , σ ) under H , (75)where x mn ( k ) is defined in (7). The distribution of the teststatistic becomes T (cid:118) (cid:40) σ X under H σ X (cid:48) ( λ ) under H , (76)where the noncentrality parameter is given by (42). Then, theprobability of false alarm is P f = P ( T > γ |H ) = e − γKMNσ , (77) and the probability of detection is P d = P ( T > γ |H ) = 1 − F χ (cid:48) ( λ ) ( 2 γKM N σ )= Q ( √ λ, (cid:114) γKM N σ ) . (78) D. Proof of Theorem 4
The HD test statistic (29) can be written as T = M (cid:88) m =1 N (cid:88) n =1 ˆ ϕ Hmn ˆ S Hn ˆ S n ˆ ϕ mn , (79)where ˆ ϕ mn = (ˆ S Hn ˆ S n ) − ˆ S Hn y mn , which is a complex Gaus-sian random vector since it is a linear transformation of y mn .Specifically, under H , ˆ ϕ mn (cid:118) CN ( ϕ mn , C ˆ ϕ mn ) , where ϕ mn = α (ˆ S Hn ˆ S n ) − ˆ S Hn S n X mn h mn , (80) C ˆ ϕ mn = σ (ˆ S Hn ˆ S n ) − . (81)As a result, we have ˆ ϕ Hmn ˆ S Hn ˆ S n ˆ ϕ mn σ / ϕ Hmn C − ϕ mn ˆ ϕ mn . (82)Next, let ϕ mn = ν mn + µ mn and ˆ ϕ mn = ˆ ν mn + ˆ µ mn ,and define the M × real vectors ε mn = [ ν Tmn µ Tmn ] T and ˆ ε mn = [ˆ ν Tmn ˆ µ Tmn ] T . Then, the test statistic is equivalent to[31, Appendix 15A]: ˆ ϕ Hmn ˆ S Hn ˆ S n ˆ ϕ mn σ / ε Tmn C − ε mn ˆ ε mn , (83)where C ˆ ε mn is the M × M covariance matrix of the realvector ˆ ε mn . Since ˆ ϕ mn (cid:118) CN ( ϕ mn , C ˆ ϕ mn ) , we have ˆ ε mn (cid:118) N ( ε mn , C ˆ ε mn ) . (84)Clearly, ˆ ε Tmn C − ε mn ˆ ε mn is central (under H ) and noncentral(under H ) chi-square distributed [32, Section 2.3]. Equiva-lently, we have ˆ ϕ Hmn ˆ S Hn ˆ S n ˆ ϕ mn σ / (cid:118) (cid:40) X M under H X (cid:48) M ( λ mn ) under H , (85)where λ mn = ε Tmn C − ε mn ε mn = 2 ϕ Hmn C − ϕ mn ϕ mn = | α | h Hmn X Hmn S Hn ˆ S n (ˆ S Hn ˆ S n ) − ˆ S Hn S n X mn h mn σ / . (86)According to the summation rule for a sum of weighted cen-tral/noncentral chi-square random variables [33], the originaltest statistic of the HD is given by Tσ / (cid:118) (cid:40) X NM under H X (cid:48) NM ( λ ) under H , (87)where λ = (cid:80) Mm =1 (cid:80) Nn =1 λ mn . Based on the above distribu-tions, the probability of false alarm and the probability of detection of the HD can be obtained by employing similarderivations as in (64) and (65): P f = Γ( N M )¯Γ( N M , γ/σ )Γ( N M ) (88) P d = Q NM (cid:16) √ λ, (cid:114) γσ (cid:17) . (89)R EFERENCES[1] J. Li and P. Stoica, “MIMO radar with colocated antennas,”
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