arXiv:1504.05727v1 [cs.IT] 22 Apr 2015 h s ftepooe ub-amr oigapoc in approach coding discussed. turbo-Hadmard also is is proposed rate channels the fading information approa DS-SS multipath of transmission -aided the use its of that The than when higher times even 5 thr about NBI, for approach of throug DS-SS types shown code-aided fixed co the is turbo-Hadamard a outperforms proposed it and approach the 32 approaches, that length simulations both extensive of for sequence allocated spreading DS bandwidth code-aided a the With sprea with compared approach. time-varying sharply of is with kind which sequences, signalling a employs spread-spectrum practical approach NBI coded coding a the decod turbo-Hadamard coding iterative propose which proposed through full in achieved we naturally low-rate approach, is Then, suppression a coding namely, preferred. i turbo-Hadamard hence NBI, there always low-rate and show of is spreading, we class approach in techniques, coded advantage special em bounding of no By a well-developed probability signatures. the of orthogonal error ing with presence the interference we the on multi-tone decoding, tradeoff in bounds we likelihood coding-spreading upper systems paper, maximum optimum derive With this first suppression. of In NBI problem studied. for extensively the c been address (DS-SS) has spread-spectrum munications direct-sequence in suppression ub-aaadcds pedsetu signalling. spread-spectrum , turbo-Hadamard inl h B inlcnas eadgtlsga,sc as such [ signal, system digital signalling multirate a a be in example, also (A for autoregressive can arise, an signal would modeled or NBI often (tone) The is signal signal. signal sinusoidal NBI a The as either [2]. despreadi to [1], prior demodulating achieve suppression NBI and be active com- of can low-rate use the gains with through performance coexistence in by caused substantial or be munications, environment may hostile which (NBI), a interference narrow-band resis inherently to is communication spread-spectrum though S ocre ihtedvlpeto ehiusfratv NBI active for techniques of development the with concerned cetfi eerhFudto fNUTudrGatNY21300 Grant under 14KJA5100 NJUPT of Grant Foundation under Research Province Unive Scientific Jiangsu Key of the Found by Project Science 61032004, Research National 61271335, the 61372123, by Grants part under in supported was work This nBodadWrls omncto n esrNtokTec Network Sensor Nanji [email protected]). , (Emails:[email protected], and and Posts of Communication University Nanjing Wireless purpo pubs-permissi Broadband to other request in any a sending for by material IEEE this the from use obtained to permission However, oigv.SraigfrNro-adInterference Narrow-Band for Spreading vs. Coding hswr a rsne npr tteIC21,Sde,Austr Sydney, 2014, ICC the at part in presented was work This oyih c 05IE.Proa s fti aeili per is material this of use Personal IEEE. 2015 (c) Copyright iouW n hnYn r ihteKyLbo iityo Educ of Ministry of Lab Key the with are Yang Zhen and Wu Xiaofu ne Terms Index Abstract nteps,asgicn muto eerhhsbeen has research of amount significant a past, the In in nbt iiinadmltr omnctos Al- communications. military and civilian both applica- in wide tions found has signalling PREAD-SPECTRUM Teueo cienro-aditreec (NBI) interference narrow-band active of use —The Nro-aditreec,cdn,spreading, coding, interference, —Narrow-band .I I. NTRODUCTION g200,China 210003, ng iouW n hnYang Zhen and Wu Xiaofu Suppression to fChina of ation 3adb the by and 03 [email protected]. st Science rsity e utbe must ses 2. la 2014. alia, n.The ing. hnology, mitted. ploy- ding ding ation om- tant -SS ch. 1]. R) ng ee h d s n eoe yeeyfre,o thslwpoaiiyof probability low has it or forces, receiv be enemy cannot it by that great so decoded of designed and be problem must a signal communic the military suppression, tions, For NBI communications. military of in purpose concern the for ing hc ed osgicn efrac mrvmn o the for improvement performance significant syste synchronous to DS-SS of leads replace structure which to coding-spreading proposed traditional was the scheme very coding APP a low turbo-Hadamard the a by rate [15], that accomplished In [16] (FHT). be Fast-Hadamard-Transform shown can APP efficient also codes was Hadamard of It decoding code. Hadamard the of lmne ihngiil rli opeiybtlaigt im- leaving efficiently but be the complexity with can trellis complexity iterative which negligible an [16], with the in plemented codes, proposed to representation Hadamard was trellis close convolutional decoder simple very component very nois Gaussian perform a of white With to additive channel. the shown (AWGN) for limit been Shannon have ultimate [16], in when concerned. tradeoff open is coding-spreading suppression remains optimum NBI it strategy the of However, a problem [15]. Such the [14], gain. for in coding considered maximized been in low- the has using implied with combined was be codes It should rate spreading [13]. and in coding multiple addressed that a [13] also was for channel tradeoff err access to coding-spreading dedicated is Optimum spreading capac entire control. ultimate the the when that achieved NBI [12] be Viterbi for can by potential early out full pointed chann was its multiple-access spread-spectrum exploit a not For may suppression. coding further channel traditional to this of coding However, channel performance. system with the combined improve be can [2] approach [11]. [10], in reported been suppress also NBI have for communications algorithms been processing (UWB) also signal New bandwidth has [5]–[9]. suppression ultra-wide NBI in years, considered N for recent out- methods In nonlinear to or suppression. shown linear previous been the has of all detector, mean perform minimum linear the (MMSE) on error based square is which method, This (DS/CD networks. multiple-access code-aid suppression code-division NBI a for direct-sequence Wang [4], in and Poor [2], by proposed In was techniques. approach techni detection filtering linear multiuser nonlinear on and including mode based [1], employs other is techniques the based one and [3] techniques, techniques processing these signal for there general, categories In two systems. spread-spectrum in suppression hspprfcsso h rbe fcdn essspread- versus coding of problem the on focuses paper This h o-aetroHdmr oe,oiial proposed originally codes, turbo-Hadamard low-rate The (DS-SS) spread-spectrum direct-sequence code-aided The utpeacs channel. multiple-access posteriori a rbblt AP decoding (APP) probability MA) ques l it el, ms, use ion are ity BI he ed ed or a- l- e 1 - - 2 interception (LPI). This is often achieved by spreading the 5) It is shown that the APP decoding of Hadamard codes signal so that its bandwidth is much larger than the true in the presence of NBI can be implemented using FHT, date rate. Hence, a low-rate coding structure for military which facilitate its practical implementation. communications is desired due to the requirement of LPI. For 6) The full coding approach for NBI suppression can also hostile jamming from potential enemy forces, the jammer often be applied to multipath fading channels. We consider monitors the transmitted signal and concentrates its power quasi-static fading channels and show that it is possible in a few frequencies within the dominant transmitted signal to develop a low-complexity iterative receiver for mul- bandwidth. Hence, the jamming signal is often regarded as tipath channels with long Hadamard sequences. NBI. The rest of the paper is organized as follows. The optimum For spread-spectrum signalling aiming at NBI suppression, coding-spreading tradeoff for NBI suppression is investigated we conjecture that the entire spreading should be again ded- in Section II. In Section-III, we give a brief description of icated to error control coding. In this paper, we first pro- low-rate full turbo-Hadamard codes without puncturing. This vide a partial theoretical justification of this conjecture under coding approach is employed for NBI suppression through maximum-likelihood decoding. Then, we propose a practical iterative decoding as shown in Section-IV. Section-V presents low-rate turbo-Hadamard coding approach with consequently simulation results and Section-VI concludes the paper. large bandwidth expansion for NBI suppression. We point out that this conjecture has also been independently observed in II. CODING VS. SPREADING [?] for partial-band jamming in frequency-hopping systems. For the purpose of NBI suppression, we observe that if a full A. System Model turbo-Hadamard codeword (without puncturing information Consider a baseband signal model with only one user at the bits for each component convolutional-) is receiver transmitted over the channel, the resulting signal presents the r(t)= Psc(t)+ i(t)+ w(t), (1) form of DS-SS signalling, but with time-varying Hadamard where c(t) is the spread-spectrump signal of the user and P sequences. Therefore, it can be well understood that the code- s is the received signal power, i(t) and w(t) represent the NBI aided approach for NBI suppression might be still useful. Due signal and the ambient channel noise, respectively. to a low-rate coding structure, the NBI suppression can be We consider two approaches for NBI suppression, namely, naturally achieved through iterative decoding. a code-aided DS-SS approach and a full coding approach, as In [17], [18], it was observed that a proper use of interleav- shown in Fig. 1. For both approaches, it is assumed that a ing in direct-sequence spreading can be beneficial for multi- block channel coding scheme is employed. At the transmitter, user interference (MUI) cancellation, where the spreading se- information bits are grouped into data frames and then input to quence is time-varying due to the introduction of interleaving. the block encoder. For a full coding approach, the coding rate The proposed turbo-Hadamard coding approach employs a should be low enough for expanding the signal spectrum. For kind of DS-SS signalling with time-varying spreading se- a coded DS-SS approach, the (channel) coding rate is often quences. Compared to [17], [18], the observed time-varying larger than 1 and the signal spectrum is mainly expanded by spreading sequences in the turbo-Hadamard coding approach 2 direct-sequence spreading. are controlled by the coding rule, and hence an extra coding It should be mentioned that the conventional coded DS- gain can be achieved. SS approach can also be viewed as a general coding ap- The main contributions of this paper are summarized as proach, where the traditional channel coding is serially con- follows: catenated with repetition coding (direct-sequence spreading) 1) By assuming a special type of NBI, namely, the multi- as described in Fig. 1. Hence, both approaches can be viewed tone interference with orthogonal signatures, we derive as a unified coding approach. upper bounds on the error probability of coded systems Denote by TF and Ti, the frame duration and the infor- in the presence of both AWGN and NBI. Then, we mation bit duration, respectively. Suppose now that every provide a partial justification of the conjecture that the K information bits are grouped as a date frame, namely, optimum coding-spreading tradeoff always favors full TF = KTi. With a unified coding view for both approaches coding for NBI suppression. shown in Fig. 1, we can consider that information bits are input 2) A practical low-rate full coding approach for NBI sup- to a low-rate encoder, which outputs coded bits (in chips) with pression is proposed, which has never been reported for duration T (chip duration). In this paper, we always assume the purpose of NBI suppression. c that T T . 3) With a matched filter receiver at chip rate, the proposed i c Assuming≫ that a block channel coding scheme is inde- turbo-Hadamard coding approach shows significantly pendently implemented over each frame, we can arrive at a higher spectral efficiency compared to the traditional signal vector representation, by projection of the signals onto DS-SS approach. a finite basis. Indeed, the received signal (column) vector r of 4) Considering that in the presence of NBI, the extrinsic length L = T /T during a frame can be obtained by passing information for iterative decoding cannot be directly F c r(t),t [0,T ) through a chip-matched filter, followed by a extracted at the positions of systematic (information) F chip-rate∈ sampler, giving bits, an approximate method is proposed and simulations show that this approximate method works well. r = Psc + i + w, (2) p 3

NBI AWGN Chip Rate

Tc T Ti Ts Tc s Ti Matched Code-aided Decoding Encoding Spreading Filter NBI Suppression

a) Code-aided DS-SS approach for NBI suppression

NBI AWGN Chip Rate T T T T i Low Rate c Matched c Decoding with NBI i Encoding Filter suppression

b) Full coding approach for NBI suppresion

Fig. 1. The block diagrams of two NBI suppression approaches. where c is a binary-phase-shift-keying (BPSK) modulated For ML decoding, one has to determine the codeword c codeword, w is a sample vector of zero-mean Gaussian noise for which the conditional probability density function p (r c) with variance σ2 and i denotes a collected column vector form reaches its maximum. | of the NBI. In general, NBI is a stochastic process with infinite memory. For the traditional coded DS-SS approach, this model still In practice, it can be well approximated by finite memory. For holds. However, the NBI suppression is often achieved via lin- ease of processing, a block-independent approximation is often ear MMSE approach with one-shot symbol duration Ts, where employed, where the NBI process i can be partitioned into T T T T n = Ts/Tc is often employed to denote the spreading factor. a sub-block vector form of i = [i1 , i2 , , iB] with each 1 ··· T Hence, the received signal vector should be further partitioned sub-vector ik of length n = Ts/Tc satisfying E ik il = as sub-vectors of length n, i.e., r = [rT , rT , , rT , , rT ]T δ R . Note that there is still some room for performance{ · } 1 2 ··· k ··· B k,l i with B = TF /Ts. Then, the kth received sub-vector during enhancement if one considers a longer memory size (> n) one symbol period Ts can be represented as for NBI, which clearly compromises with increased decoding complexity. rk = Psck + ik + wk, (3) Then, p (r c) can be factored as | where c is the data-bearingp ( 1 modulated) spreading- k ± B sequence vector, wk is a Gaussian noise vector and ik denotes p (r c)= p (r c ) . (6) | k| k a NBI vector, which is assumed to be wide-sense stationary k=1 with mean zero and covariance matrix Ri. Y With a linear MMSE approach for NBI suppression, the To evaluate p (r c ), it requires to know the probability k| k detector simply makes a symbol-by-symbol decision as ˆbk = distribution of ik. Here, we assume that ik is a Gaussianly- T sgn u rk , where u is chosen to minimize the mean-square distributed vector, namely, ik (0, Ri). If the NBI is error generated by an AR process, this∼ is N just a precise assumption.  If the NBI is modeled as a sinusoidal signal with multiple MSE , E uT r sgn uT r . (4) k − k tones, or the digital NBI signal, this is approximately satisfied.   With this assumption, (6) can be clearly represented as

B. Maximum-Likelihood Decoding under Gaussian- B distributed NBI 1 T 1 p (r c) exp (rk ck) Q− (rk ck) , (7) | ∝ −2 − − ! As shown in Fig. 1, both approaches can be considered as a kX=1 block coding scheme, where a linear binary (L,K) where P is assumed to 1 and Q = R + σ2I. C is employed in the sense of chip-rate transmission with the s i coding rate given by Hence, an ML decoder only needs to choose a coded sequence c which maximizes the metric K Tc Rc = = . (5) B L Ti η = c , y + λ(c). (8) h k ki Hence, one can bound the error probability of this unified k=1 coding system in the presence of NBI under maximum- X likelihood (ML) decoding to show the fundamental tradeoff 1It is assumed that the size of memory coincides well with the symbol of coding versus spreading. period in the DS-SS approach for ease of processing. 4

1 where yk = Q− rk and With simple union bounding technique [19], the word error B probability can be bounded as 1 T 1 λ(c)= c Q− c . (9) − 2 k k F k=1 γl X Pw AhQ 2γs 1 h , (16) ≤ v − 1+ nγl  h hmin u l=0 ! C. Error Probability under Multi-tone Interference with Or- ≥X u X t  thogonal Signatures where Ah denotes the number of codewords of weight h in The pair-wise error probability (PEP), which represents the C. Note that the union bound on bit error probability can also probability of choosing the coded sequence ˆc when the coded be obtained by employing the input-output weight enumerator sequence c was transmitted, is the fundamental expression for [19]. the construction of the union or other upper bounds on the average error probability performance of a system. D. Coding vs. Spreading In general, it is difficult to derive the PEP in the presence of NBI. To gain essential insights, we, however, focus on a special class of NBI process, namely, multi-tone interference 6 with orthogonal signatures [2]. This class of NBI process is Full Coding approximately Gaussian-distributed when the number of tones 5 Coded Spreading is large thanks to the central limit theorem. Hence, the metric 4 (8) still holds with ML decoding. Letting η (or ηˆ) denote the metric (8) for the correct (or 3 incorrect) coded sequence c (or ˆc), then the PEP of choosing , dB 0 2 the coded sequence ˆc instead of the actual transmitted coded /N b sequence c is given by E 1

P (c cˆ)= Pr η η<ˆ 0 c . (10) 0 → −  

The metric difference can be computed as −1

B −2 0 0.1 0.2 0.3 0.4 0.5 η ηˆ = ck ˆck, yk + λ(c) λ(ˆc), (11) − h − i − Equivalent Coding Rate kX=1 where Fig. 2. Comparison of minimum threshold of full coding vs. coded spreading B for random codes as L →∞. 1 T 1 λ(c) λ(ˆc)= (ck cˆk) Q− (ck + ˆck) . (12) − −2 − In this subsection, we still focus on the multi-tone inter- Xk=1 For analytic tractability, we should further introduce a series ference with orthogonal signatures, as it admits a closed- of approximations as did in the Appendix. For conciseness, the form expression for the upper bound on the pair-wise error derivation is summarized in the Appendix. probability (15). Let h = dH (ˆc, c) be the between cˆ and With BPSK modulation, it is clear that γs = RcEb/N0, where is the information bit energy and is the c, and γl,l =1, , F , be the lth tone’s power-to-noise ratio. Eb N0/2 In the Appendix,··· it is shown that two-sided power spectral density of the white Gaussian noise process at the receiver. F F γl 2 γl Define κi =1 , δ = h/L,µ(δ) = (ln Ah)/L µz = E ηˆ η c 2σ− 1 h, (13) − l=0 1+nγl − ≈ − 1+ nγl ! and n o Xl=0 P and 2µ(δ) 1 δ c0(Divsalar) = max 1 e− − . (17) F δ − 2δ 2 2 γl    σz = D ηˆ η c 4σ− 1 h (14) In [19], Divsalar derived a simple upper bound and further − ≈ − 1+ nγl l=0 ! 1 n o showed that, if Eb/N0 > i c c0(Divsalar), the simple upper X κ R for sufficiently large B. bound goes to 0 as the block size L goes to infinity. Ps We now include the effect of Ps. Defining γs = 2σ2 , we For the set of random linear (L,K) block codes of rate Rc, have it is well known that

F γl µ(δ)= H(δ) (1 Rc) ln 2, (18) P (c cˆ) Q 2γs 1 h , (15) − − → ≈ v − 1+ nγl  u l=0 ! where H(δ)= δ ln δ (1 δ) ln(1 δ). u X − − − − t Now, we are in a position to discuss the optimum coding- where Q( ) is the Gaussian integral function  · spreading tradeoff for NBI suppression. With a unified cod- + 2 1 ∞ t ing view, a coded DS-SS approach can be viewed as a Q(x)= exp dt. √2π − 2 serially-concatenated coding approach, where direct-sequence Zx   5

(of length n) spreading can be regarded as a simple (n, 1) A length-2r bi-orthogonal Hadamard code carries r + 1 inner repetition coding scheme. information bits. Therefore, the rate of a length-2r Hadamard r+1 It is well known that the serial concatenation of any outer code is Rh = 2r . For such a code, the codeword set is j code with an inner repetition code can not produce any good formed by the columns of Hn, denoted by h , j = r ± {± code (namely, the weight distribution Ah,h 0 is not 0, 1, , 2 1 . In a systematic encoding, the bit indexes ≥ ··· − } r 1 desirable). For example, let us consider a powerful rate-1/2 = 0, 1, 2, 4, , 2 − are used as information bit posi- 1 I { ··· } outer code with µo(δ)= H(δ) 2 ln 2, which is concatenated tions. with an (n, 1) inner repetition− code. Then, the concatenated code has 1 . By employing the simple bound µ(δ) = n µo(nδ) B. Convolutional-Hadamard Codes [19], the minimum Eb/N0 threshold conditioned on a given To encode a convolutional-Hadamard code, the information value of κi can be computed as (L ) → ∞ bit stream is first segmented into blocks of r bits. The kth E 1 b = c (Divsalar). (19) block can be written as N κ R 0  0 threshold i c T dk = [dkr, dkr+1, , dkr+r 1] , (21) − In the case of F = 1,γl = 20 dB, the minimum Eb/N0 ··· threshold versus the coding rate is plotted in Fig. 2 for where dkr+j 0, 1 , j = 0, 1, , r 1. Then, each block ∈ { } ··· − both NBI suppression approaches. For the coded spreading is employed to produce a parity-check bit. For the kth block, approach, we have employed a rate-1/2 random code as the produced parity-check bit is the outer code, which is further concatenated with a (n, 1) pk = dkr dkr+1 dkr+r 1. (22) repetition code (spreading). Therefore, an equivalent coding ⊕ ⊕···⊕ − 1 rate of 2n is obtained. For the full coding approach, the rate- The parity bits of information blocks pk are further input 1 { } 2n random code is employed. As shown in Fig. 2, the optimum to a rate-1 recursive to produce the coded coding-spreading tradeoff always favors negligible spreading bits qk . The simplest rate-1 recursive convolutional code when L . is a{ two-state} convolutional code with generator 1/(1 + D), → ∞ Therefore, we arrive at a conclusion that the entire band- namely, the binary accumulator or the differential encoder, width expansion should be dedicated to error control for NBI where the kth output coded bit is suppression and a low-rate full coding approach designed with qk = qk 1 pk. (23) the maximized coding gain could be better than the code-aided − ⊕ DS-SS approach 2. In the remainder of this paper, we show that The information blocks dk are then appended with qk this conclusion is also supported in practice through extensive { } T T { } to form the augmented input blocks [dk , qk] . The augmented simulations for three types of NBI. blocks, each with r +1 bits, are input to the encoder of a Hadamard code, producing n =2r coded bits for each block. III. LOW-RATE FULL TURBO-HADAMARD CODES For the purpose of NBI suppression, we employ a low-rate C. Full Turbo-Hadamard Codes full turbo-Hadamard code for spreading the signal bandwidth. A full turbo-Hadamard code is constructed by con- When such a codeword is transmitted over a channel with catenating several convolutional-Hadamard codes in paral- BPSK signalling, it can be viewed as the employment of lel. Let d d d be a group of infor- coded spread-spectrum signalling with time-varying spreading D = [ 0, 1, , L 1] mation bits, which are··· input to− the encoder for produc- sequences, which competes well with the traditional DS-SS ing a full turbo-Hadamard codeword. As shown in Fig. signalling for NBI suppression. It should be mentioned that the 3, ( (0)) and its interleaved versions (m) original turbo-Hadmard codes proposed in [16] use punctured D = D D ,m = are employed by the component convolu- Hadamard sequences for each codeword, which gives an 1, ,M 1 tional··· Hadamard− codes to produce Hadamard codewords obstacle for developing iterative decoding in the presence of M (m) c c c NBI. Without puncturing, it also benefits a potential signal C = [ mL, mL+1, , mL+L 1],m = 0, ,M 1 of size r . Hence,··· a full turbo-Hadamard− codeword··· − can acquisition mechanism for the proposed full coding approach 2 L be obtained× by collecting all component codewords, namely, (with time-varying spreading sequences). (0) (1) (M 1) C = [C , C , , C − ] = [c0, c1, , cML 1], where ··· ··· − r ck, k =0, 1, ,ML 1, is the column vector of length 2 . A. Hadamard Codes Compared··· to the original− turbo-Hadamard codeword [16], a An n n with n =2r can be recursively full turbo-Hadamard codeword consists of M sub-codewords, constructed× as produced by M component convolutional-Hadamard codes without puncturing. Hence, information bits can be repeatedly +Hn/2 +Hn/2 Hn = , (20) observed in the final codeword. The employment of full com- +Hn/2 Hn/2  −  ponent Hadamard codewords can facilitate NBI suppression and H1 = [+1]. without sacrificing much bandwidth since the rate of each convolutional-Hadamard code is very low. With a full turbo- 2In essence, this section provides just a partial justification of this claim due to a series of approximations and the restriction on multi-tone interference. Hadamard code using BPSK modulation, the transmitted sig- Hence, a rigourous proof is still missing. nal can be seen as a kind of direct-sequence spread-spectrum 6

D ^`dk NBI Conv. Hadamard Encoder 0

D )1( Interleaver Conv. Hadamard Encoder 1 Modulator Channel 噯 噯 噯 噯 D M  )1( Interleaver Conv. Hadamard Encoder M-1

Fig. 3. System Model. signalling with time-varying spreading-sequences (Hadamard B. APP Decoding of Hadamard Code in the Presence of NBI r sequences) of fixed length n =2 . Assume now that all of the Hadamard codewords have The rate of a full turbo-Hadamard code is simply given by independent and equal probabilities of occurrence. Let ck = T r [ck[0],ck[1], ,ck[n 1]] be the transmitted kth Hadamard R = , (24) ··· − F M2r codeword and rk be the noisy observation of ck (3). The APP decoding of a Hadamard code over +1, 1 involves while the original turbo-Hadamard code is of rate evaluating the logarithm-likelihood ratio (LLR){ − } r R = . (25) p (r c ) T H r + M(2r r) k| k − Pr(ck[i]=+1 rk) ck[i]=+1 Lk[i] = log | = log X , Hence, the ratio is Pr(ck[i]= 1 rk) p (r c ) − | k| k r ck[i]= 1 RF M2 (M 1)r X− (26) r = − r − , i =0, 1, , 2 1. (27) RT H M2 ··· − Let us define which quickly approaches 1 as r increases. j j 1 j T 1 j ψ( h ) = exp h , y exp h Q− h , ± ± k · −2 IV. ITERATIVE DECODINGINTHE PRESENCE OF BOTH j   = exp h , yk + λj , (28) NBI AND AWGN ± where  We have shown in Section-II the potential of a full coding 1 j T 1 j λ = h Q− h . (29) approach in NBI suppression with ML decoding. However, the j −2 complexity issue makes an ML decoder impractical. In this Then, (27) can be expanded as shown in the top of the next section, we develop a practical iterative decoding algorithm page. in the presence of both NBI and AWGN for a low-rate full 1 1 In the absence of NBI, it can be shown that λj = 2 n σ2 turbo-Hadamard coding approach. 2 j T 1 j j 2 − since σ h Q− h = h = n. Efficient techniques for evaluating (28) and (30)k werek proposed in [16] using FHT. In the presence of NBI, we now show that the FHT-based A. Channel Model efficient techniques still work with the assumption that the With a vector representation of the channel model (3), we NBI process is stationary. assume that the NBI is a block-independent random process. For a stationary NBI process, the covariance matrix Q does For convenience, it is always assumed that the block memory not change for a sufficiently long period. Hence, λj , j = length of NBI is the same as the length of component 0, 1, , 2r 1 (29) can be pre-computed and the received r ··· − Hadamard codeword (n =2 ). signal vectors rk are properly modified to obtain yk = 1 { } { Therefore, when a full turbo-Hadamard codeword C = Q− rk . } j r [c0, c1, , cML 1] is transmitted, the received equivalent The inner products h , yk , j = 0, 1, , 2 1 are ··· − ··· − complex base-band signal vector in the duration of Hadamard elements of Hnyk and thus can be efficiently evaluated by an r codeword nTc can be represented as (3), where ck is now FHT of size n =2 . The results are then modified using the the kth (component) Hadamard codeword of a full turbo- pre-computed λj and 2n exponential functions are performed Hadamard codeword. for obtaining ψ( hj ), j =0, 1, , 2r 1. ± ··· − In what follows, three types of NBI signals [2], multi- By defining two 2r 1 sized compound vectors a and b × tone signal, autoregressive (AR) signal, and digital signal, with are considered. For conciseness, we still use P = 1 in this ψ(+hj ) s a[j]= , j =0, 1, , 2r 1 (31) section. ψ( hj ) ··· −  −  7

1 j T 1 j 1 j T 1 j exp r h Q− r h + exp r + h Q− r + h −2 k − k − −2 k k H[i,j]=+1   H[i,j]= 1   Lk[i] = log X   X−   , 1 j T 1 j 1 j T 1 j exp r h Q− r h + exp r + h Q− r + h −2 k − k − −2 k k H[i,j]= 1   H[i,j]=+1   X−   X   ψ( hj ) ± H[i,j]= 1 = log X± . (30) ψ( hj ) ± H[i,j]= 1 X∓

and For iterative decoding, it is essential to compute the extrinsic j information for systematic information bits. For an AWGN H[i,j]= 1 ψ( h ) r b[i]= ± ± j ,i =0, 1, , 2 1, (32) channel, the extrinsic information in the log-likelihood form H[i,j]= 1 ψ( h ) ··· −  P ∓ ±  at the positions of information bits can be computed as it can be againP shown [16] that e a Lk(i)= Lk(i) Lk(i) Lc rk(i),i , (39) b = Qna, (33) − − · ∈ I √Ps where Qn is a n n matrix obtained from Hn by the following where Lc =2 σ2 . replacements: × In the presence of NBI, the last term in (39) cannot be 1 0 0 1 directly employed as it was ruined by NBI. Hence, we need +1 , 1 . (34) to estimate this term, which, however, is not easy in general → 0 1 − → 1 0     in the presence of NBI. Here, we propose an approximation The derivation process is the same as that of [16], and thus by simply omitting this term, giving omitted here. e a As shown in [16], the computation of (33) can be efficiently Lk(i) Lk(i) Lk(i),i , (40) carried out using an n-point APP-FHT. ≈ − ∈ I for the extrinsic information at the positions of information C. Soft-In/Soft-Out Decoding of Convolutional-Hadamard bits. Indeed, due to very low-rate coding, we have L, |I| ≪ Codes and the contribution of Lc rk(i) is also very limited. Hence, · For soft-in soft-out (SISO) decoding of convolutional- this approximation is justified. Simulations show that this Hadamard codes, the Bahl, Cocke, Jelinek, and Raviv (BCJR) approximation method does work well. algorithm can be employed. It is based on the trellis de- scription of convolutional-Hadamard codes as shown in [16]. D. The Overall Decoder A simple 1/(1 + D) convolutional code has 2 states and With the SISO decoder for each component convolutional- r r 2 branches per state. Therefore, there are 2 parallel edges Hadamard code, the overall decoder structure is the same as connecting sk (state at the kth block-wise time epoch) to sk+1 that of Fig. 7 in [16], and thus omitted here. and the branch metric function can be computed as

γk(sk,sk+1)= ψ(ck), (35) V. EXTENSION TO MULTIPATH FADING CHANNELS

ck (sk,sk+1) ∈CX In practical wireless communication systems, an AWGN where (sk,sk+1) is the set of possible Hadamard codewords channel model may not be applicable. Instead, multipath C connecting sk to sk+1. fading channels are often considered. The BCJR algorithm can be characterized by the following In general, the multipath scenario suffers several difficulties. forward and backward recursions: Firstly, low-rate turbo-Hadamard codes perform well in an AWGN channel, which, however, may perform badly for αk+1(sk+1)= αk(sk)γk(sk,sk+1), (36) sk some multipath channels (or inter-symbol interference (ISI) X channels). Secondly, with the introduction of fading, practi- βk(sk)= βk+1(sk+1)γk(sk,sk+1). (37) cal wireless communications essentially see an ensemble of

sk+1 multipath (or ISI) channels, which constitutes a main obstacle X Finally, the log-likelihood ratio at the ith bit-wise time epoch for code design. Indeed, design of “good” codes for multipath for the kth Hadamard codeword is computed as fading channels is still a big challenge. In this section, we do not discuss the code design problem. ψ(ck)αk(sk)βk+1(sk+1) Instead, we focus on the decoding of low-rate turbo Hadamard ck[i]=+1 codes in the presence of both NBI and multipath fading. Quasi- Lk(i) = log X . (38) ψ(ck)αk(sk)βk+1(sk+1) static fading is considered, where the multipath taps remain ck[i]= 1 fixed during the block of a codeword, but independently X− 8 varying from one block to another. We assume that the receiver code instead of the powerful for the code-aided input is preceded by a matched filter, and then followed DS-SS approach, due to the use of short block length for fast by a noise whitening filter. Hence, a discrete-time complex simulations. In our simulation, the information bit length of baseband representation of the multi-path channel with P +1 200 is employed for the convolutional code of constraint length non-zero taps is considered, where the delay of tap l is denoted 7, and Viterbi decoding is employed for the NBI suppressed T as τl, and the coefficient of tap l is fl. The channel output in MMSE (soft) outputs u rk, k =1, , F (please refer to (4)). its complex form can be written as With this short information bit length,··· we notice that turbo

P codes cannot show much advantage over the convolutional ˜ code of constraint length 7 in performance. r˜k = f0ck + fvck τl + ik +w ˜k, (41) − l=1 In the past years, various NBI suppression methods were X widely employed such as in the literature [3], [5]–[7], [9]– where ˜i is a complex NBI sequence, and w˜ is a complex k k [11]. Although various NBI suppression techniques may result AWGN{ sequence.} { } in different performance, we do not compare these techniques Suppose the length of Hadamard sequence is large enough with our work, since much of these works focus on the signal so that max τ , , τ n. By collecting n received 1 P processing techniques. By employing the upper bounds for samples (41){ (similar··· to} (3)), ≤ the received signal in a vector evaluating the ML performance as in Section-II, the traditional form of length n can be written as DS-SS approach shows its performance weakness compared to P a full coding approach for NBI suppression, no matter what- ˜ ˜rk = f0ck + flc¯k τl + ik + w˜ k, (42) ever signal processing technique is employed. This theoretical − Xl=1 performance weakness is mainly caused by its spreading +τl (n τl) τl (repetition coding) structure, which leaves limited room for where c¯k τl = ck +c−k 1− , ck± denotes ck shifting down − − performance improvement with advanced signal processing (+) or up (-) by τl positions and the vacant positions are filled with zeros. techniques. For simplicity, it is assumed that both fl and τl are For a full turbo-Hadamard code, we use N for the in- perfectly estimated and available at the receiver.{ } Given{ } f terleaver length, M for the number of component codes, R { l} and τl , we denote the ISI part in (42) as for the code rate, r for the order of the included Hadamard { } code, g(D) for the generator polynomial of the employed L k , convolutional code and κ for the iteration number. Unless ¯ck 1 ϕ (ck, ck 1)= f0ck + flc˜k τl , (43) − − − specified otherwise, κ is set to 10. l=1 X For ease of comparison, it is supposed that the chip rates which is a function of both c and c . k k 1 observed over the channel for both approaches are just the In the presence of both NBI and AWGN,− one can show that same. For the code-aided DS-SS approach, the spreading k † ˜ 1 k sequence is a Hadamard sequence with length 32 and the rate- log p (˜rk ck, ck 1) ˜rk c¯k 1 Q− ˜rk c¯k 1 (44) − | ∝− − − − − 1/2 convolutional code with generators G1 = 171, G2 = 131 ˜ ˜ ˜ where Q = E[ikik† ]+E[w˜kw˜ k† ] with ( ) denoting the complex is employed. For the turbo-Hadamard coding approach, the pa- conjugate of the transpose. · † 1 rameters are set as follows: M =2, r =5 and g(D)= 1+D . The SISO decoding of component convolutional Hadamard Hence, the Hadamard sequences are of the same length 32 for codes can now be again implemented using BCJR algorithm, both approaches. but with expanded super-state trellis. The super state at time For fast simulations, N = 200 is assumed unless specified epoch k is formed by joining the state pair of the component otherwise. With various parameters described as above, the convolutional Hadamard code at time epoches of k and k 1, r 5 rate of the full turbo-Hadamard code is R = r = . If sup − F M2 64 namely, s = (sk 1,sk). With max τ1, , τL n, the k − { ··· } ≤ information bits are input at a rate of fb bits/s, the resulting multi-path fading essentially introduces the symbol memory transmission chip rate observed over the channel should be of 2 with symbol denoting the vector of Hadamard sequence. r equal to fc = fb/R = fbM2 /r. For the rate-1/2 convo- Let S denote the number of states for each component convo- lutional coded DS-SS system, the transmission chip rate is lutional Hadamard code. Then, the number of super-states is r equal to fc = 2fb2 . For APP decoding of Hadamard codes 2S. Therefore, the decoding complexity is almost independent in the presence of NBI, the covariance matrix Q is required. of P thanks to the use of long Hadamard sequences. T In simulations, Q = E rkrk is pre-computed during the With SISO decoding of component convolutional Hadamard training mode. codes, iterative decoding of turbo Hadamard codes in the pres-  With an equal channel bandwidth allocated for both ap- ence of both NBI and multipath fading can be implemented proaches, it is clear that the information transmission rate for in a similar manner as discussed in Section-IV. the turbo-Hadamard coding approach can be r = 5 times higher than that of the traditional DS-SS approach with rate- VI. SIMULATION RESULTS 1/2 convolutional coding. In what follows, we show that In this section, the proposed turbo-Hadamard coding ap- with such a significantly-higher transmission rate, the turbo- proach is compared with the code-aided DS-SS approach with Hadamard coding approach can still perform better than the convolutional coding. We employ the traditional convolutional traditional code-aided DS-SS approach. 9

A. Tone Interference multi-tone interference with orthogonal signatures are much For tone interference, its complex version takes the form of less harmful than the general multitione interference. F ˜ j(2πflk+φl) ik = Ple , (45) B. Autoregressive (AR) Interference l=1 X p Suppose now that the NBI signal can be modeled as a p-th where F is the number of tones, P and f are the power l l order AR process, where p N, i.e., and normalized frequency of the lth sinusoid, and φl are ≪ independent random phases uniformly distributed over{[0,}2π). p For the real channel model employed in (3), it simply takes ik = aj ik j + ek, (46) − − the real part of ˜i , namely, i = ˜i . j=1 k k ℜ k X 2  where ek is a white Gaussian process with variance ε . −1 10

code−aided DS−SS + conv. −1 proposed turbo−Hadamard 10 turbo−Hadamard without NBI union bound code−aided DS−SS + conv. −2 10 proposed turbo−Hadamard −2 10 Bit Error Rate −3 10 Bit Error Rate −3 10

−4 10 −4 1 1.5 2 2.5 3 3.5 10 Eb/N0, dB 1.6 1.8 2 2.2 2.4 2.6 2.8 3 Eb/N0,dB

Fig. 4. Comparison of the BER performance for the two approaches with multi-tone interference (orthogonal signatures). Fig. 5. Comparison of the BER performance for the two approaches with AR interference. In [20], we have shown the performance of the proposed turbo Hadmard approach under the general multi-tone inter- In our simulation, the NBI is generated by a second-order AR model with both poles at 0.99, i.e., a = 1.98 and a = ference with randomly-generated frequencies. Here, we focus 1 − 2 on the multi-tone interference with orthogonal signatures. 0.98. The total interference power is kept constant at 25 dB Simulation conditions are set as follows: F =3, the interfering relative to a unity power signal (Ps =1). tones have the same power, and the total interference power It is shown in Fig. 5 that the proposed turbo-Hadamard is kept constant at 25 dB relative to a unity power signal coding approach outperforms the traditional code-aided DS- Ps =1. To ensure orthogonal signatures, fl fk is chosen as SS approach although its transmission rate is about 5 times r − the multiple of 1/n = 2− for all l = k, and φl is randomly higher than the code-aided DS-SS approach. picked from [0, 2π). All of conditions6 remain the same for both approaches. C. Digital Interference Simulations have been carried out to show the capability of both approaches against the multi-tone interference with An NBI signal can also be a digital communication signal orthogonal signatures. It is shown in Fig. 4 that the proposed with a data rate much lower than the spread-spectrum chip turbo-Hadamard coding approach still outperforms the tradi- rate. The digital NBI model can find its applications when the tional code-aided DS-SS approach although its transmission spread-spectrum communication system works concurrently rate is about 5 times higher than the code-aided DS-SS with some narrow-band digital communication systems. Here, approach. In Fig. 4, we also plotted the decoding performance we assume a fixed relationship between the data rates of the for turbo Hadamard codes with perfect knowledge of NBI, as two users, i.e., q bits of the narrow-band user occur for each a benchmark. As shown, the proposed turbo-Hadmard coding bit of the SS user. approach performs very close to the decoding counterpart with In general, the digital NBI signal and the SS signal are perfect knowledge of NBI, and about 0.2 dB in SNR loss is asynchronous. Let t0 be the fixed time lag between the SS bit observed. and the nearest previous start of an NBI bit. For multi-tone interference with orthogonal signatures, the The digital NBI is generated by a base-band digital BPSK fc union upper bound on the bit error probability similar to (16) is signalling with rate fI = q and a fixed lag t0 = 13Tc. The also shown in Fig. 4. By referring to (16), the loss in Eb/N0 total interference power is kept constant at 20 dB relative to is about 0.4 dB due to multi-tone interference. Clearly, the a unity power signal (Ps =1). 10

−1 10 −1 10 code−aided DS−SS + conv. M=2, N=200 proposed turbo−Hadamard M=2, N=1000

M=3, N=200 −2 10 −2 10 M=3, N=1000 Bit Error Rate −3 Bit Error Rate 10 −3 10

−4 10 −4 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 10 0 0.5 1 1.5 2 2.5 3 Eb/N0, dB Eb/N0,dB

Fig. 6. Comparison of the BER performance for the two approaches with Fig. 7. The BER performance of the turbo-Hadamard coding approach for digital interference. different values of M and N with AR interference.

−1 Assuming that the digital NBI signal has a waveform of 10 rectangular pulse, the autocorrelation function of the chip- r=5, turbo−Hadamard r=6, turbo−Hadamard sampled NBI signal is then r=7, turbo−Hadamard r=5, code−aided DS−SS Pi k (1 | | ), k L r=6, code−aided DS−SS L L r=7, code−aided DS−SS Ri(k)= − | |≤ (47) −2 0, k >L 10  | | Although the digital NBI is far from Gaussianly-distributed, the proposed algorithm does work and even outperforms the

traditional code-aided DS-SS approach as shown in Fig. 6. Bit Error Rate −3 10 D. Impacts of M, N and r The performance of turbo-Hadamard coding approach can be enhanced if either the interleave length N or the number of −4 10 component codes M increases. This is well explained in Fig. 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 7 for the AR interference with the same setting as before. It Eb/N0, dB should be noted that the increase of M can reduce the spectral efficiency. As shown, the turbo-Hadamard coding approach Fig. 8. The BER performance of the two approaches for different values of performs excellent with M =3 and N = 1000 for strong AR r with AR interference. interferences.

0 With the increase of r, the length of spreading sequence 10 r is exponentially expanded (n = 2 ), resulting into improved AWGN processing gain. However, the BER performance of the code- ISI−Ch1 ISI−Ch2 −1 aided DS-SS approach with convolutional coding cannot be 10 ISI−Ch3 improved as shown in Fig. 8. The reason is that the capability Multipath fading of NBI suppression can be achieved with relatively short spreading sequence and its BER performance is largely deter- −2 10 mined by the employed convolutional coding scheme, which remains almost fixed with the increase of r. With the proposed Bit Error Rate turbo-Hadamard coding approach, the BER performance can −3 10 be steadily improved with the increase of r, which means that a significant coding gain can be obtained with a lower coding rate. −4 10 0 2 4 6 8 10 12 14 16 18 E. Multipath Fading Channels Eb/N0, dB

To gain an essential insight into the performance of the Fig. 9. The BER performance of turbo-Hadamard coding approach in proposed full coding approach under multipath fading chan- both NBI and multipath (fading) channels, where ISI-Ch1: [f0,f1] = [p3/4, p1/4]; ISI-Ch2:[f0,f1] = [p1/2, p1/2]; ISI-Ch3:[f0,f1] = nels, a simple two-tap channel model with τ1 = n is em- ployed. Hence, this ISI channel model can be characterized [p1/4, p3/4]. 11

TABLE I: Decoding Complexity Per Information Bit 2,κ = 10). The proposed turbo-Hadmard coding approach has Approach Mul./Div. Exp./Log. Add./Comp. significantly higher complexity, which is mainly due to the full Proposed turbo-Hadamard 340 276 824 Code-aided DS-SS 2 0 192 coding approach and the employment of iterative decoding.

VII. CONCLUSION AND FUTURE WORK by [f ,f ] if no fading is considered. We consider three ISI 0 1 In this paper, we have derived upper bounds on the error channel models, namely, ISI-Ch1: [f ,f ] = [ 3/4, 1/4], 0 1 performance of coded systems in the presence of both AWGN ISI-Ch2:[f ,f ] = [ 1/2, 1/2], and ISI-Ch3:[f ,f ] = 0 1 p 0 p1 and multi-tone interference with orthogonal signatures, which [ 1/4, 3/4], respectively. p p can be employed to evaluate the system performance. By using The complex AR interference is assumed, and its real (or p p tight bounding techniques, we have provided a partial justifi- imaginary) part takes the same setting as before. As shown in cation that the optimum coding-spreading favors full coding Fig. 9, the decoding performance under ISI channels perform for the purpose of NBI suppression. It should be emphasized worse than that under the AWGN channel, and deteriorates that a rigourous proof of this justification is difficult due to when the second tap f becomes more dominant. 1 mathematical intractability for general classes of NBI. With quasi-static fading, we consider that both f and f 0 1 More practically, we have proposed a low-rate full turbo- are independently generated according to complex Gaussian Hadamard coding approach for NBI suppression. With proper distribution with zero mean and variance of 1 . Also shown 2 iterative processing, it was shown that the proposed low-rate in Fig. 9, there is an obvious error floor for multipath fading coding approach performs better than the traditional coded DS- channel. The error floor comes from some bad realizations SS approach for general classes of NBI with noticeably higher of multipath fading channel, namely, max f , f 1 or 1 0 spectral efficiency. For iterative decoding Hadamard codes in f f . {| | | |} ≪ | 1| ≫ | 0| the presence of NBI, we found that the fast FHT-based APP algorithm can still work, which makes this approach attractive F. Complexity Issue in practice. The proposed full coding approach can also be In this subsection, we summarize the computational com- adapted to multipath fading channels, and a low-complexity plexity of the proposed turbo-Hadamard coding approach in iterative receiver exists if the length of underlying Hadamard the AWGN channel, which is compared with that of the code- sequences is not smaller than the multipath time. aided DS-SS approach with convolutional coding. For ease of With a full turbo-Hadamard coding approach, the employ- comparison, we assume that the length of spreading sequence ment of DS-SS signalling with time-varying spreading se- is of size 2r, as adopted in our simulation. quences can be assumed, which could be potentially employed With the introduction of NBI, both approaches require to for signal acquisition. Indeed, there are only n =2r spreading 1 r modify the received signal vectors as yk = Q− rk, where sequences for an n n Hadamard matrix with n =2 , while 1 × Q− can be efficiently updated using the recursive least- the space of binary spreading sequences of length n is of squares (RLS) adaptive algorithms. Compared to the code- size 2n. Therefore, the receiver may search n =2r spreading aided DS-SS approach, the proposed approach requires to sequences in parallel for the acquisition of the incoming full r further pre-compute λj , j = 0, , 2 1 (29). As the cost turbo-Hadmard coded signal, where FHT could be efficiently in this pre-computation stage is··· almost− the same, we do not employed to make this search process faster. As this topic consider it for final comparison. is rather involved, much work remains open. Indeed, we can The decoding complexity of the proposed turbo-Hadamard expect a prospect application of full coding approach in future coding approach can be summarized as follows. With itera- military communications if the acquisition of low-rate full tive decoding, the decoding complexity is determined by the coding signal can be efficiently implemented. number of iterations κ and the operations required for M Finally, the multiuser scenario deserves investigation in the component APP decoders in each decoding iteration. Each future. convolutional-Hadmard APP decoder is implemented section- by-section over S-state trellis and it was estimated in [16] ACKNOWLEDGMENT that for each section, about 10 2r 2r +4S 12 addi- The authors would like to thank Prof. Xiang-Gen Xia and tions, 2 2r +4+8S multiplications/divisions,× − −2 2r + r Prof. Wei-Ping Zhu for their helpful comments. exponential/logrithm× functions are required. × The decoding complexity of the code-aided DS-SS approach with convolutional coding is evident [2]. Whenever the refined APPENDIX signal vector yk is computed, the linear MMSE detector can EVALUATION OF THE PEP be directly implemented by vector multiplications. With rate- In this appendix, we consider a special class of NBI process 1/2 convolutional coding, it requires 2 multiplications per with F tones fl,l = 1, 2, , F , as defined by (45). For information bit. The decoding complexity of convolutional ease of derivation, we further··· assume orthogonal signatures, code is well known, which is determined by the constraint namely, length (Viterbi algorithm). H g gk = nδl,k, l, k [1, F ] (A.1) The decoding complexities per information bit are summa- l ∀ ∈ j2πfl j4πfl j2π(n 1)fl T rized in Table. I for both approaches (r = 5,M = 2,S = where g = [1,e ,e , ,e − ] . l ··· 12

It has been shown in [2] that Thus,

B F F P 2 γl 1 2 l g gH (A.2) µz = 2σ− hk ξk,l Q− = σ− I 2 l l , − 1+ nγl − σ + nPl k=1 l=0 ! l=0 ! X X X F B 2 γl Then, we have = 2σ− h ξk,l − 1+ nγl ! Xl=0 kX=1 B F F B 1 γl 2 2 γl T 2σ− h E ξ , λ(c) = 1 ck gl k,l 2 ≈ − 1+ nγl · { } −2σ − 1+ nγl ! l=0 k=1 ! k=1 l=0 X X X F X B (A.9) B γ 1 2 l cT g = 2 1 k l B −2σ − 1+ nγl · B where h = hk denotes the Hamming distance between l=0 k=1 ! k=1 XF X c and cˆ. B γ 2 P l cT g Due to the summation of ξk,l over blocks (k), we can use 2 1 E k l ≈ −2σ − 1+ nγl in place of , where is the expectation with l=0 ! E ξk,l ξk,l E X n o { } c {·} F respect to k for a given hk, leading to B nγl (A.3) 2 = 2 1 , T −2σ − 1+ nγl E ξk,l = E u gl,hk l=0 ! { } { hk } X uT g gH v +E hk l,hk E l,n hk n hk where γ = P /σ2, E is taken with respect to c and the { }· { − − } l l k = hk. (A.10) last step is due to {·} Hence, we get T 2 T H E ck gl = tr E ck ck glgl 2 F γl { } µ 2σ− 1 h. (A.11) z ≈ − l=0 1+nγl n o = n.  (A.4) For computing the variance, P we note that

In the above derviation, we have used the approximation B 2 2 1 B T T 2 2 that k=1 ck gl E ck gl for large B and the E (ˆη η) c = E ˆc c , y c B ≈ { − } |h k − k ki| k ergodicity of the sub-vector c over the column space of H . k=1 P nk o n X n o Hence, B T 1 T 1 = (ˆck ck) Q− ckck + Q Q− (ˆck ck) λ(c) λ(ˆc) 0, (A.5) − − − ≈ k=1 X B  2 T 1 and the metric difference can be simplified as =E ηˆ η c + (ˆc c ) Q− (ˆc c ). − k − k k − k n o k=1 B X

η ηˆ ck ˆck, yk . (A.6) Then, we get − ≈ h − i k=1 X 2 , 2 2 σz D ηˆ η c = E (ˆη η) c E ηˆ η c Next, we proceed to derive both the mean and the variance − − − − Bn o n o n o of η ηˆ assuming that the codeword c is transmitted. It is T 1 = (ˆc c ) Q− (ˆc c ) straightforward− to show k − k k − k k=1 X F B 2 γl , T 1 4σ− 1 h. (A.12) µz E η ηˆ c = (ck ˆck) Q− ck. (A.7) ≈ − 1+ nγl − − l=0 ! n o k=1 X X REFERENCES Without loss of generality, we assume that ck differs from ˆck at hk head positions. Hence, we employ a sub-vector [1] H. V. Poor and L. A. Rusch, “Narrowband interference suppression in T T T spread spectrum CDMA,” IEEE Personal Commun. Mag., vol. 1, pp. decomposition of both ck and cˆk, namely, ck = [uhk , vn hk ] T T T T T − T 14–27, Aug. 1994. and ˆck = [ uhk , vn hk ] . Thus, ck cˆk = [2uhk , 0n hk ] . [2] H. V. Poor and X. D. Wang, “Code-aided interference suppression for − − − − It follows that DS/CDMA communications - Part I: Interference suppression capabil- ity,” IEEE Trans. Commun., vol. 45, pp. 1101–1111, 1997. F [3] L. M. Milstein, “Interference rejection techniques in spread spectrum T 1 2 γl communications,” Proc. IEEE, vol. 76, pp. 657–671, Jun. 1988. (ck ˆck) Q− ck =2σ− hk ξk,l , (A.8) [4] H. V. Poor and X. D. Wang, “Code-aided interference suppression for − − 1+ nγl l=0 ! DS/CDMA communications - Part II: Parallel blind adaptive implemen- X tations,” IEEE Trans. Commun., vol. 45, pp. 1112–1122, 1997. where uT g 2 uT g gH v and g [5] H. Xiong, W. Zhang, Z. Du, B. He, and D. Yuan, “Front-end narrowband ξk,l = hk l,hk + hk l,hk l,n hk n hk l = interference mitigation for DS-UWB receiver,” IEEE Trans. Wireless − − [gT , gT ]T . Commun., vol. 12, pp. 4328–4337, 2013. l,hk l,n hk − 13

[6] N. Song, R. C. de Lamare, M. Haardt, and M. Wolf, “Adaptive widely Zhen Yang received his B.Eng. and M.Eng. degrees linear reduced-rank interference suppression based on the multistage from Nanjing University of Posts and Telecommu- wiener filter,” IEEE Trans. Signal Process., vol. 60, pp. 4003–4016, nications in 1983 and 1988, respectively, and his Aug. 2012. Ph.D. degree from Shanghai Jiao Tong University in [7] A. Gomaa and N. Al-Dhahir, “A sparsity-aware approach for NBI 1999, all in electrical engineering. He was initially estimation in MIMO-OFDM,” IEEE Trans. Wireless Commun., vol. 10, employed as a Lecturer in 1983 by Nanjing Univer- pp. 1854–1862, Jun. 2011. sity of Posts and Telecommunications, where he was [8] H. Shao and N. C. Beaulieu, “Direct sequence and time-hopping promoted to Associate Professor in 1995 and then a sequence designs for narrowband interference mitigation in impulse Full Professor in 2000. He was a visiting scholar at radio UWB systems,” IEEE Trans. Commun., vol. 59, pp. 1957–1965, Bremen University, Germany during 1992-1993, and 2011. an exchange scholar at Maryland University, USA [9] A. Giorgetti, M. Chiani, and M. Z. Win, “The effect of narrowband in- in 2003. His research interests include various aspects of signal processing terference on wideband wireless communication systems,” IEEE Trans. and communication, such as communication systems and networks, cognitive Commun., vol. 53, pp. 2139–2149, Dec. 2005. radio, spectrum sensing, speech and audio processing, compressive sensing [10] J. J. Perez-Solano, S. Felici-Castell, and M. A. Rodriguez-Hernandez, and wireless communication. He has published more than 200 papers in “Narrowband interference suppression in frequency-hopping spread academic journals and conferences. spectrum using undecimated wavelet packet transform,” IEEE Trans. Prof. Yang serves as Vice Chairman and Fellow of Chinese Institute of Vech. Tech., vol. 7, pp. 1620–1629, May 2008. Communications, Chairman of Jiangsu Institute of Communications, Vice [11] R. C. de Lamare and R. Sampaio-Neto, “Adaptive reduced-rank process- Director of Editorial Board of The Journal on Communications and China ing based on joint and iterative interpolation, decimation and filtering,” Communications. He is also a Member of Editorial Board for several other IEEE Trans. Signal Process., vol. 57, pp. 2503–2519, 2009. journals such as Chinese Journal of Electronics, Data Collection and Pro- [12] A. J. Viterbi, “Very low rate convolutional codes for maximum theoret- cessing et al, the Chair of APCC (Asian Pacific Communication Conference) ical performance of spread-spectrum multiple-access channels,” IEEE Steering Committee during 2013 2014. Jour. Select. Areas in Comm., vol. 8, pp. 641–649, Jun. 1990. [13] S. Verd and S. Shamai, “Spectral efficiency of CDMA with random spreading,” IEEE Trans. Inform. Theory, vol. 45, p. 622C640, Mar. 1999. [14] P. Frenger, P. Orten, and T. Ottosson, “Code-spread CDMA using maximum free distance low-rate convolutional codes,” IEEE Trans. Commun., vol. 48, pp. 135–144, Jan. 2000. [15] L. Ping, L. Liu, K. Y. Wu, and W. K. Leung, “Approach the capacity of multiple access channels using interleaved low-rate codes,” IEEE Commun. Lett., vol. 8, pp. 4–6, Jan. 2004. [16] L. Ping, W. K. Leung, and K. Y. Wu, “Low-rate turbo-hadamard codes,” IEEE Trans. Inform. Theory, vol. 49, p. 3213C3224, Dec. 2003. [17] C. Schlegel, “CDMA with partitioned spreading,” IEEE Commun. Lett., vol. 11, no. 12, pp. 913–915, Dec. 2007. [18] Y. Cai, R. C. de Lamare, and R. Fa, “Switched interleaving techniques with limited feedback for interference mitigation in DS-CDMA sys- tems,” IEEE Trans. Commun., vol. 59, no. 7, pp. 1946–1956, 2011. [19] D. Divsalar, “A simple tight bound on error probability of block codes with application to turbo codes,” JPL, TMO Progr. Rep. 42 - 139, Nov. 1999. [20] X. Wu, Z. Yang, J. Yan, and J. Cui, “Low-rate turbo-hadamard coding approach for narrow-band interference suppression,” in Proc. of IEEE -ICC, Sydney, Australia, 2014.

Xiaofu Wu was born in Jan. 1975. He received the B.S. and M.S. degrees in electrical engineering from the Nanjing Institute of Communications Engineer- ing, Nanjing, China, in 1996 and 1999, respectively, and the Ph.D. degree in electrical engineering from the Peking University, Beijing, China, in 2005. From 2005 to 2007, he was with the Southeast University as a Post-Doctoral researcher at the National Mobile Communication Research Laboratory. Since 2012, he has been with the Nanjing University of Posts & Telecommunications, where he is currently a Professor. His research interests are in coding and , information- theoretic security and mobile computing. He has contributed over 25 IEEE/IEE jounal papers.