A Low-Delay Low-Complexity EKF Design for Joint Channel and CFO Estimation in Multi-User Cognitive Communications
aa r X i v : . [ c s . I T ] N ov A Low-Delay Low-Complexity EKF Design forJoint Channel and CFO Estimation in Multi-UserCognitive Communications
Pengkai Zhao
Electrical Engineering, UCLA, CA, USA
Cong Shen
Qualcomm Inc., San Diego, CA, USA
Abstract —Parameter estimation in cognitive communicationscan be formulated as a multi-user estimation problem, which issolvable under maximum likelihood solution but involves highcomputational complexity. This paper presents a time-sharingand interference mitigation based EKF (Extended Kalman Filter)design for joint CFO (carrier frequency offset) and channelestimation at multiple cognitive users. The key objective is torealize low implementation complexity by decomposing high-dimensional parameters into multiple separate low-dimensionalestimation problems, which can be solved in a time-sharedmanner via pipelining operation. We first present a basic EKFdesign that estimates the parameters from one TX user to oneRX antenna. Then such basic design is time-shared and reusedto estimate parameters from multiple TX users to multiple RXantennas. Meanwhile, we use interference mitigation moduleto cancel the co-channel interference at each RX sample. Inaddition, we further propose adaptive noise variance trackingmodule to improve the estimation performance. The proposeddesign enjoys low delay and low buffer size (because of its onlinereal-time processing), as well as low implementation complexity(because of time-sharing and pipeling design). Its estimationperformance is verified to be close to Cramer-Rao bound.
I. I
NTRODUCTION
Cognitive communication system is widely accepted asa perspective way in increasing the spectrum efficiency ofwireless networks, where primary links and secondary linkscan usually co-exist in the network, resulting in an interferencelimited environment. Parameter estimation in cognitive com-munications is a challenging problem because of (i) the exis-tence of co-channel interference, and (ii) the high-dimensionalparameters from multiple TX users to multiple RX anten-nas. In particular, note that different TX users often haveindependent carrier frequency offset (CFO) values (includingboth oscillator offsets and doppler offsets), which usuallyintroduce serious nonlinear components within the observedsignal, complicating the estimation problem. Meanwhile, chan-nel responses from multiple TX users to multiple RX antennascan result in a set of high-dimensional parameters, which arealso difficult to estimate. Finally, due to the existence of multi-user interference, CFO and channel parameters usually haveto be treated together and be estimated in a joint way so asto approach the optimal performance, which further increasesthe estimation complexity.Without loss of generality, this paper assumes OrthogonalFrequency Division Multiplexing (OFDM) system, which is an overwhelming choice for modern wireless systems. Theclassical CFO and channel estimation method in a single-userOFDM system is based on two repeated training symbols[1]. It has low implementation complexity and near-optimalperformance, but only applies to a single-user scenario . Inmulti-user OFDMA systems with unique subcarrier set perTX user, CFO and channel parameters can be recovered byexploiting distinct subcarrier structures among TX users ([2],[3]). But this method requires separate subcarrier allocation fordifferent users. Consequently, in a general multi-user cognitivesystem without specific subcarrier allocation per user and withoverlapped training symbols, ML and EM related methodsseem to be the only applicable choice, where all TX users’parameters have to be formulated into a maximum likelihood(ML) estimation problem [4], which is solvable under Ex-pectation Maximization (EM) method [5] in an iterative way.However, since the entire OFDM block is stored offline andis iteratively processed multiple times, these ML and EMapproaches often require high computational complexity andhigh processing delay.Based on above considerations, this paper will focus onusing Kalman filter structure to estimate the CFO and channelparameters in multi-user cognitive communications. Our majorobjective is to achieve low-complexity and low-delay estima-tion performance in cognitive systems. In general, Kalmanfilter is a good candidate for low delay and low complexity pa-rameter estimation primarily due to its real-timing processingproperty. It has been conventionally used for CFO and channelestimation in multi-user OFDM systems, e.g., the FFT-BlockEKF design in [6], the parallel EKF design in [7], and theparticle filter design in [7]. However, these existing designsinherently suffer from multiple issues related to complexity,delay and buffer size as follows:1) Block EKF design in [6] operates on an FFT-block basis,which grows increasingly complex as FFT size becomeslarge (e.g., 2048 FFT size). Also, parameters estimatedin this method are handled in a high dimension manner.2) Parallel EKF design in [7] also operates on an FFT-blockbasis, marking it complex under large FFT size. Parame-ters in this design are jointly estimated by calculating the It is also applicable for multi-user scenario with non-overlapping trainingsymbols, but this is not the case considered in this paper. ovariance information between different users, leadingto a high matrix dimension.3) Beyond FFT size and parameter dimension issues similarto item 1 and 2, particle filter design in [7] needs torepeat the Kalman operation at multiple particle samples,yielding a multiplicative effect on complexity.To summarize, the major challenge in implementing a low-complexity EKF design lies in the factors of: (i) multipleTX users; (ii) multiple RX antennas; (iii) high parameterdimension; and (iv) large FFT size (e.g., 2048 size).With low-complexity and low-delay requirement in mind,this paper will present a time-sharing and interference miti-gation based Extended Kalman Filter (EKF) design for multi-user cognitive communications, which can estimate the CFOand channel parameters from multiple TX users to multiple RXantennas in a time-sharing manner. Here low delay property isachieved by using Kalman filter estimation at each RX samplein a real-time manner, and low complexity property is achievedby reusing a single user EKF design in a time-sharing andpipeling way. We first present a fundamental EKF design thatestimates the CFO and channel parameters from one TX userto one RX antenna. Then such basic EKF design is reused in atime-shared way to estimate the parameters from multiple TXusers to multiple RX antennas. Meanwhile, at each RX sample,an interference mitigation strategy is developed to estimate andremove the expected multi-user interference. In addition, weprovide an adaptive noise variance tracking module to furtherenhance the estimation performance. Because of the usageof EKF structure, our design is essentially different from theparticle filters in [7] and the EM method in [5]. Our designis also different from the Parallel-EKF design in [7] and theFFT-Block EKF design in [6] at the following perspectives:(i) our design runs at each time domain RX sample, not at anFFT-block basis; (ii) our design treats each user separately, notjointly; (iii) our design can be implemented in a time-sharingway, which is less considered in [6] and [7]; (iv) systemmodel in our design is different from the ones in [6], [7] byintegrating CFO parameter into channel response (see Eqn.(5) in section II). Analysis and simulations results validatethat our proposed design can closely approach the Cramer-Rao bound, and has lower computational complexity than theones in [6], [7]. Finally, although cognitive communication isa typical application scenario for our proposed design, it isalso applicable in many other multi-user systems that satisfythe conditions presented in section II.II. S YSTEM M ODEL
A. Problem Formulation
We consider a total of Q TX users in the cognitive network.One of them is the primary TX user (i.e., base station),and the rest are all secondary TX users. Primary TX user’stransmission is based on a time division MAC protocol, wheretime is divided into different time frames with equal duration. Time-sharing in this paper indicates that the same hardware module canbe reused by different processes at separate time slots.
Secondary users can maintain time synchronization with theprimary TX user by learning and synchronizing with its timeframes. Each secondary RX user is equipped with N A multipleantennas to decode the packets. Without loss of generality,we assume that every TX user has only one spatial stream,and there exists Q ≤ N A . Also, every TX user has a distincttraining symbol s q ( n ) with ≤ q ≤ Q and ≤ n ≤ N F − .Here N F is the FFT size of OFDM system.Each TX user has an independent carrier frequency offset(CFO) that is caused by both the oscillator offset and thedoppler offset. Denote TX user q ’s CFO value as ε q . Fora given secondary RX user, the channel from TX user q tothe m th RX antenna of this secondary user is denoted as h q,m ( p q,ml ) , ∀ ≤ l ≤ L max . Here L max is the number of timedomain paths in the channel response, and p q,ml is an integervalue representing the relative delay of the l th path. We assumethat all p q,ml values have already been determined at an earlystage (e.g., using PN sequences at coarse synchronization).The received signal at the m th RX antenna is derived as: y m ( n ) = Q X q =1 exp (cid:18) j πε q nN F (cid:19) L max X l =1 h q,m ( p q,ml ) s q [( n − p q,ml ) N F ]+ z m ( n ) , ≤ n ≤ N F − (1)where ( n − p q,ml ) N F = { ( n − p q,ml ) mod N F } is circularshift, and z m ( n ) is the background noise at the m th RXantenna.The task in this work is to estimate CFO parameter ε q andchannel parameter h q,m ( p q,ml ) for all users ( ≤ q ≤ Q ) andall antennas ( ≤ m ≤ N A ). Obviously, the optimal estimationis the solution to this maximum likelihood (ML) problem: min N A X m =1 (cid:12)(cid:12) y m ( n ) − Q X q =1 exp (cid:18) j π b ε q nN F (cid:19) L max X l =1 b h q,m ( p q,ml ) s q [( n − p q,ml ) N F ] (cid:12)(cid:12) (2)where b ε q and b h q,m ( p q,ml ) represent the estimated values. Thereare a total of ( L max N A + 1) Q parameters in Eqn. (2), whichconstitutes a high-dimensional parameter estimation problem. B. State-Space Formulation
ML solution can generally approach the optimal perfor-mance but it requires huge computations, which are highlyundesirable in most systems. Instead, this paper proposesan EKF design for the estimation of the CFO and channelparameters, which can sequentially update the estimationresults at each RX sample, resulting in low buffer size andlow estimation delay. Initially, it is straightforward to directlyapply an EKF design at Eqn. (2) by building all CFO andchannel parameters into one state vector, whose dimension isas high as ( L max N A + 1) Q . This method will significantlyincrease the complexity of the derived Kalman filter. With This unique training symbol can be determined according to either theunique user ID in the network, or the access order in the current time frame. uch complexity consideration in mind, we first propose a low-dimensional EKF design that can estimate the parameters fromone TX user to one RX antenna, which has only ( L max + 1) parameters. Then we reuse this fundamental EKF designin a time-shared manner to estimate the parameters frommultiple TX users to multiple RX antennas. In this way, high-dimensional parameters are estimated by sequentially reusinga low-dimensional estimator, which reduces the complexity ofthe proposed EKF design.We first present an RX signal formulation from the perspec-tive of TX user q and the m th RX antenna as: y q,m ( n ) = exp (cid:18) j πε q nN F (cid:19) L max X l =1 h q,m ( p q,ml ) s q [( n − p q,ml ) N F ]+ z q,m ( n ) , (3)here y q,m ( n ) is extracted from y m ( n ) with the aid of interfer-ence mitigation module, and z q,m ( n ) represents the residualnoise at TX user q and the m th RX antenna, which includesboth the residual co-channel interference from other TX usersand the background noise at the m th RX antenna. Additionally,the initial value of y q,m ( n ) without any interference mitigationis set to y q,m ( n ) = y m ( n ) . Details about the interferencemitigation module will be given in section III.Now we define the associated state vector as: X q,m ( n ) = [ ε q , H q,m ( n )] T , (4) H q,m ( n ) = exp (cid:18) j πnε q N F (cid:19) (cid:2) h q,m ( p q,m ) , ..., h q,m ( p q,mL max ) (cid:3) . (5)The state-space model for TX user q and the m th RX antennacan be derived from (3) as: X q,m ( n ) = f { X q,m ( n − } = D q,m ( ε q ) X q,m ( n − , (6) D q,m ( ε q ) = (cid:20) × L max L max × exp ( j πε q /N F ) I L max × L max (cid:21) (7) y q,m ( n ) = G q,m ( n ) X q,m ( n ) + z q,m ( n ) , (8) G q,m ( n ) = (cid:2) , s q [( n − p q,m ) N F ] , s q [( n − p q,m ) N F ] , ...,s q (cid:2) ( n − p q,mL max ) N F (cid:3) (cid:3) . (9)Finally, it is worth mentioning that the derived state-spaceformulation is a nonlinear model, since there is a nonlinearcomponent exp ( j πε q /N F ) in the matrix D q,m ( ε q ) .III. I NTERFERENCE M ITIGATION BASED
EKF D
ESIGN
This section sequentially describes (i) the basic EKF designthat aims at only one TX user and one RX antenna, (ii)the interference mitigation module that cancels co-channelinterference at each RX sample, (iii) the proposed adaptivenoise variance tracking module, and (iv) the overall paradigmof the proposed design.
A. Fundamental EKF Design
The key idea behind the EKF design is using Jacobianderivation to linearize the nonlinear matrix D q,m ( ε q ) at local estimates: F q,m ( n −
1) = ∂f ( X q,m ( n − ∂ X q,m ( n − (cid:12)(cid:12)(cid:12)(cid:12) b X q,m ( n − | n − = (cid:20) × L max α ( n − b H Tq,m ( n − | n −
1) exp ( α ( n − I L max × L max (cid:21) (10) α ( n −
1) = j π b ε q,m ( n − | n − /N F (11)where b X q,m ( n − | n − represents the estimated state vec-tor after processing the ( n − th RX sample. Based on (10),the prediction steps in our fundamental EKF design are: b X q,m ( n | n −
1) = D q,m ( b ε q,m ( n − | n − b X q,m ( n − | n − , (12) P q,m ( n | n −
1) = F q,m ( n − P q,m ( n − | n − F Hq,m ( n − . (13)And the updating steps are as follows: K q,m ( n ) = P q,m ( n | n − G Hq,m ( n ) × (cid:2) G q,m ( n ) P q,m ( n | n − G Hq,m ( n ) + σ q,m ( n ) (cid:3) − , (14) P q,m ( n | n ) =[ I − K q,m ( n ) G q,m ( n )] P q,m ( n | n − , (15) b X q,m ( n | n ) = b X q,m ( n | n − K q,m ( n ) h y q,m ( n ) − G q,m ( n ) b X q,m ( n | n − i . (16)Here σ q,m ( n ) represents the variance of the observation noise z q,m ( n ) in Eqn. (3). Also, CFO estimate b ε q,m ( n | n ) in the statevector b X q,m ( n | n ) should only use its real part as b ε q,m ( n | n ) =Real [ b ε q,m ( n | n )] . Finally, the EKF design presented above isonly used for the parameter estimation of one TX user andone RX antenna. This basic EKF design is then iterated in atime-shared manner to estimate the parameters of all TX users( ≤ q ≤ Q ) and all RX antennas ( ≤ m ≤ N A ). B. Interference Mitigation and Refined CFO Estimation
Before describing the interference mitigation module, wefirst look at the refinement of the CFO estimates. AlthoughTX user q has only one CFO parameter, our proposed EKFdesign can result in N A different estimates that are derivedfrom N A RX antennas, which are denoted as b ε q,m ( n | n − , ≤ m ≤ N A . As a result, we can use these N A differentestimates to get a refined result b ε q ( n | n − , which is calculatedas: b ε q ( n | n −
1) = N A X m =1 / P q,m ( n | n − (1 , P N A r =1 / P q,r ( n | n − (1 , b ε q,m ( n | n − , (17)where P q,m ( n | n − (1 , denotes P q,m ( n | n − ’s elementlocated at the st row and st column.Recall that y q,m ( n ) involved in (3) and (16) is extractedfrom the original RX signal y m ( n ) with the help of aninterference mitigation strategy. Having derived the refined ulti-UserInterferenceMitigation Rx samples@multipleantennas EKF for User 1, Rx antenna1EKF for User 2, Rx antenna1EKF for User Q-1, Rx antenna N A EKF for User Q, Rx antenna N A … ... Noise variance trackingNoise variance trackingNoise variance trackingNoise variance tracking … ... Fig. 1. Block diagram of our proposed design.
CFO estimate b ε q ( n | n − , now the interference mitigationprocess can be applied at y q,m ( n ) as follows: y q,m ( n ) = y m ( n ) − Q X u =1 ,u = q exp (cid:18) j π b ε u ( n | n − N F (cid:19) · G u,m ( n ) b X u,m ( n | n − . (18) C. Adaptive Noise Variance Tracking
Since y q,m ( n ) is extracted from y m ( n ) via interferencemitigation module, the variance of z q,m ( n ) , (i.e., σ q,m ( n ) usedin Eqn. (14)), is varying during the convergence process of theinterference mitigation module, which has to be adaptivelytracked. Such variance tracking is based on the followingobservation: E (cid:12)(cid:12)(cid:12) y q,m ( n ) − G q,m ( n ) b X q,m ( n | n − (cid:12)(cid:12)(cid:12) ≈ G q,m ( n ) P q,m ( n | n − G Hq,m ( n ) + σ q,m ( n ) . (19)Using (19), noise variance σ q,m ( n ) can be tracked as: σ q,m ( n ) = (cid:20) − − b − b n +1 (cid:21) · σ q,m ( n − − b − b n +1 · { max [ e q,m ( n ) , } , (20) e q,m ( n ) = (cid:12)(cid:12)(cid:12) y q,m ( n ) − G q,m ( n ) b X q,m ( n | n − (cid:12)(cid:12)(cid:12) − G q,m ( n ) P q,m ( n | n − G Hq,m ( n ) , (21)where b = 0 . is the decay factor used to exponentiallyweight the history values. D. Block Diagram
The complete functional diagram of our proposed designis shown in Fig. 1. In this paradigm, received samples at allRX antennas are first processed in the interference mitigationmodule. Then the resultant samples are sequentially processedin the basic EKF module and noise variance tracking module.For the ease of description, all components in Fig. 1 are de-picted in a parallel manner. However, in practice these designcomponents can be implemented in a time-shared manner, andonly one single EKF module is physically required.
TABLE IC
OMPLEXITY C OMPARISON (N UMBER OF C OMPLEX M ULTIPLICATIONS ) Design Name Number of Complex Multiplications
Proposed Design ≈ L + 10 L + 14 L max + 2 Full-State EKF O (cid:8) Q ( L max N A + 1) (cid:9) FFT-Block EKF [6] O{ N A N F ( QN A ) ( L max + 1) + QN A ( N A N F ) ( L max + 1) +( QN A ) ( L max + 1) } Parallel EKF [7] O{ Q N F ( L max + 1) + QN F ( L max + 1) + QN F ( L max + 1) } Particle Filter [7] O{ N P Q N F ( L max + 1) + N P QN F ( L max + 1) + N P QN F ( L max + 1) } N P is the number of particle samples.EM method [5] O{ N L QN F L max + N L QN F L } N L is the number of iterations. IV. S
IMULATIONS AND D ISCUSSIONS
A. Parameters Setup
The OFDM system built in the simulation has a bandwidthof 20MHz and a FFT size N F = 2048 . Each TX user’sCFO value is independently and randomly generated withinthe range of [-2, 2] . Wireless channels are generated usingthe SUI-3 channel model [8], which has L max = 3 non-zeropaths at the time domain. SNR in this paper is defined asthe ratio of the signal power to the noise power at one RXantenna, i.e., SNR = σ R /σ Z where σ R is the total receivedsignal power at one RX antenna that is coming from allTX users, and σ Z is the power of the background noise. Inthe simulation, CFO and channel parameters are estimatedusing one OFDM training symbol with N F = 2048 samples.Estimation results are validated via the mean square error(MSE) performance. Specifically, MSE for channel estimationis defined as a normalized version: MSE( h q,m ) = P L max l =1 (cid:12)(cid:12)(cid:12)b h q,m ( p q,ml ) − h q,m ( p q,ml ) (cid:12)(cid:12)(cid:12) P L max l =1 | h q,m ( p q,ml ) | . (22)Cramer-Rao bounds for the MSE results of CFO and channelestimation can be derived according to [9] as: CRB
CFO (SNR) = 3 Q π · N F · SNR · N A , (23) CRB
Chan (SNR) = (cid:18) L max N F + 32 N F (cid:19) Q SNR . (24) B. Simulation Results
We consider a cognitive network with one primary linkand three secondary links (a total of 4 links). We investigatethe CFO and channel estimations at one secondary RX userwith N A = 4 RX antennas. Without loss of generality, weassume that this secondary RX user has the same receivedpower from all TX users. We plot the MSE results of CFOand channel estimation in Fig. 2 and Fig. 3, respectively. The Since integer frequency offsets can generally be estimated during thecoarse synchronization stage, CFO value at fine synchronization stage isusually between -0.5 and 0.5. But here we use range 2 to demonstrate ourdesign’s estimation performance. esults show that our proposed design can closely approachthe Cramer-Rao bounds. In addition, we repeat our simulationby disabling interference mitigation module, or noise variancetracking module. The corresponding results (Fig. 2 and Fig.3) indicate that without interference mitigation, the estimationperformance can be dramatically degraded. And without noisevariance tracking module, there could be an error floor athigh SNR values because of the inaccurate tracking of noisevariance information. Using the values in Fig. 2 and Fig. 3,it is feasible to further investigate the BER/PER performance.But such discussions heavily depend on the designed receiverstructure, which is omitted here for page limitation.
C. Delay, Buffer Size and Complexity Analysis
This subsection evaluates the issues of complexity, delayand buffer size in the considered designs. We first look atthe complexity issue. In particular, we count the number ofcomplex multiplications involved in our proposed EKF design,which is listed in Table I. And for comparison, in that table, wealso list the complexity results of Full-State EKF, FFT-BlockEKF [6], Parallel EKF [7], Particle filter [7], and EM method[5]. Here Full-State EKF represents the EKF that builds all ( L max N A + 1) Q states in (2) into one state vector, yieldinghigh state dimension. We can see that our proposed designenjoys the lowest computation complexity, which is only atthe order of L . But Full-State EKF’s complexity is around Q N A higher than our design. Moreover, FFT-Block EKF,Parallel EKF, Particle Filter, and EM method’s complexities all rely on FFT size N F , which is significantly large in ourcase ( N F = 2048 ).Now we further look at the delay and buffer size in theproposed design. Since our EKF scheme updates the Kalmanestimate at each RX sample (not at each FFT block) in anonline and real-time manner, it has low estimation delay andrequires low buffer size. However, Particle Filter [7], ParallelEKF [7], and EM approach [5] all operate at an FFT-blockbasis with buffer size N F = 2048 samples, resulting in both alarge delay and a large buffer size. Even worse, particle filterand EM method both need to process the FFT-block multipletimes (e.g., particle samples in particle filter, and iterations inEM method), leading to additional estimation delay.V. C ONCLUSION
This paper has presented a low-delay and low-complexityEKF design that can estimate the CFO and channel parametersin multi-user cognitive communications. We first present afundamental EKF design that works at one TX user andone RX antenna. Then this basic EKF design is reused ina time-shared way to estimate the parameters for multipleTX users at multiple RX antennas. Besides, an interferencemitigation strategy is proposed to estimate and cancel themulti-user interference at each RX sample. Moreover, adaptivenoise variance tracking module is further employed to furtherenhance the estimation performance. Compared with existing Particle filters in [7] and EM designs in [5] have even higher complexitybecause of either the number of particle samples, or the number of iterations. −10 −8 −6 −4 −2 SNR (dB) M SE o f C F O E s t i m a t i on Proposed Designw/o interference mitigationw/o noise variance trackingCramer−Rao bound
Fig. 2. CFO estimation’s MSE results under different SNR values. −8 −6 −4 −2 SNR (dB) M SE o f C hanne l E s t i m a t i on Proposed Designw/o interference mitigationw/o noise variance trackingCramer−Rao bound
Fig. 3. Channel estimation’s MSE results under different SNR values. related designs (FFT-Block EKF [6], Parallel EKF [7], Particlefilter [7], and EM method [5]), our proposed design enjoyslow computation complexity (because of pipelining and time-sharing design), low delay and low buffer size (due to its onlineand run-time estimation). Besides, its estimation performancecan closely approach the Cramer-Rao bound.R
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IEEE Transactions onCommunications , vol. 51, no. 11, pp. 1910–1917, Nov. 2003. r X i v : . [ c s . I T ] N ov A Low-Delay Low-Complexity EKF Design forJoint Channel and CFO Estimation in Multi-UserCognitive Communications
Pengkai Zhao
Electrical Engineering, UCLA, CA, USA
Cong Shen
Qualcomm, Inc., San Diego, CA, USA
Abstract —Parameter estimation in cognitive communicationscan be formulated as a multi-user estimation problem, which issolvable under maximum likelihood solution but involves highcomputational complexity. This paper presents a time-sharingand interference mitigation based EKF (Extended Kalman Filter)design for joint CFO (carrier frequency offset) and channelestimation at multiple cognitive users. The key objective is torealize low implementation complexity by decomposing high-dimensional parameters into multiple separate low-dimensionalestimation problems, which can be solved in a time-sharedmanner via pipelining operation. We first present a basic EKFdesign that estimates the parameters from one TX user to oneRX antenna. Then such basic design is time-shared and reusedto estimate parameters from multiple TX users to multiple RXantennas. Meanwhile, we use interference mitigation moduleto cancel the co-channel interference at each RX sample. Inaddition, we further propose adaptive noise variance trackingmodule to improve the estimation performance. The proposeddesign enjoys low delay and low buffer size (because of its onlinereal-time processing), as well as low implementation complexity(because of time-sharing and pipeling design). Its estimationperformance is verified to be close to Cramer-Rao bound.
I. I
NTRODUCTION
Cognitive communication system is widely accepted asa perspective way in increasing the spectrum efficiency ofwireless networks, where primary links and secondary linkscan usually co-exist in the network, resulting in an interferencelimited environment. Parameter estimation in cognitive com-munications is a challenging problem because of (i) the exis-tence of co-channel interference, and (ii) the high-dimensionalparameters from multiple TX users to multiple RX anten-nas. In particular, note that different TX users often haveindependent carrier frequency offset (CFO) values (includingboth oscillator offsets and doppler offsets), which usuallyintroduce serious nonlinear components within the observedsignal, complicating the estimation problem. Meanwhile, chan-nel responses from multiple TX users to multiple RX antennascan result in a set of high-dimensional parameters, which arealso difficult to estimate. Finally, due to the existence of multi-user interference, CFO and channel parameters usually haveto be treated together and be estimated in a joint way so asto approach the optimal performance, which further increasesthe estimation complexity.Without loss of generality, this paper assumes OrthogonalFrequency Division Multiplexing (OFDM) system, which is an overwhelming choice for modern wireless systems. Theclassical CFO and channel estimation method in a single-userOFDM system is based on two repeated training symbols[ ? ]. It has low implementation complexity and near-optimalperformance, but only applies to a single-user scenario . Inmulti-user OFDMA systems with unique subcarrier set perTX user, CFO and channel parameters can be recovered byexploiting distinct subcarrier structures among TX users ([ ? ],[ ? ]). But this method requires separate subcarrier allocation fordifferent users. Consequently, in a general multi-user cognitivesystem without specific subcarrier allocation per user and withoverlapped training symbols, ML and EM related methodsseem to be the only applicable choice, where all TX users’parameters have to be formulated into a maximum likelihood(ML) estimation problem [ ? ], which is solvable under Ex-pectation Maximization (EM) method [ ? ] in an iterative way.However, such ML and EM approaches often require highcomputational complexity and high processing delay, becausein these methods, the entire OFDM block is stored offline andis iteratively processed multiple times.Based on above considerations, this paper will focus onusing Kalman filter structure to estimate the CFO and channelparameters in multi-user cognitive communications. Our majorobjective is to achieve low-complexity and low-delay estima-tion performance in cognitive systems. In general, Kalmanfilter is a good candidate for low delay and low complexity pa-rameter estimation primarily due to its real-timing processingproperty. It has been conventionally used for CFO and channelestimation in multi-user OFDM systems, e.g., the FFT-BlockEKF design in [ ? ], the parallel EKF design in [ ? ], and theparticle filter design in [ ? ]. However, these existing designsinherently suffer from multiple issues related to complexity,delay and buffer size as follows:1) Block EKF design in [ ? ] operates on an FFT-blockbasis, which grows increasingly complex as FFT sizebecomes large (e.g., 2048 FFT size). Also, parametersin this method are jointly estimated with high parameterdimension.2) Parallel EKF design in [ ? ] also operates on an FFT-blockbasis, marking it complex under large FFT size. Parame- It is also applicable for multi-user scenario with non-overlapping trainingsymbols, but this is not the case considered in this paper. ers in this design are jointly estimated by calculating thecovariance information between different users, leadingto a high matrix dimension.3) Beyond FFT size and parameter dimension issues similarto item 1 and 2, particle filter design in [ ? ] needs torepeat the Kalman operation at multiple particle samples,yielding a multiplicative effect on complexity.To summarize, the major challenge in implementing a low-complexity EKF design lies in the factors of: (i) multipleTX users; (ii) multiple RX antennas; (iii) high parameterdimension; and (iv) large FFT size (e.g., 2048 size).With low-complexity and low-delay requirement in mind,this paper will present a time-sharing and interference miti-gation based Extended Kalman Filter (EKF) design for multi-user cognitive communications, which can estimate the CFOand channel parameters from multiple TX users to multiple RXantennas in a time-sharing manner. Here low delay property isachieved by using Kalman filter estimation at each RX samplein a real-time manner, and low complexity property is achievedby reusing a single user EKF design in a time-sharing andpipeling way. We first present a fundamental EKF design thatestimates the CFO and channel parameters from one TX userto one RX antenna. Then such basic EKF design is reused in atime-shared way to estimate the parameters from multiple TXusers to multiple RX antennas. Meanwhile, at each RX sample,an interference mitigation strategy is developed to estimate andremove the expected multi-user interference. In addition, weprovide an adaptive noise variance tracking module to furtherenhance the estimation performance. Because of the usageof EKF structure, our design is essentially different from theparticle filters in [ ? ] and the EM method in [ ? ]. Our designis also different from the Parallel-EKF design in [ ? ] and theFFT-Block EKF design in [ ? ] at the following perspectives:(i) our design runs at each time domain RX sample, not at anFFT-block basis; (ii) our design treats each user separately, notjointly; (iii) our design can be implemented in a time-sharingway, which is less considered in [ ? ] and [ ? ]; (iv) systemmodel in our design is different from the ones in [ ? ], [ ? ] byintegrating CFO parameter into channel response (see Eqn.(5) in section II). Analysis and simulations results validatethat our proposed design can closely approach the Cramer-Rao bound, and has lower computational complexity than theones in [ ? ], [ ? ]. Finally, although cognitive communication isa typical application scenario for our proposed design, it isalso applicable in many other multi-user systems that satisfythe conditions presented in section II.II. S YSTEM M ODEL
A. Problem Formulation
We consider a total of Q TX users in the cognitive network.One of them is the primary TX user (i.e., base station),and the rest are all secondary TX users. Primary TX user’stransmission is based on a time division MAC protocol, where Time-sharing in this paper indicates that the same hardware module canbe reused for different processes at separate time slots. time is divided into different time frames with equal duration.Secondary users can maintain time synchronization with theprimary TX user by learning and synchronizing with its timeframes. Each secondary RX user is equipped with N A multipleantennas to decode the packets. Without loss of generality,we assume that every TX user has only one spatial stream,and there exists Q ≤ N A . Also, every TX user has a distincttraining symbol s q ( n ) with ≤ q ≤ Q and ≤ n ≤ N F − .Here N F is the FFT size of OFDM system.Each TX user has an independent carrier frequency offset(CFO) that is caused by both the oscillator offset and thedoppler offset. Denote TX user q ’s CFO value as ε q . Fora given secondary RX user, the channel from TX user q tothe m th RX antenna of this secondary user is denoted as h q,m ( p q,ml ) , ∀ ≤ l ≤ L max . Here L max is the number of timedomain paths in the channel response, and p q,ml is an integervalue representing the relative delay of the l th path. We assumethat all p q,ml values have already been determined at an earlystage (e.g., using PN sequences at coarse synchronization).The received signal at the m th RX antenna is derived as: y m ( n ) = Q X q =1 exp (cid:18) j πε q nN F (cid:19) L max X l =1 h q,m ( p q,ml ) s q [( n − p q,ml ) N F ]+ z m ( n ) , ≤ n ≤ N F − (1)where ( n − p q,ml ) N F = { ( n − p q,ml ) mod N F } is circularshift, and z m ( n ) is the background noise at the m th RXantenna.The task in this work is to estimate CFO parameter ε q andchannel parameter h q,m ( p q,ml ) for all users ( ≤ q ≤ Q ) andall antennas ( ≤ m ≤ N A ). Obviously, the optimal estimationis the solution to this maximum likelihood (ML) problem: min N A X m =1 (cid:12)(cid:12) y m ( n ) − Q X q =1 exp (cid:18) j π b ε q nN F (cid:19) L max X l =1 b h q,m ( p q,ml ) s q [( n − p q,ml ) N F ] (cid:12)(cid:12) (2)where b ε q and b h q,m ( p q,ml ) represent the estimated values. Thereare a total of ( L max N A + 1) Q parameters in Eqn. (2), whichconstitutes a high-dimensional parameter estimation problem. B. State-Space Formulation
ML solution can generally approach the optimal perfor-mance but it requires huge computations, which are highlyundesirable in most systems. Instead, this paper proposesan EKF design for the estimation of the CFO and channelparameters, which can sequentially update the estimationresults at each RX sample, resulting in low buffer size andlow estimation delay. Initially, it is straightforward to directlyapply an EKF design at Eqn. (2) by building all CFO andchannel parameters into one state vector, whose dimension isas high as ( L max N A + 1) Q . This method will significantly This unique training symbol can be determined according to either theunique user ID in the network, or the access order in the current time frame. ncrease the complexity of the derived Kalman filter. Withsuch complexity consideration in mind, we first propose a low-dimensional EKF design that can estimate the parameters fromone TX user to one RX antenna, which has only ( L max + 1) parameters. Then we reuse this fundamental EKF designin a time-shared manner to estimate the parameters frommultiple TX users to multiple RX antennas. In this way, high-dimensional parameters are estimated by sequentially reusinga low-dimensional estimator, which reduces the complexity ofthe proposed EKF design.We first present an RX signal formulation from the perspec-tive of TX user q and the m th RX antenna as: y q,m ( n ) = exp (cid:18) j πε q nN F (cid:19) L max X l =1 h q,m ( p q,ml ) s q [( n − p q,ml ) N F ]+ z q,m ( n ) , (3)here y q,m ( n ) is extracted from y m ( n ) with the aid of interfer-ence mitigation module, and z q,m ( n ) represents the residualnoise at TX user q and the m th RX antenna, which includesboth the residual co-channel interference from other TX usersand the background noise at the m th RX antenna. Additionally,the initial value of y q,m ( n ) without any interference mitigationis set to y q,m ( n ) = y m ( n ) . Details about the interferencemitigation module will be given in section III.Now we define the associated state vector as: X q,m ( n ) = [ ε q , H q,m ( n )] T , (4) H q,m ( n ) = exp (cid:18) j πnε q N F (cid:19) (cid:2) h q,m ( p q,m ) , ..., h q,m ( p q,mL max ) (cid:3) . (5)The state-space model for TX user q and the m th RX antennacan be derived from (3) as: X q,m ( n ) = f { X q,m ( n − } = D q,m ( ε q ) X q,m ( n − , (6) D q,m ( ε q ) = (cid:20) × L max L max × exp ( j πε q /N F ) I L max × L max (cid:21) (7) y q,m ( n ) = G q,m ( n ) X q,m ( n ) + z q,m ( n ) , (8) G q,m ( n ) = (cid:2) , s q [( n − p q,m ) N F ] , s q [( n − p q,m ) N F ] , ...,s q (cid:2) ( n − p q,mL max ) N F (cid:3) (cid:3) . (9)Finally, it is worth mentioning that the derived state-spaceformulation is a nonlinear model, since there is a nonlinearcomponent exp ( j πε q /N F ) in the matrix D q,m ( ε q ) .III. I NTERFERENCE M ITIGATION BASED
EKF D
ESIGN
This section sequentially describes (i) the basic EKF designthat aims at only one TX user and one RX antenna, (ii)the interference mitigation module that cancels co-channelinterference at each RX sample, (iii) the proposed adaptivenoise variance tracking module, and (iv) the overall paradigmof the proposed design.
A. Fundamental EKF Design
The key idea behind the EKF design is using Jacobianderivation to linearize the nonlinear matrix D q,m ( ε q ) at local estimates: F q,m ( n −
1) = ∂f ( X q,m ( n − ∂ X q,m ( n − (cid:12)(cid:12)(cid:12)(cid:12) b X q,m ( n − | n − = (cid:20) × L max α ( n − b H Tq,m ( n − | n −
1) exp ( α ( n − I L max × L max (cid:21) (10) α ( n −
1) = j π b ε q,m ( n − | n − /N F (11)where b X q,m ( n − | n − represents the estimated state vec-tor after processing the ( n − th RX sample. Based on (10),the prediction steps in our fundamental EKF design are: b X q,m ( n | n −
1) = D q,m ( b ε q,m ( n − | n − b X q,m ( n − | n − , (12) P q,m ( n | n −
1) = F q,m ( n − P q,m ( n − | n − F Hq,m ( n − . (13)And the updating steps are as follows: K q,m ( n ) = P q,m ( n | n − G Hq,m ( n ) × (cid:2) G q,m ( n ) P q,m ( n | n − G Hq,m ( n ) + σ q,m ( n ) (cid:3) − , (14) P q,m ( n | n ) =[ I − K q,m ( n ) G q,m ( n )] P q,m ( n | n − , (15) b X q,m ( n | n ) = b X q,m ( n | n − K q,m ( n ) h y q,m ( n ) − G q,m ( n ) b X q,m ( n | n − i . (16)Here σ q,m ( n ) represents the variance of the observation noise z q,m ( n ) in Eqn. (3). Also, CFO estimate b ε q,m ( n | n ) in the statevector b X q,m ( n | n ) should only use its real part as b ε q,m ( n | n ) =Real [ b ε q,m ( n | n )] . Finally, the EKF design presented above isonly used for the parameter estimation of one TX user andone RX antenna. This basic EKF design is then iterated in atime-shared manner to estimate the parameters of all TX users( ≤ q ≤ Q ) and all RX antennas ( ≤ m ≤ N A ). B. Interference Mitigation and Refined CFO Estimation
Before describing the interference mitigation module, wefirst look at the refinement of the CFO estimates. AlthoughTX user q has only one CFO parameter, our proposed EKFdesign can result in N A different estimates that are derivedfrom N A RX antennas, which are denoted as b ε q,m ( n | n − , ≤ m ≤ N A . As a result, we can use these N A differentestimates to get a refined result b ε q ( n | n − , which is calculatedas: b ε q ( n | n −
1) = N A X m =1 / P q,m ( n | n − (1 , P N A r =1 / P q,r ( n | n − (1 , b ε q,m ( n | n − , (17)where P q,m ( n | n − (1 , denotes P q,m ( n | n − ’s elementlocated at the st row and st column.Recall that y q,m ( n ) involved in (3) and (16) is extractedfrom the original RX signal y m ( n ) with the help of aninterference mitigation strategy. Having derived the refined ulti-UserInterferenceMitigation Rx samples@multipleantennas EKF for User 1, Rx antenna1EKF for User 2, Rx antenna1EKF for User Q-1, Rx antenna N A EKF for User Q, Rx antenna N A … ... Noise variance trackingNoise variance trackingNoise variance trackingNoise variance tracking … ... Fig. 1. Block diagram of our proposed design.
CFO estimate b ε q ( n | n − , now the interference mitigationprocess can be applied at y q,m ( n ) as follows: y q,m ( n ) = y m ( n ) − Q X u =1 ,u = q exp (cid:18) j π b ε u ( n | n − N F (cid:19) · G u,m ( n ) b X u,m ( n | n − . (18) C. Adaptive Noise Variance Tracking
Since y q,m ( n ) is extracted from y m ( n ) via interferencemitigation module, the variance of z q,m ( n ) , (i.e., σ q,m ( n ) usedin Eqn. (14)), is varying during the convergence process of theinterference mitigation module, which has to be adaptivelytracked. Such variance tracking is based on the followingobservation: E (cid:12)(cid:12)(cid:12) y q,m ( n ) − G q,m ( n ) b X q,m ( n | n − (cid:12)(cid:12)(cid:12) ≈ G q,m ( n ) P q,m ( n | n − G Hq,m ( n ) + σ q,m ( n ) . (19)Using (19), noise variance σ q,m ( n ) can be tracked as: σ q,m ( n ) = (cid:20) − − b − b n +1 (cid:21) · σ q,m ( n − − b − b n +1 · { max [ e q,m ( n ) , } , (20) e q,m ( n ) = (cid:12)(cid:12)(cid:12) y q,m ( n ) − G q,m ( n ) b X q,m ( n | n − (cid:12)(cid:12)(cid:12) − G q,m ( n ) P q,m ( n | n − G Hq,m ( n ) , (21)where b = 0 . is the decay factor used to exponentiallyweight the history values. D. Block Diagram
The complete functional diagram of our proposed designis shown in Fig. 1. In this paradigm, received samples at allRX antennas are first processed in the interference mitigationmodule. Then the resultant samples are sequentially processedin the basic EKF module and noise variance tracking module.For the ease of description, all components in Fig. 1 are de-picted in a parallel manner. However, in practice these designcomponents can be implemented in a time-shared manner, andonly one single EKF module is physically required.
TABLE IC
OMPLEXITY C OMPARISON (N UMBER OF C OMPLEX M ULTIPLICATIONS ) Design Name Number of Complex Multiplications
Proposed Design ≈ L + 10 L + 14 L max + 2 Full-State EKF O (cid:8) Q ( L max N A + 1) (cid:9) FFT-Block EKF [ ? ] O{ N A N F ( QN A ) ( L max + 1) + QN A ( N A N F ) ( L max + 1) +( QN A ) ( L max + 1) } Parallel EKF [ ? ] O{ Q N F ( L max + 1) + QN F ( L max + 1) + QN F ( L max + 1) } Particle Filter [ ? ] O{ N P Q N F ( L max + 1) + N P QN F ( L max + 1) + N P QN F ( L max + 1) } N P is the number of particle samples.EM method [ ? ] O{ N L QN F L max + N L QN F L } N L is the number of iterations. IV. S
IMULATIONS AND D ISCUSSIONS
A. Parameters Setup
The OFDM system built in the simulation has a bandwidthof 20MHz and a FFT size N F = 2048 . Each TX user’sCFO value is independently and randomly generated withinthe range of [-2, 2] . Wireless channels are generated usingthe SUI-3 channel model [ ? ], which has L max = 3 non-zeropaths at the time domain. SNR in this paper is defined asthe ratio of the signal power to the noise power at one RXantenna, i.e., SNR = σ R /σ Z where σ R is the total receivedsignal power at one RX antenna that is coming from allTX users, and σ Z is the power of the background noise. Inthe simulation, CFO and channel parameters are estimatedusing one OFDM training symbol with N F = 2048 samples.Estimation results are validated via the mean square error(MSE) performance. Specifically, MSE for channel estimationis defined as a normalized version: MSE( h q,m ) = P L max l =1 (cid:12)(cid:12)(cid:12)b h q,m ( p q,ml ) − h q,m ( p q,ml ) (cid:12)(cid:12)(cid:12) P L max l =1 | h q,m ( p q,ml ) | . (22)Cramer-Rao bounds for the MSE results of CFO and channelestimation can be derived according to [ ? ] as: CRB
CFO (SNR) = 3 Q π · N F · SNR · N A , (23) CRB
Chan (SNR) = (cid:18) L max N F + 32 N F (cid:19) Q SNR . (24) B. Simulation Results
We consider a cognitive network with one primary linkand three secondary links (a total of 4 links). We investigatethe CFO and channel estimations at one secondary RX userwith N A = 4 RX antennas. Without loss of generality, weassume that this secondary RX user has the same receivedpower from all TX users. We plot the MSE results of CFOand channel estimation in Fig. 2 and Fig. 3, respectively. The CFO value at fine synchronization stage is generally between -0.5 and0.5, because any integer frequency offset can be estimated during the coarsesynchronization stage. But here we use range 2 to demonstrate our design’sestimation performance. esults show that our proposed design can closely approachthe Cramer-Rao bounds. In addition, we repeat our simulationby disabling interference mitigation module, or noise variancetracking module. The corresponding results (Fig. 2 and Fig.3) indicate that without interference mitigation, the estimationperformance can be dramatically degraded. And without noisevariance tracking module, there could be an error floor athigh SNR values because of the inaccurate tracking of noisevariance information. Using the values in Fig. 2 and Fig. 3,we can further investigate the BER/PER performance in thesystem. But such discussions heavily depend on the designedreceiver structure, which is omitted here for page limitation.
C. Delay, Buffer Size and Complexity Analysis
This subsection evaluates the issues of complexity, delayand buffer size in the considered designs. We first look atthe complexity issue. In particular, we count the number ofcomplex multiplications involved in our proposed EKF design,which is listed in Table I. And for comparison, in that table, wealso list the complexity results of Full-State EKF, FFT-BlockEKF [ ? ], Parallel EKF [ ? ], Particle filter [ ? ], and EM method[ ? ]. Here Full-State EKF represents the EKF that builds all ( L max N A + 1) Q states in (2) into one state vector, yieldinghigh state dimension. We can see that our proposed designenjoys the lowest computation complexity, which is only atthe order of L . But Full-State EKF’s complexity is around Q N A higher than our design. Moreover, FFT-Block EKF,Parallel EKF, Particle Filter, and EM method’s complexities all rely on FFT size N F , which is significantly large in ourcase ( N F = 2048 ).Now we further look at the delay and buffer size in theproposed design. Since our EKF scheme updates the Kalmanestimate at each RX sample (not at each FFT block) in anonline and real-time manner, it has low estimation delay andrequires low buffer size. However, Particle Filter [ ? ], ParallelEKF [ ? ], and EM approach [ ? ] all operate at an FFT-blockbasis with buffer size N F = 2048 samples, resulting in both alarge delay and a large buffer size. Even worse, particle filterand EM method both need to process the FFT-block multipletimes (e.g., particle samples in particle filter, and iterations inEM method), leading to additional estimation delay.V. C ONCLUSION
This paper has presented a low-delay and low-complexityEKF design that can estimate the CFO and channel parametersin multi-user cognitive communications. We first present afundamental EKF design that works at one TX user and oneRX antenna. Then this basic EKF design is reused in a time-shared way to estimate the parameters for multiple TX usersat multiple RX antennas. Besides, an interference mitigationstrategy is proposed to estimate and cancel the multi-userinterference at each RX sample. Moreover, adaptive noisevariance tracking module is also employed to further enhancethe estimation performance. Compared with existing related Particle filters in [ ? ] and EM designs in [ ? ] have even higher complexitybecause of either the number of particle samples, or the number of iterations. −10 −8 −6 −4 −2 SNR (dB) M SE o f C F O E s t i m a t i on Proposed Designw/o interference mitigationw/o noise variance trackingCramer−Rao bound
Fig. 2. CFO estimation’s MSE results under different SNR values. −8 −6 −4 −2 SNR (dB) M SE o f C hanne l E s t i m a t i on Proposed Designw/o interference mitigationw/o noise variance trackingCramer−Rao bound
Fig. 3. Channel estimation’s MSE results under different SNR values. designs (FFT-Block EKF [ ? ], Parallel EKF [ ? ], Particle filter[ ? ], and EM method [ ??