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Amplifyandforward relay identification using joint Tx/Rx I/Q imbalancebased device fingerprinting
EURASIP Journal on Wireless Communications and Networking volumeÂ 2019, ArticleÂ number:Â 64 (2019)
Abstract
Relay identification is necessary in many cooperative communication applications such as detecting the presence of malicious relays for communication security, selecting the intended relays for signal forwarding, and tracing a specific relay. However, this identification task becomes extremely challenging for amplifyandforward (AF) relaying systems since AF relays usually have no capability of adopting traditional identification methods implemented above the physical layer. This paper proposes a physicallayer AF relay identification scheme based on the exploitation of the devicespecific inphase and quadraturephase imbalance (IQI) feature. Given that IQI estimation is mandatory in most present receivers for compensation, it is costeffective to make use of these estimation results for fingerprinting AF relays. A generalized likelihood ratio testbased fingerprint differentiation technique is adopted to detect the minor difference between two rangelimited IQI fingerprints. Using this differentiation technique, a whitelistbased identification algorithm consisting of fingerprint registration, update, and identification is proposed. Furthermore, the optimal training signals that lead to the maximal detection probability are derived for the typical quadrature amplitude modulation and phaseshift keying modulation schemes. The simulation results validate our derivations and confirm that the proposed method can accurately identify AF relays.
1 Introduction
Relaying technique is a promising part of 5G and beyond cellular networks since it works toward broadening the communications coverage area, improving the communication reliability, and increasing the throughput of the network. In an ideal scenario, the identity of an intermediate relay is inessential for the destination device as long as the relay can provide satisfactory quality of service without any malicious behavior. However, identifying relay nodes is of great importance in practice. Signal relaying can potentially introduce new communication security threats, such as the ghostandleech attack [1], the maninthemiddle attack [2], and the denial of service attack [3]. Some relay selection schemes require preclassifying relays into trusted or untrusted groups for cooperation [4â€“6]. In these cases, relay identification is necessary either to detect the malicious relay nodes or to classify those intended relays. In addition, the applications of relay identification can be extended to wireless device localization and tracking [7, 8].
Although a crosslayer decodeandforward relay identification was investigated in [8], this method cannot be applied to amplifyandforward (AF) relay identification since AF nodes only work in the physical layer without any content modifications to the forwarded signals. This implies that all existing upperlayer identification methods are not applicable to the case of AF relaying, thus resulting in the need to fingerprint AF relays in the physical layer. It has been reported that a wireless transmitter can be identified by its unique hardware imperfections [9, 10]. At the transmitter side, all transmitted signals are inevitably affected by the devicespecific RF frontend impairments. At the receiver side, both channel and RF imperfection estimations are usually mandatory for signal reception, and thus it is costeffective to use these estimation results for wireless transmitter identification.
1.1 Related work
As a typical RF imperfection, transmitting (Tx) IQI has been studied for fingerprinting wireless transmitters [11â€“13]. The authors of [11] studied Tx IQI measurements of a large group of transmitters and concluded that IQI can uniquely represent the identity of a wireless transmitter. In [12], IQI fingerprinting was experimentally validated in the scenario of multiple 802.11n multipleinput multipleoutput transmitters. The authors of [13] proposed a distance test (DT) method in which the Euclid distance between two Tx IQI fingerprints is used for device differentiation.
However, most current methods, including [11â€“13], are not suitable for AF relay identification. Unlike the regular transmitters, an AF must receive signals before the signal retransmission process. Given this fact, both receiving (Rx) and Tx IQI need to be taken into account in fingerprinting AF relays [14, 15]. The authors of [16, 17] proposed a varying fingerprint (VF) comprising both channel gain component and Tx/Rx IQI parameter component, where only the IQI parameter is useful for relay identification. However, the performance of using VF is inadequate in the case of small IQI values. In practice, the manufacturing market requires that the hardwarelevel imperfections (e.g., IQI parameter) must be small and within a limited range [18â€“20]. Due to the inevitable channel variation and estimation noise, the uniqueness, stability, and distinguishability of such small IQI parameters are remarkably weakened, thereby leading to degraded identification accuracy. To address these problems, an enhanced AF relay fingerprinting approach and high accuracy fingerprint differentiation technique with optimal performance design are urgently necessary.
1.2 Our methods and contributions
This paper proposes an accurate AF relay identification method using physicallayer IQIbased fingerprinting without the aid of any upperlayer identification technique. We consider small IQI values in a dualhop AF relaying system in which the source node transmits signals to the destination node via an AF relay. Based on existing Rx/Tx IQI estimation results, we propose an improved AF relay fingerprint without channel gains interleaving compared to the VF fingerprinting method. This new fingerprint is used in our whitelistbased fingerprint registration and identification algorithms for relay identification. Moreover, we propose a generalized likelihood ratio test (GLRT)based fingerprint differentiation technique and optimal signal designs to further improve the fingerprint identification accuracy under small IQI values. The main contributions of this paper are summarized as follows.

We propose a new fingerprint of AF relay and analyze its dynamic range and features in order to reveal the main technical challenges in relay identification small IQI values. These general analysis results can also be useful in other IQIrelated AF relaying performance studies.

An accurate GLRTbased AF relay identification algorithm is proposed. The numerical results demonstrate that the introduced identification algorithm outperforms other identification methods [13, 17] in terms of distinguishing delicate IQI fingerprint differences, even with fewer symbols.

Optimal and suboptimal training signals for quadrature amplitude modulation (QAM) and phaseshift keying (PSK) modulations are designed to maximize the capability of detecting an intended AF relay.
The remainder of this paper is organized as follows. Section 2 introduces the system model of our IQIbased AF relay identification. In Section 3, the features of IQI fingerprint are analyzed. A GLRTbased fingerprint differentiation technique is presented in Section 4. In Section 5, the fingerprint registration, identification, and update algorithms are described. In Section 6, the optimal and suboptimal signal designs are presented. Section 7 presents the numerical results and discussions. Finally, this paper is concluded in Section 8.
Notations: (Â·)^{âˆ—}, \(\mathbb {E}[\cdot ]\), Â·, (Â·)^{T}, and (Â·)^{H} denote conjugate, expectation, absolute value, transpose, and conjugate transpose operations, respectively. Bold lowercase and uppercase letters represent vectors and matrices, respectively. For vector a, we use a_{k} to show its kth element. I denotes the identity matrix. det(A) is the determinant of matrix A. â„œ{x} and I{x} denote the real part and imaginary part of x, respectively.
2 System model
This section mainly describes the system model for the AF relayâ€™s IQI fingerprint, the dualhop AF relaying system, and the whitelistbased AF relay identification scheme. Finally, the objective of this study is presented.
2.1 IQI fingerprint model for AF relay
In an AF relay, all signal processing is accomplished in the physical layer. Specifically, the inband received signal is downconverted, amplified, upconverted, and finally forwarded toward the destination node [14, 21]. We thus model the AF relay as the cascade of an Rx component, an amplifier, and a Tx component, as shown in Fig. 1. We consider the asymmetric IQI model [15, 22] in which the signal is affected by the Rx/Tx gain imbalance Î±_{rx}/Î±_{tx} and the Rx/Tx phaseshift imbalance Î¸_{rx}/Î¸_{tx}. Since the frequencydependent IQI plays a minor role compared to the dominant effect caused by the frequencyindependent IQI [23], the former is not considered in this model. After the imperfect RF frontend, the passband signal y_{p}(t)=â„œ{y(t)e^{jÏ‰t}} is forwarded to the destination, where y(t) denotes the baseband equivalent signal as
where
and \({\mu _{\text {rx}}=\frac {1}{2}\left [1+(1+\alpha _{\text {rx}})e^{j\theta _{\text {rx}}}\right ]}\), \(\nu _{\text {rx}}=\frac {1}{2}\left [1(1+\alpha _{\text {rx}})\right. \left.e^{j\theta _{\text {rx}}}\right ]= 1\mu _{\text {rx}}^{*}\), \(\mu _{\text {tx}}=\frac {1}{2}\left [1+(1+\alpha _{\text {tx}})e^{j\theta _{\text {tx}}}\right ]\) and \(\nu _{\text {tx}}=\frac {1}{2}\left [1(1+\alpha _{\text {tx}})e^{j\theta _{\text {tx}}}\right ]=1\mu _{\text {tx}}.\) Also, x(t) is the baseband signal of the input x_{p}(t)=â„œ{x(t)e^{jÏ‰t}}.
In this paper, g_{1} and g_{2} are the joint Tx/Rx IQI parameters and play the role of fingerprinting AF relays. Since IQI is a stable hardware feature, it can be steady in a time scale much longer than an identification session [24]. Hence, we consider that g_{1} and g_{2} are constants in the identification procedure. In addition, all of the following analyses are considered in the baseband domain.
2.2 System model for dualhop AF relaying
We consider a dualhop AF relay system consisting of a source node (S), an AF relay node (R), and a destination node (D). It is assumed that S, R, and D are equipped with a single antenna and operate in halfduplex mode. Thus, S first transmits the signal to R in the first phase, and then, R retransmits the amplified signal to D in the second phase. It is considered that D can only receive signals via R. The two individual channels Sâ€“ R and Râ€“D experience independent block fading so that the instantaneous complex channel gains h_{SR} and h_{RD} are independent and fixed for a block [25â€“29]. n_{SR} and n_{RD} represent the receiving noises at R and D, which are circularly symmetric complex Gaussian (CSCG) random variables as \(n_{\text {SR}}\sim CN\left (0,\sigma _{1}^{2}\right)\) and \(n_{\text {RD}}\sim CN\left (0,\sigma _{2}^{2}\right)\).
As shown in Fig. 2, D is able to identify R through analyzing g_{1} and g_{2}, which are extracted from the received signals. To avoid additional implementation costs, our identification method makes use of the channel and IQI estimation results, where such estimations are usually mandatory in most present receivers (i.e., D). Thus, we assume that D has the knowledge of h_{SR} and h_{RD} before performing the proposed identification method. Similar to [15], we consider the IQIfree S and D and focus on the IQI caused by the AF relays since it plays the important role of fingerprinting AF relays. Given that the IQI estimation is usually carried out using training signals, we consider the training signals in our identification method.
In the Sâ€“ R phase, an NÃ—1 training symbol block s is transmitted from S to relay R. The kth received symbol at R can be represented as
where s_{k} is the kth transmitted symbol and has the average power \(P=\mathbb {E}\left [s_{k}^{*}s_{k}\right ]\), k=1,2,â‹¯,N. In the Râ€“D phase, using (1) and (4), the received Tx/Rx IQI distorted signal at D is
where \(n_{k}=g_{1}h_{\text {RD}} n_{\text {SR}k}+g_{2}h_{\text {RD}} n_{\text {SR}k}^{*}+ n_{\text {RD}k}\). Let \(\mathbf {H}_{s}=[ \mathbf {s} \quad \mathbf {s}^{*}]\text {diag}\left (h_{\text {SR}}h_{\text {RD}}, h_{\text {SR}}^{*}h_{\text {RD}}\right)\) and g=[g_{1} g_{2}]^{T}, the observed block is
The LS estimator, as a typical IQI estimator [19, 23], is then employed to estimate the IQI vector g, which can be given by
where the estimation error is \(\mathbf {n}_{g}=\left (\mathbf {H}_{s}^{H} \mathbf {H}_{s}\right)^{1}\mathbf {H}_{s}^{H} \mathbf {n}\). It is noteworthy that the Bayesian estimator is not suitable for our study since g is an unknown constant instead of a random variable.
2.3 Whitelistbased AF relay identification
Fig. 2 illustrates our whitelistbased AF relay identification based on the IQI estimation results [30]. We first whitelist the valid IQI fingerprints using the proposed fingerprint registration. After that, once a new relay tries to communicate with the destination node, the relay identification can be accomplished by comparing this new relayâ€™s fingerprint with the preregistered fingerprints.
Our objective is to achieve accurate AF relay identification even with small IQI values through analyzing g.
3 Device fingerprinting analysis
In this section, we analyze the important fingerprint (i.e., g_{1} and g_{2}) to reveal the technical challenges in accurately identifying AF relays with small IQI values.
3.1 Fingerprint range analysis
Applying Eulerâ€™s formula to (2) and (3), we get
Without the loss of generality, the ranges of amplitude and phaseshift imbalances are assumed to be Î¸_{tx}â‰¤Î¸_{m1},Î¸_{rx}â‰¤Î¸_{m2},Î±_{tx}â‰¤Î±_{m1}, and Î±_{rx}â‰¤Î±_{m2}. The ranges of â„œ{g_{1}}, I{g_{1}}, â„œ{g_{2}} and I{g_{2}} can be derived as
where Î±_{min}=(1âˆ’Î±_{m1})(1âˆ’Î±_{m2}), Î±_{max}=(1+Î±_{m1}) (1+Î±_{m2}), and Î¸_{max}=Î¸_{m1}+Î¸_{m2}. To see the individual effect of amplitude imbalance and phaseshift imbalance on g_{1} and g_{2}, we can set Î±_{min}=Î±_{max}=1 to remove the amplitude imbalance, while set Î¸_{max}=0 to remove the phase shift imbalance in (10). It is notable that we consider a=1 in g_{1} and g_{2} in this study. For the cases aâ‰ 1, we can use g_{1}/a and g_{2}/a to replace g_{1} and g_{2} in our analysis.
As reported in [31], the realistic IQI can result in an image rejection ratio down to 2040 dB, which is equivalent to Î¸_{rx/tx}â‰¤14.4212Â°, Î±_{rx/tx}â‰¤0.22. This is consistent with the IQI values used in [13, 19, 32]. Therefore, we reasonably define (5Â°<Î¸_{m1},Î¸_{m2}<14.5Â°,0.05<Î±_{m1},Î±_{m2}<0.22) as the large IQI case, define (0<Î¸_{m1},Î¸_{m2}<5Â°,0<Î±_{m1},Î±_{m2}<0.05) as the small IQI case in our study, and give the numerical ranges of â„œ{g_{1}}, I{g_{1}}, â„œ{g_{2}}, and I{g_{2}} in Table 1.
3.2 SignaltoIQIdistortion ratio analysis
To mathematically represent the relative amount of g_{1} and g_{2}, the signaltoIQIdistortion ratio (defined in [18]) of our system is given by
where Îº=(1+Î±_{tx})(1+Î±_{rx}). The range of Î³ is further derived as
where \(A_{m}=(1+\alpha _{m1})(1+\alpha _{m2})+\frac {1}{(1+\alpha _{m1})(1+\alpha _{m2})}\), \(B_{m}=(1\alpha _{m1})(1\alpha _{m2})+\frac {1}{(1\alpha _{m1})(1\alpha _{m2})}\). The infinity in (12) denotes the IQIfree case.
Proof
See Appendix 1. â–¡
3.3 Challenges of identifying small and rangelimited fingerprints
Substituting the large IQI and small IQI in (12), we obtain 10 log10(Î³)âˆˆ [11.142 dB,âˆž) and [14.854 dB,âˆž), respectively, which implies that g_{2} is always much less than g_{1}. Table 1 and the numerical result of (12) indicate that the ranges of â„œ{g_{1}},I{g_{1}},â„œ{g_{2}}, and I{g_{2}} are extremely limited in small IQI case, which holds that â„œ{g_{1}}â‰ˆ1, and I{g_{1}},â„œ{g_{2}},I{g_{2}}â‰ˆ0 [18â€“20, 33, 34]. Therefore, the accurate detection of the differences between such rangelimited IQI fingerprints g in the presence of estimation error term (i.e., n_{g}) becomes further difficult.
In the following, we propose GLRTbased fingerprint differentiation technique and optimal signal designs to handle this challenging AF relay identification.
4 Generalized likelihood ratio testbased fingerprint differentiation
In this section, GLRT is applied to decide whether a fingerprint belongs to a preregistered AF relay or not.
We assume that \(\hat {\mathbf {g}}_{v}=\mathbf {g}_{v}+\mathbf {n}_{v}\) is one of the preregistered valid fingerprints, where g_{v} denotes the actual value of this IQI fingerprint, and n_{v} denotes the estimation error. Since hardwarelevel IQI fingerprints are stable, the amount of n_{v} can be significantly reduced based on enough samples in the fingerprint registration phase, such as by using equalgain combining, maximalratio combining, or selection combining techniques [16]. The fingerprint comparison between \(\hat {\mathbf {g}}_{\text {LS}}\) and \(\hat {\mathbf {g}}_{v}\) is
where Î”g=gâˆ’g_{v}.
A binary hypothesis testing can be modeled by
where hypothesis \(\mathcal {H}_{0}\) means that the two compared fingerprints belong to the same AF relay, implying that g=g_{v}. Otherwise, hypothesis \(\mathcal {H}_{1}\) is accepted.
According to Appendix 2, the elements of n can be modeled as CSCG variables. Therefore, their linear combination n_{g} is CSCG as well. The mean vector and covariance matrix of g_{off} can be given by
where
where Ïƒ^{2} is the variance of n_{g}, and
In (16), \(\mathbb {E}\left [\mathbf {n}_{g}\mathbf {n}_{v}^{H}\right ]=\mathbb {E}\left [\mathbf {n}_{g}\right ]\mathbb {E}\left [\mathbf {n}_{v}^{H}\right ]=0\) and \(\mathbb {E}\left [\mathbf {n}_{v}\mathbf {n}_{g}^{H}\right ]=\mathbb {E}[\mathbf {n}_{v}]\mathbb {E}\left [\mathbf {n}_{g}^{H}\right ]=0\) since n_{g} and n_{v} are independent variables. The expression of Î£_{v} depends on the specific processing in the fingerprint registration phase. Taking equalgain combining as an example, \(\mathbf {n}_{v}=\frac {1}{Z}\sum _{i}^{Z}\mathbf {n}_{v,i}\), where n_{v,i} denotes the ith estimation error. Similar to n_{g}, n_{v,i} are independent and identically distributed circularly symmetric complex Gaussian variables. Given \(\mathbb {E}\left [\mathbf {n}_{v,i}\mathbf {n}_{v,j}^{H}\right ]=0, i\neq j\), Î£_{v} can be given by
where each \(\boldsymbol {\Sigma }_{v,i}=\mathbb {E}[\mathbf {n}_{v,i}\mathbf {n}_{v,i}^{H}]\). Consequently, we can get g_{off}âˆ¼CN(Î”g,Î£). Given that g_{off} is a zeromean and nonzero mean complex normal vector under \(\mathcal {H}_{0}\) and \(\mathcal {H}_{1}\), respectively, the likelihood functions of g_{off} under \(\mathcal {H}_{0}\) and \(\mathcal {H}_{1}\) can be given by [35]
where Î”gâ‰ 0 in Eq. (21).
Due to the presence of noise in (13), Î”g is not directly available. We apply the GLRT ([36], eq. 6.12) to determine \(\mathcal {H}_{0}/\mathcal {H}_{1}\) as
where Î· is a real positive number. According to (21), the numerator of (22) is maximized when the exponential argument (g_{off}âˆ’Î”g)^{H}Î£^{âˆ’1}(g_{off}âˆ’Î”g) is minimized. Since Î£ and Î£^{âˆ’1} are positive definite matrices, we have \(\mathbf {x}^{H}\boldsymbol {\Sigma }^{1}\mathbf {x}\geqslant 0\) for any x and the equality holds if and only if x=0 ([37], Definition 1(i)). Hence, Î”g=g_{off} maximizes the numerator of (22), and produces the logarithmic GLRT as
If A is larger than the threshold T, hypothesis \(\mathcal {H}_{1}\) is supported. Otherwise, \(\mathcal {H}_{0}\) is supported. We use NeymanPearson lemma [36, Theorem 3.1] to determine the best T with a given false alarm probability by solving the following equation
where \(p_{A}(\cdot \mathcal {H}_{0})\) denotes the probability density function (PDF) of A under \(\mathcal {H}_{0}\), and P_{FA} is the false alarm probability. Appendix 3 shows that the PDFs of A under hypotheses \(\mathcal {H}_{0}\) and \(\mathcal {H}_{1}\) are
where I_{1}(Â·) is the first order modified Bessel function of the first kind ([38], eq. 9.6.19). The parameter Î² in (26) is defined as
where Î»_{1} and Î»_{2} are the eigenvalues of Î£^{âˆ’1}, b_{1} and b_{2} are the elements of b that can be given by
where Q is a 2Ã—2 matrix whose ith column is the eigenvector of Î£^{âˆ’1} corresponding to Î»_{i}, i=1,2, and satisfies QQ^{H}=I. Substituting (25) into (24), P_{FA} can be obtained as
and T for a given false alarm rate can be calculated by solving (29).
Finally, the detection probability for this fingerprint comparison can be derived as
where \(\mathbf {C}= \left [\begin {array}{cc} C_{1}+\epsilon _{1} & C_{2}+\epsilon _{2} \\ C_{3}+\epsilon _{3} & C_{4}+\epsilon _{4} \end {array}\right ] \), Q_{m}(Â·,Â·) represents the mth order Marcum Qfunction ([39], eq. 4.59), and
where \(\gamma _{1}=\frac {\sum _{i=1}^{N}s_{i}^{2}}{\sigma ^{2}}\), \(\gamma _{2}=\frac {\sum _{i=1}^{N}s_{i}^{2}}{\sigma ^{2}}\). Based on our fingerprint registration process (see Algorithm 1), \(\boldsymbol {\Sigma }_{v}= \left [\begin {array}{cc} \epsilon _{4} & \epsilon _{2}\\ \epsilon _{3} & \epsilon _{1} \end {array}\right ] \) and Îµ_{1}=Îµ_{4}, \(\epsilon _{2}=\epsilon _{3}^{*}\).
In most practical cases, it can be observed from (19) that the effect of Î£_{v} in Î£ can nearly vanish when large Z is used, which means Îµ_{1},Îµ_{2},Îµ_{3},Îµ_{4}â‰ˆ0. Hence, P_{D} can be further simplified as
where \(\mathbf {D}= \left [\begin {array}{cc} h_{\text {SR}}h_{\text {RD}}^{2}\gamma _{1} & h_{\text {SR}}^{*2}h_{\text {RD}}^{2}\gamma ^{*}_{2} \\ h_{\text {SR}}^{2}h_{\text {RD}}^{2}\gamma _{2} & h_{\text {SR}}h_{\text {RD}}^{2}\gamma _{1} \\ \end {array}\right ]\). By using (32) instead of (30), we are able to quickly calculate the instantaneous detection probability for every fingerprint comparison, which may enable the timely adjustment of T in (23) for ensuring the desired detection performance.
In summary, using the GLRT presented in (22) and (23), two fingerprints can be differentiated with the detection probability derived in (30) given a given false alarm rate. In practical implementation, the detection probability can be quickly computed using (32) instead of (30).
5 Whitelistbased AF relay identification algorithm
Based on our above derived results, this section presents the AF relay identification algorithm consisting of the fingerprint registration, update, and identification.
5.1 Fingerprint registration
We consider that a new relay, R_{AF}, joins in the network. The fingerprint of R_{AF} (i.e., \(\hat {\mathbf {g}}_{v,{U}}\)) needs to be extracted and stored in the whitelist.
As shown in Algorithm 1, the average of the Z estimated IQI fingerprint is used to improve the accuracy of the registered fingerprint. Then, \(\hat {\mathbf {g}}_{v,{U}}\) is registered as the Uth valid fingerprint in the whitelist.
5.2 Fingerprint identification and update
If this R_{AF} is not claimed as a newly joined relay, the fingerprint identification algorithm should be executed to verify its identity. In some practical applications, such as an authentication application, R_{AF} may actively claim its identity, which can be indexed to one of the valid fingerprints in the whitelist. In this case, we only need to compare the estimated fingerprint of R_{AF} (i.e., \(\hat {\mathbf {g}}_{\text {LS}}\)) with the claimed fingerprint. Without the loss of generality, we assume that R_{AF} does not claim its identity. As presented in Algorithm 2, the proposed GLRTbased differentiation is performed to compare R_{AF}â€™s \(\hat {\mathbf {g}}_{\text {LS}}\) with all U prerecorded G_{v} onebyone, which leads to three possible cases: Case #1 (l=0): R_{AF} is not identified as any candidate in the whitelist, implying all U GLRT claim \(\mathcal {H}_{1}\). Case #2 (l=1): R_{AF} is identified as one candidate, implying that only one GLRT claims \(\mathcal {H}_{0}\). Case #3 (l>1): There are a number of l candidates that have the similar fingerprints to R_{AF}.
To handle Case #3, we use the maximum likelihood (ML) technique and choose the relay with its A_{j} closest to 1 as the final decision, since that A=1 can lead to the ML in (25). If R_{AF} is not identified, alarm is set to 1 to give an alarm. Otherwise, the R_{AF}â€™s identity is obtained in ID.
In addition, although stable in the time scales of hours and days, the hardware fingerprint may slowly change over time, and thus, it may have the chance to affect the identification performance. In fact, it is practically impossible to model and predict the random minor changes of our fingerprint in a short time duration. As reported in [24], it may take up to 30 days to observe the distinguishable hardware changes, which is usually unnecessary to be taken into account in our short time identification. In our case, it is more efficient to consider the fingerprint update by taking the advantage of every signal reception since our IQI fingerprint can be extracted from any relayed signals. As shown in line 22 of Algorithm 2, after identifying R_{AF}, Algorithm 1 can be optionally used to update the R_{AF}â€™s fingerprint in the whitelist. Moreover, the fingerprint registration can be periodically executed if needed, for example, after receiving a certain number of signals.
6 Optimal signal design for maximizing identification performance
Unlike the easily programmable upperlayer identities (e.g., medium access control address) or fast varying channelbased characteristics, IQI is fixed on the hardware level for a relatively long period. Therefore, once the presence of a relayâ€™s IQI fingerprint is detected, this fingerprint can be used as preknowledge associated with this relay for a period. This enables us to design the best signal for intentionally detecting this relay again with a maximized detection probability in the future, which is useful in many practical applications. Taking authentication as an example, we assume that a malicious AF relay R^{â€²} is detected by an authentication, and its fingerprint is thereby stored. In practice, this R^{â€²} can repeatedly attempt to impersonate other legitimate relays through changing its upperlayer identity. Given the stability of IQI, the already stored fingerprint of R^{â€²} can be used as known in Eq. (13) to obtain g_{off} and the preknowledge of Î”g can be obtained as well according to our previous GLRT analysis. With this preknowledge, we are able to derive the optimal s_{i} to intentionally maximize the detection probability for the future authentication of R^{â€²}.
We first assume that the Î”g associated with an intended AF relay is known. In (30), the two arguments of P_{D} are Î² and T, where the value of T is to hold a fixed false alarm, thus implying P_{D} can only be maximized by adjusting Î² in practice. According to [40], P_{D} is a strictly increasing function with respect to Î². Thus, maximizing P_{D} is equivalent to maximizing Î²=2Î”g^{H}(Î£_{g}+Î£_{v})^{âˆ’1}Î”g>0 through adjusting s_{i}. However, the mathematical expression of Î£_{v} is unavailable in general as it is only dependent on the specific sample processing method of the past fingerprint registration phase. Hence, the adjustment of s_{i} cannot affect Î£_{v}. In practice, Î£_{v} always approaches a null matrix with the proper diversitycombining techniques and enough samples, as discussed in Eq. (32). Also, using Lemma 1 below, we get \(\Delta \mathbf {g}^{H}\boldsymbol {\Sigma }_{g}^{1}\Delta \mathbf {g}\geqslant \Delta \mathbf {g}^{H}(\boldsymbol {\Sigma }_{g}+\boldsymbol {\Sigma }_{v})^{1}\Delta \mathbf {g}\), where the equality holds if and only if Î£_{v}=0. Therefore, we practically consider Eq. (32) with Î£_{v}=0 in the maximization of Î².
Lemma 1
Given two NÃ—N positive definite Hermitian matrices U and V, and a NÃ—1 complex vector x, the following inequality holds
Proof
See Appendix 4. â–¡
Using (16), (27), (28), and setting Î›=diag(Î»_{1},Î»_{2}), we can get
where h_{1}=h_{SR}h_{RD} and \(h_{2}=h_{\text {SR}}^{*}h_{\text {RD}}\). It can be observed that maximizing Î² is also equivalent to maximizing J for a given Ïƒ^{2}. Since s_{i} is subject to the specific constellation patterns, QAM and PSK as two basic modulation schemes are considered in deriving the optimal s_{i}.
6.1 QAM modulation case
In QAM modulation, the signal can be represented as s_{i}=a_{i}+jb_{i}. Let l=h_{1}Î”g_{1}^{2}+h_{2}Î”g_{2}^{2}, \(c=\Re \left \{h_{2}^{*}\Delta g_{2}^{*}h_{1}\Delta g_{1}\right \}\), and \(d=\Im \left \{h_{2}^{*}\Delta g_{2}^{*}h_{1}\Delta g_{1}\right \}\), then J can be expressed as
where J_{i} is defined as
Since s_{i} are independent, we are able to separately design s_{i} to maximize J_{i}, which leads to the maximal J. The rule of optimal signal design under QAM modulations is summarized in Proposition 1.
Proposition 1
If d<0, the optimal s_{i} is the point at the angles of the rectangular constellation in the first and third quadrants of the I/Q coordinate plane.
If d>0, the optimal s_{i} is the point at the angles of the rectangular constellation in the second and fourth quadrants of the I/Q coordinate plane.
If d=0, the optimal s_{i} is the point at any angle of the rectangular constellation of the I/Q coordinate plane.
Proof
See Appendix 5. â–¡
In our optimal design, the value of the optimal symbol only has two options when d>0 or d<0. This may occasionally result in a singular matrix \(\mathbf {H}_{s}^{H} \mathbf {H}_{s}\) that makes the matrix inversion difficult, e.g., the calculation of Î£^{âˆ’1}. For practical implementation purpose, we give a more robust suboptimal signal design as follows:
If d>0 or d<0, the suboptimal signals are the two constellation points that are the closest to the optimal signal.
Figure 3a shows the optimal symbols (red) and suboptimal symbols (green) in a 64QAM constellation. It is noteworthy that the optimal and suboptimal symbols are not required to be located in the same quadrant. For instance, if d<0, an optimal symbol has been chosen from the first quadrant, while the suboptimal symbol can be either in the first or the third quadrant.
6.2 PSK modulation case
For a general MPSK modulation, the constellation diagram is a circle such that the signal can be represented by \(s_{i}=A_{s}e^{j\theta _{i}}\phantom {\dot {i}\!}\), where A_{s} denotes the constant signal amplitude and Î¸_{i} denotes the phaseshift. The rule of optimal signal design for PSK is proposed in Proposition 2.
Proposition 2
The optimal s_{i} should satisfy \(\theta _{i}=\frac {\phi }{2}\) or \(\theta _{i}=\frac {\phi }{2}+\pi \), where
Proof
See Appendix 6. â–¡
Since Ï• can be any degree, whereas the values of Î¸_{i} are subject to the specific PSK constellation diagram, we need to choose the MPSK symbol having the closest angle to Î¸_{i} as the optimal s_{i} in practice.
Similarly, we also propose a suboptimal design to solve the problem of a singular matrix, \(\mathbf {H}_{s}^{H} \mathbf {H}_{s}\). Since the degree of derived âˆ’Ï• is twice that of Î¸_{i}, we first create a \(\frac {M}{2}\)PSK constellation. As shown in Fig. 3b, P_{k} denotes the square of the kth constellation point of MPSK, i.e., \(P_{k}=s_{k}^{2}=A_{s}^{2}e^{j2\theta _{k}}\). As shown in this newly defined \(\frac {M}{2}\)PSK constellation, the phase shift between two neighboring points (e.g., P_{k} and P_{kâˆ’1}) is \(\frac {4\pi }{M}\). The phase shift for areas A_{1}, A_{2}, A_{3}, and A_{4} are \(\frac {\pi }{M}\), and the phase shift for A_{5} and A_{6} are \(\frac {2\pi }{M}\). Without the loss of generality, it is assumed that âˆ’Ï• is closer to P_{k} than P_{kâˆ’1} and P_{k+1}, which means that it falls in one of the sectors of A_{1}, A_{2} (red sectors), A_{3}, or A_{4} (gray sectors). Herein, we take a combination of two signals (s_{i},s_{i+1}) into account to approach the desired phase âˆ’Ï• and accordingly present the rules of suboptimal approach as follows:

1
If âˆ’Ï• falls in A_{1} or A_{2}, we set s_{i}=s_{k+n},s_{i+1}=s_{kâˆ’n},n=0,1,2â‹¯.

2
If âˆ’Ï• falls in A_{3}, we set s_{i}=s_{k+n},s_{i+1}=s_{k+1âˆ’n}.

3
If âˆ’Ï• falls in A_{4}, we set s_{i}=s_{k+n},s_{i+1}=s_{kâˆ’1âˆ’n}.
In practical implementation, Proposition 1/2 can be first considered to design the optimal QAM/PSK signals. Then, a portion of them can be flexibly replaced by applying the suboptimal approaches.
7 Numerical results
In this section, numerical results are presented to validate our derivations and evaluate the performance of the proposed IQIbased AF relay identification method.
We consider a=1, Z=500, 16QAM, and 16PSK modulations. Rayleigh fading model is used to generate h_{SR}âˆ¼CN(0,2) and h_{RD}âˆ¼CN(0,2), unless otherwise noted. SNR=\(10\log _{10}(\frac {P}{\sigma ^{2}})\) is defined as used in [14, 41, 42], where P=10. To evaluate the identification performance under the challenging small IQI condition, the amplitude imbalance and phaseshift imbalance are randomly chosen within [âˆ’â€‰0.05,0.05] and [âˆ’â€‰5Â°,5Â°]. As an example of the contrast, to achieve satisfactory identification performance, the method proposed in [13] requires large IQI values [âˆ’â€‰0.3,0.3] and [âˆ’â€‰15Â°,15Â°], which is rarely the case in practice. All simulation results are based on the average of 10^{5} independent realizations of our system.
7.1 Numerical results for IQI device fingerprint
The dynamic range of the IQI device fingerprints and the parameter Î³ derived in Section 3 are first evaluated. Figure 4 shows the real and imaginary values of g_{1} and g_{2} vs. Î±, where Î±=(1+Î±_{rx})(1+Î±_{tx}). The Rx and Tx IQIs are set as Î¸_{rx}=5Â°, Î¸_{tx}=1Â°, and Î± varies between 0.81 and 1.21. The curves between two vertical lines confirm that â„œ{g_{1}}â‰ˆ1, and I{g_{1}},â„œ{g_{2}},I{g_{2}}â‰ˆ0 under the small IQI values since sin(Î¸_{tx}âˆ’Î¸_{rx}), sin(Î¸_{tx}+Î¸_{rx}), and 1âˆ’(1+Î±_{tx})(1+Î±_{rx}) cos(Î¸_{tx}+Î¸_{rx}) approximate zero in (8) and (9). In addition, these 4 simulated parametersâ€™ ranges also validate our analytical results derived in (10).
Figure 5 depicts 10 log10Î³ in terms of Î±. The figure shows that the peak values of all four simulated 10 log10Î³ appear at Î±=1, which is the case of the absence of phaseshift imbalance. It is shown that the black curve (Î¸_{rx}=Î¸_{tx}=0Â°) can go to infinity when Î±=1, thus confirming the result of the infinity case in Eq. (12). We then substitute Î±=0.64âˆ’1.44 in (12) to calculate the lower limit of 10 log10Î³. As expected, the calculated analytical results (11.9032 dB, 12.5637 dB, 12.8595 dB, and 13.1708 dB) can match the simulated results.
The above numerical results confirm our IQI fingerprint derivations in Eq. (8)â€“(12).
7.2 Performance evaluation of proposed AF relay identification
In Fig. 6, the detection probability P_{D} in terms of the false alarm probability P_{FA} is shown. It can be seen that the analytical P_{D} calculated by Eq. (30) is well confirmed by the simulation results. Threshold T, which is calculated by solving (29), decreases with P_{FA} varying from 10^{âˆ’3} to 1.
In Fig. 7, P_{D} for different instantaneous channel gains h_{SR} and h_{RD} is evaluated. We set a fixed T for holding P_{FA}=5%, and randomly choose a fixed fingerprint offset Î”g=[0.0306+j0.0669 âˆ’0.0186âˆ’j0.0712]^{T}, fixed s and noise variance Ïƒ^{2}=0.1 to ensure constant Î³_{1} and Î³_{2}. We consider h_{SR}âˆ¼CN(0,w_{SR}) and h_{RD}âˆ¼CN(0,w_{RD}) to respectively generate h_{SR} and h_{RD}, where w_{RD}=ww_{SR}, and show P_{D} vs w_{SR} with different w. The fingerprint estimation can become more precise under good channel condition, thus improving the accuracy of the fingerprint differentiation. As expected, P_{D} increases with the growth of w_{SR} and w_{RD}.
In Fig. 8, P_{D} using the optimal signal described in Propositions 1 and 2 is compared with the suboptimal and nonoptimal signals. We substitute the optimal signal derived by Propositions 1 and 2 into (30) to obtain the analytically maximal P_{D}, and call it the upper bound. The nonoptimal signals are randomly chosen symbols. The results show that our suboptimal methods have a detection probability loss of only 2.59% and 1.68% on average in QAM and PSK, respectively, compared to the optimal methods. On the other hand, our suboptimal methods also show on average a 34.54% and 23.43% higher P_{D} than the nonoptimal cases. Therefore, the suboptimal solution is capable of significantly improving the identification performance in the small IQI condition, and furthermore it can be more robust than the optimal solution.
At last, the performance of the identification algorithm is evaluated. Specifically, the algorithm can either choose the correct identity from the four preregistered candidates for the relay under test or give a timely alarm if the relay is not identified. The aforementioned DT [13] and VF methods [17] are simulated and compared. The correct identification rate (CIR) is defined as the ratio of the number of correct identity claims and correct alarms to the total number of identification attempts. Figure 9 shows the CIR performance vs. SNR. Since the estimation error n_{g} is reducing with the increase of SNR, the detection accuracy and CIR go higher. It can also be seen that our method is superior to the VF and DT schemes in all simulated cases. Compared to VF, our fingerprint removes most effects of the channel variation, and thus, the uniqueness and stability of our fingerprint are improved. Regarding DT, the distance between different fingerprints g is usually too short to differentiate due to the limited range of IQI, as discussed in Section 3. Meanwhile, in our method, the proposed GLRTbased differentiation and optimal signal designs can significantly improve the accuracy in differentiating minor Î”g. Additionally, our identification can achieve better performance but use fewer training symbols. For instance, it can be seen that the CIR of our method with N=32 and P_{FA}=1% is even higher than VF with N=512 and P_{FA}=1%, which also implies lower Tx power consumption by using our method.
8 Conclusion
A physicallayer AF relay identification scheme was proposed to accurately identify relays based on the examination of their unique IQIs. Since IQI estimation and compensation are usually mandatory in most present wireless receivers, we direct use the LS estimation results to generate the IQI fingerprint of an AF relay node. Comprehensive analyses including the features of this IQI fingerprint, the probabilities of detection and false alarm in differentiating between two relays with small IQI values, and whitelistbased relay identification algorithms were presented. In addition, we proposed optimal and suboptimal training signal designs for the purpose of maximizing the detection probability. The simulation results validated our analytical results and showed that the proposed identification method could accurately identify AF relays even under the small IQI condition.
9 Appendix 1: Proof for the range of Î³ in Eq. (12)
From (11), it is found that the ideal IQIfree condition, i.e., Î¸_{m1}=Î¸_{m2}=0,Î±_{m1}=Î±_{m2}=0, can result in a zero denominator and a real positive numerator. In this case, Î³ is positive infinity.
We then summarize the derivations for the minimum value of Î³ as an optimization problem, which is given by
Using the constraints in (37b), the range of Î± can be determined as
Using the addition and subtraction theorems of trigonometric functions, (37a) can be simplified as
Note that both the numerator and denominator are divided by Î± cosÎ¸_{tx} cosÎ¸_{rx} in deriving (39). This is reasonable as the practical amplitude mismatch and phase mismatch are small enough to ensure Î±>0 and cosÎ¸_{tx} cosÎ¸_{rx}>0. According to the inequality of arithmetic and geometric means [43], we can obtain
where \(A_{m}=(1+\alpha _{m1})(1+\alpha _{m2})+\frac {1}{(1+\alpha _{m1})(1+\alpha _{m2})}\), \(B_{m}=(1\alpha _{m1})(1\alpha _{m2})+\frac {1}{(1\alpha _{m1})(1\alpha _{m2})}\). It is notable that the monotonicity of \(\alpha +\frac {1}{\alpha }\) is considered in (40) to find its upper bound. Further, we can also refer to the monotonicity of cosine and tangent, and get
Based on (40)(42), the lower bound of (39) can be calculated as
Consequently, the range of Î³ can be obtained as shown in (12).
10 Appendix 2: Analysis for the PDF of n
We first define n=n^{â€²}+n_{RD}, where the kth element of n^{â€²} is \(n^{\prime }_{k}=g_{1}h_{\text {RD}} n_{\text {SR}k}+g_{2}h_{\text {RD}} n_{\text {SR}k}^{*}\). Generally, n^{â€²} is an improper complex Gaussian vector as it depends on both of n_{SR} and \(\mathbf {n}^{*}_{\text {SR}}\), and the PDF of n^{â€²} is [44, 45]
where n_{t}=[n^{â€²}^{T} n^{â€²}^{H}]^{T}, and R is a covariance matrix of n_{t} given by \(\mathbf {R}=\mathbb {E}\left [\mathbf {n}_{t}\mathbf {n}_{t}^{H}\right ]\).
However, in the case of IQI caused \(n^{\prime }_{k}\), the conjugate term \(g_{2}h_{\text {RD}} n_{\text {SR}k}^{*}\) nearly vanishes since g_{2}â‰ˆ0. Substituting â„œ{g_{1}}â‰ˆ1, and I{g_{1}},â„œ{g_{2}},I{g_{2}}â‰ˆ0 [18â€“20, 33, 34] in \(n^{\prime }_{k}\), we can get \(\Re \left \{n'_{k}\right \}, \Im \left \{n'_{k}\right \} \sim N\left (0,\left (a_{1}^{2}+a_{2}^{2}\right)\frac {\sigma _{1}^{2}}{2}\right)\), where a_{1}=â„œ{g_{1}h_{RD}},a_{2}=I{g_{1}h_{RD}}. Also, since \(\mathbb {E}[\Re \{n_{\text {SR}k}\}\Im \{n_{\text {SR}k}\}]=\mathbb {E}[\Re \{n_{\text {SR}k}\}]\mathbb {E}[\Im \{n_{\text {SR}k}\}]=0\) and \(\mathbb {E}[\Re \{n_{\text {SR}k}\}^{2}]=\mathbb {E}[\Im \{n_{\text {SR}k}\}^{2}]=\frac {\sigma _{1}^{2}}{2}\), we can get \(\mathbb {E}[\Re \{n'\}\Im \{n'\}]\approx a_{1}a_{2}\left (\mathbb {E}\left [\Re \{n_{\text {SR}k}\}^{2}\right ]\mathbb {E}\left [\Im \{n_{\text {SR}k}\}^{2}\right ]\right)=0\). Given that n_{RDk} is circularly symmetric complex Gaussian random variable, n_{k}=nkâ€²+n_{RDk} can be modeled as a circularly symmetric complex Gaussian variable since its real and imaginary parts are independent and identically distributed Gaussian.
11 Appendix 3: Derivations for PDFs in Eqs. (25) and (26)
This appendix derives the PDFs of A under \(\mathcal {H}_{0}\) in (25) and \(\mathcal {H}_{1}\) in (26).
The eigendecomposition of matrix Î£^{âˆ’1} is
where Î›=diag(Î»_{1},Î»_{2}) with its elements Î»_{i}>0,i=1,2 denoting the ith eigenvalue of Î£^{âˆ’1}; Q is a 2Ã—2 matrix whose ith column is the eigenvector of Î£^{âˆ’1} corresponding to Î»_{i}. Substituting (45) in metric A, we can obtain
where d=Qg_{off}=[d_{1},d_{2}]^{T}. Given (13) (16) (28) and the orthogonal matrix Q, the covariance matrix of d can be computed as
Since cov(d_{1},d_{2})=cov(d_{2},d_{1})=0, d_{1}, d_{2} are two independent random variables. Also, we can obtain \(d_{1}\sim CN\left (b_{1}, \frac {1}{\lambda _{1}}\right), d_{2}\sim CN\left (b_{2}, \frac {1}{\lambda _{2}}\right)\), where b=QÎ”g as defined in (28). Since b_{i}=0 under \(\mathcal {H}_{0}\) and b_{i}â‰ 0 under \(\mathcal {H}_{1}\), A follows scaled central/noncentral chisquared distributions under \(\mathcal {H}_{0}\) and \(\mathcal {H}_{1}\), respectively. Using (46), we define K=2A as
where k_{1},k_{2},k_{3} and k_{4} are normalized, and defined by \(k_{1}=\sqrt {2\lambda _{1}}\Re \{d_{1}\}, k_{2}=\sqrt {2\lambda _{1}}\Im \{d_{1}\}, k_{3}=\sqrt {2\lambda _{2}}\Re \{d_{2}\}\), \(k_{4}=\sqrt {2\lambda _{2}}\Im \{d_{2}\}\).
In this case, the PDF of K under \(\mathcal {H}_{0}\) is a central chisquared PDF with 4 degrees of freedom as [38]
While under \(\mathcal {H}_{1}\), K follows the standard noncentral chisquared distribution with 4 degrees of freedom and noncentrality Î²_{1} and its PDF can be given by
where the noncentrality is defined as
It can be seen that Î²_{1}=Î², where Î² is defined in (27).
Finally, we are able to obtain the PDF of A under \(\mathcal {H}_{0}\) and \(\mathcal {H}_{1}\) by first replacing x with 2x in (49) and (50) and then multiplying the two PDFs by 2, which produces (25) and (26), respectively.
12 Appendix 4: Proof for Lemma 1
Since U and V are positive definite Hermitian matrices, U and V are invertible, and (U+V) is invertible Hermitian matrix as well. Using Woodbury matrix identity, we get
and thus
Since U^{âˆ’1}(U^{âˆ’1}+V^{âˆ’1})^{âˆ’1}U^{âˆ’1} is positive definite, we can obtain x^{H}U^{âˆ’1}(U^{âˆ’1}+V^{âˆ’1})^{âˆ’1}U^{âˆ’1}x>0. Therefore, x^{H}U^{âˆ’1}x>x^{H}(U+V)^{âˆ’1}x.
13 Appendix 5: Proof for Proposition 1
According to (35), we can do some manipulations and express J_{i} as a summation of two quadratic components, which is given by
We set h_{1}Î”g_{1}=c_{1}+jd_{1} and h_{2}Î”g_{2}=c_{2}+jd_{2}. Given that l=h_{1}Î”g_{1}^{2}+h_{2}Î”g_{2}^{2}, \(c=\Re \{h_{2}^{*}\Delta g_{2}^{*}h_{1}\Delta g_{1}\}\), and \(d=\Im \left \{h_{2}^{*}\Delta g_{2}^{*}h_{1}\Delta g_{1}\right \}\), the l,c and d can be expressed as \(l=c_{1}^{2}+c_{2}^{2}+d_{1}^{2}+d_{2}^{2}, c=c_{1}c_{2}+d_{1}d_{2}\) and d=d_{1}c_{2}âˆ’d_{2}c_{1}.
Hence, the representation of coefficient l+2c is
and the second coefficient is
Eqs. (55) and (56) show that the coefficients of the two quadratic components are positive. Therefore, the maximum value of J_{i} can be achieved only if the maximum values of \(\left (a_{i}\frac {2d}{l+2c}b_{i}\right)^{2}\) and \(b_{i}^{2}\) can be simultaneously achieved. Given that l+2c>0, it can be seen that the values of \(\left (a_{i}\frac {2d}{l+2c}b_{i}\right)^{2}\) and \(b_{i}^{2}\) depend upon the sign of d and the maximum modulus of a_{i}, b_{i}. Without loss of generality, we assume the maximum values of a_{i} and b_{i} are a_{max}>0 and b_{max}>0, respectively, under the current QAM modulation. For the square constellation case, a_{max}=b_{max}.
If d=0, (54) reduces to
In this case, J_{i} can be maximized when a_{i}=a_{max} and b_{i}=b_{max}, which implies the four angles of the constellation diagram.
If d<0, it results in \(\frac {2d}{l+2c}>0\). Thus, the quadratic components \(\left (a_{i}\frac {2d}{l+2c}b_{i}\right)^{2}\) and \(b_{i}^{2}\) can be maximized when a_{i}=a_{max},b_{i}=b_{max} or a_{i}=âˆ’a_{max},b_{i}=âˆ’b_{max}, which corresponds to the angles of constellation diagram in the first and third quadrants.
If d>0, then \(\frac {2d}{l+2c}<0\). In this case, a_{i} and b_{i} are required to have opposite signs and satisfy a_{i}=a_{max}, b_{i}=âˆ’b_{max} or a_{i}=âˆ’a_{max},b_{i}=b_{max}, which corresponds to the angles of constellation diagram in the second and fourth quadrants.
14 Appendix 6: Proof for Proposition 2
As per Eq. (33), it can be seen that \(\sum _{i=1}^{N} s_{i}^{2}\) in the first item \(\left (h_{1}\Delta g_{1}^{2}+h_{2}\Delta g_{2}^{2}\right)\sum _{i=1}^{N} s_{i}^{2}\) is fixed in PSK and thereby it is not adjustable. Hence, the maximization of J depends upon whether or not the second item \(2R=2\Re \left \{h_{2}^{*}\Delta g_{2}^{*}h_{1}\Delta g_{1}\sum _{i=1}^{N} s_{i}^{2}\right \}\) can be maximized through adjusting s_{i}. First, we apply Eulerâ€™s formula and set
where Ï• is defined as
According to (58) and \(s_{i}=A_{s}e^{j\theta _{i}}\phantom {\dot {i}\!}\), we can obtain
Since Î¸_{i} is independent, we can separately adjust Î¸_{i} to maximize all R_{i}, where R_{i} is defined as
As a result, R can be maximized because \(R=\sum _{i=1}^{N}R_{i}\). Considering the (âˆ’Ï€,Ï€] range limit of Î¸_{i} and Ï•, R_{i} can achieve its maximum value when \(\theta _{i}=\frac {\phi }{2}\) or \(\theta _{i}=\frac {\phi }{2}+\pi \).
Abbreviations
 AF:

Amplifyandforward
 CSCG:

Circularly symmetric complex Gaussian
 DT:

Distance test
 GLRT:

Generalized likelihood ratio test
 IQI:

Inphase and quadraturephase imbalance
 ML:

Maximum likelihood
 PDF:

Probability density function
 PSK:

Phaseshift keying
 QAM:

Quadrature amplitude modulation
 Rx:

Receiving
 SNR:

Signaltonoise ratio
 Tx:

Transmitting
 VF:

Varying fingerprint
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Peng Hao (phao5@uwo.ca), Xianbin Wang (xianbin.wang@uwo.ca), and Aydin Behnad (abehnad@uwo.ca) are with the Department of Electrical and Computer Engineering, Western University, N6A 5B9 London, Ontario, CA.
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Hao, P., Wang, X. & Behnad, A. Amplifyandforward relay identification using joint Tx/Rx I/Q imbalancebased device fingerprinting. J Wireless Com Network 2019, 64 (2019). https://doi.org/10.1186/s1363801913695
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DOI: https://doi.org/10.1186/s1363801913695