An iALM-ICA-based Anti-Jamming DS-CDMA Receiver for LMS Systems
Hyoyoung Jung, Jaewook Kang, Tae Seok Lee, Suil Kim, Kiseon Kim
11 An iALM-ICA-based Anti-Jamming DS-CDMAReceiver for LMS Systems
Hyoyoung Jung,
Student Member, IEEE , Jaewook Kang,
Member, IEEE , Tae Seok Lee,
Student Member, IEEE ,Suil Kim,
Member, IEEE , and Kiseon Kim,
Senior Member, IEEE
Abstract —We consider a land mobile satellite communicationsystem using spread spectrum techniques where the uplink isexposed to MT jamming attacks, and the downlink is corruptedby multi-path fading channels. We proposes an anti-jamming re-ceiver, which exploits inherent low-dimensionality of the receivedsignal model, by formulating a robust principal component anal-ysis (Robust PCA)-based recovery problem. Simulation resultsverify that the proposed receiver outperforms the conventionalreceiver for a reasonable rank of the jamming signal.
Index Terms —Land mobile satellite communications, directsequence spread spectrum, code division multiple access, anti-jamming, robust principal component analysis.
I. I
NTRODUCTION L AND mobile satellite (LMS) communications facilitate amyriad of applications such as Global Navigation Satel-lite System (GNSS) and commercial broadcasting systems,e.g. DVB-RCS and S-DMB [1]. Due to the openness ofLMS communications, both uplink and downlink channels areeasily interrupted by unexpected interferences of surroundingcommunications as well as intentional jammers as reported in[2]. Furthermore, the frequency selectivity caused by multi-path scatters at the receivers’ side makes it difficult to recoverthe source data under LMS applications operating at high-frequency bands [3].Spread spectrum (SS) techniques are data modulation meth-ods that spread the bandwidth of the signal over the bandwidthactually required. Systems adopting SS techniques have beeneffectively utilized for the suppression of interferences, alle-viation of multi-path fading effects, and the resilience againstjamming signals [4]. Code division multiple access (CDMA)provides multiple access capability and helps increasing sys-tem throughput by applying the SS notion. CDMA is classifiedinto time-hopping (TH), frequency-hopping (FH), and directsequence (DS) SS, among which DS-CDMA has been mostlystudied in the literature and widely applicable in reality dueto its low complexity and implementation cost [5], [14]. Theperformance of DS-CDMA is limited by jamming signalsand interferences since they often exceed the anti-jammingcapability of SS techniques.To mitigate the effect of jamming/interference signals,the most common method is to filter the received signal
Authors’ addresses: H. Jung and K. Kim, School of Electrical Engi-neering and Computer Science, Gwangju Institute of Science and Technol-ogy, Gwangju, South Korea (e-mail: { rain, kskim } @gist.ac.kr); T. S. Lee,Telecommunications Technology Association, Seongnam, South Korea ([email protected]) S. Kim, Agency for Defense Development, Daejeon, SouthKorea ([email protected]) J. Kang, Soundly corp., Seoul, South Korea([email protected]). in space, time, and frequency domains [6–10]. Space-timeadaptive processing can mitigate wideband and narrowbandjamming, but it requires additional antennas [6], [7]. Time-frequency filtering can alleviate narrowband and continuous-wave jamming, it however requires some prior informationabout jamming signals [8–10]. A main weakness of afore-mentioned filtering methods is an extreme degradation ofthe anti-jamming performance when the jamming signals arecoming from the same direction with the source signals, andsubsequently high jamming-to-signal ratio (JSR).In this context, blind source separation (BSS) using in-dependent component analysis (ICA) was proposed to relaxrequirements [12]. BSS with ICA separates multiple sourcesignals by analyzing the statistical independences using higherorder statistics with the assumption that signals from differentsources are statistically independent [11]. BSS-ICA provides awide applicability, including blind multiuser detection, whichis to recover the source bit sequence from a received mixturewithout any knowledge of the user spreading code [5], andjamming suppression in CDMA communications [12–14]. Onelimitation of BSS-ICA is that it requires a number of observa-tions equal to or larger than a number of sources that we wantto separate. Additionally, the anti-jamming capability degradeswhen the jamming signals are varying in both the time andfrequency domain, and the source is already corrupted byjamming in the uplink channel.In this work, we investigate the anti-jamming behavior ofCDMA-based LMS communications, where the satellite actsas a simple amplify-and-forward (AnF) relay. We considerthe uplink jamming scenario, which is frequently used inelectronic warfare because it is efficient to impair all receiverscritically at once [2]. The uplink jammer is assumed asa multi-tone (MT) jamming with frequency hopping (FH),which is one of the principal categories of intelligent jam-ming strategies [15]. We observe that the jamming signalsactually rely on a few number of jamming frequencies and anumber of hopping occurrences. With these observations, thematrix representation of the jamming signal can be modeledas a low-rank matrix having low-dimensionality. Low-rankjamming/interference can be easily found in many emergingapplications, including communication and network systems[16], [17]. Based on the scenario we discussed, descriptionsof the signals and the system, including jamming signals aredetailed in Subsection II . A . We also remark that the numberof active users is often much less than the multi-user capacityof systems for many applications, including CDMA [18–20]. This low activity thus implies sparse DS-CDMA signals a r X i v : . [ c s . I T ] M a r having low-dimensionality property.The present paper fruitfully exploits low-dimensionalityattributes to recover the source signal from the received signalwhere an MT-FH jamming was interfering in the uplinkchannel. The approach we propose in this paper is to modelthe DS-CDMA signal and the jamming signal into matrixrepresentations and to formulate the recovery problem intoa matrix decomposition problem. To decompose the receivedsignal by utilizing their low-dimensionality, we suggest ananti-jamming DS-CDMA receiver, including robust principalcomponent analysis (Robust PCA) in addition to ICA basedreceiver. Robust PCA recovers a matrix L from highly cor-rupted measurements Q = L + R , where L and R are low-rank and sparse matrices, respectively [22]. In contrast toGaussian noise in classical PCA, the entries in a sparse matrix R can have larger magnitudes which are unknown. Extensivesimulations show that Robust PCA performs better than ICAonly under the assumptions of the low-rank jamming signaland the sparsity of a transmitted DS-CDMA signal when anumber of users are less than the length of a spreading code.Even in the other cases, the proposed receiver guarantees acomparable anti-jamming performance to ICA only.This paper is organized as follows. Section II formulatesthe system model, uplink scenario, and downlink scenarioseparately. Section III suggests a recovery problem usingmatrix decompositions, Robust PCA and ICA, with algorithmsto solve the optimization problems. Section IV presents nu-merical simulation results to justify the anti-jamming ability ofthe proposed receiver structure. Finally, section V summarizesthe paper. II. S YSTEM M ODEL
The system model considered in this paper consists of atransmitter, a land-based jammer, a satellite, and a receiver.We divide the system model into two subsections: uplinkscenario and downlink scenario. In the uplink scenario, thetransmitted signal and jamming signal models are providedin matrix forms. In the downlink scenario, the LMS channelis formulated as a circulant matrix, and the received signalmodel is given. In what follows, the system model is explainedbased on the block diagram of proposed anti-jamming CDMAstructure depicted in Fig. 1.
A. Uplink Scenario
On the satellite uplink, for synchronous CDMA transmis-sions by K multi-users at the base station, the input data X ∈ R K × N is given in a matrix form where K users have N bits. The input data X can be divided into N column vectorsas follows: X = [ x , · · · , x n , · · · , x N ] ∈ R K × N , (1)where x n = [ x ,n , · · · , x k,n , · · · , x K,n ] is a column vectorthat is a collection of n th bits of K users, and x k,n is the n th bit of the k th user. The transmitted DS-CDMA signal s ( t ) isrepresented as [5]: s ( t ) = K (cid:88) k =1 N (cid:88) n =1 M (cid:88) m =1 x k,n c k ( t − nT b − mT c ) , (2) Input data (cid:1798) ∈ (cid:1337) (cid:3012)(cid:3400)(cid:3015)
DS-CDMA Spreading (cid:1819) (cid:3041) ← (cid:1777) (cid:3041) (cid:1824) (cid:3041) ∈ (cid:1337) (cid:3014)(cid:3400)(cid:2869) in Base stationTransmittedsignal (cid:1793) ∈ (cid:1337) (cid:3014)(cid:3400)(cid:3015) (cid:3397)
UplinkMT-FH Jammer (cid:1784) ∈ (cid:1337) (cid:3014)(cid:3400)(cid:3015)
Uplink ChannelStrong LOS
Satellite with AnF Downlink Multipath ChannelAWGNchannelAnti-Jamming DS-CDMA Receiverin LMS mobilesReceived signal (cid:1799) ∈ (cid:1337) (cid:3014)(cid:3400)(cid:3015)
Output data (cid:1798)(cid:3553) ∈ (cid:1337) (cid:3012)(cid:3400)(cid:3015)
AWGNchannel (cid:1782) (cid:3048)(cid:3043) ∈ (cid:1337) (cid:3014)(cid:3400)(cid:3014) (cid:1782) (cid:3031)(cid:3042)(cid:3050)(cid:3041) ∈ (cid:1337) (cid:3014)(cid:3400)(cid:3014)
Fig. 1. Block diagram of the LMS communication systems with the anti-jamming DS-CDMA receiver where c k ( · ) is the k th user spreading code, T c is chip duration, T b = M T c is the bit duration, and M is the length of thespreading code. Sampling by T c , the encoding of DS-CDMA(2) can be formulated into a matrix representation using the n th spreading code matrix C ( n ) ∈ R M × K . The N numbers ofspreading code matrices are generated at every bit index n =1 , . . . , N , and each spreading code matrix randomly chooses K column vectors from Walsh code W ∈ R M × M . Walsh codeis adopted out of Gold code, maximal length code, and Walshcode due to its orthogonality and simplicity.The transmitted signal matrix S ∈ R M × N , which is acollection of samples s m,n = s ( mT c + nT b ) is the outputof spreading block and is formulated as: S = [ s , · · · , s n , · · · , s N ] ∈ R M × N , (3)where the n th column vector of S is generated by: s n = C ( n ) x n ∈ R M × . (4)The transmitted signal S is jammed by uplink jammingsignals. Typically, jamming signals are characterized by fre-quency parameters such as jamming frequency bandwidthin partial band jamming and jamming frequencies in MTsjamming. In our system model, the MT-FH jamming signalis given as: j ( t ) = (cid:115) P J M p M (cid:88) m =1 δ m ( t )exp[ i πf m t + φ m ] , (5) where P J is the power of the jamming signal, the quantity δ m ( t ) is equal to 1 when the m th frequency is jammed attime t with a probability p , f m is the m th frequency, φ m isthe phase of the m th tone jammer. It is noted that the powerof the jamming signal is divided by M p for the normalization.The Fourier transform of the jamming signal j ( t ) , during n th bit duration [ nT b , nT b + ( m − T c ] , can be representedin a vector j (cid:48) n ∈ R M × . The m th frequency element of j (cid:48) n isformed as: j (cid:48) m,n = δ m ( n ) Z (cid:115) P J M p , (6)where δ m ( n ) is 1 when the m th frequency is jammed at n th bit duration with a probability p , and Z is a random variablethat is distributed normally, i.e., Z ∼ N (0 , .Using the function δ m ( n ) , various types of jamming signals,including narrowband, MT, and wideband jamming, can begenerated by adjusting non-zero frequency components. Thetime domain representation of j (cid:48) n is obtained by inverse Fouriertransformation, i.e., j n = F − { j (cid:48) n } . The jamming signal J ∈ R M × N for entire bit durations is given as: J = [ j , · · · , j n , · · · , j N ] ∈ R M × N . (7)In the case of a typical MT jammer without FH, columnvectors of the jamming signal J are same during the wholeset of bit durations, which is represented as j = j = ... = j N .In other words, the jammer attacks the same frequency com-ponents of all column vectors that is j (cid:48) = j (cid:48) n ∀ n = 1 , . . . , N ,and thus, we obtain the jamming signal J , which is a rank-1matrix. In addition, we consider an MT-FH jamming signalthat the jammer hops the jamming frequency componentsseveral times. Consequently, the jammer also changes jammingvectors j n depending on their frequency vectors j (cid:48) n . If thenumber of hops increases in the MT-FH jamming signal, therank of the jamming signal also increases. For instance, ifthe jammer hops four times, then it generates four jammingfrequency vectors randomly and the jammer transmits theinverse Fourier transform of each jamming frequency vectoruntil the jammer hops their jamming frequencies. Finally, thejamming signal J has four distinct parts and can be calculatedas a rank-4 matrix. Let rank- r denote the rank of the jammingsignal and r represent the number of hopping events. Signal-to-jamming ratio (SJR) is defined as follows:SJR [dB] = 20 log (cid:107) S (cid:107) F (cid:107) J (cid:107) F [dB] , (8)where (cid:107) S (cid:107) F = (cid:113)(cid:80) Mm =1 (cid:80) Nn =1 | s m,n | is the Frobeniousnorm of a matrix S , which represents the signal energy.The received signal in the satellite, which is jammed by thejamming signal J ∈ R M × N , can be expressed as H up ∗ ( S + J ) + V ∈ R M × N , where H up defines the uplink channelassuming that there always exists a highly strong line-of-sight(LOS) path due to the aid of directional antennas pointingto the satellite [2]. V is a simple additive white Gaussiannoise (AWGN). We now consider the satellite as a simple AnFrelay which amplifies signals by an amplifying gain G AnF andtransmits the outcome to the receiver.
ICA (cid:1798)(cid:3561) → (cid:1798)(cid:3553)
Robust PCA (cid:1797) (cid:3021) (cid:1778) →(cid:1797) (cid:3021) (cid:1793)(cid:3552), (cid:1797) (cid:3021) (cid:1784)(cid:4632)
Channel Estimation (cid:1782) (cid:3031)(cid:3042)(cid:3050)(cid:3041) (cid:1799) (cid:1778) (cid:3404)(cid:1782) (cid:3031)(cid:3042)(cid:3050)(cid:3041)(cid:2879)(cid:2778) (cid:1799) (cid:1793)(cid:3552)
DS-CDMA Depreading (cid:1819)(cid:3548) (cid:3041) → (cid:1824)(cid:3556) (cid:3041)
Anti-Jamming DS-CDMA Receiverin LMS mobiles (cid:1798)(cid:3561) (cid:1798)(cid:3553)(cid:1777) (cid:3041)
Fig. 2. Details of the anti-jamming DS-CDMA receiver blocks from Fig. 1.
B. Downlink Scenario
We assume that the corresponding downlink receiver mustbe designed with consideration of the LMS characteristicsdue to the mobility of receivers. The conventional LMSliterature states that such a satellite downlink is representedas a frequency-selective channel consisting of a LOS path and2 to 4 clustered diffuse paths with high path-loss [3]. When weexpress the frequency-selective channel in a discrete form, thechannel impulse responses are divided into three components:a direct path, near echoes and far echoes. We mathematicallymodel the downlink frequency-selective channel using a cir-culant matrix [21] as given below: H down = CM[ h , h , · · · , h l , · · · , h L − ] ∈ R M × M . (9)Let h = [ h , · · · , h l , · · · , h L − ] be the equivalent discretetime channel impulse response, CM[ · ] indicates the circulantmatrix that begins with the index h , and L denotes thenumber of paths of the downlink channel. Discrete channelimpulse response h l is a complex Gaussian random variablerepresenting fading channel environments ( l = 0 , ..., L − .Finally, the received DS-CDMA signal is modeled as: Y = H down ∗ { G AnF ∗ H up ∗ ( S + J ) } + V ∈ R M × N , (10)where V denotes the sum of both uplink and downlink AWGNchannels whose elements are i.i.d. Signal-to-noise ratio (SNR)is defined as a ratio between the average powers of the signaland the AWGN noise as follows:SNR [dB] = 20 log (cid:107) S (cid:107) F (cid:107) V (cid:107) F [dB] . (11)Fig. 2 details the proposed recovery block which comprisesof four specific blocks: Channel Estimation, Robust PCA, De-spreading, and ICA. This paper deals with practical and diversedownlink channel models representing urban with/without aLOS path, and rural with/without a LOS path environments,which are specified in [3]. We assume that the receiver hasa perfect knowledge of the spreading code W ∈ R M × M . Inaddition, perfect channel estimation of the downlink channelmatrix (cid:98) H down = H down ∈ R M × M is also considered. III. R
ECOVERY P ROBLEM & WITH M ATRIX D ECOMPOSITION
In this section, we describe a recovery problem of thereceived signal of (10), and formulate the recovery problemas a matrix decomposition problem. Our approach then is todecompose the received signal Y into the transmitted signal S and the jamming signal J by utilizing their inherent low-dimensionality features. A. Low-Dimensionality Properties & Channel Estimation
We demonstrate low-dimensionality properties of the trans-mitted DS-CDMA signal matrix and the uplink jamming signalmatrix that they can be often represented as a sparse matrixand a low-rank matrix in many emerging applications [16–20].First, the transmitted DS-CDMA signal S ∈ R M × N haslow-dimensionality, since the number of active users in CDMAsystems is often much lower than the spreading gain ( K (cid:28) M ) [18], [19]. This low activity is observed in a wide range ofapplications. In typical tactical communications, active usersare usually very small because the spreading gain of militarysystems focuses mainly on the anti-jamming capability. Dueto emerging 5G and IoT technologies, numerous devices areinactive most of the time but occasionally communicate forminor updates [20]. If the receiver has a priori informationof spreading codes as the transmitter, each column in matrix W T S ∈ R M × N has only K numbers of non-zero componentsdue to column vectors of Walsh code that are generatedindependently. Therefore, the low-dimensionality of the DS-CDMA signal matrix can be represented by the sparse matrix W T S ∈ R M × N .Second, we also remark that low-rank jamming signals arepresent and studied in [16], [17] and references therein. Withthis observation, the MT-FH uplink jamming signal matrix J ∈ R M × N can be assumed to have the low-dimensionality.Aforementioned in Subsection II . A , the typical MT-FH jam-ming signal matrix J ∈ R M × N is modeled as the low-rankmatrix. The term “low-rank matrix” refers to a rank of thematrix that is small compared to the largest possible rank.Moreover, since rank ( AB ) ≤ min { rank ( A ) , rank ( B ) } , amatrix W T J ∈ R M × N is also a low-rank matrix.Depending on the above descriptions, our objective is topropose an anti-jamming DS-CDMA recovery structure, whichis depicted in Fig. 2, exploiting the low-dimensionality of thetransmitted signal and the jamming signal. We assume thatthe AnF gain of the satellite compensates the uplink channel,i.e., G AnF ∗ H up = I M ∈ R M × M , where I M is an identitymatrix with size M . This assumption is due to the strong LOSpath in the uplink channel. With the assumption of the perfectestimation H down , Robust PCA decomposes W T S and W T J from W T D , where D = H − down Y . The input and outputsignals of the Robust PCA block are D ∈ R M × N and (cid:98) S ∈ R M × N , respectively. The Despreading block then despreads (cid:98) S ∈ R M × N with the known spreading code matrices for allbits C ( n ) ∈ R M × K ∀ n = 1 , . . . , N . Finally, ICA reconstructsthe original signal (cid:98) X ∈ R K × N from (cid:101) X ∈ R K × N by usingindependence inherently contained in the received signal. Algorithm 1: iALM for Robust PCA problem
Data: W T D ∈ R M × N , λ = 1 / √ M Result: W T (cid:98) J ← L k , W T (cid:98) S ← R k Λ ← W T D / max (cid:0) (cid:107) W T D (cid:107) , λ − (cid:107) W T D (cid:107) ∞ (cid:1) . R ← , µ ← . / (cid:107) W T D (cid:107) , k ← . while not converged do // Solve L k +1 = arg min L L ( L , R k , Λ k , µ k ) [ U , P , V ] ← svd( W T D − R k + µ − k Λ k ) . L k +1 ← U · Th[ P : µ − k ] · V T . // Solve R k +1 = arg min R L ( L k +1 , R , Λ k , µ k ) R k +1 ← Th[ W T D − L k +1 + µ − k Λ k : λµ − k ] . Λ k +1 ← Λ k + µ k ( W T D − L k +1 − R k +1 ) . Update µ k to µ k +1 . k ← k + 1 . end In subsections
III . B and III . C , we delineate the function-ality of the recovery block regarding matrix decomposition. Toimplementing Robust PCA and ICA for our anti-jamming DS-CDMA receiver, we modify iALM and Fast ICA algorithmsfor the system model of this paper. B. The iALM Algorithm for Robust PCA
We now consider a matrix decomposition problem to re-cover the sparse DS-CDMA signal W T S and the low-rankjamming signal W T J by solving the following convex opti-mization problem: min W T J , W T S (cid:13)(cid:13) W T J (cid:13)(cid:13) ∗ + λ (cid:13)(cid:13) W T S (cid:13)(cid:13) , subject to W T D = W T J + W T S , (12)where λ is a weighting parameter, (cid:107) A (cid:107) := (cid:80) m,n | a m,n | denotes the (cid:96) -norm (i.e., the sum of absolute values of allentries of the matrix A ), and (cid:107) A (cid:107) ∗ := (cid:80) i σ i ( A ) the nuclearnorm of the matrix A (i.e., the sum of singular values of A ).The optimization problem (12) simply minimizes a weightedcombination of the nuclear norm and (cid:96) -norm is referred to asRobust PCA [22]. Robust PCA can recover sparse componentsof the signal matrix even though the matrix are entirelycorrupted by a low-rank matrix. The weighting parameter λ controls the balance of regularization between the sparsityand the low-rank constraints. Based on prior knowledge tothe solution, a choice of λ often improves performance. Forexample, if we know that W T S is very sparse, it is possibleto recover matrices W T J of larger rank by increasing λ .However, λ = 1 / √ M is recommended for the existence andthe uniqueness of the solution in practical problems [22]. Wealso choose λ = 1 / √ M in this paper.In this paper, we have chosen to solve the Robust PCA prob-lem (12) using an augmented Lagrangian multiplier (ALM)algorithm introduced in [23]. ALM has been proved to con-verge to the exact optimal solution in fewer iterations [24].In practical applications, it works stably across a wide rangeof problem settings with no parameter tuning [22]. The ALM method operates on the augmented Lagrangian function of theRobust PCA optimization (12) L ( W T J , W T S , Λ , µ ) . = (cid:13)(cid:13) W T J (cid:13)(cid:13) ∗ + λ (cid:13)(cid:13) W T S (cid:13)(cid:13) + (cid:68) Λ , W T D − W T ( S + J ) (cid:69) + µ (cid:13)(cid:13)(cid:13) W T D − W T ( S + J ) (cid:13)(cid:13)(cid:13) F , (13)where (cid:104) A, B (cid:105) = tr( A T B ) and µ is a positive scalar with aLagrange multiplier matrix Λ . A generic ALM algorithm isto solve (12) by repeatedly solving ( W T J k , W T S k ) = arg min W T J , W T S L ( W T J , W T S , Λ k , µ k ) , (14)and then update the Lagrangian multiplier matrix by Λ k +1 = Λ k + µ k ( W T D − W T ( S + J )) . (15)For the low-rank and sparse decomposition problem, thesolution of a complex optimization of (14) can be obtainedby solving very simple calculations sequentially as follows: W T S k +1 = arg min W T S L ( W T J k , W T S , Λ k , µ k )= Th (cid:104) W T D − W T J + µ − k Λ k : λµ − k (cid:105) , (16) W T J k +1 = arg min W T J L ( W T J , W T S k , Λ k , µ k )= U · Th (cid:2) P : µ − k (cid:3) · V T , (17)where Th [ a : µ ] = sgn( a ) max( | a | − µ, is the shrinkageoperator and extend it to matrices by applying it to eachelement, and UPV T = (cid:2) W T D − W T S k − µ − k Λ k (cid:3) is anysingular value decomposition. In (17), the rank of W T J k +1 is minimized by thresholding corresponding singular values.Additionally, In (16), reliable sparse components remain afterthresholding elements values.Algorithm 1 describes procedures to solve Robust PCA withproper initialization. Algorithm 1 is referred to as inexact ALM(iALM) since it inexactly solves (14) by updating (16) and(17) iteratively. Finally, the sparse DS-CDMA signal W T S and the low-rank jamming signal W T J are decomposed byapplying iALM. The initialization of Λ in the algorithm isselected to make the objective function (13) reasonably large.The most important implementation detail of the algorithmis the choice of { µ k } . The choice is directly related to theconvergence of the algorithm. It is known that Algorithm 1converges to the optimal solution of Robust PCA if { µ k } isnondecreasing and (cid:80) + ∞ k =1 µ − k = + ∞ [24]. We have chosen µ = 1 . / (cid:107) W T D (cid:107) and µ k +1 = min(1 . µ k , µ ) , where (cid:107) A (cid:107) = max i σ i ( A ) is a 2-norm of a matrix A i.e., the largestsingular value of the matrix A .After recovering the transmitted signal (cid:98) S = WR k fromthe received signal by using Robust PCA approach. The MT-FH uplink jamming signal can be removed effectively. Then, (cid:101) X ∈ R K × N is obtained by despreading (cid:98) S with the spreadingcode matrix C ( n ) in Fig. 2. Algorithm 2:
Fast ICA for ICA problem
Data: (cid:101) X := ( (cid:101) x i,j ) ∈ R K × N (where i = 1 , . . . , K , and j = 1 , . . . , N ) Result: (cid:98) X ← ω T (cid:101) X (cid:101) x i,j ← (cid:101) x i,j − N N (cid:80) j =1 (cid:101) x i,j // Centering the data [ Q , Γ ] ← eig(cov( (cid:101) X )) (cid:101) X ← QΓ − / Q T (cid:101) X // Whitening the data To find initial (random) weight vector ω ; k = 0 while not converged do ω k ← E { (cid:101) X g ( ω Tk (cid:101) X ) T } − E { g (cid:48) ( ω Tk (cid:101) X ) } ω k // where E {·} means averaging over// all column vectors of matrix (cid:101) X ) ω k +1 ← ω k / (cid:107) ω k (cid:107) k ← k + 1 end C. Fast ICA Algorithm for ICA
The next step of our AJ receiver structure is the ICAblock which reconstructs the final estimate of the input data (cid:98) X ∈ R K × N from a mixed observation (cid:101) X ∈ R K × N . BSSusing ICA here cannot only detect multi-user signals, but alsosuppress multi-user interferences, inter-symbol interferences,and intentional jamming signals in CDMA systems [12], [14].Authors of [12] evaluate the anti-jamming performance of thereceiver and show via numerical results that 5dB SJR gains interms of bit-error-ratio (BER) of − under AWGN channelwhen signal-to-noise-ratio (SNR) is fixed to 20dB. In ourscenario, ICA reconstructs the original signal (cid:98) X ∈ R K × N from (cid:101) X ∈ R K × N , which is also shown in Fig. 2.To extract independent components from the mixture ma-trix, we adapt the Fast ICA algorithm [11] which is basedon a fixed-point iteration. For computational simplicity andfast convergence, many studies (also in [5], [12]) considerFast ICA, which is the most popular ICA algorithm thus far.The Fast ICA algorithm used to restructure (cid:98) X ∈ R K × N is described in Algorithm 2, where g ( a ) = tanh( a ) and g (cid:48) ( a ) = 1 − tanh ( a ) . The notation cov( A ) symbolizes the co-variance matrix of A . A procedure [ Q , Γ ] = eig( A ) performseigendecomposition of a matrix A = QΓQ − , where Q isthe square matrix whose columns vectors are eigenvectors of A , and Γ is the diagonal matrix whose diagonal elements arethe corresponding eigenvalues. Fast ICA effectively separatesthe input data (cid:98) X by finding an inverse transformation ω T (cid:101) X that maximizes the statistical independence.In the next section, we perform extensive simulations toverify the anti-jamming ability of the proposed receiver.IV. S IMULATION R ESULTS AND D ISCUSSIONS
The anti-jamming DS-CDMA receivers using matrix de-composition methods such as Robust PCA and ICA areassessed through simulations for the following two receivertypes: • Receiver-Type1 : The conventional anti-jamming DS-CDMA receiver using ICA without Robust PCA,
Fig. 3. BER versus SJR with SNR fixed to 5dB and 10dB under urbanenvironments (LOS).Fig. 4. BER versus SJR with SNR fixed to 5dB and 10dB under urbanenvironments (nLOS). • Receiver-Type2 : The proposed anti-jamming DS-CDMAreceiver using both, ICA and Robust PCA approaches.The DS-CDMA transmitted signals are generated by fol-lowing system parameters: K = 30 users, N = 1000 bits,and M = 1024 spreading code length of the Walsh code. Thesystem transmits M chips within each bit duration bearingthe information of K users. We consider various types of thedownlink channels such as urban environments with a LOSpath and without a LOS path (nLOS), and rural environmentswith LOS/nLOS. It is known that the downlink channel inLMS communications is a frequency selective channel dueto its multi-path propagation consisting of a direct path, nearechoes, and far echoes. Parameter sets including a numberof taps, delays, and channel gains are set to the measure-ment data of the LMS International Telecommunication Union(ITU) model [3]. In the MT-FH uplink jamming scenario, theprobability p that the m th frequency is jammed at the n th bitduration is set to be 0.1 i.e., p = 0 . . Simulations also considera range of rank- r MT-FH jamming signals such as 1, 100,
Fig. 5. BER versus SJR with SNR fixed to 5dB and 10dB under ruralenvironments (LOS).Fig. 6. BER versus SJR with SNR fixed to 5dB and 10dB under ruralenvironments (nLOS). r = 1 is a typical MT jamming without hopping, and r = 1000 isan MT-FH jamming with hopping every bit duration. We run1000 Monte Carlo simulations to observe a reliable BER levelof − with SJR = [ − , dB and SNR = 5 and dB, asused in [12], [14]. It is worthy noted that a broad-band noisejamming can be more effective than the MT-FH jamming asthe SJR is too low. However, in the paper, we focus on theMT-FH jamming in order to discuss the effects of the jammingrank r on the performance of the proposed receiver.Fig. 3, 4, 5 and 6 show BER performances of the Receiver-Type1 and the Receiver-Type2 versus SJR values with variousranks of the MT-FH jamming signal under four differentchannel scenarios. The simulation results in Fig. 3 and 4consider urban environments with the 5 paths frequency selec-tive downlink channels, while Fig. 5 and 6 present the BERperformance in rural environments with 3 paths. Furthermore,Fig. 3 and 5 consider the presence of a LOS path, whereas SJR = −25dB
SJR = −20dB
SJR = −15dB
SJR = −10dB
Fig. 7. BER versus r (rank of jamming signal) change with SNR fixed to10dB under urban environments (LOS). SJR = −25dB
SJR = −20dB
SJR = −15dBSJR = −10dB
Fig. 8. BER versus r (rank of jamming signal) change with SNR fixed to10dB under urban environments (nLOS). Fig. 4 and 6 do not.Fig. 3 presents the anti-jamming performance of the afore-mentioned two receivers versus SJR under the urban environ-ments including a LOS path with SNR = 5 and dB on theleft and right figures, respectively. Each subfigure considersthe MT-FH uplink jammer with different jamming ranks of 1,100, and 1000. The blue curves are for the Receiver-Type2,and the red curves are for the Receiver-Type1. The resultsshow that the Receiver-Type2 outperforms the Receiver-Type1in most cases and the BER performance of the Receiver-Type2increases as rank decreases. Especially, it is noted that theReceiver-Type2 completely decomposes the transmitted DS-CDMA signal matrix and the MT-FH jamming signal matrixwith the rank r = 1 , when the signal power is larger thanan SJR level of -20dB with a fixed SNR level of 10dB.This implies that the typical MT jamming without FH canbe easily separated by the Robust PCA even with very highSJR value. It is also noteworthy that typical uplink jammers inGPSs are commonly simple single tone pulse generators with SJR = −25dB
SJR = −20dB
SJR = −15dBSJR = −10dB
Fig. 9. BER versus r (rank of jamming signal) change with SNR fixed to10dB under rural environments (LOS). SJR = −25dBSJR = −20dBSJR = −15dBSJR = −10dB
Fig. 10. BER versus of r (rank of jamming signal) change with SNR fixedto 10dB under rural environments (nLOS). a high power amplifier, which can be effectively mitigated byusing the proposed receiver. In the case that MT-FH jammingsignals hop every bit duration, whose the rank increases up to r = 1000 , the Receiver-Type2 for SNR=10dB still guaranteesa comparable anti-jamming performance compared to its coun-terpart. In another case of SNR=5dB, although the Receiver-Type1 outperforms the Receiver-Type2 for r = 1000 , theReceiver-Type2 performs better for MT-FH jamming signalswith low hopping rates. Simulation results also remark that theBER results of the Receiver-Type1 are almost the same andindependent with respect to the rank of the jamming signalfor both SNR=5dB and 10dB. This implies that ICA does notutilize low-dimensionality to decompose the signals.In Fig. 4, we simulate the BER of the two receivers insimilar conditions of Fig.3 except that the urban environmentwith nLOS path is considered. From the Fig. 4, we observe thatthe BER performance of the Receiver-Type2 increases whenthe jammer decreases its hopping rate (the rank r ). However,it should be noticed that MT-FH jammers, which require a B it E rr o r R a t e ( ▲ ) R un ti m e s ( ● ) [ s ] Number of users K
Runtimes and BER of Rx1 and Rx2 versus K
Runtime of Rx1 Runtime of Rx2BER of Rx1 BER of Rx2 (a) Runtimes and BER of the Receiver-Type1/2 ver-sus the number of users K , with M = 128 , N =100 B it E rr o r R a t e ( ▲ ) R un ti m e s ( ● ) [ s ] Spreading code length M
Runtimes and BER of Rx1 and Rx2 versus M
Runtime of Rx1 Runtime of Rx2BER of Rx1 BER of Rx2 (b) Runtimes and BER of the Receiver-Type1/2versus the spreading code length M with K =3 , N = 100 B it E rr o r R a t e ( ▲ ) R un ti m e s ( ● ) [ s ] Number of bits N
Runtimes and BER of Rx1 and Rx2 versus N
Runtime of Rx1 Runtime of Rx2
BER of Rx1 BER of Rx2 (c) Runtimes and BER of the Receiver-Type1/2 ver-sus the number of bits N with K = 6 , M = 128 Fig. 11. Runtimes and BER of the Receiver-Type1/2 versus the number of users K , the spreading code length M , and the number of bits N . Simulationconsiders the rural environment with nLOS path, SNR=5dB, SJR=-10dB, and the jamming rank is N/ . high-rank r , are not common due to their high complexity andhardware costs in practical satellite communication systems.The figure also shows that the BERs of the Receiver-Type1/2are saturated to . · − and · − as SJR increases whenSNR=5dB and 10dB, respectively. This result is explained bythe effects of the LMS channel under the urban environmentwith nLOS that implies a highly fading channel. Similar toFig. 3, the worst BER performance of the Receiver-Type2 isobserved when r = 1000 .Fig. 5 and 6 plot the BERs of the Receiver-Type1/2 underthe rural environment with LOS/nLOS. The BERs of theReceiver-Type1 under the rural environments is almost equalto the BERs under the urban environments. One differenceis that the BERs under the rural environment with nLOS arenot saturated within the simulated SJR region. For the caseof r = 1 (no hopping) of the Receiver-Type2, the BERsapproach roughly − at SJR=-20dB and SNR=10dB whilethe BER of the Receiver-Type2 under the urban environmentwith LOS is − . One reason, why the BER of the Receiver-Type1/2 are not saturated and the Receiver-Type2 gives betterAJ performance, is that the rural LMS channels are measuredby fewer paths and long delay channel impulse responsescompared to the urban environments.In Fig. 7, 8, 9, and 10, we compare the anti-jammingperformances of the Receiver-Type1/2 versus the rank of thejamming signal with SJR = − , − , − , and − dB underthe MT-FH uplink jamming for four different environments.Fig. 7 and 8 plot the BERs versus the rank- r under urban en-vironments with LOS/nLOS and Fig. 9 and 10 are under ruralenvironments with LOS/nLOS. Red dotted curves correspondto the BERs of the Receiver-Type1 for different SJRs, and bluecurves are for the BERs of the Receiver-Type2. X-axis repre-sents the rank of the jamming r = (1 , , , , ,which the minimum r corresponds to no hopping and themaximum r is for the case of hopping every bit duration.Overall Fig. 7, 8, 9, and 10, as the rank of the jammingdecreases, the anti-jamming capacity of the Receiver-Type2 increases. On the other hand, the Receiver-Type1 does notimprove the BER performance although the rank of thejamming decreases. At a low-rank range of r < , whichrepresents less hopping MT-FH jammers, the Receiver-Type2significantly outperforms the Receiver-Type1 for a wide rangeof SJRs. Moreover, at high-rank jammings, the Receiver-Type2performs equally well or slightly worse than the Receiver-Type1 depending on SJR. The BER differences betweenthe Receiver-Type1/2 for the high-rank jamming decrease asSJR decreases. In addition, ranges of rank values, wherethe Receiver-Type2 performs better than the Receiver-Type1,become wider as SJR decreases–in other words, the jammingpower increases. It is observed that the range of rank valuesthat Receiver-Type2 outperforms Receiver-Type1 is smallerwhen the LMS downlink channel becomes severe. Simulationresults conclusively remark that the proposed Receiver-Type2is more effective than the conventional Receiver-Type1 forlow-rank ( r < ), high power jammers, and less-severemulti-path environments. In addition, even for high-rank andmore-severe multi-path channels, the Receiver-Type2 is com-petitive to the Receiver-Type1.The CPU runtimes of the MATLAB implementations andBER performances of the Receiver-Type1 and the Receiver-Type2 with respect to various DS-CDMA system parametersare summarized in Fig. 11. The subfigures for the numberof users K , the spreading code length M , and the numberof bits N are presented in Fig. 11a, Fig. 11b, and Fig.11c, respectively. The rural environment with nLOS path isassumed, and SNR and the jamming rank are set to 5dBand N/ . In addition, BERs are measured at SJR of -10dB. Generally, the results show that the computational timeof the Receiver-Type2, combining Robust PCA and ICA, iscomparable to that of the Receiver-Type1 using ICA only. TheFig. 11a implies that the computational time of the Receiver-Type1 increases linearly as the number of users K increases,while the gap between the runtime of the Receiver-Type2 andthat of the Receiver-Type1 reduces. It is also seen increasing K degrades the BER performances of both the Receiver-Type1and the Receiver-Type2. The Fig. 11b shows that the spreadingcode length M only linearly increases the CPU runtime of theReceiver-Type2, while the BER performance of the Receiver-Type2 is exponentially improving. Moreover, in the Fig. 11c,we observe that the runtimes of both the Receiver-Type1 andthe Receiver-Type2 increase as the number of bits N increases.It is also noted that the additional time for combining RobustPCA algorithms on the Receiver-Type1, when the number ofbits N is less than 400, is less than the computational time ofthe Receiver-Type1 itself.V. C ONCLUSION
In this paper, we considered the anti-jamming problem ofDS-CDMA receivers against the presence of uplink jammersunder LMS communication systems. We developed an anti-jamming DS-CDMA receiver that decomposes the receivedsignal into the transmitted signal and the unintended uplinkjamming signal by exploiting the fact that they are typi-cally low-dimensionality. Utilizing their low-dimensionalityattributes, we suggested the integration of Robust PCA andICA approaches, which are implemented by iALM and FastICA algorithms.Anti-jamming performances of Receiver-Type1 (the con-ventional receiver using only ICA without Robust PCA) andReceiver-Type2 (the proposed receiver using both Robust PCAand ICA) were assessed in the scenarios that consider the MT-FH uplink jammer and practical downlink channels includingurban and rural environments. Simulation results show thatRobust PCA in Receiver-Type2 achieves significant perfor-mance improvement as compared with the Receiver-Type1 fora wide range of the rank of the MT-FH jamming signal. Thisimplies that Robust PCA separates various jamming signalsmore effectively than ICA only. The performance improvementincreases as the rank decreases. For ranks lower than 200that represent MT-FH jamming signals with less hopping,Receiver-Type2 outperforms Receiver-Type1. Even for largeranks that signify frequent hopping jamming, Receiver-Type2shows a comparable performance to its counterpart. In conclu-sion, our proposed receiver has potential applications in DS-CDMA based LMS systems under various uplink jammers.A
CKNOWLEDGMENT
The authors gratefully acknowledge the support from Elec-tronic Warfare Research Center (EWRC) at Gwangju Instituteof Science and Technology (GIST), originally funded byDefense Acquisition Program Administration (DAPA) andAgency for Defense Development (ADD).R
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Mathe-matical Programming , submitted, 2009. Hyoyoung Jung received the B.S. and M.S. degreesfrom Inha University and the Gwangju Institute ofScience and Technology (GIST), South Korea, in2011 and 2013, respectively. He is currently pur-suing the Ph.D. degree with GIST. His research in-terests include statistical signal processing, machinelearning, and anti-jamming communication systems.
Jaewook Kang received the B.S. degree in informa-tion and communications engineering (2009) fromKonkuk University, South Korea, and the M.S. andPh.D. degrees in information and communicationsengineering (2010,2015) from the Gwangju Instituteof Science and Technology (GIST), South Korea.He is currently working in Soundlly Inc. as a signaland data processing engineer. His recent researchinterests include signal processing based on low-dimensionality, low-complex ultrasound receiver de-sign, and machine learning techniques for IoT de-vices.
Tae Seok Lee received the B.S. degree in con-trol engineering from Kwangwoon University, SouthKorea, in 2015 and M.S. degree in electrical andcomputer engineering from GIST, South Korea, in2017. He is currently working in Telecommunica-tions Technology Association (TTA), as a seniorresearch engineer. His research interests include thesatellite communications and data processing underbig data system.
Suil Kim received the B.S. and M.S. degrees in elec-trical engineering from Soongsil University, SouthKorea, in 1986 and 1988, respectively, and thePh.D. degree in electrical engineering and computerscience from the Korea Advanced Institute of Sci-ence and Technology, South Korea, in 2000. He iscurrently a principal researcher in the Agency forDefense Development (ADD) and a professor in theUniversity of Science and Technology in Daejeon,South Korea. His research interests are on anti-jamming modems and interference cancellation forwireless communication systems.