Novel Detectors for Passive Radar Sensing With I/Q Imbalance and Additive Distortion

Passive radar has been recognized as one of the most attractive and promising approaches to solving the critical challenge of spectrum scarcity. However, the ubiquitous hardware impairment (HWI) might severely degrade the passive radar's performance. This article develops maximum likelihood estimation algorithms considering I/Q imbalance (IQI) and additive distortion (AD) to propose the generalized likelihood ratio test (GLRT)-based detectors. For single-target cases, the novel GLRT detectors correspond to different knowledge levels of the transmitted signals and channel state information. Then, the single-target scenarios are extended to the two-target case to investigate the effects of transmitter–receiver distance, bistatic range and range resolution brought by the interfering target. Specifically, we propose the HWI mitigation algorithm to approximate and simplify the complex GLRT expressions due to the interfering target, thereby counteracting the effects of IQI and AD. Finally, simulation results demonstrate that both amplitude mismatch and phase mismatch have degrading impact on the overall detection probability of single-target passive radar, and the two-target detector has a very high-range resolution compared with classical works in DVB-T. Also, the range resolution is proven to be robust to HWI.


I. INTRODUCTION
S PECTRAL scarcity has been a significant technical chal- lenge in wireless communications.There have been several effective solutions in recent decades to alleviate this problem.One approach is to utilize the unlicensed industrial, scientific, and medical (ISM) bands.Near-field communications are such a technology to use the ISM, thus mitigating the challenge of spectrum scarcity [1].To address the mismatch between optimal beamformers at the center frequency and their poor performance at distinct radio frequencies within the wide bandwidth for millimeter wave and terahertz communications using massive phased arrays, a low-complexity technique InFocus was proposed to mitigate the misfocus effect in the near-field system [2].Different transmit antenna structures for near-field signaling, including fully digital architectures, hybrid phase shifter-based precoders, and the emerging dynamic metasurface antenna architecture for massive multiple-input multiple-output (MIMO) arrays were studied to explore the potential of feasible beams [3].Long-term evolution (LTE) access to the unlicensed spectrum is also helpful in solving spectrum scarcity [4].The performance of coexistence between LTE and Wi-Fi systems was evaluated by focusing on a system-level simulation analysis to assess the network performance in an office scenario [5].The computing offloading problem for private industrial Internet-of-Things (IoT) enabled by LTE over unlicensed spectrum technology was investigated to figure out the channel availability and task arrivals via the proposed constrained deep Q-learning-based task scheduling algorithm with provable convergence [6].Another approach is dynamic spectrum access, which mainly employs the cognitive radio (CR) to dynamically find unused spectrum bands and temporarily utilize them for secondary users [7].In [8], the authors boosted energy efficiency for data transmission in licensed channels via a two-way information exchange dynamic spectrum sensing algorithm in clustered CR-IoT networks.A cache-enabled unmanned aerial vehicles cooperation scheme in the CR network was proposed to reduce the redundant traffic load and enhance the transmission capability of CR network in the work of [9].Besides, there is also a choice to use the spectrum bands at extremely high frequencies, such as visible light communications (VLC).For example, a new field of view geometry-based single bounce model for VLC channels was proposed to mitigate the effects of channel distortions [10].The authors in [11] developed a comprehensive VLC channel modeling and characterization based on ray tracing which was capable of receiving channel impulse responses for any nonideal sources.
Different from the above-mentioned solutions, joint radar and communications (JRC) attempts to achieve the spectrum sharing of radar system bands that fall into the S-band (2-4 GHz) and C-band (4-8 GHz) with communications systems according to three categories: codesign, coexistence, and cooperation [12].On the one hand, the networking of multiple radars requires complex communications techniques to improve the overall radar performance [13].On the other hand, the communications systems could be aided by radar to discover a neighbor more quickly [13].Therefore, JRC has been under intensive investigation and becomes one of the most successful methods to solve spectrum scarcity.In particular, codesign achieves the spectrum sharing by jointly designing a dual-functional radar communications system.Liu et al. [14] proposed an integrated signal model that simultaneously communicated with downlink users and detected radar targets by optimizing the transmit 1937-9234 © 2023 IEEE.Personal use is permitted, but republication/redistribution requires IEEE permission.
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beamforming to match the desired radar beampattern while satisfying the communications performance requirements.An optimization problem of the weighted summation of radar signal and communications signal was formulated, and solved by semidefinite relaxation algorithm based on zero-forcing technique in [15].To cancel the interference of active waveforms from radars, two algorithms were proposed to jointly estimate the waveform and demodulate the received data by implementing an iterative process [16].The authors of [17] investigated the effect of interference from an unaltered radar system on the performance of both single-carrier and multicarrier communications systems, then they proposed the complex-valued constellation design problems to either maximize the transmission rate or minimize the error rate.The cooperative spectrum sharing and joint sampling scheme between a MIMO-matrix completion radar and a communications system were devised to reduce the radar receiver's effective interference power further without compromising the capacity and transmit power to the communications system below a certain level [18].In [19], a hybrid active passive MIMO radar network was designed by employing target returns from both the radar transmitters and communications transmitters, while the communications system could extract useful information not only from communications receiver but also from radar echo signals from the target.
The interest in passive radar is excellent among all the cooperative JRC regimes.Passive radar can utilize the radio frequency (RF) signals from a communications system as the illuminator of opportunity (IO) to detect potential targets in a bistatic passive setting.In this setting, a typical passive radar system often receives the signal transmitted directly from the IO in the reference channel (RC) and the signal reflected by the potential target in the surveillance channel (SC).Thus, passive radar requires no emission of radar signals and avoids revealing the position of its own sensor [20].Due to its complete covertness and passiveness, frequency allocation is not required to deploy the passive radar receiver.Therefore, the passive radar is able to operate in densely populated areas with a lot of IO signals at various frequencies [20].Besides, this passive setting has some extra advantages over conventional radars, including lower cost, smaller space, and much higher portability [21].Therefore, it has become an interesting research topic for JRC designs.
There have been a number of works on passive radar.For example, the normalized cross ambiguity function (NCAF) was proposed to combine information contaminated by the Gaussian noise from both RC and SC, and the cumulative distribution function of NCAF was derived in [22].Direct-path interference in multistatic passive radar was considered using a generalized likelihood ratio test (GLRT) detector in [23].The extensive cancellation algorithm was modified to a sliding window version subject to limitations, and its counteracting effect against these limitations was demonstrated by estimating the filter coefficients in [24].In [25], a proper reference signal was reconstructed by analyzing the basis expansion model for the channel and using antenna diversity and channel estimation to alleviate the severe distortion of digital video broadcasting-terrestrial (DVB-T) orthogonal frequency division multiplexing (OFDM) signals in an airborne passive radar system.In [26], the authors considered the performance variability for different signal-to-noise ratios (SNRs) in RC and SC, and proposed several GLRT detectors to achieve significant performance improvement over the conventional cross-correlation detectors.In [27], the analytical expressions of the cross-correlation detector's false alarm probability and detection probability were given to explore the extent to which the interference and noise shall be suppressed, and the superiority of cross-correlation detectors over matched filter detectors was shown.Direct signal interference mitigation was examined to identify the primary factors affecting interference suppression for various spectrally and spatially diverse digital television (DTV) waveforms under realistic situations in [28].The multistatic passive radar based on single-antenna DTV processing was proposed to analyze the detection range in the drone detection scenario [29].
Most of the works mentioned previously have assumed that the transceiver hardware is ideal.However, communications devices are known to suffer from hardware impairment (HWI), such as power amplifier nonlinearity, phase noise and in-phase and quadrature imbalance (IQI).These HWI will affect the system performance by distorting image signals, causing phase and amplitude mismatch, and raising the noise floor [30].Several works have examined the effect of HWI on communications systems.For example, the analytical expression for the outage probability (OP) at the amplify-and-forward dual-hop relay suffering from IQI was derived over nonidentical and independent Nakagami-m fading channels [31].In [32], the authors derived the analytical symbol error rate for MIMO OFDM systems over Rayleigh channels.They showed that receive (RX) IQI has a more considerable impact than transport (TX) IQI and that extra RX antennas could mitigate RX IQI.Full-duplex cooperative nonorthogonal multiple access relaying systems suffering from imperfect successive interference cancellation and IQI were studied regarding the ergodic sum rate and the OP [33].
Besides IQI, additive distortion (AD) is another common type of HWI.The analytical expression for OP and the optimal beamforming for a dual-hop MIMO amplify-and-forward relay were studied with AD in [34].Considering both IQI and AD direct-conversion radio, closed-form expressions for the false alarm and detection probabilities were derived and extended to CR networks in [35].The improper Gaussian signaling (IGS) scheme was compared with the traditional proper Gaussian signaling scheme in terms of achievable rate and OP, and IGS was found to counteract AD efficiently [36].
All of the above-mentioned works have shown that HWI is an important factor limiting communications performances.However, they have not considered HWI in a passive radar setting that uses communications signals.There have been works on HWI for active radar systems.For example, a 160 GHz frequency-modulated continuous-wave (FMCW) radar was investigated with an HWI mitigation method in [37].In [38], a parametric radar model considering chirp-sequence and HWI was proposed.In [39], the signal-to-distortion-plus-noise ratio and range-Doppler sensitivity with constant false alarm rate were studied for a 77 GHz FMCW automotive radar with both IQI and phase noise.Nevertheless, active radars and passive radars Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
are very different in terms of hardware, setup, and IO.Thus, these results cannot be used for passive radars.
Motivated by the above-mentioned observations, our work focuses on novel detectors considering IQI and AD in passive radar sensing.The major contributions of this article are summarized as follows.
1) We present a practical passive radar framework to solve the spectrum scarcity.This system model considers the HWI of IQI and AD at both the TX (IO) and the passive radar's RX.
2) The detector design problem is formulated as a parameter estimation problem for GLRT.For the single-target cases, the closed analytical forms of four novel detectors are derived via maximum likelihood estimate method corresponding to four scenarios with different knowledge levels of the transmitted signals and channel state information.
3) The single-target GLRT problem is extended to the twotarget GLRT problem to investigate the performance of operation range limit and range resolution in the presence of an interfering target.To tackle this issue, we propose the HWI mitigation algorithm to approximate and simplify the GLRT formulas, which avoids processing the complicated HWI terms.4) Numerical results show that IQI has a significant degrading impact on the detection probability of the single-target passive radar detectors.Especially the amplitude mismatch has more impact than the phase mismatch.Also, the proposed HWI mitigation algorithm is proven to provide stable range resolution regardless of IQI.The rest of this article is organized as follows.Section II presents the passive radar model with HWI.Section III derives the new detectors in various realistic single-target scenarios.Section IV gives the detector with an interfering target to investigate the range limit and range resolution.Section V provides simulation results of the impact of HWI on the performance of the proposed detectors.Finally, Section VI concludes this article.
Notation: The italic letter denotes a scalar and the lower case boldface letter represents a vector.CN (•, •) denotes a complex Gaussian random variable.E(•) represents the expectation operation.Re(•) and Im(•) represent the real part and the imaginary part of a complex number, respectively.

II. SYSTEM MODEL
Similar to that in [26], it is assumed that both the IO and the passive radar receiver have a single antenna.The RC is the direct-path from the IO to the passive radar receiver.The SC is the path where the transmitted signal is reflected by the object to be detected at the passive radar receiver.Assume that the transmitted information signal is s(t) with E(|s(t)| 2 ) = P .For the ith transmitted sample with a sampling frequency f s at the time instants t = i/f s , i = 1, . .., I, we use subscript i to represent it as s i .This sample first suffers from the IQI at the transmitter to become where , a t and φ t are the amplitude and phase mismatch at the transmitter due to the IQI, respectively.Then, the AD d i due to the imperfect power amplification is added to s t,i as At the IO's transmitter, there is an AD d t,i .Also, there is an AD d r,i at the passive radar's receiver.By considering the aggregate effect of the d t,i and d r,i , it is equivalent to be an AD d i at the transmitter.Therefore, d i has been modeled as complex Gaussian with d i ∼ CN (0, (σ 2 t + σ 2 r )P ) where and σ 2 r are the variances of the AD at the transmitter and the receiver, respectively, in all previous works.This is motivated by the central limit theorem from various RF stages and the details of the explanation can be found in [40], [41].Under different hypotheses, one has the received signal x r,i from the RC and the received signal x s,i from the SC as where h r and h s are the channel gains in the RC and SC, respectively, w r,i and w s,i are the additive white Gaussian noise (AWGN) in the RC and SC, respectively.They are independent and identically distributed complex Gaussian random variables with CN (0, σ 2 w ).Similarly, x r,i and x s,i also suffer from the IQI at the receiver with , where a r and φ r are the amplitude and phase mismatch at the receiver, respectively.In our work, we assume that a t = a r = a and φ t = φ r = φ for convenience.Therefore, the received signals after considering all HWI are Substituting ( 2) and ( 3) into (4), one has where In the next section, we will propose the corresponding detectors for various situations.

III. NEW DETECTORS
For the detection at the passive radar receiver, we take I received samples as given in (5).At the IO transmitter, the transmitted information signal sample vector is s = [s 1 , s 2 , . .., s I ].At the passive radar receiver, the received signal sample vectors are y r = [y r,1 , y r,2 , . .., y r,I ] and y s = [y s,1 , y s,2 , . .., y s,I ] in RC and SC, respectively.For a specific sample y r,i in y r and y s,i in y s , the joint likelihood function of (y r,i , y s,i ) under hypothesis where l r,H 0 and l s,H 0 are the likelihood functions of y r,i and y s,i under H 0 , respectively.Similarly, under hypothesis H 1 , the joint likelihood function of (y r,i , y s,i ) is l H 1 ,i (y r,i , y s,i ) = l r,H 1 (y r,i ) • l s,H 1 (y s,i ), where l r,H 1 and l s,H 1 are the likelihood function of y s,i under H 1 .Note that y r,i is the same for both H 0 and H 1 .Thus, let l r,H 0 (y r,i ) = l r,H 1 (y r,i ) = l r (y r,i ) in the following.Due to the introduction of I/Q imbalance coefficients K 1 and [40].Therefore, z r,i is an improper Gaussian variable, and so is z s,i .Using the probability density function of the improper Gaussian variable in [42], one has where and Then, one has where l H 0 (y r , y s ) and l H 1 (y r , y s ) are the joint likelihood functions under H 0 and H 1 , respectively.According to the Neyman-Pearson criterion, the detection is performed by taking the ratio of the likelihood function l H 1 (y r , y s ) over the likelihood function l H 0 (y r , y s ) with an appropriate detection threshold as where γ denotes the detection threshold.To further simplify the detection, one can take the logarithm of (11) as From ( 5) and ( 6), one sees that l H 0 (y r , y s ) is dependent of h r and s while l H 1 (y r , y s ) is dependent of h r , h s , and s, where h r , h s , and s might be unknown and need to be estimated.We denote the MLEs of h r , h s , and s under H 1 as ĥr,H 1 , ĥs,H 1 , and where Âri,H where Âri,H 0 is the estimated value of A ri under H 0 .It is a function of ĥr,H 0 and ŝH 0 .Different detectors will be derived based on ( 14) and (15) in the following sections.

A. Coherent Detector
In this detector, we assume that h r , h s , and s are all known so that ĥr,H 1 = ĥr,H 0 = h r , ĥs,H 1 = h s , and ŝH 1 = ŝH 0 = s.This is the case for integrated radar-communications designs when the radar function and the communications function use the same device to share information between them.Let h r = Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE II COMPARATIVE STUDY OF THE DETECTORS TABLE III SIMULATION PARAMETER SETUP
|h r |e iθ r , h s = |h s |e iθ s , and θ = θ r − θ s .To get the desired detector, we first tackle the terms of A ri and A si,H 1 in ( 8) and ( 9) by substituting and where and with y R s,i = Re(y s,i ), y R r,i = Re(y r,i ), y I s,i = Im(y s,i ), and where 20)-( 22), one has ln Therefore, (13) becomes Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where the proposed coherent detector Λ 0 is denoted as the CH-GLRT ("CH" means "coherent") detector and the detection threshold T 0 will be determined by the predetermined false alarm probability.

B. GLRT With Known "s"
In this case, we only know the transmitted information vector s while h r and h s are unknown, so that ŝH 1 = ŝH 0 = s, while h s and h r can be independently estimated under H 1 and H 0 , respectively.Using ( 15) and ( 21), one has with which the MLEs of θ r as θr,H 0 and |h r | as | h r,H 0 | under H 0 are obtained, respectively.Similarly, the MLEs θs,H 1 and | h s,H 1 | under H 1 can also be obtained.Thus, one has where D s (s) are given in Appendix A. The derived detector Λ 1 is denoted as the KS-GLRT ("KS" means "known s") detector.The derivation details are given in Appendix A.

C. GLRT With Known h r and h s
In this case, we know the channel gains h r and h s while the transmitted signal s is unknown.Thus, ĥr,H 1 = ĥr,H 0 = h r and ĥs,H 1 = h s .Since each transmitted symbol s i is independent, we can get the MLE of s i for each sample separately.For H 0 , we use ( 15) and ( 21) to have For H 1 we use ( 14), (21), and (22) to have By letting the derivatives of ( 28) and ( 29) to be 0, one has the MLE of s i under H 0 as ŝi,H 0 and the MLE of s i under H 1 as ŝi,H 1 .Substituting ŝi,H 0 and ŝi,H 1 into ( 14) and ( 15), one has where and This detector Λ 2 is denoted as the KCI-GLRT ("KCI" means "known channel information of h s and h r ") detector.The derivation details are given in Appendix B.

D. GLRT With Known h r
In this case, we know the channel gain h r while the transmitted signal s and h s are unknown, so ĥr,H 1 = ĥr,H 0 = h r .The RC could use reference symbols for channel estimation and therefore h r is easier to obtain than h s .Also, the MLE of s i under H 0 as ŝi,H 0 and the MLE of s i under H 1 as ŝi,H 1 are the same as those in KCI-GLRT.
Then, one can obtain the MLE of 14) and ( 15), the log-likelihood ratio finally becomes The detector Λ 3 is denoted as the KR-GLRT ("KR" means "known h r ") detector.The derivation details are given in Appendix C. To indicate the necessity of considering HWI, we also provide the corresponding conventional detectors ignoring the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
existence of IQI and AD for each novelly proposed detector as benchmarks in Appendix D.

IV. DETECTION WITH INTERFERING TARGET
In this section, the novel detector with an interfering target is also derived based on GRLT.For the RC, it receives a direct arrival signal with a time delay of τ r = L/c, where L is the transmitter-receiver distance and c is the transmission velocity of radio.Thus, (3) becomes x r,i = h r s l,i + w r,i (35) with s l,i = s d,i−l 0 , where , where R is the bistatic range of the desired target and δ R is the difference of bistatic range between the interfering target and the desired target and h s,2 is the SC channel gain of the interfering target.Under H 1 the SC signal contains the reflection of the desired signal, so where l 1 = τ s f s and l 2 = δ s f s .Under H 0 the SC signal consists of the reflection of one interfering target and the AWGN in the SC.Thus, where h s,1 is the channel gain of the desired target in SC.Similarly, x r,i and x s,i suffer from HWI at the receiver to become (4).
Observing from ( 36) and (37), x s,i can be expressed by and where α = h s,1 /h r and β = h s,2 /h r .According to [43], it is reasonable to assume that α 1 and β 1 since the signal is severely attenuated in the SC compared with the RC.Therefore, one can approximate (38) as and Thus, only the likelihood of X s = (x s,1 , . .., x s,I ) is required to make the detection.To explore the impact of L, R, and δ R , these parameters are predefined and known to the passive radar.MLE is used to estimate α and β.The likelihood of X s is where and By ( 39) and ( 40), we mitigate the effect of HWI.σ w is first assumed to be fixed and known, then one obtains that where and 41) is the MLEs of α and β under H 1 .Similarly, one has the MLE of β under H 0 as and the MLE of Using ( 45) and ( 46), the GLRT is expressed as (47) From (47), one has Denote X i = (x r,1−l 1 −l 2 , . .., x r,I−l 1 −l 2 ) T and X 0 = (x r,1−l 1 , . . ., x r,I−l 1 ) T .The projection matrix that projects a vector onto is the orthogonal projection matrix that projects a vector onto the space orthogonal to that spanned by X i .Using the definitions of X i , X 0 , M, and M ⊥ , one has Therefore, according to [43], the detector Λ 4 becomes The above-mentioned equation can be regarded as the square of correlation parameter between the projection of the received vector X s onto the signal subspace and the desired target delayed vector X 0 onto the signal subspace.This statistic Λ 4 is denoted as RANGE-GLRT detector.

V. NUMERICAL RESULTS AND DISCUSSION
In this section, the new detectors are compared with the conventional detectors as benchmarks in the detection probability P d .The transmitted signal s from IO is generated using 64-quadrature amplitude modulation (QAM) scheme unless specified otherwise.The comparison is performed using Monte Carlo simulation to demonstrate the superiority of the proposed detectors over the conventional ones.Each curve .The threshold for each proposed detector is determined by fixing the false alarm probability to 0.01, which gives a threshold of 3, 15, 30, and 35 for CH-GLRT, KS-GLRT, KCI-GLRT, and KR-GLRT, respectively.We fix σ 2 w = 1, SNR r = 10 dB, σ 2 = σ 2 t + σ 2 r = 1, and P = 1 while changing SNR s , a, φ, and I.
The sample size I is studied first.We set SNR r = 10 dB, SNR s = −7 dB, a = 0.9, φ = 10 • , and σ 2 = 1.0.All four proposed detectors are compared in terms of the detection performance.It is illustrated in Fig. 1 that all detectors improve as I increases because more samples give more accurate detection.As expected, CH-GLRT detector is the best since it knows all the necessary parameters in the detection.KCI-GLRT outperforms KR-GLRT by having extra information of h s .KS-GLRT is the worst when only the transmitted signal s is known.Thus, knowing s is less valuable than knowing the channel gains in passive radar detection.Next, we let σ 2 = 1 and φ = 0 • while a  is varying.Other parameters are set as SNR r = 10 dB, σ 2 = 1.0, φ = 0 • , and I = 100 to examine the impact of amplitude mismatch a on the detection performance in Fig. 2. One can observe that a has a significant impact on the overall performance.A smaller a represents a more severe amplitude mismatch.The effect of phase mismatch φ is examined in Figs. 3 and 4. In this case, we vary φ from 1°to 10°with a = 1.0 while other parameters are the same as those in Fig. 2. The result in Fig. 3 shows that φ has little impact on the detection performance when the transmitted signal uses 64-QAM modulation.Considering that the QAM method employs both amplitude and phase to differentiate the information symbols, a pure phase mismatch might not affect the detection of the target much.To verify this, Fig. 4 provides the performance comparison for phase-shift keying (PSK), where all parameters are the same as those in Fig. 3.In Fig. 4, φ has a considerable impact on the detection performance since the phase mismatch of 10 • exceeds the approximate phase interval of 5.62°for two neighboring information symbols  in 64-PSK modulation and severely distorts the signal.In this regard, QAM is more robust to phase mismatch than PSK.We also note that the phase mismatch φ has less impact on the detection performance than the amplitude mismatch a, as a causes larger performance degradation by comparing Figs. 2  and 4.
Next, we compare the performances of the proposed detectors considering HWI and the corresponding conventional detectors ignoring HWI.The case with negligible IQI is represented by a = 0.99 and φ = 1 • .With a sample size I = 100, σ 2 = 1.0, and SNR r = 10 dB, we show the comparison in Fig. 5.It can be seen that the proposed detectors outperform their corresponding conventional ones, which indicates that the detector is very sensitive to the nonideality of the transceiver and that this sensitivity is different for different detectors.Therefore, it is pretty necessary to derive the new detectors for HWI.
Different from the four new detectors presented previously, RANGE-GLRT uses the DVB-T signals.It must consider the time delay and the path loss due to the long bistatic range and interfering target.Therefore, P and I are set to be 100 and 1000, respectively.SNR s,1 and SNR s,2 are fixed to be -7 dB to study the effects of L, R, and δ R .It is illustrated in Fig. 6 that a range limit of about 33 km exists for the RANGE-GLRT detector.Such a range limit is the sum of the bistatic range and the transmitter-receiver distance, which is also the sum of the distance from the target to the transmitter and the distance from the target to the receiver.The passive radar can no longer detect the desired target when the range is beyond the limit.When L is fixed, increasing R within its range limit is always feasible for better detection performance.Interestingly, some periodic range intervals perform poorly in detecting the desired target, indicating that the detection probability is not linearly correlated with R.Only the green dotted line of L = 15 km keeps the ideal detection probability of 1 before R reaches its maximum of 18 km.Thus, these different performance patterns make the green line "invisible" as it overlaps with the other three lines.This phenomenon is consistent with [44].Finally, we investigate the range resolution of RANGE-GLRT in Fig. 7. Fix L = 10 km and R = 15 km, the range resolution is measured by the interval of δ R where the desired target is "covered" or "dissembled" by the interfering target so that the passive radar cannot detect it.The range resolution is about 30 m, which is very close to the theoretical one of DVB-T [20].Also, the range resolution is not affected by the level of HWI, which proves that our proposed HWI mitigation algorithm is effective.

VI. CONCLUSION
Several novel passive radar detectors have been proposed based on different channel and signal information knowledge considering both IQI and AD.For the single-target passive radar, numerical results have demonstrated that HWI does have a significant impact on the overall detection performance and that the destructive effects of IQI should not be ignored even when the amplitude mismatch and the phase mismatch are small.The amplitude mismatch has a much more degrading impact than the phase mismatch in general.Furthermore, the HWI mitigation algorithm has been proposed to develop the RANGE-GLRT detector to investigate the effect of range and range resolution.This detector is demonstrated to have robust range resolution regardless of the IQI level.Finally, the sample size can improve the overall detection performance.

APPENDIX A DERIVATION OF KS-GLRT DETECTOR
According to (26), by solving the equation ∂f 0 (|h r |,θ r ) ∂θ r = 0, we obtain the MLE of θ r under H 0 as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. where and where θr,H 0 is given in (50).The MLE of h r under H 0 and H 1 is the same, i.e., ĥr,H 1 = ĥr,H 0 .Similarly, we can obtain ĥs,H where and Therefore, by substituting (54) and (55) into ( 14) and ( 15), one has (27).

APPENDIX D CONVENTIONAL DETECTORS IGNORING HWI
Here, we derive the corresponding conventional detectors ignoring the existence of IQI and AD.These detectors are used as benchmarks for comparison.The system model in (3) ignoring HWI becomes H 0 : y r,i = h r s i + w r,i y s,i = w s,i , H 1 : y r,i = h r s i + w r,i y s,i = h s s i + w s,i .( When all parameters are known, it can be shown that the conventional coherent detector ignoring HWI is which is denoted as the CVCH-GLRT ("CVCH" means the "conventional coherent") detector.When only s is known, the conventional detector from [26] is which is denoted as the CVKCI-GLRT ("CVKCI" means the "conventional known channel information of h r and h s ") detector.
When only h r is known, the conventional detector ignoring HWI can be shown as (76) This is denoted as the CVKR-GLRT ("CVKR" means the "conventional known h r ") detector.