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IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1

A Blind Likelihood-Based Approach for OFDMSpectrum Sensing in the Presence of I/Q Imbalance

Ahmed ElSamadouny, Ahmad Gomaa, and Naofal Al-Dhahir, Fellow, IEEE

Abstract—We investigate the spectrum sensing problem inorthogonal frequency-division multiplexing-based cognitive ra-dio networks under I/Q imbalance. We start by deriving thelikelihood ratio test in the presence of I/Q imbalance at theanalog front-ends of both the primary and secondary users.In addition, we derive closed-form expressions for the proba-bilities of detection and false alarm and the receiver operatingcharacteristics and examine their dependence on both transmitand receive I/Q imbalance levels. Furthermore, we compare theperformance of the likelihood ratio test with that of the energydetector and demonstrate the superiority of the former over thelatter. Next, we generalize our analysis to the blind case wherewe derive simple closed-form expressions for the generalizedlikelihood ratio test and its false alarm probability as a functionof the received signal only, i.e. without requiring any knowledgeof the primary-to-secondary channel response, noise statistics,or I/Q imbalance parameters. Our results demonstrate that thecorrelation properties of the primary user’s signal induced bytransmit I/Q imbalance are signal features that can be exploitedin a blind fashion at the secondary user to enhance the detectionprobability significantly compared to the conventional energydetector.

Index Terms—I/Q imbalance, Spectrum sensing, LRT, GLRT,OFDM.

I. INTRODUCTION

COGNITIVE radio (CR) technology is an attractive solu-tion to the spectrum under-utilization problem inherent

in static spectrum allocations [1]. A CR secondary user(SU) attempts to utilize the free resources unoccupied bythe primary user (PU) through cognitive techniques that aimat protecting the PU network from service interruption andinterference. Reliable spectrum sensing [2], [3] is a key CRfunctionality for efficient spectrum utilization where spectrumholes are identified before SUs can start their transmissions.

Orthogonal Frequency Division Multiplexing (OFDM) iswidely adopted as the transmission scheme of choice foralmost all broadband wireless standards (including WLAN,WiMAX, LTE, DVB, etc.) required to provide higher ratesand better reliability at low cost and power consumption.Therefore, reliable spectrum sensing of OFDM signals isan important practical problem and is the subject of thispaper. However, unlike most of the previous literature on

Manuscript received July 12, 2013; revised October 29, 2013. The editorcoordinating the review of this paper and approving it for publication was H.Arslan.

This paper was made possible by NPRP grant # NPRP 09−062−2−035from the Qatar National Research Fund (a member of Qatar Foundation). Thestatements made herein are solely the responsibility of the authors. This paperwas presented in part at GLOBECOM 2012 and GLOBECOM 2013.

The authors are with The University of Texas at Dallas, USA (e-mail:[email protected], aarg [email protected], [email protected]).

Digital Object Identifier 10.1109/TCOMM.2014.030814.130536

this subject, we investigate this problem in the presence ofa key performance-limiting analog front-end (AFE) circuitimperfection; namely Inphase/Quadrature imbalance (IQI) [4],[5].

IQI refers to amplitude and phase mismatches between thein-phase (I) and quadrature (Q) branches of a transceiver.Ideally, the I and Q branches of the mixer should have equalamplitude and 90o phase shift. However, in practice, AFEcomponents (such as mixers, phase shifters, analog filters,and amplifiers) suffer from imperfections due to fabricationprocess variations which are difficult to control, increase withfabrication technology area down-scaling, and cannot be effi-ciently compensated in the analog domain due to power-area-delay tradeoffs. Understanding the IQI impact on performancehas received increased attention recently [6] due to growingindustry trends to use direct-conversion transceivers instead ofconventional super-heterodyne transceivers. Although direct-conversion transceivers enable low-power and low-cost CMOSimplementations, they are more sensitive to I/Q imbalanceeffects [7].

OFDM performance degradation in the presence of receive(Rx) IQI at the SU’s AFE was investigated in early pub-lications such as [8], [9]. More recently, OFDM spectrumsensing performance in the presence of Rx IQI was studied in[10] and [11] where the authors showed by simulations thatRx IQI at the SU’s AFE does not impact the performanceof the energy and cyclo-stationary detectors in OFDM spec-trum sensing. The effect of IQI on OFDMA and multi-bandspectrum sensing was studied in [12], [13]. The authors in[12] considered an OFDMA system where the SU is able tosense a specific subcarrier and decide among three hypotheses;namely, whether the PU is present, interference from theimage subcarrier affects the sensed subcarrier due to IQI, orthere is only noise. This work considers an OFDM systemwhere the sensing decision is made over the whole OFDMsymbol. In addition, our approach does not assume perfectknowledge of the system parameters like [12]. The authors in[13] proposed a sample based multichannel energy detectorusing an energy correction method to compensate for IQIperformance degradation. However, our approach exploits theIQI in the primary OFDM signal to enhance its detectionperformance at the secondary user.

Unlike previous work, in this paper, we derive the optimumlikelihood-based spectrum sensing rule (also known as theNeyman-Pearson test [14]) for OFDM PU signals in thepresence of IQI. Spectrum sensing performance is measuredby the probability of correct detection (PD), which is theprobability that the SU correctly detects the PU signal, andthe probability of false alarm (PFA), which is the probability

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2 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION

that the SU erroneously decides that the PU is present. Weinvestigate the impact of IQI on PD for a given PFA [14]. TheReceiver Operating Characteristics (ROC) curve, at a givensignal to noise ratio (SNR), summarizes the performance ofthe likelihood-based detector where each point on the ROCcurve corresponds to an optimal choice of (PD, PFA) for agiven detection threshold.

The likelihood ratio test (LRT) is the optimal detectionalgorithm for spectrum sensing [14] assuming perfect knowl-edge of the system parameters at the receiver. The LRTdecision statistic for OFDM spectrum sensing in the presenceof IQI was derived in the frequency domain in [15]. Althoughthe LRT detector appears to be unrealistic (since systemparameters knowledge is generally difficult to obtain at the SUreceiver), we introduce it in the first part of the paper for twomain reasons. First, the LRT detector is optimal and, hence,serves as a performance benchmark (upper-bound). Second,we use the optimal LRT detector to convey the key messageof this paper which is that the presence of IQI at the PUimproves its detection probability at the SU. Hence, a reliabledetection algorithm should extract and use key features of thereceived PU signals to perform reliable spectrum sensing. Forexample, the main features exploited in [16] were the cyclic-prefix (CP) structure of an OFDM signal and the multi-pathfading nature of the channel. The key feature (fingerprint) ofour model comes from the PU transmitter IQI which createscorrelation between each OFDM subcarrier and its image [6]that does not exist in the noise-only hypothesis. In CR settings,it is generally infeasible for the SU to estimate the unknownsystem parameters using pilots transmitted from the PU due tothe absence of legitimate communication between the licensedPU and the cognitive SU. Furthermore, pilots result in athroughput loss which motivates this paper on blind spectrumsensing using the Generalized Likelihood-Ratio Test (GLRT)[14].

Blind spectrum sensing using the GLRT approach was stud-ied in several papers including [16], [17]. The CP, multi-pathnature, and non-white spectrum are the main features in thereceived PU signal which were exploited in [16], [17] to deriveefficient PU detection algorithms. The fundamental differencebetween this work and [16], [17] is that the assumption of acircularly-symmetric complex Gaussian received signal is nolonger valid for our problem because the received PU signalin the presence of IQI is a linear combination of the IQI-free signal and its complex conjugate [6]. Hence, the receivedsignal from the PU will have an improper (i.e. circularly-asymmetric) complex Gaussian distribution which requiresmore sophisticated analysis than the circularly-symmetric case[18], [19]. This is due to the fact that the real and imaginaryparts of the received signal will be correlated [20], [21] and thecorrelation level will depend on the IQI level. Early key resultson GLRT detection under the assumption of an impropercomplex signal were presented in [22], [23]. In this paper,we utilize the special structure of the covariance matrix of theimproper received signal due to the PU IQI. In our analysis, weassume no Rx IQI at the SU thanks to Rx IQI calibration [6]before attempting to blindly sense the spectrum. However, weverify, by simulations, the performance gains of our proposedGLRT detector even in the presence of residual Rx IQI.

We consider the practical and challenging scenario where allsystem parameters are unknown at the SU. The channel stateinformation, PU’s IQI level, PU’s signal variance, and receivernoise variance are all treated as unknown parameters. Reliablespectrum sensing becomes a much more challenging task asthe number of unknown parameters increases, especially atlow SNR. Under this general scenario, we derive the GLRTblind spectrum sensing rule, which is based on the Neyman-Pearson test [14], and analyze PFA as a function of the decisionthreshold. We also investigate the effect of the PU’s transmit(Tx) IQI level on PD for a given PFA [14].

The key contributions of this paper are summarized asfollows. First, we derive exact closed-form expressions forPD, PFA, and ROC for OFDM spectrum sensing under thegeneral joint Tx/Rx IQI model. Second, we derive simplifiedasymptotic approximations of these expressions and verifytheir accuracy numerically. Third, we derive a closed-formexpression for the number of OFDM symbols required toachieve a desired (PD, PFA) pair. Fourth, for comparison, wederive the analogous PD and PFA expressions for the popularenergy detector (ED) to compare its performance with ourlikelihood-based detector. Fifth, we demonstrate the somewhatsurprising result that Tx IQI at the PU AFE improves PD,which has not been previously reported in the literature tothe best of our knowledge. Finally, we derive a closed-formexpression for the GLRT as a blind decision statistic for theTx-IQI-impaired OFDM PU and analyze its PFA as a functionof the decision threshold.

This paper is organized as follows. The system model ofjoint Tx/Rx IQI is described in Section II. In Section III, weformulate the LRT and analyze its performance under the gen-eral model and several special cases. In addition, a closed-formexpression for the minimum sensing time needed to achievea target performance level is derived. Performance analysisof the ED in the presence of IQI is derived in Appendix C.The modified system model of PU Tx IQI for GLRT spectrumsensing is described in Section IV. Furthermore, a closed-formGLRT expression is derived as a blind decision statistic andits performance is analyzed. Simulation results are presentedin Section V and conclusions are drawn in Section VI.

Notations: Unless otherwise stated, lower and upper casebold letters denote vectors and matrices, respectively. Thematrices I, 0 and F denote, respectively, the identity, all-zeroand Fast Fourier Transform (FFT) matrices and their subscriptsdenote theirs sizes. The middle row of F corresponds to theDC subcarrier. For matrices, we define A# � IA∗I, whilefor vectors, a# � Ia∗ where I=FFT is the reversal (image)permutation matrix. Furthermore, ( )H , ( )∗, and ( )T denotethe matrix complex-conjugate transpose, complex conjugate,and transpose operations, respectively, and ( )−H = (( )H)−1.The operators E [ ] and | | denote the statistical expectationand absolute value, respectively. The notation x ∼ CN (m,R)means that x is a circularly-symmetric complex Gaussianrandom vector with mean m = E[x] and covariance matrixR = E[(x − m)(x − m)H ]. The matrix trace and deter-minant are denoted by tr() and det(), respectively. Finally,diag (x0, .., xN−1) denotes a diagonal matrix with entriesx0, .., xN−1. A summary of the key variables used in thispaper is given in Table I.

ELSAMADOUNY et al.: A BLIND LIKELIHOOD-BASED APPROACH FOR OFDM SPECTRUM SENSING IN THE PRESENCE OF I/Q IMBALANCE 3

TABLE IKEY VARIABLES USED THROUGHOUT THE PAPER

Variable Definition Variable DefinitionL Number of OFDM subcarriers (OFDM size) N Number of OFDM symbolss L× 1 original signal vector before Tx IQI y L× 1 received signal vector before Rx IQIz L× 1 received signal vector after Rx IQI v L× 1 received noise vector before Rx IQIσ2s Signal variance σ2

v Noise variancehk CFR coefficient at subcarriers k η Decision thresholdH L× L channel frequency response matrix H L× L TD circulant channel matrix

μt/r, νt/r Tx/Rx IQI parameters (2) ak, bk, ck Functions of the received signal (29)

{λk}k=L/2k=−L/2 Eigenvalues of R11 (4) {δk}k=L/2

k=−L/2 Eigenvalues of R12 (4)

II. SYSTEM MODEL AND ASSUMPTIONS

We consider a SU sensing an OFDM-based PU in thepresence of joint Tx/Rx IQI. We assume an overlay CRsystem in which the SU should not interfere with the PU.The PU suffers from Tx IQI while the SU suffers from RxIQI as shown in Fig 1. We consider frequency-independent1

IQI caused by the gain and phase mismatches between thebranches connecting the oscillator signals to the I and Qmixers of the AFE [4]. The gain and phase mismatches aredenoted, respectively, by εt/r and θt/r where the subscriptt/r denotes Tx/Rx, respectively. Under both Tx IQI at the PUand Rx IQI at the SU, we consider the case where the SUsenses N OFDM symbols. The received time-domain (TD)baseband-equivalent signal of the nth OFDM symbol at theSU, assuming that the PU is present (Hypothesis H1), afterCP removal is [4], [6]

zn = μryn + νry∗n (1)

where

yn = μtHsn + νtHs∗n + vn (2)

μt= cos (θt/2)+j εtsin (θt/2), νt= εtcos (θt/2)+j sin (θt/2)

μr=cos (θr/2)−j εrsin (θr/2), νr=εrcos (θr/2)+j sin (θr/2)

where j �√−1, sn = FH

L sn and sn ∼ CN (0L×1, σ2sIL) is

the L× 1 transmitted frequency-domain (FD) IQI-free signalvector of the nth OFDM symbol and the interference from s∗narises due to IQI effects. In addition, yn = FH

L yn and yn isthe L×1 received FD signal vector of the nth OFDM symbolbefore applying the Rx IQI effect. In the IQI-free scenario,εt/r = 0 and θt/r = 0; hence, μt/r = 1 and νt/r = 0.Furthermore, vn = FH

L vn and vn ∼ CN (0L×1, σ2vIL) is the

L × 1 complex FD additive white Gaussian noise (AWGN)vector with variance σ2

v , and H is the L × L circulantTD channel matrix whose first column contains the zero-padded channel impulse response (CIR) vector. The fadingchannel is assumed to be frequency-selective. Hence, H iscirculant thanks to the CP and the assumption that the channelis quasi-static over the OFDM symbol duration. We writeH = FH

LHFL where H is the L×L diagonal matrix with thechannel frequency response (CFR) vector on its main diagonal.We assume that the CIR memory is less than the CP lengthLp.

1The extension to frequency-dependent IQI is straightforward following theapproach in [24]

I

Q

90 -

Oscillator

Primary Transmitter AFE Secondary Receiver AFE

90 -

h

t r

t r

t

t r

r Spectrum

Sensing

Fig. 1. Cognitive radio system block diagram under joint Tx/Rx IQI.

III. LIKELIHOOD-BASED OFDM SPECTRUM SENSING

UNDER I/Q IMBALANCE

A. The Likelihood Ratio Test (LRT)

The received TD signal vector of the nth OFDM symbol znunder Hypothesis H1 in (1) is a widely-linear transformationof the transmitted signal vector sn and the noise vector vn

because it depends linearly not only on sn and vn but alsoon their complex conjugates [20], [21]. Hence, zn underHypothesis H1 is an improper (i.e. not circularly symmetric)complex Gaussian random vector [22] with probability densityfunction (PDF)

p(zn|H1) =1

πL|det(RH1)|1/2exp

(−1

2zHntR

−1H1

znt

)(3)

where znt = [zTn , zHn ]T denotes an augmented vector formed

by stacking zn on top of its conjugate z∗n and RH1 is the2L× 2L augmented covariance matrix of znt defined by

RH1 � E[zntzHnt] �

[R11 R12

R∗12 R∗

11

](4)

where R11 � E[znzHn ] is the Hermitian positive semi-definite

covariance matrix of zn and R12 � E[znzTn ] is the com-

plex symmetric complementary covariance matrix of zn [20].Hence, the joint PDF of the N independent and identicallydistributed (i.i.d.) received OFDM symbols under HypothsisH1 is given by

p(Z|H1)=1

πNL|det(RH1)|N/2exp

(−

N∑n=1

1

2zHntR

−1H1

znt

)(5)

where Z = [z1z2 . . . zN ] is the L×N TD matrix formed byhorizontally stacking the N received OFDM symbols.


Proposition 1: The LRT decision statistic T has the fol-lowing compact form

T�p(Z|H1)

p(Z|H0)=

N∑n=1

L/2∑k=1

yHn,k

(I2−(Ck+I2)

−1)yn,k

H1

≷H0

η (6)

where yn,k =

[μr νrν∗r μ∗

r

]−1

zn,k =

[yn(k)y∗n(−k)

], for k =

{1, . . . , L/2}, is the 2 × 1 received FD signal vector of thenth OFDM symbol at subcarriers k and −k before applyingthe Rx IQI effect (see (59) in Appendix A), and

Ck�σ2s

σ2v

[(|μt|2+|νt|2) |hk|2 2μtνthkh−k

2μ∗t ν

∗t h

∗kh

∗−k

(|μt|2+|νt|2) |h−k|2

](7)

where hk and h−k are the CFR coefficients at subcarriers kand −k, respectively, and σ2

v is the noise variance, and σ2s is

the PU signal variance.Proof: See Appendix A.

Knowing the CFR, signal variance, noise variance, and IQIparameters, we can easily compute the LRT decision statisticas in (6).

B. Performance Analysis

The probabilities of detection and false alarm are definedas follows

PD � Pr (T > η;H1) =

∫ ∞

t=η

fT (t;H1)dt (8)

PFA � Pr (T > η;H0) =

∫ ∞

t=η

fT (t;H0)dt (9)

where fT (t;H1) and fT (t;H0) are the PDFs of T under H1

and H0, respectively.1) Exact Analysis: In Appendix B, we show that, regard-

less of the hypothesis type (H1 or H0), the LRT decisionstatistic T can be written as the sum of NL independentexponential random variables. To derive fT (t;Hi), we firstcompute its characteristic function as follows

φT (ω;Hi) =

NL/2∏k=1

φwn(k)(ω;Hi)φwn(−k)(ω;Hi)

=

NL/2∏k=1

(1

1− jωE[w(k)]i× 1

1− jωE[w(−k)]i

)where φwn(k)(ω;Hi) and E[w(k)]i are, respectively, the char-acteristic function and statistical expectation of wn(k) underHypothesis Hi. Using partial fractions, the PDF of T is givenby

fT (t;Hi)=F−1{φT (ω;Hi)} (10)

=

NL/2∑k=1

⎛⎝Ai(k)exp(

−TE[w(k)]i

)E[w(k)]i

+Ai(−k)exp

(−T

E[w(−k)]i

)E[w(−k)]i

⎞⎠where F−1 denotes the inverse Fourier transform operator and

Ai(k) �NL/2∏

m=1,m �=k

(E[w(k)]i

E[w(k)]i − E[w(m)]i

)×

NL/2∏m=1

(E[w(k)]i

E[w(k)]i − E[w(−m)]i

)(11)

Ai(−k) �NL/2∏m=1

(E[w(−k)]i

E[w(−k)]i − E[w(m)]i

)×

NL/2∏m=1,m �=k

(E[w(−k)]i

E[w(−k)]i − E[w(−m)]i

)(12)

Proposition 2: PD and PFA for the LRT are given, respec-tively, by

PD =

NL/2∑k=−NL/2,k �=0

A1(k) exp

( −η

E[w(k)]1

)

PFA =

NL/2∑k=−NL/2,k �=0

A0(k) exp

( −η

E[w(k)]0

) (13)

where Ai(k) for i = 0, 1 is computed from (11) and (12), andE[w(k)]i and E[w(−k)]i are given in (68) of Appendix B fori = 0, 1. Proposition 2 follows directly by substituting from(10) into (8) and (9).

2) Asymptotic Analysis: For large L, we apply the centrallimit theorem (CLT) and approximate the distribution of T ,given in (67), by a Gaussian distribution whose mean and vari-ance under Hypothesis Hi (i = 0, 1) are given, respectively,by

mT,i = N

L/2∑k=1

{E[w(k)]i + E[w(−k)]i}

σ2T,i = N

L/2∑k=1

{(E[w(k)]i)

2+ (E[w(−k)]i)

2} (14)

and the correct detection and false alarm probabilities aregiven by

PD = Q

(η −mT,1

σT,1

); PFA = Q

(η −mT,0

σT,0

)(15)

where Q(x) = 1√2π

∫∞x

exp(−t2/2)dt is the well-known Q-function. Using the asymptotic expressions in (15), we derivea closed-form expression for the threshold η

η = mT,1 + σT,1Q−1(PD) = mT,0 + σT,0Q

−1(PFA) (16)

where Q−1() is the inverse Q-function. From (15), we derivethe following closed-form relation between PD and PFA whichdefines the ROC function

PD = Q

(mT,0 −mT,1 + σT,0Q

−1(PFA)

σT,1

)(17)

C. Effect of the Number of Processed OFDM Symbols (Sens-ing Time)

In this Subsection, we derive a closed-form expression forthe minimum number of processed OFDM symbols, denotedby N , required to achieve a certain (PD, PFA) pair anddemonstrate the fundamental tradeoff between the sensingtime length and the achieved performance. We re-write PD

and PFA in (15) as follows

PD = Q

(η −NmT,1√

NσT,1

); PFA = Q

(η −NmT,0√

NσT,0

)(18)


mT,i =

L/2∑k=1

{E[w(k)]i + E[w(−k)]i}

σT,i2 =

L/2∑k=1

{(E[w(k)]i)

2+ (E[w(−k)]i)

2} (19)

where mT,i and σT,i2, i = 0, 1 are the mean and variance

obtained in (14) for a single OFDM symbol (N = 1). Solvingfor η in (19), we get the following relation between PD andPFA characterizing the ROC curve

PD = Q

(NmT,0 −NmT,1 +

√NσT,0Q

−1(PFA)√NσT,1

)(20)

Thus, the minimum number of OFDM symbols required toachieve a certain (PFA, PD) pair is

N =

⌈(σT,1Q

−1(PD)− σT,0Q−1(PFA))

2

(mT,1 −mT,0)2

⌉(21)

where⌈x⌉

is the ceil of x defined as the smallest integergreater than or equal to x.

D. Special Cases

1) IQI-Free Case: For the case of no IQI at both the PUtransmitter and SU receiver, i.e, μt = μr = 1 and νt = νr = 0,the LRT and ED decision statistics given, respectively, in (6)and (69) reduce to

TLRT =N∑

n=1

L/2∑k=−L/2

σ2s

σ2v|hk|2

σ2s

σ2v|hk|2 + 1

|yn(k)|2

TED =

N∑n=1

L/2∑k=−L/2

|yn(k)|2(22)

It can be seen from (22) that the performance superiority of theLRT over ED is mainly due to exploiting the channel and SNRknowledge. It is clear from (22) that the LRT converges to theED at very high SNR; however this is an unlikely scenario inthe context of spectrum sensing as discussed in Section V.

2) Effect of Rx IQI: The LRT performance (in terms ofPD and PFA) as given by (13) depends only on the eigenvaluesof the matrix Ck in (7) which, in turn, do not depend onthe Rx IQI parameters as shown in Appendix B. Hence,interestingly, the LRT performance is not affected by IQI atthe SU receiver. This is not totally surprising because RxIQI results in multiplication of the received signal and noisevectors by the matrix B defined in (45) of Appendix A.Intuitively, since the LRT is basically a ratio test, the effectof B cancels out.

3) Effect of Tx IQI: Unlike the SU Rx IQI, the PU Tx IQIaffects the LRT performance because the eigenvalues of thematrix Ck (γ(k) and γ(−k) in (64) of Appendix B) dependon the Tx IQI parameters. Setting μt = 1 and νt = 0 in

(64), we get γ (k)(−k)

=σ2s

2σ2v

(bk ±

√b2k − 4ck

), where bk =

|h−k|2 + |hk|2 and ck = |hkh−k|2, i.e., γ(k) = σ2s

σ2v|hk|2 and

γ(−k) =σ2s

σ2v|h−k|2. Hence, the detection performance for Rx-

only IQI is the same as the IQI-free case.

E. Is Tx IQI A Friend or Foe for LRT Spectrum Sensing ?

We compare the LRT performance with and without Tx IQI.Starting with PD as defined in (15), mT,1 and σ2

T,1 as definedin (14), are given by

mT,1 = N

L/2∑k=1

(γ(k) + γ(−k)) = N

L/2∑k=1

σ2s

σ2v

bk

σ2T,1 = N

L/2∑k=1

(γ2(k) + γ2(−k)) = N

L/2∑k=1

σ4s

σ4v

(b2k − 2ck)

where γ(k) and γ(−k) are given in (64). Inspecting theexpressions of bk and ck in (65), we find that bk increases withthe Tx gain imbalance parameter εt since |μt|2+ |νt|2=1+ε2t .Hence, mT,1 increases with the Tx IQI level. Furthermore,ck decreases with both the Tx gain and phase imbalancelevels since |μt|2 − |νt|2=(1− ε2t )cosθt. Hence, the quantity(b2k − 2ck) increases with the IQI level and so does σ2

T,1.According to (15), since both mT,1 and σ2

T,1 increase withthe Tx IQI level, we find that PD shows the same behaviorfor a fixed threshold η. Moreover, mT,0 and σ2

T,0 in (14) are

functions of E[w(k)]0=γ(k)

γ(k)+1 and (E[w(k)]0)2= γ2(k)

(γ(k)+1)2 ,respectively, whose rates of change with γ(k) are small. Theexpressions for E[w(k)]0 and (E[w(k)]0)

2 involve normaliza-tion where any change of γ(k) in the numerator is opposedby a corresponding change in the denominator reducing theoverall change in the ratio. Hence, the behavior of PFA withthe Tx IQI level is mainly determined by the threshold η whichincreases with the Tx IQI level to keep PD fixed as explainedin the above discussion on PD. Therefore, PFA decreases withthe Tx IQI level for fixed PD. To summarize, Tx IQI improvesboth PD and PFA, as it will be demonstrated by the simulationresults in Section V.

IV. BLIND SPECTRUM SENSING USING THE GENERALIZED

LIKELIHOOD RATIO

Starting from this section, we consider a more challengingscenario where all system parameters are unknown at the SU.The channel matrix H between the PU and the SU, the PU’ssignal variance σ2

s , the noise variance σ2v , and the Tx IQI

parameters (μt and νt) are all unknown system parameters.To simplify the analysis, we assume no Rx IQI at the SUwhich is assumed to calibrate its receiver chain [6] beforesensing the spectrum. However, we will examine the effect ofRx IQI on the performance by simulations in Section V.

The received TD IQI-impaired baseband signal of the nth

OFDM symbol at the SU, assuming that the PU is present(Hypothesis H1), after CP removal is given by

zn = μtHsn + νtHs∗n + vn (23)

which can be obtained from (1) by substituting μr = 1 andνr = 0. The PDF of zn is given by (3) and RH1 can bewritten as a block-circulant matrix as follows

RH1 = E[zntzHnt] = σ2

sKKH + σ2vI2L (24)

where K is defined in (45) of Appendix A. According to(4), R11 =

(|μt|2 + |νt|2)σ2sHH

H+ σ2

vIL is a circulant

Hermitian matrix, and R12 = 2μtνtσ2sHH

Tis a circulant


complex symmetric matrix. The received TD signal vector ofthe nth OFDM symbol under Hypothesis H0 is simply theAWGN noise vector zn = vn distributed as follows

p(zn|H0) =1

πL|det(RH0)|1/2exp

(−1

2zHntR

−1H0

znt

)(25)

where RH0 = σ2vI2L is the noise covariance matrix.

When all system parameters are known, the optimal deci-sion statistic for the spectrum sensing hypothesis test is theLRT [15] derived in Section III. However, if any of the systemparameters is unknown to the SU, the LRT decision statisticcan not be computed leading us to the GLRT.

A. The Generalized Likelihood Ratio Test (GLRT)

Without loss of generality, we consider the case where theSU senses N OFDM symbols. The received signal vector ofthe nth OFDM symbol under Hypotheses H1 and H0 is given,respectively, by

H1 : zn = μtHsn + νtHs∗n + vn

H0 : zn = vn

(26)

The GLRT approach first computes the maximum likelihoodestimates (MLE) of the unknown parameters under both Hy-potheses H0 and H1, then, it computes the LRT by replacingthe unknown parameters by their MLEs [14]. Let Θ0 andΘ1 be the vectors of unknown parameters under H0 and H1,respectively. Their MLEs are defined as follows

Θ0 = argmaxΘ0

p(Z|Θ0, H0)

Θ1 = argmaxΘ1

p(Z|Θ1, H1)(27)

where Z = [z1z2 . . . zN ] is the L×N TD matrix formed byhorizontally stacking the N received OFDM symbols. Thus,the GLRT decision statistic can be written as follows

GLRT =p(Z|RH1 , H1)

p(Z|RH0 , H0)=

N∏n=1

p(zn|RH1 , H1)

N∏n=1

p(zn|RH0 , H0)

=

N∏n=1

[ |det(RH0)|1/2|det(RH1)|1/2

exp(− 1

2 zHntR

−1H1

znt

)exp

(− 1

2 zHntR

−1H0

znt

)]

where znt = [zTn zHn ]T , and RH0 and RH1 are the MLEsof RH0 and RH1 , respectively. In Appendix D, we show thatthe log-likelihood function of the N received OFDM symbolsunder Hypothesis H1 is given by

log p(Z|RH1 , H1)=−NL logπ−NL/2∑k=1

log |λkλ−k−|δk|2|

−NL/2∑k=1

akλ−k+bkλk−2Re{ckδ∗k}λkλ−k − |δk|2 (28)

where λk and δk for k = {−L/2, . . . , L/2} are the eigenval-ues of R11 and R12, respectively as defined in (82) and (83)

of Appendix D, and

ak � 1

N

N∑n=1

|zn(k)|2 ; bk � 1

N

N∑n=1

|zn(−k)|2

ck � 1

N

N∑n=1

(zn(k)zn(−k))

(29)

where zn(k) is the FD received signal at subcarrier k of thenth OFDM symbol.

To calculate log p(Z|RH1 , H1), we substitute for λk and δkin (28) by their MLEs. Thanks to the block-diagonal structureof RH1 , shown in (81), the log likelihood function of the Nreceived OFDM symbols in (84) is equal to the sum of thelog likelihood functions of the received FD pair of subcarriers{zn(k), zn(−k)} for k = {1, . . . , L/2} as given below

log p(Z|RH1 , H1) =log (π)−NL−NL/2∑k=1

log det(Gk)

−N∑

n=1

L/2∑k=1

[z∗n(k) zn(−k)

]G−1

k

[zn(k)z∗n(−k)

](30)

where Gk=

[λk δkδ∗k λ−k

]is the covariance matrix of

[zn(k)z∗n(−k)

],

i.e., λk=E[|zn(k)|2] and δk=E[zn(k)zn(−k)]. It was shownin [17] that the MLE of Gk is the sample covariance matrix,hence, the MLE of λk and δk can be written as follows

λk=1

N

N∑n=1

|zn(k)|2

δk= δ−k=1

N

N∑n=1

(zn(k)zn(−k))

, k = {−L/2, . . . ,L/2} (31)

and the ML estimate of RH1 is easily obtained using (31) asa sole function of the observations.

The likelihood function analysis under Hypothesis H0 (ab-sence of the PU signal) is easier for two reasons. First, thereis no PU IQI which causes correlation between each OFDMsubcarrier and its image. Second, the noise is white anduncorrelated. Hence, the only unknown parameter is the noisevariance σ2

v which we need to estimate to apply the GLRT.The log-likelihood function of the N independent receivedOFDM symbols under Hypothesis H0 will be similar to (84)with RH1 and H1 replaced by RH0 and H0, respectively.Substituting RH0 = σ2

vI2L, the log-likelihood function isgiven by

log p(Z|RH0 , H0) = −NL logπ −NL logσ2v

− 1

σ2v

N∑n=1

L/2∑k=−L/2

|zn(k)|2 (32)

where the MLE of σ2v is given by

σ2v =

1

NL

N∑n=1

L/2∑k=−L/2

|zn(k)|2 (33)


Using (30) and (32), the logarithm of the GLRT is calculatedas follows

log (GLRT)� log p(Z|RH1 , H1)− log p(Z|RH0 , H0)

=L log[ 1L

L/2∑k=1

(ak+bk)]−

L/2∑k=1

log |akbk−|ck|2|(34)

which leads us to the following propositionProposition 3: The GLRT decision statistic for the model

in (26) is given by

GLRT =

1L

L/2∑k=1

(ak + bk)[L/2∏k=1

|akbk − |ck|2|]1/L

=1L tr(G)

[det(G)]1/L

(35)

where G=diag{G1, . . . ,GL/2

}, Gk=

[ak ckc∗k bk

], and ak, bk,

and ck are defined in (29). Hence, The GLRT decision statisticis given by the ratio of the arithmetic to geometric means ofthe eigenvalues of the L × L block-diagonal matrix G. Thismatrix completely defines the effect of the PU’s Tx IQI onthe covariance matrix of the received signal and constitutesan RF fingerprint for the IQI-impaired PU.

B. Performance Analysis

For blind spectrum sensing, the probabilities PD and PFA

are defined as follows

PD � Pr (GLRT > η;H1) ; PFA � Pr (GLRT > η;H0)(36)

The decision threshold η is determined based on a systempredefined PFA, which is then used to evaluate PD. To derivea closed-form expression for PFA and η, we need to find theCDF of the GLRT under Hypothesis H0. We show numericallythat the Arithmetic mean (AM) and the Geometric Mean (GM)of the eigenvalues of the matrix G can be approximated ascorrelated Gaussian random variables and the approximationbecomes more accurate as N increases. The mean and varianceof the AM are denoted by mAM and σ2

AM , respectively, whilemGM and σ2

GM are the mean and variance of the GM whichare given by

mAM = mGM = σ2v ; σ2

AM = σ2GM =

1

NLσ4v (37)

Hence, the joint PDF of the two-dimensional random vector[AM GM] is approximated by a bivariate Gaussian distributionfunction with a correlation coefficient (ρ) given by

ρ =E[(AM −mAM )(GM −mGM )]

σAMσGM(38)

The CDF of the ratio of two dependent Gaussian randomvariables is given by [25]

F (η) = Ω

{mGMη −mAM

σAMσGMβ(η)

}(39)

β(η) =

(η2

σ2AM

− 2ρη

σAMσGM+

1

σ2GM

)0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PFA

PD

SNR= −4dB (closed form) SNR= −4dB (simulation)SNR= −6dB (closed form)SNR= −6dB (simulation)SNR= −8dB (closed form)SNR= −8dB (simulation)

Fig. 2. Comparison between exact analytical and simulated ROC curvesusing LRT for L = 32 at different SNR levels, N = 1, θt = 1o, and εt = 1dB.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PFA

PD

SNR= −8 dB (closed form)

SNR= −8 dB (simulation)

SNR= −10 dB (closed form)


SNR= −12dB (closed form)


Fig. 3. Comparison between approximate analytical and simulated ROCcurves using LRT for L = 128 at different SNR levels, N = 1, θt = 1o,and εt = 1 dB.

where Ω{} is the CDF of a standard Gaussian random variable.

Proposition 4: PFA for the GLRT in (35) is the complementof the standard Gaussian CDF function and is given by

PFA = 1− F (η) = Q

(mGMη −mAM

σAMσGMβ(η)

)(40)

Using (40), we derived the following closed-form expressionfor η as a function of the required PFA and the receivedobservations only (since we assume blind spectrum sensing)by replacing unknown quantities with their ML estimates in(41) on the top of the next page where τ = Q−1(PFA) and ρ,m, and σ2 are the sample correlation coefficient, the samplemean, and sample variance, respectively, which are given by(see (29))

mAM=1

L

L/2∑k=1

(ak+bk) ; mGM=

⎡⎣L/2∏k=1

|akbk−|ck|2|⎤⎦1/L

(42)

σ2AM =

1

NLm2

AM ; σ2GM =

1

NLm2

GM (43)


η =mAMmGM − τ2ρσAM σGM

m2GM − τ2σ2

GM

+τ√m2

AM σ2GM + m2

GM σ2AM + (ρ2 − 1)τ2σ2

AM σ2GM − 2mAMmGM ρσAM σGM

m2GM − τ2σ2

GM

(41)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PFA

PD

LRT using uniform PDP

LRT using Vehicular−A channel

ED

Fig. 4. Comparison between ROC curves using LRT using uniform PDP, LRTusing ITU Vehicular-A channel, and ED for L = 128 at SNR = −10dB,N = 1, θt = 1o, and εt = 1 dB.

−12 −11 −10 −9 −8 −7 −6 −50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

PD with Tx IQI (P

FA=0.1)

PD IQI−free (P

FA=0.1)

PFA

with Tx IQI (PD=0.9)

PFA

IQI−free (PD=0.9)

Fig. 5. PD and PFA versus SNR level for LRT under Tx IQI with θt = 1o

and εt = 1 dB compared with an IQI-free system for L = 128 and N = 1.

We conclude this section by emphasizing that the GLRT stilloutperforms the ED for the IQI-free scenario as it will beillustrated in Fig. 8 of Section V. In this case, the GLRT

decision statistic is equal to

1L

L/2∑k=−L/2

ak

[L/2∏

k=−L/2

ak

]1/L where ak, defined

in (29), is the average energy received at subcarrier k. Thisratio measures the energy spread of the received signal acrossall subcarriers compared to the ED which only measures thereceived signal energy.

V. NUMERICAL RESULTS

In our simulations, we consider two different channel mod-els. First, we assume a multi-path CIR with 8 independentand identically distributed taps where each tap is a zero-meancomplex Gaussian random variable. Second, we consider the

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PFA

PD

LRTGLRTED with noise power knowledgeED without noise power knowledge

Fig. 6. Comparison between ROC curves using LRT, GLRT, and ED forL = 128, N = 8 at SNR = −10dB, θt = 1o, and εt = 1 dB.

popular ITU Vehicular-A channel [26] and compare its resultswith the first model. The noise is assumed to be AWGN. TheSNR is defined as the ratio between signal variance and noisevariance. The amplitude imbalance εt is defined in log-scaleas εt = 10 log10(1+εt) dB so that 0 dB corresponds to no am-plitude imbalance. In Fig. 2, we compare the exact analyticaland simulated ROC curves using LRT for L = 32 at differentSNR levels. In Fig. 3, we compare the approximate analytical(using CLT) and simulated ROC curves for L = 128. Theaccuracy of our derived expressions is evident in Figs. 2 and3. In Fig. 4, we compare the performances of the LRT andED using the two considered channel models. As shown inFig. 4, our LRT-based approach significantly outperforms theconventional ED approach for both channel models. Fig. 5shows that the presence of Tx IQI at the PU’s AFE in factimproves PD (at fixed PFA) and PFA (at fixed PD). In Fig.6, we compare the performances of the GLRT, LRT, andED. The LRT represents an ideal performance upper-boundon the GLRT since the former assumes perfect knowledge ofall system parameters. We note that the GLRT performanceis relatively close to that of the LRT even with a relativelysmall number of processed OFDM symbols of N = 8. Fig.6 also shows that our GLRT-based approach significantlyoutperforms the conventional ED which does not exploit signalcorrelation due to IQI. It is well-known [27] that the EDis sensitive to noise variance knowledge uncertainty. In Fig.7, we show the effect of the number of processed OFDMsymbols on the detection performance. The ROC curves in thisfigure show the significant performance improvement achievedeven by slightly increasing the number of processed OFDMsymbols. Assuming that PFA = 0.1, the time required for theSU to perform reliable spectrum sensing is only 5 to 8 OFDMsymbols, where PD ranges from 0.65 and 0.96 as shown inFig. 7. Fig. 8 shows the effect of the Tx IQI level on PD at afixed PFA for the LRT, the GLRT, and the ED. It is clear from


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PFA

PD

N = 5

N = 8

N = 11

Fig. 7. Comparison between ROC curves with GLRT using different numberof processed OFDM symbols for L = 128 at SNR = −10dB, θt = 1o, andεt = 1 dB.

this figure that the performance of the GLRT is superior tothe ED performance over all considered transmit IQI levels.The GLRT performance is significantly improved as the PUtransmit IQI level increases. This performance enhancementis mainly due to two reasons; First, the LRT and the GLRTexploit the correlation between each OFDM subcarrier andits image which arises due to IQI. Second, even when thereis no IQI, the LRT and the GLRT outperform the ED sincethey exploit the channel knowledge (or its estimate based onthe received signal in case of GLRT) while the ED uses onlythe energy of the received signal. Fig. 8 also shows that evenin the presence of a residual Rx IQI level of 0.3 dB and 3degrees, the GLRT outperforms the ED. Finally, in Fig. 9,we demonstrate the excellent match between the closed-formGLRT false alarm probability expression and the simulationresults. We conclude this section by noting that the GLRTperformance can be further improved by integrating it withSU cooperative sensing techniques [28] and by exploiting theCP in OFDM waveforms [16].

VI. CONCLUSION

In this paper, we started by deriving the LRT for OFDM-based spectrum sensing in the presence of IQI at both the PUand SU AFEs. We showed that, unlike Tx IQI, Rx IQI at theSU does not impact the LRT performance. We derived thecorrect detection and false alarm probabilities and the ROCcurve for the LRT and the ED. Furthermore, we showed thatTx IQI at the PU improves the LRT detection probabilityat a fixed false alarm probability, i.e., Tx IQI improves theROC performance. Simulation results for typical SNR levelsshowed that the LRT outperforms the ED under Tx IQI. Next,we generalized our analysis to scenarios where the channelresponse, signal characteristics, noise characteristics, and theIQI parameters are not known at the SU. We derived the GLRTdetector for blind spectrum sensing of IQI-impaired OFDMsignals and showed that its decision statistic is given by theratio of the arithmetic to geometric means of the eigenvaluesof a block diagonal Hermitian matrix which depends on thereceived signal only. Each 2×2 block in this matrix represents

the sample covariance matrix of the received signal at eachOFDM subcarrier and its image calculated across all processedOFDM symbols. The GLRT exploits the correlation betweeneach PU OFDM signal subcarrier and its image, inducedby IQI, to improve the PU signal’s detection probability atthe SU compared to the ED. We also showed that the SU’swaiting time for a reliable blind spectrum sensing is relativelyshort. Hence, the GLRT is an efficient blind spectrum sensingscheme with an attractive sensing-throughput tradeoff thatturns IQI at the PU from a foe to a friend !

APPENDIX APROPOSITION 1 PROOF : LRT DECISION STATISTIC

DERIVATION

Starting with (1), the 2L× 1 augmented signal vector zt isdefined as follows

znt �[znz∗n

]=

[μrμtH+ νrν

∗t H

∗μrνtH+ νrμ

∗tH

∗

ν∗rμtH+ μ∗rν

∗t H

∗ν∗rνtH+ μ∗

rμ∗tH

∗] [

sns∗n

]+

[μrIL νrILν∗r IL μ∗

rIL

] [vn

v∗n

](44)

� B(Ksnt + vnt) (45)

where B=

[μrIL νrILν∗r IL μ∗

rIL

], K=

[μtH νtH

ν∗t H∗

μ∗tH

∗], snt =

[sns∗n

],

and vnt =

[vn

v∗n

]. Hence, we have

RH1 � E[zntzHnt] = B(σ2

sKKH + σ2vI2L)B

H (46)

Hence, using (45) and (46), the Gaussian distribution exponentin (3) is given by (47) on the top of the next page, whereynt � Ksnt+ vnt is the received TD signal vector of the nth

OFDM symbol before applying Rx IQI effects. The invertedmatrix in (47) is a block-circulant matrix with the followingeigenvalue decomposition

σ2sKKH+σ2

vI2L�[FH

L 0L

0L FHL

][Λ11 Λ12

ILΛ∗12IL ILΛ11IL

][FL 0L

0L FL

](48)

where Λ11 and Λ12 are L× L diagonal matrices given by

Λ11=(|μt|2 + |νt|2

)σ2sHH∗+σ2

vIL ; Λ12=2μtνtσ2sHILHIL

(49)

and H = FLHFHL is the L×L diagonal matrix with the CFR

on its main diagonal. Substituting from (48) into (47), we get

zHntR−1H1

znt = yHnt

[Λ11 Λ12


]−1

ynt

�[yHn y#H

n

] [V11 V12

V21 V22

] [yn

y#n

](50)

= 2[yHn V11yn +Re{yH

n V12y#n }] (51)

where ynt =

[FL 0L

0L FL

]ynt =

[FL 0L

0L FL

][yn

y∗n

]�

[yn

y#n

]and

V11, V12, V21, and V22 are calculated using the 2×2 block-matrix inversion formula2 and the diagonal structure of each

2[A BC D

]−1=

[(A − BD−1C)−1 −(A − BD−1C)−1BD−1

−D−1C(A − BD−1C)−1 (D − CA−1B)−1

].


zHntR−1H1

znt = (Ksnt + vnt)H(σ2

sKKH + σ2vI2L)

−1(Ksnt + vnt)

� yHnt

[(|μt|2 + |νt|2)σ2sHH

H+ σ2

vIL 2μtνtσ2sHH

T

2μ∗t ν

∗t σ

2sH

∗H

H (|μt|2 + |νt|2)σ2sH

∗H

T+ σ2

vIL

]−1

ynt (47)

0 0.5 1 1.5 2 2.5

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

εt (dB)

PD

LRT

GLRT with no Rx IQI

GLRT with residual Rx IQI (εr=0.1, θ

r=1o)

GLRT with residual RX IQI (εr=0.3, θ

r=3o)

ED

Fig. 8. Comparison between PD versus amplitude mismatch of IQI usingLRT, GLRT, and ED for L = 128 and N = 8 at θt = 2o, SNR = −12dB,PFA = 0.1.

block matrix

V11=(Λ11 − |Λ12|2ILΛ−1

11 IL)−1

; V22=ILV11IL (52)

V12=−Λ12

(Λ11ILΛ

−111 IL−|Λ12|2

)−1; V21=V∗

12 (53)

The matrices Vij for i, j = {1, 2}, are L × L diagonalmatrices whose main diagonal elements are denoted by vij(k)for k = {−L/2, . . . , L/2} defined in (54).

The received TD signal vector of the nth OFDM symbolunder Hypothesis H0 is given by zn = μrvn + νrv

∗n.

Hence, the joint PDF of the N received OFDM symbolsZ = [z1z2 . . . zN ] under Hypothesis H0 is given by

p(Z|H0)=1

πNL|det(RH0)|N/2exp

(−

N∑n=1

1

2zHntR

−1H0

znt

)(55)

Hypothesis H1 results can be used by substituting for σs = 0in (49) and (50) resulting in

zHntR−1H0

znt=[yHn y#H

n

][σ2vIL 0L

0L σ2vIL

]−1[yn

y#n

]=

2

σ2v

yHn yn

(56)

From (5) and (55), the likelihood ratio (LR) is given by

LR=p(Z|H1)

p(Z|H0)=|det(RH0)|N/2

|det(RH1)|N/2

exp

(−

N∑n=1

12 z

HntR

−1H1

znt

)exp

(−

N∑n=1

12 z

HntR

−1H0

znt

)(57)

Note that the determinants ratio in (57) is a known constantlumped with the decision threshold η. Substituting from (51)

0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

threshold (η)

PF

A

Closed Form (η=0.94)

Simulations

Fig. 9. PFA for GLRT versus the decision threshold for L = 128, N = 8at SNR = −10dB, θt = 1o, and εt = 1 dB.

and (56) into (57) and taking the logarithm, we get the LRT

T �N∑

n=1

[zHntR

−1H0

znt − zHntR−1H1

znt]

�N∑

n=1

[yHn

[1

σ2v

−V11

]yn−Re{yH

n V12y#n }

]H1

≷H0

η (58)

All matrices in the above equation are diagonal, hence, theLRT can be written as a sum of the terms involving eachsubcarrier k and its image subcarrier −k as follows

T =

N∑n=1

L/2∑k=1

[(1

σ2v

− v11(k))|yn(k)|2

+ (1

σ2v

− v11(−k))|yn(−k)|2−2Re{v12(k)y∗n(k)y∗n(−k)}]

where yn = FLyn = [yn(−L/2) . . . yn(L/2)]T is the L × 1

received FD signal vector of the nth OFDM symbol beforeRx IQI effects as given in (50). Finally, defining the 2 × 1column vector

yn,k � [yn(k) y∗n(−k)]T (59)

and applying straightforward algebra, we arrive at (6) and (7)in Proposition 1.

APPENDIX BPROOF THAT THE LRT IS A SUM OF INDEPENDENT

EXPONENTIAL RANDOM VARIABLES

Using the matrix inversion lemma, we have

(Ck + I2)−1

= I2 −(I2 +C−1

k

)−1(60)


v11(k) =

(|μt|2 + |νt|2)σ2s |h−k|2 + σ2

v[(|μt|2 + |νt|2)σ2

s |hk|2 + σ2v

][(|μt|2 + |νt|2)σ2

s |h−k|2 + σ2v

]− 4σ4s |μtνthkh−k|2

; v22(k) = v11(−k)

v12(k) =−2μtνtσ

2shkh−k[

(|μt|2 + |νt|2)σ2s |hk|2 + σ2

v

][(|μt|2 + |νt|2)σ2

s |h−k|2 + σ2v

]− 4σ4s |μtνthkh−k|2

; v21(k) = v∗12(k)(54)

Hence, the LRT in (6) is rewritten as follows

T =N∑

n=1

L/2∑k=1

yHn,kCk (Ck + I2)

−1 yn,k (61)

Using the eigenvalue decomposition Ck � UkΓkUHk , we

write

T =

N∑n=1

L/2∑k=1

yHn,kUkΓk (Γk + I2)

−1UH

k yn,k

�N∑

n=1

L/2∑k=1

xHn,kΓk (Γk + I2)

−1xn,k (62)

=

N∑n=1

L/2∑k=1

(γ(k)

γ(k) + 1|xn(k)|2+ γ(−k)

γ(−k) + 1|xn(−k)|2

)H1

≷H0

η

(63)

where xn,k � UHk yn,k � [xn(k) xn(−k)]T ,UH

k Uk = I2and Γk is the eigenvalue matrix of Ck given in (7)

Γk=

[γ(k) 00 γ(−k)

]; γ (k)

(−k)

=σ2s

2σ2v

(bk±

√b2k − 4ck

)(64)

where γ(k) and γ(−k) are real and nonnegative, and

bk =(|h−k|2 + |hk|2

) (|μt|2 + |νt|2)

ck = |hk|2|h−k|2(|μt|2 − |νt|2

)2 (65)

Note that xn,k in (62) is an affine transformation ofyn,k; hence, it is also Gaussian distributed, i.e., xn,k ∼CN (UkE[yn,k],U

Hk E[yn,ky

Hn,k]Uk). The PDFs of xn,k un-

der H0 and H1 are

H0 : xn,k ∼ CN (02×1, I2)

H1 : xn,k ∼ CN (02×1,Γk + I2)(66)

For both hypotheses, xn,k is a zero-mean circularly-symmetricGaussian random vector with a diagonal auto-correlation ma-trix. Hence, xn(k) and xn(−k) are independent and theirabsolute squares (|xn(k)|2 and |xn(−k)|2) are independentand exponentially distributed. Thus, the LRT is given by

T =

N∑n=1

L/2∑k=1

(wn(k) + wn(−k))H1

≷H0

η (67)

where wn(k)=γ(k)|xn(k)|2

γ(k)+1 and wn(−k)= γ(−k)|xn(−k)|2γ(−k)+1 are

independent and exponentially distributed random variableswith the following means

H0 : E[w(k)]0=γ(k)

γ(k) + 1, E[w(−k)]0=

γ(−k)

γ(−k) + 1

H1 : E[w(k)]1=γ(k), E[w(−k)]1=γ(−k)

(68)

We emphasize that γ(k) and γ(−k) do not depend on the RxIQI parameters μr and νr. Note that the subscript n of wn(k)in (68) is omitted since the means are independent of n.

APPENDIX CTHE ENERGY DETECTOR PERFORMANCE ANALYSIS IN

THE PRESENCE OF IQI

For comparison with LRT, we analyze the performance ofthe ED in the presence of joint Tx/Rx IQI. The ED decisionrule is given by

TED �N∑

n=1

L/2∑k=1

zn,k �N∑

n=1

L/2∑k=1

zHn,kzn,k

=N∑

n=1

L/2∑k=1

(|zn(k)|2 + |zn(−k)|2)H1

≷H0

ηED (69)

where zn,k = [zn(k) zn(−k)∗]T is the received FD signalvector at subcarriers k and −k after Rx IQI effect. The ele-ments of the set {zn,k}L/2

k=1 are statistically independent since{yn,k}L/2

k=1 are also independent (see Appendix B). Hence, weapply the CLT and approximate TED by a Gaussian randomvariable whose mean and variance are given, respectively, by

mED=

NL/2∑k=1

E[zn,k] ; σ2ED=

NL/2∑k=1

E[z2n,k]− (E[zn,k])2 (70)

In the sequel, we derive E[zn,k] and E[z2n,k] as follows

E[zn,k]=E[|zn(k)|2]+E [|zn(−k)|2]=Rk

z(1, 1)+Rkz(2, 2)

(71)

where Rkz(i, j) is the (i, j)th entry of the 2× 2 matrix Rk

z

Rkz � E[zn,kz

Hn,k] = σ2

v (Ck + I2) (72)

where Ck is defined in (7). Furthermore,

E[z2n,k]=E[|zn(k)|4]+E [|zn(−k)|4]+2E [|zn(k)zn(−k)|2]

(73)

Since zn,k is a circularly-symmetric Gaussian random vector,then |zn(k)|2 and |zn(−k)|2 are both exponentially-distributedand correlated because the matrix Rk

z is not diagonal. Hence,

E[|zn(k)|4] =(E[|zn(k)|2])2

E[|zn(−k)|4] = (E[|zn(−k)|2])2 (74)

E[|zn(k)zn(−k)|2]=Ezn(−k)

[|zn(−k)|2 E[|zn(k)|2 |zn(−k)]]

(75)

where E[|zn(k)|2 | zn(−k)

]is the expectation of |zn(k)|2

conditioned on zn(−k). Since zn(k) and zn(−k) are jointlyGaussian, we have [29]

E[|zn(k)|2 | zn(−k)

]=

|Rkz(1, 2)|2

(Rkz(2, 2))

2 |zn(−k)|2

+Rk

z(1, 1)Rkz(2, 2)− |Rk

z(1, 2)|2Rk

z(2, 2)(76)


Substituting from (76) into (75), we get

E[|zn(k)zn(−k)|2] = |Rk

z(1, 2)|2 +Rkz(1, 1)R

kz(2, 2) (77)

Finally, substituting from (77) into (73) and then into (70), weget

mED =

NL/2∑k=1

Rkz(1, 1) +Rk

z(2, 2)=

NL/2∑k=1

tr(Rk

z

)(78)

σ2ED =

NL/2∑k=1

(Rk

z(1, 1))2+(Rk

z(2, 2))2+2|Rk

z(1, 2)|2

=

NL/2∑k=1

tr(Rk

z(Rkz)

H)

(79)

The ED probabilities of correct detection (P EDD ) and false

alarm (P EDFA ) are related by

P EDD = Q

(mED,0 −mED,1 + σED,0Q

−1(P EDFA )

σED,1

)(80)

where mED,i and σED,i are computed from (78) and (79),respectively, based on Hypothesis Hi.

APPENDIX DGENERALIZED LIKELIHOOD FUNCTION UNDER

HYPOTHESIS H1

From (24), RH1 is a block-circulant matrix. Hence, toestimate the Hermitian circulant covariance matrix, R11 (orthe complementary circulant covariance matrix R12), it issufficient to estimate Lp values (equal to CP length) for eachmatrix. Therefore, RH1 has the form

RH1 =

[R11 R12

R∗12 R∗

11

]=

[FH

L 0L

0L FHL

][Λ11 Λ12


][FL 0L

0L FL

](81)

where Λ11 = diag{λ−L/2, . . . , λL/2} is a diagonal matrixwhose main elements are the FFT of the first column of R11;i.e.

λk = ΣLl=1rl,1e

−j 2πklL , k = {−L/2, . . . , L/2} (82)

where rl,1 is the element in row l and the first column of R11.We note that λk is real since R11 is a circulant Hermitianmatrix. Similarly, Λ12 = diag{δ−L/2, . . . , δL/2} is also adiagonal matrix whose main diagonal elements are the FFTof the first column of R12; i.e.

δk = ΣLl=1ql,1e

−j 2πklL , k = {−L/2, . . . , L/2} (83)

where ql,1 is the element in row l and the first column ofR12. Note that δk = δ−k since R12 is a circulant symmetricmatrix. Using the MLE invariance theorem [29], we computethe MLEs of R11 and R12 by substituting for λk and δk bytheir MLEs (λk and δk) in (81). Hence, the ML estimationproblem under Hypothesis H1 reduces from

RH1 = maxRH1

log p(Z|H1,RH1)

to [λk δk

]T= arg max

λk,δklog p(Z|H1,RH1)

The log-likelihood function of the N independent receivedOFDM symbols under H1 is

log p(Z|RH1 , H1) = −NL logπ − N

2log det(RH1)

− 1

2

N∑n=1

zHntR−1H1

znt (84)

Using the eigenvalue decomposition of RH1 in (81), we have

zHntR−1H1

znt =[zHn z#H

n

] [Λ11 Λ12

Λ∗12 ILΛ11IL

]−1[znz#n

]=

[zHn z#H

n

] [V11 V12

V21 V22

] [znz#n

](85)

where zn = FLzn and z#n = FLz∗n, and V11, V12, V21, and

V22 are given by (53). The matrices Vij , i, j = {1, 2}, areL × L diagonal matrices whose main diagonal elements aredenoted by vij(k), k = {−L/2, . . . , L/2} where

v11(k) = v22(−k) =λ−k

λkλ−k − |δk|2

v12(k) = v∗21(k) =−δk

λkλ−k − |δk|2(86)

Substituting from (86) into (85), we get

[zn]Ht R−1

H1[zn]t =

L/2∑k=1

[ 2λ−k

λkλ−k − |δk|2 |zn,k|2

+2λk

λkλ−k − |δk|2 |zn,−k|2

+ 4Re{ −δkλkλ−k − |δk|2 z

∗n,kz

∗n,−k}

](87)

where zn,k is the FD received signal at subcarrier k of the nth

OFDM symbol. The determinant of RH1 is simplified using(81) as follows

det(RH1) =

L/2∏k=1

(λkλ−k − |δk|2)2 (88)

Substituting from (87) and (88) into (84), we arrive at (28).

REFERENCES

[1] S. Haykin, “Cognitive radio: brain-empowered wireless communica-tions,” IEEE J. Sel Areas Commun., vol. 23, no. 2, pp. 201–220, 2005.

[2] T. Yucek and H. Arslan, “A survey of spectrum sensing algorithms forcognitive radio applications,” IEEE Commun. Surveys Tutorials, vol. 11,no. 1, pp. 116–130, 2009.

[3] Y.-C. Liang, Y. Zeng, E. Peh, and A. T. Hoang, “Sensing-throughputtradeoff for cognitive radio networks,” IEEE Trans. Wireless Commun.,vol. 7, no. 4, pp. 1326–1337, 2008.

[4] F. Horlin and A. Bourdoux, Digital Compensation for Analog Front-Ends. Wiley, 2008.

[5] A. Gomaa and N. Al-Dhahir, “Multi-user SC-FDMA systems under IQimbalance: EVM and subcarrier mapping impact,” in Proc. 2011 IEEEGLOBECOM, 2011.

[6] A. Tarighat and A. H. Sayed, “Joint compensation of transmitterand receiver impairments in OFDM systems,” IEEE Trans. WirelessCommun., vol. 6, no. 1, pp. 240–247, Jan. 2007.

[7] B. Razavi, RF Microelectronics. Prentice Hall, 1998.


[8] A. Baier, “Quadrature mixer imbalances in digital TDMA mobile radioreceivers,” in 1990 International Zurich Seminar on Digital Comm.

[9] C.-L. Liu, “Impacts of I/Q imbalance on QPSK-OFDM-QAM detec-tion,” IEEE Trans. Consum. Electron., vol. 44, no. 3, pp. 984–989, 1998.

[10] J. Verlant-Chenet, J. Renard, J.-M. Dricot, P. De Doncker, and F. Horlin,“Sensitivity of spectrum sensing techniques to RF impairments,” in 2010IEEE Vehicular Technology Conference — Spring.

[11] A. Zahedi-Ghasabeh, A. Tarighat, and B. Daneshrad, “Cyclo-stationarysensing of OFDM waveforms in the presence of receiver RF im-pairments,” in 2010 IEEE Wireless Communications and NetworkingConference.

[12] O. Semiari, B. Maham, and C. Yuen, “Effect of IQ imbalance on blindspectrum sensing for OFDMA overlay cognitive radio,” in Proc. 2012IEEE Communications in China.

[13] A. Gokceoglu, S. Dikmese, M. Valkama, and M. Renfors, “Enhanced en-ergy detection for multi-band spectrum sensing under RF imperfections,”in Proc. 2013 International Conference on Cognitive Radio OrientedWireless Networks.

[14] S. M. Kay, Fundamentals Of Statistical Signal Processing: DetectionTheory. Prentice Hall, 1998.

[15] A. ElSamadouny, A. Gomaa, and N. Al-Dhahir, “Likelihood-basedspectrum sensing of OFDM signals in the presence of Tx/Rx I/Qimbalance,” in 2012 IEEE GLOBECOM.

[16] S. Bokharaiee, H. Nguyen, and E. Shwedyk, “Blind spectrum sensingfor OFDM-based cognitive radio systems,” IEEE Trans. Veh. Technol.,vol. 60, no. 3, pp. 858–871, Mar. 2011.

[17] R. Zhang, T. Lim, Y. Liang, and Y. Zeng, “Multi-antenna based spectrumsensing for cognitive radios: a GLRT approach,” IEEE Trans. Commun.,vol. 58, no. 1, pp. 84–88, Jan. 2010.

[18] L. Anttila, M. Valkama, and M. Renfors, “Circularity-based I/Q im-balance compensation in wideband direct-conversion receivers,” IEEETrans. Veh. Technol., vol. 57, no. 4, pp. 2099–2113, July 2008.

[19] L. Anttila and M. Valkama, “On circularity of receiver front-end signalsunder RF impairments,” in Proc. 2011 European Wireless Conf.

[20] P. Schreier and L. Scharf, Statistical Signal Processing of Complex-Valued Data. Cambridge University Press, 2010.

[21] B. Picinbono, “Second-order complex random vectors and normaldistributions,” IEEE Trans. Signal Process., vol. 44, no. 10, pp. 2637–2640, Oct. 1996.

[22] P. Schreier, L. Scharf, and C. Mullis, “Detection and estimation ofimproper complex random signals,” IEEE Trans. Inf. Theory, vol. 51,no. 1, pp. 306–312, Jan. 2005.

[23] P. Schreier, L. Scharf, and A. Hanssen, “A generalized likelihood ratiotest for impropriety of complex signals,” IEEE Signal Process. Lett.,vol. 13, no. 7, pp. 433–436, July 2006.

[24] B. Narasimhan, D. Wang, S. Narayanan, H. Minn, and N. Al-Dhahir,“Digital compensation of frequency-dependent joint Tx/Rx I/Q imbal-ance in OFDM systems under high mobility,” IEEE J. Sel. Topics SignalProcess., vol. 3, no. 3, pp. 405–417, 2009.

[25] D. Hinkley, “On the ratio of two correlated normal random variables,”Oxford J. Biometrika, vol. 56, no. 3, 1969.

[26] “High speed downlink packet access: UE radio transmission and recep-tion (FDD),” 3GPP TR 25.890, 2002-2005.

[27] R. Tandra and A. Sahai, “SNR walls for signal detection,” IEEE J. Sel.Topics Signal Process., vol. 2, no. 1, pp. 4–17, Feb. 2008.

[28] J. Lunden, V. Koivunen, A. Huttunen, and H. Poor, “Collaborativecyclostationary spectrum sensing for cognitive radio systems,” IEEETrans. Signal Process., vol. 57, no. 11, pp. 4182–4195, Nov. 2009.

[29] S. M. Kay, Fundamentals Of Statistical Signal Processing: EstimationTheory. Prentice Hall, 1993.

Ahmed ElSamadouny received BSc degree inelectronics and communications engineering fromAlexandria University, Egypt, in 2008 and MSc fromNile University, Egypt, in 2010. He is currentlyworking toward the Ph.D. degree at the University ofTexas at Dallas, USA. His research interests includecognitive radio spectrum sensing, RF impairmentsequalization at the baseband, and power line com-munication.

Ahmad Gomaa received his B.S and M.S degreesin electronic and communication engineering fromCairo University, Egypt, in 2005 and 2008, respec-tively. He received his Ph.D. degree in electricalengineering from the University of Texas at Dallas,USA, in 2012. He is currently with BroadcomCorporation, USA. His research interests includecompressive sensing applications in digital commu-nications, sparse filter design, channel equalizationand RF impairments compensation at the baseband.

Naofal Al-Dhahir earned his Ph.D. degree in Elec-trical Engineering from Stanford University in 1994.From 1994 to 1999, he was a senior member of thetechnical staff at GE R&D Center in NY where heworked on satellite communication systems designand anti-jam GPS receivers. From 1999 to 2003, hewas a principal member of technical staff at AT&TShannon Laboratory in NJ where he worked onMIMO algorithms for broadband wireless systems.In 2003, he joined UT-Dallas as a tenured AssociateProfessor, became a full Professor in 2007, and an

Erik Jonsson Distinguished Professor in 2010.He has authored over 260 journal and conference papers with over 6200

citations and is a co-inventor of 33 issued US patents. He is an Area Editor forIEEE TRANSACTIONS ON COMMUNICATIONS and Technical Program Co-Chair for IEEE ISPLC 2015. He is co-recipient of the IEEE VTC Fall 2005best paper award, the 2005 IEEE signal processing society young author bestpaper award, and the 2006 IEEE Donald G. Fink best journal paper award.He was elected an IEEE fellow in November 2007 for “contributions to highdata rate communication over broadband channels.”

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