On the Diversity of Noncoherent Distributed Space-Frequency Coded Relay Systems With Relay Censoring
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(2) 3492. in [12] showed that a full diversity of order two can be achieved in a single-relay network. Perfect instantaneous channel state information (CSI) was assumed in the aforementioned work [4]–[8], [11], [12] to enable coherent detection at the destination. To obtain perfect CSI, an expensive effort is required to perform accurate channel estimation, which is challenging in general and especially in multi-hop wireless networks and/or fast-fading environments. Therefore, there has been much interest in communications that do not require the instantaneous CSI, which has been termed as noncoherent communications in [10] and [17]–[23]. Noncoherent communication in flat-fading MIMO systems was studied by Hochwald and Marzetta [17], where unitary STC was proposed to achieve the same diversity order as in coherent communication. The existence of such unitary STC and their construction were investigated by Tarokh and Kim [18], where an elegant geometric interpretation of ML decoding with unitary modulation was also presented. Noncoherent space–frequency coded MIMO-OFDM systems were studied by Borgmann and Bolcskei [19], where the diversity of the noncoherent MIMO-OFDM system was shown identical to its coherent counterpart. Besides point-to-point communication, noncoherent communication was also studied in the context of wireless relay networks, e.g., for AF relaying [20]–[22] and DF relaying [10], [23], [24]. For AF relaying, Souryal [20] proposed a noncoherent generalized likelihood ratio test (GLRT)-based receiver that requires local noise power information but not instantaneous/statistical CSI, and showed that near secondorder diversity can be achieved for the single-relay scenario. Lin et al. [21] proposed a noncoherent GLRT-based sequence detector that requires neither noise power information nor CSI, and demonstrated that near full diversity can be achieved for three special relaying scenarios with at most two relays. Farhadi and Beaulieu [22] developed a completely noncoherent receiver based on maximum energy selection (MES) and showed that the noncoherent MES detection can promise full spatial diversity with any number of relays. For DF relaying, Chen and Laneman [10] showed that, with noncoherent detection at the destination and with multiple relays that always transmit, the diversity order is only about half of the number of relays due to error propagation. Chang et al. [23] proposed a noncoherent orthogonal space–time block coded (OSTBC) OFDM scheme to enable simultaneous transmission (as opposed to orthogonal transmission in [20]–[22]) and showed that the proposed distributed OSTBC scheme can achieve full spatial diversity when perfect decoding at the relays is established. Valenti et al. [24] studied the noncoherent receiver design that does not require the phase information for physical-layer network coded relay systems for the cases of known and unknown channel amplitudes. A. Motivations Noncoherent communication in relay networks has not been as well studied for frequency-selective fading channels (e.g., [23]) as for flat fading channels (e.g., [10], [20]–[22]). In particular, the combination of noncoherent distributed SFC and. IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 10, OCTOBER 2014. relay censoring for DF relaying has not been investigated. We are therefore motivated to conduct an analytical study of the maximum achievable diversity in noncoherent distributed SFC systems with multiple censored DF relays by considering both perfect and imperfect censoring. The design and analysis of noncoherent distributed SFC systems with relay censoring are not direct extensions from point-to-point communication systems (e.g., [19]) because: 1) For DF relaying with multiple censored relays, the relay decoding status information (i.e., whether a relay has successfully decoded and will participate in cooperative transmission) is not available at the destination especially in noncoherent settings; and 2) The error propagation effect due to censoring errors dramatically complicates the diversity analysis. B. Contributions In this paper, we perform the diversity analysis of a noncoherent, distributed space–frequency coded OFDM-based DF relaying system under frequency-selective fading environments. We consider a network comprised of a single source, multiple relays, and a single destination. Similar to [8], [11], and [25], censoring is performed before each relay decodes and forwards the source’s information toward the destination. We derive the achievable diversity analytically for both cases of perfect and imperfect relay censoring. The main contributions of this paper are summarized below: • With perfect relay censoring, we consider cases with and without partial information of channel statistics and relay decoding status at the destination. When partial information is available, we show that a diversity of order R · min{Ls , L} can be achieved with the ML decoder, where R is the number of cooperating relays, and Ls and L are the channel order (number of delay paths) between the source and each relay and between each relay and the destination, respectively. When partial information is not available, i.e., a completely noncoherent receiver, we show that both the ML decoder and a proposed suboptimal correlator-based decoder can still yield the same diversity of order R · min{Ls , L}. The correlator-based decoder was proposed for analytical tractability to verify the diversity result for the (completely noncoherent) ML decoder as well as for its own merits, since 1) it does not require instantaneous/statistical CSI and the relay decoding status information, and 2) it allows a simple closed-form decoding rule at the destination. The derived diversity results are numerically verified. • With imperfect relay censoring, i.e., taking into account the possible mismatch between the censoring result and the actual state of decoding at each relay, we characterize the relationship between the censoring error probabilities and the achievable diversity of the system analytically. We show that when the likelihood of two types of censoring errors increases, the suboptimal correlator-based decoder may not attain the full diversity of order R · min{Ls , L}. Numerical results demonstrate an increased performance gap between the ML decoder and the correlator-based decoder with imperfect relay censoring. The results suggest.
(3) CHIU et al.: DIVERSITY OF NONCOHERENT DISTRIBUTED SFC RELAY SYSTEMS WITH RELAY CENSORING. 3493. We call a relay in the first two cases an active node and in the latter two cases a silent node. The probability of each of the four events is determined by the relay decoding error probability prs as well as the two types of relay censoring errors: Δ. pr0|1 = P(relay Rr is active | Rr decodes incorrectly ) , Δ. pr1|0 = P(relay Rr is silent | Rr decodes correctly ) , Fig. 1. A two-hop wireless relay system with a single source, R relays, and a single destination.. that the noncoherent distributed system can offer a comparable diversity as the conventional MIMO-OFDM system provided that relay censoring is carefully implemented. The outline of this paper is as follows. Section II presents the system model. The diversity analysis is conducted in Section III with perfect relay censoring and Section IV with imperfect relay censoring. Simulation results and discussions are presented in Section V. Conclusion is given in Section VI. Notations: diag{d1 , . . . , dD } denotes the diagonal matrix with d1 , . . . , dD as the diagonal entries where di , 1 ≤ i ≤ D, . can be a vector or scalar. f1 (ρ) = f2 (ρ) represents the diversity equivalence relation, i.e., f1 and f2 have the same exponent, or f1 (ρ) f2 (ρ) = limρ→∞ loglog mathematically limρ→∞ loglog ρ ρ . II. S YSTEM M ODEL Consider a two-hop wireless relay network consisting of a single source, R relays, and a single destination, as depicted in Fig. 1. All nodes are each equipped with a single antenna. The information is transmitted from the source to the destination in two time slots via the cooperative DF communication protocol. In Slot I, the source node broadcasts the information message to the relays. We assume that in Slot I there is no direct link from the source to the destination for ease of analytical development, similar to [8]. In Slot II, with the source node silent, each relay first adopts a censoring scheme (e.g., an SNR threshold-based approach [11]) to assess its decoding ability and determine whether it should participate in cooperative transmission in Slot II. Only relays that pass the censoring (e.g., the received SNR is greater than a threshold) will decode the received signals and then cooperatively and concurrently forward the re-encoded signals toward the destination. Depending on the censoring result and the actual state of decoding at each relay, we have four possibilities for each relay: 1) Useful relay: The relay passes the censoring and thus will decode and forward in Slot II. The relay indeed decodes correctly. 2) Harmful relay: The relay passes the censoring and thus will decode and forward in Slot II. The relay however decodes incorrectly. 3) Useless relay: The relay fails the censoring and thus will not decode and forward in Slot II. However, the relay would decode correctly if decoding were performed. 4) Controlled relay: The relay fails the censoring and thus will not decode and forward in Slot II. Indeed, the relay would decode incorrectly if decoding were performed.. where the superscript r (r = 1, . . . , R) indicates that these probabilities are relay-dependent and are specific to the rth relay Rr . The probabilities 1 − pr1|0 , pr0|1 , pr1|0 , and 1 − pr0|1 are related to the four events above for relay Rr (i.e., useful, harmful, useless, and controlled, in that order), respectively. The probability prs that relay Rr erroneously decodes the transmitted signal in Slot I depends on the coding/decoding scheme and the channel condition between source and relay Rr . The values of pr0|1 and pr1|0 depend on the censoring scheme employed at Rr . For example, in the SNR threshold-based relay censoring considered in [11] and [12] with flat Rayleigh faded sourcerelay channel and BPSK signaling, a closed-form expression of pr0|1 can be derived using [12, Eqs. (4) and (6)], the error decoding probability prs , and P(relay Rr is active). Then, pr1|0 can be obtained from pr0|1 by the Bayes’ rule. All relay nodes in the network employ OFDM-based transmission with N subcarriers. The baseband frequency-selective fading channel between the rth relay and the destination node has L independent delay paths hr (l), 0 ≤ l ≤ L − 1. The channel gain hr (l) is modeled as complex Gaussian random variable (statistically independent across 2both r and l) with zero 2 , where L−1 mean and variance σr,l l=0 σr,l = 1, 1 ≤ r ≤ R for channel power normalization. In Slot II transmission, an active relay node re-encodes its decoded message using a mapping, i.e., space–frequency coding. Let the constellation alphabet with size K used in Slot I be S = {s0 s1 · · · sK−1 }. Further, let the decoded signal at the rth active relay be ˆsr . Suppose that the rth relay has decoded the received signal as ˆsr = sk . Then, it will re-encode the message into an N × 1 OFDM symbol vector crk for transmission according to the mapping sk → crk . The superscript r indicates that different relays may re-encode the message differently. Let Ci = [c1i c2i · · · cR i ] represent the codeword matrix constructed when all relays have si as the decoded message. The codebook C for Slot II transmission is therefore defined as C = {C0 C1 · · · CK−1 }. In Slot II, the destination receives the sum of signals from all active relays. The received signal rd at the destination node after Fast Fourier Transform (FFT) can be expressed as rd =. √. ρ. . Hr cr + n,. (1). r: active. where ρ is the power scaling factor for each relay, cr ∈ {cr0 , . . . , crK−1 } is the transmitted codeword from the rth relay, Hr is a diagonal matrix with the nth diagonal entry (Hr )nn being the frequency response of the channel gain from the rth relay to the destination at the nth subcarrier, and n ∼ CN (0, IN ) is complex Gaussian noise vector. The matrix Hr and the l channel gain are related according to Hr = L−1 l=0 hr (l)D ,.
(4) 3494. IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 10, OCTOBER 2014. N −1. where D = diag{1, e−j2π N , . . . , e−j2π N }. Substituting Hr into (1) and reordering the summation give 1. rd =. L−1 √ l ρ D hr (l)cr + n. l=0. (2). r: active. To characterize the relays’ activities, we define a relay status matrix Sm = diag{Sm,1 , . . . , Sm,R } where 1, if Rr is active and has decoded as ˆsr = sm , Sm,r = 0, otherwise. The Sm,r signifies the activity of the rth relay with respect to the mth information message. Assuming that the transmitted signal from the source node in Slot I is si , we have ⎛ ⎞ L−1 K−1 √ l⎝ D Ci Si h(l) + Cm Sm h(l)⎠ + n, rd = ρ l=0. for 1 ≤ r ≤ R. Our analysis is conducted for two separate cases depending on whether partial information is available at the destination. In Section III-A, we consider the case where the 2 , ∀r, l) destination has the (long-term) channel statistics (σr,l and the relay decoding status S information, as in [8]. In Section III-B, we consider the previously unexamined case where the destination has neither the channel statistics nor the relay censoring information, i.e., a completely noncoherent receiver. A. Noncoherent Receiver With Known Channel Statistics and Relay Status. m =i T. with h(l) = [h1 (l) h2 (l) . . . hR (l)] . Since the relay status matrix Si will nullify the terms that correspond to the relays that either did not participate or did not decode correctly as si , we can rewrite rd as √ ˆ i h + √ρ ˆ j h + n. ρEi S Ej S K−1. rd =. ˆ = IL ⊗S with S = diag{S1 , where E ∈ {E0 , . . . , EK−1 } and S . . . , SR } and 1, if Rr is active, Sr = 0, otherwise,. (3). j =i. where Ei = [Ci DCi · · · DL−1 Ci ] is the pseudo-codeword maˆ i = IL ⊗ Si is the stacked relay trix [19] associated with Ci , S status matrix with IL being the identity matrix of order L and ⊗ being the Kronecker product, and h = [hT (0) · · · hT (L − 1)]T is the stacked channel vector. Note that the second term in (3) corresponds to the interference (error propagation) contributed by the harmful relays. Definition: The diversity order is defined as log PEP(ρ) − limρ→∞ log ρ , i.e., the asymptotic exponent in ρ of the pairwise error probability PEP(ρ). III. D IVERSITY A NALYSIS W ITH P ERFECT R ELAY C ENSORING In this paper, we analytically study the diversity order of the system when the receiver does not know the instantaneous CSI (h), i.e., noncoherent detection, by considering perfect (this section) and imperfect (Section IV) relay censoring. By perfect relay censoring we mean that there is a perfect agreement between the censoring result and the actual state of decoding at each relay [6], [8]; specifically, pr1|0 = 0 and pr0|1 = 0 for all r = 1, . . . , R, i.e., the cases of useless and harmful relays are precluded. The analysis in this section is of theoretical usefulness, which allows us to explore the core ideas of this work. We will relax the perfect censoring assumption later in ˆm = 0 Section IV. As a result of this consideration we have S for m = i when the transmitted message from the source is si . Thus, √ ˆ rd = ρESh + n,. 1) Maximum Likelihood Decoding: We first evaluate the minimum achievable error probability with ML decoding. With known channel statistics and relay decoding status, the likeliˆ is multivariate complex Gaussian. hood function p(rd |Ci , S) Given that the transmitted codeword matrix is Ci , the covariΔ ˆ 2 EH , where Σ2 = ance matrix of rd is Λi = IN + ρEi SΣ i E[hhH ] = diag{σ 20 , . . . , σ 2L−1 } is the covariance matrix of h 2 2 2 T with σ 2l = [σ1,l σ2,l . . . σR,l ] for l = 0, . . . , L − 1. Note that ˆ and Σ2 is diagonal in ˆ2 = S we have exploited the fact that S the expression of Λi . The conditional density of the received signal is then given by. .
(5) 1. −1 ˆ = (4) exp −rH p rd Ci , S d Λi rd . N π det(Λi ) ˆ ML = Consequently, we have the ML decoding rule C −1 Λ r + ln det Λ ). arg minCi ∈C (rH d i i d We restrict ourselves to the case of a unitary codebook, i.e., EH i Ei = IRL for all i, for analytical tractability. This consideration first appeared in [17] and has since been widely adopted in the design and analysis of STC and SFC systems [5], [8], [19]. Applying the matrix inversion lemma to Λ−1 i , the ML decoding rule is simplified as. ˆ 2 )−1 EH rd , (5) ˆ 2 (IRL +ρSΣ ˆ MLA = arg max rH Ei SΣ C d i Ci ∈C. where the subscript MLA represents the ML decoding rule considered in Section III-A (to distinguish from the one considered in Section III-B). The ML decoding rule here requires ˆ and the channel knowledge of the relay decoding status S 2 statistics Σ at the destination node. 2) Conditional Pairwise Error Probability: Based on the ML decoding in (5), we derive the pairwise error probability (PEP) of deciding on Cj at the destination given that Ci was actually transmitted. The PEP conditioned on the relay status ˆ is given by matrix S . . ˆ =P v > 0|Ci , S ˆ ≤ E esv |Ci , S ˆ ,∀s > 0, P Ci → Cj |Ci , S (6).
(6) CHIU et al.: DIVERSITY OF NONCOHERENT DISTRIBUTED SFC RELAY SYSTEMS WITH RELAY CENSORING. where the inequality follows from the Chernoff bound and v = ˆ 2 −1 H ˆ 2 −1 H rH d (Ej SΣ Θ Ej − Ei SΣ Θ Ei )rd with Θ = IRL + 2 ˆ . Following a similar algebraic approach in [19], a diρSΣ versity preserving bound can be obtained as L−1 i ρ −nSi L nS ˆ ˆ 2 P(Ci → Cj |Ci , S) < λ−1 n (SΣ Q), 4 n=0. where the relation li ≥ l1 , i = 1, . . . , n, is exploited in deriving the asymptotic behavior. Lemma 1 states that if the average PEP assumes the form of f (ρ) in Lemma 1, the diversity order is determined by the dominant exponent. Proposition 1: With perfect relay censoring, the maximum achievable diversity order is R · min{Ls , L} with the noncoherent ML decoder in (5). Proof: The average PEP at the destination can be expressed as . ˆ i P Ci → Cj |Ci , S ˆ P S|C (8) P(Ci → Cj |Ci ) = ˆ S. ˆ is upper bounded in (7), and where P(Ci → Cj |Ci , S) ˆ P(S|Ci ) = j∈Ru|i (1 − pjs ) · k∈Rs|i pks with Ru|i being the set of useful relays (with cardinality nSi ) and Rs|i the set of silent relays (with cardinality R − nSi ) given that the transmitted message in Slot I is si . There is no harmful relay due to perfect relay censoring. The average PEP is P (Ci → ⎛ Cj |Ci ) ⎞ ⎞⎛ . ⎝ (1 − pjs )⎠ · ⎝ pks ⎠ · γnSi ρ−nSi L , (9) = ˆ S. j∈Ru|i. k∈Rs|i. where the diversity equivalence follows from the diversity prenSi L−1 Δ ˆ −1 Σ2 Q) λ n (S serving bound in (7), and γnSi = 4nSi L n=0 is irrelevant to ρ. According to Lemma 1, to obtain the diversity order we only need to find the dominant exponent and can . discard γnSi in (9). As ρ → ∞ and with prs = ρ−Ls , we have. (7). where nSi = R r=1 Sr is the number of active relays with deˆ 2 Q) represents the nth eigenvalue of SΣ ˆ 2 Q, coded si , λn (SΣ H H and Q = IRL − Ej Ei Ei Ej . In (7), the decay rate of the conditional PEP scales with ρ−nSi L when Q is full rank [19]. 3) Average Pairwise Error Probability: We obtain the avˆ erage PEP at the destination node by averaging over all S. Diversity order can be derived from the obtained average PEP. With perfect relay censoring, the state of the rth relay node Sr is a Bernoulli random variable, i.e., Sr = 0 with probability prs and Sr = 1 with probability 1 − prs . Hereafter, we consider that the relay decoding error probability prs scales with ρ−Ls (i.e., . prs = ρ−Ls ), where Ls is the number of delay paths between the source and each relay. This scaling law is achievable at each relay through proper coding schemes. To analyze the diversity order of the system, we first provide the following lemma. Lemma: Let f (ρ) = ni=1 αi ρ−li , where l1 ≤ l2 ≤ . . . ≤ ln are nonnegative integers and αi is a positive constant irrele. vant of ρ for i = 1, . . . , n. Then, we have f (ρ) = ρ−l1 . Proof: As ρ → ∞, we have n 1 1 −l1 −(li −l1 ) log f (ρ) = log ρ + log αi ρ log ρ log ρ i=1 1 log ρ−l1 → log ρ. 3495. −. log k −nS L ps · ρ i log ρ k∈Rs|i. ⎛⎛ ⎞ ⎞ −1 ⎝⎝ → log pks ⎠ + log ρ−nSi L ⎠ log ρ k∈Rs|i. = (R − nSi )Ls + nSi L.. (10). From (9), (10), and Lemma 1, it follows that the diversity order is min0≤nSi ≤R ((R − nSi )Ls + nSi L) = R · min{Ls , L}. B. Completely Noncoherent Receiver 1) Maximum Likelihood Decoding and Correlator-Based Decoding: To obtain the likelihood function without ˆ we take the expectation of (4) over all knowledge of S, ˆ which yields a Gaussian mixture p(rd |Ci ) = possible S, ˆ ˆ The optimal ML decoding ˆ P(S|Ci )p(rd |Ci , S). S corresponds to ˆ MLB = arg max p (rd |Ci ) , C. (11). Ci ∈C. where the subscript MLB represents the ML decoding rule considered in Section III-B. The error probability and diversity analysis based on (11) is analytically intractable since there exists no closed-form decision rule in (11). To overcome this obstacle, we consider a low-complexity, completely noncoˆ = IRL (i.e., no relay herent receiver. Specifically, we let S decoding status is needed) and Σ2 = IRL (i.e., no statistical CSI is needed) to yield a suboptimal correlator-based decoder which exploits only the correlation structure of the codeword matrices, i.e.,
(7). ˆ subopt = arg max rH Ei EH rd . C d i Ci ∈C. (12). Note that this suboptimal decoder approximates the ML decoder in (11) well in the high SNR regime since all relays with a proper censoring mechanism tend to be useful relays. In what follows, we first analyze the achievable diversity order with the suboptimal decoder and then deduce from the result about the achievable diversity order with the ML decoding in (11). 2) Conditional Pairwise Error Probability: The conditional PEP based on the decoding rule in (12) can be shown to be upper bounded by ˆ = P(w > 0|Ci , S) ˆ ≤ E esw |Ci , S ˆ , P(Ci → Cj |Ci , S) ∀s ≥ 0,. (13).
(8) 3496. IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 10, OCTOBER 2014. Δ. H H H where w = rH d Ej Ej rd − rd Ei Ei rd . Substituting rd = √ ˆ ρEi Sh + n into w and conditioning on the channel yield. . ˆ h E esw Ci , S,.
(9). ˆ EH Ej EH Ei − IRL Sh ˆ = exp sρhH S i. j. .
(10) H × E exp snH (Ej EH j − Ei Ei )n. √
(11) ˆ H Ej EH − I N n ρhH SE · exp 2s i j. . ˆ h (14). Ci , S,.
(12) ˆ EH Ej EH Ei − IRL Sh ˆ ≤ exp sρhH S i. j. 12
(13) .
(14) H ˆ C n − E E , S, h × E exp 2snH Ej EH. i i i j . √
(15) ˆ H Ej EH − I N n ρhH SE × E exp 4s i j. 12. ˆ h. Ci , S,. − N2. 1 ρˆ 2
(16) H IRL − EH det−1 IRL + SΣ E E E ≤ j i i j 3 9 − N2 di,j 1 ρ −di,j −1 ˆ 2 λk (SΣ Q), ≤ 3 9. (18). k=1. H where Q = IRL − EH j Ei Ei Ej as defined in (7), and di,j is 2 ˆ Q. In (18), the decay rate of the conditional the rank of SΣ PEP scales with nSi L when Q is full rank, similar to (7). 3) Average Pairwise Error Probability: Following a similar argument as in Section III-A3, we can obtain. . P(Ci → Cj |Ci ) = ρ−R min{Ls ,L} .. (19). (15) With the result of Proposition 1 and (19) we have. where {·} denotes the real part of its argument, and the inequality follows from the Cauchy–Schwarz inequality. The first expectation in (15) is related to the MGF of a Hermitian quadratic form of complex Gaussian random variables, for which the closed-form solution can be found in [26]. The second expectation in (15) can be evaluated using the moment generating function (MGF) of a Gaussian random variable. Carrying out these two expectations we have. . ˆ h E esw Ci , S,.
(17) ˆ EH Ej EH Ei − IRL Sh ˆ ≤ exp (s − 2s2 )ρhH S i j
(18) 1
(19) H , · det− 2 IN − 2s Ej EH j − Ei Ei. Taking expectation over h on both sides and using the MGF formula in [26] again, it follows that. ˆ P w > 0 Ci , S. (16). where the bound is valid for any s satisfying 0 < s < 12 for a H positive definite IN − 2s(Ej EH j − Ei Ei ). Note that while it is possible to obtain the optimal s that minimizes the bound, any s satisfying 0 < s < 12 is sufficient to provide a valid upper bound in (16). This is because any fixed s within 0 < s < 12 just constitutes a constant scaling factor in (16) and does not alter the exponent in ρ. Consequently, any legitimate s will yield the same diversity order as in (18) and (19) shown later. We adopt s = 13 , which leads the determinant in (16) to be bounded by − N2. 1 1 2
(20) H Ej EH − E E , (17) det− 2 IN − ≤ i i j 3 3 where we have used the Weyl’s inequality [27, Theorem 4.3.1] H H H that λ(Ej EH j − Ei Ei ) ≤ λmax (Ej Ej ) − λmin (Ei Ei ) = 1. Substituting (17) into (16) and then (16) into (13) leads to. ˆ h P w > 0 Ci , S, − N2 ρ.
(21) 1 ˆ EH Ej EH Ei − IRL Sh ˆ hH S · exp ≤ . i j 3 9. ρ−R min{Ls ,L} . . = PEPMLA ≤ PEPMLB ≤ PEPsubopt = ρ−R min{Ls ,L} , (20) where PEPMLA , PEPMLB , and PEPsubopt represent the PEP under the decision rule in (5), (11), and (12), respectively. In (20), the first diversity equivalence has been developed in Section III-A. The first inequality follows from the fact that the MLA decoder described in Section III-A has additional information, i.e., the relay decoding status, the second inequality follows directly from the optimality of the MLB decoder as compared to the suboptimal decoder, and the third inequality comes from (19). By the squeeze theorem we conclude that . . PEPMLA = PEPMLB = PEPsubopt . The first diversity equivalence indicates that the maximum achievable diversity is the same R min{Ls , L} with or without knowledge of the channel statistics and the relay status at the destination. That is, the destination can still retrieve full diversity even without knowledge of the channel statistics and the relay decoding status. The second diversity equivalence indicates that both the optimal ML decoding and the suboptimal correlator-based decoding yield the same maximum achievable diversity for a completely noncoherent receiver. Remark 1: As seen in (7) and (18), the decaying rate of the error probability scales with ρ−RL when Q is full rank for an arbitrary pair of distinct pseudo-codeword matrices Ei and Ej . This full-rank requirement is analogous to the requirement in noncoherent SFC codeword construction for conventional MIMO-OFDM [19]. This suggests that the codeword matrices constructed based on the coding criteria in [19] can be adopted here to achieve full diversity of the system..
(22) CHIU et al.: DIVERSITY OF NONCOHERENT DISTRIBUTED SFC RELAY SYSTEMS WITH RELAY CENSORING. IV. D IVERSITY A NALYSIS W ITH I MPERFECT R ELAY C ENSORING In this section, we analyze the maximum achievable diversity of the noncoherent system with imperfect relay censoring. We present interesting new results of how error probabilities prs , pr1|0 , and pr0|1 affect the system’s achievable diversity at the destination node when the destination does not have the relay censoring status information. Without loss of generality, we assume that the transmitted message s from the source in Slot I is si . The noncoherent ML decoding at the destination with no information of the relay status is ˆ ML = arg max p (rd |Ci ) C Ci ∈C ˆ K−1 , = arg max ESˆ K−1 p rd |Ci , S 0 Ci ∈C. (21). 0. where rd is given in (3) with the possible existence of harmful Δ ˆ ˆ K−1 = ˆ 1, . . . , relays due to imperfect censoring, and S (S 0 , S 0 ˆ S K−1 ) collects the relay status with respect to all possible transmitted signals. We first obtain the conditional probability ˆ K−1 in the following. mass function (PMF) of S 0 Lemma 2: Suppose that the following condition holds: . P(Rj active, ˆsj = sp |s = si ) = P(Rj active, ˆsj = sq |s = si ) for all relays Rj , j = 1, . . . , R and any p = q = i. Then, the conˆ K−1 |Ci ) is ditional PMF P(S 0. ˆ K−1 |Ci ) P(S 0. .
(23) 1 − pjs 1 − pj1|0 = j∈Ru|i. ×.
(24). 1−. pks. p1|0 + 1 −. pk0|1. pks. k∈Rs|i. ×. . pl0|1 pls ,. (22). l∈Rh|i Δ. where Ru|i = {r : Si,r = 1|s = si } is the set of useful relays, Δ. Rs|i = {r : Sm,r = 0, ∀m|s = si } is the set of silent relays, Δ. and Rh|i = {r : Sm,r = 1, m = i|s = si } is the set of harmful relays, when the transmitted message in Slot I is si . Proof: See Appendix A. . Note that the condition P(Rj active, ˆsj = sp |s = si ) = P(Rj active, ˆsj = sq |s = si ) in Lemma 2 states that the probabilities that Rj erroneously decodes as sp and as sq decay at the same rate. This condition holds when the distances from si to sp and to sq follow si − sp = η · si − sq , where η is irrelevant to ρ. Most modulation schemes satisfy this distance relation. To facilitate further mathematical development without loss of analytical insights, we hereafter consider a common prs , pr1|0 , and pr0|1 for all relays (thus, the superscript r will be dropped). Then, (22) simplifies to K−1
(25)
(26) n n
(27) . ˆ p0|1 ps Sm P S |Ci = (1 − ps ) 1 − p1|0 Si × 0 m =i.
(28) n
(29) × (1 − ps )p1|0 + ps 1 − p0|1 0 ,. (23). 3497. Δ where nSm = R r=1 Sm,r is the number of active relays that Δ have decoded sm (m = i), and n0 = R − K−1 m=0 nSm is the number of silent relays. For a noncoherent decoding rule without knowledge of the relay status (i.e., the noncoherent ML decoder in (21) or the correlator-based decoder in (12)), the PEP can be expressed as. P (Ci → Cj |Ci ). K−1 K−1 ˆ ˆ P C . = P S |C → C |C , S i i j i 0 0. (24). ˆ K−1 S 0. Note that, from Lemma 1, we only need to consider the dominant term in (24) to analyze the achievable diversity order. To characterize the scaling law of the PEP, we first examine the ˆ K−1 |Ci ) as the scaling behavior of the conditional PMF P(S 0 SNR goes to infinity. ˆ K−1 |Ci ) in (23) is Lemma 3: In high SNR regimes, P(S 0 diversity-equivalent to K−1
(30) n nSm
(31) . ˆ m=i = p p1|0 + ps 0 . (25) |C p P S i 0|1 s 0 Proof: See Appendix B. As suggested by (25), the achievable diversity order depends on the error probabilities ps , p1|0 , and p0|1 . In what follows, we specifically characterize this relationship in three propositions. Proposition 2: If p1|0 is a positive constant that does not scale with ρ, the diversity order is zero with the noncoherent ML decoder. Proof: We aim to show that . P (Ci → Cj |Ci ) = ζ1. (26). for some constant ζ1 irrelevant to ρ. To see this, conˆ K−1 = sider the dominant summation term in (24) where S 0 Δ ˆ 0 = IRL and n0 = (0RL , 0RL , . . . , 0RL ) = O. This leads to S R, i.e., no relay helps forward the information from the source to the destination and the destination node receives only noise in . ˆ K−1 = O) = ζ2 for Slot II. Thus, we have P(Ci → Cj |Ci , S 0 some constant ζ2 . This fact, coupled with (25) in Lemma 3, leads to K−1. ˆ ˆ K−1 = O P S = O|Ci P Ci → Cj |Ci , S 0 0 R R .
(32) .
(33) = p1|0 + ps ζ2 = p1|0 ζ2 ,. (27). where the second diversity equivalence follows from the fact . that ps = ρ−Ls and that p1|0 is a constant. As the diversity is determined by the dominant term in (24), we reach (26). Proposition 2 demonstrates that, if p1|0 does not scale with ρ, the decoding error will be dominated by the event that all relays are useless relays in high SNR regimes. Proposition 3: If p0|1 is a positive constant that does not scale with ρ, the maximum achievable diversity order is upper bounded by R2 Ls with the noncoherent ML decoder. Proof: Since the diversity is determined by the dominant term in (24) as shown in Lemma 1, it suffices to examine whether there exists one term in (24) that has.
(34) 3498. IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 10, OCTOBER 2014. the diversity R2 Ls for the upper bound. We consider R 2 1 s. ˆ j = IL ⊗ ˆ i = IL ⊗ diag{1, . . . , 1, 0, . . . , 0}, S the case that S R 2 1 s. TABLE I T HE C ODEBOOK C FOR S LOT II T RANSMISSION , 2π 2π W HERE Φ = diag{ej K u0 , . . . , ej K u(N −1) } W ITH 0 ≤ uk ≤ K − 1 FOR k = 0, 1, . . . , N − 1, AND fi I S THE iTH C OLUMN OF AN N × N FFT M ATRIX. ˆ m = 0RL for m = i, j. For diag{0, . . . , 0, 1, . . . , 1}, and S notational brevity, we denote the relay status collectively as ˆ K−1 = Vij . Since the decision at the destination has no S 0 ˆ K−1 = preference for Ci over Cj , we have P(Ci → Cj |Ci , S 0 Vij ) = 12 . Using (25) we can obtain . K−1 ˆ K−1 = Vij ˆ = Vij |Ci P Ci → Cj |Ci , S P S 0 0 R
(35) R−2 R2 . 1
(36) = p0|1 ps 2 p1|0 + ps 2 R R 1
(37) p0|1 ps 2 (ps )R−2 2 ≥ 2 R R . = ρ−RLs + 2 Ls = ρ− 2 Ls , where the last diversity equivalence follows from the fact that . ps = ρ−Ls and p0|1 is a constant. It is worthwhile to note that by setting p1|0 = 0 and Ls = 1, Proposition 3 reduces to the special case considered previously [10] for noncoherent wireless DF relaying with binary frequency shift keying (BFSK) modulation over flat fading channels. Finally, we analyze how the scaling laws of p0|1 and p1|0 affect the maximal achievable diversity order when the receiver employs the noncoherent correlator-based decoder in (12) with imperfect relay censoring. Lemma 4: The conditional PEP for the correlator-based decoder has the property. . ˆ K−1 ∈ Uij = ζ3 , P Ci → Cj |Ci , S 0 Δ ˆ K−1 ˆ ˆ j = 0RL , and S ˆ m = 0RL where Uij = {S : S i = 0RL , S 0 for all m = i, j} represents the relay status where there is at least one useful relay and at least one harmful relay and where the harmful relay(s) all decoded the messages as sj , and ζ3 is a constant irrelevant of ρ. Proof: See Appendix C. . Proposition 4: If p0|1 and p1|0 scale with ρ as p0|1 = ρ−L0|1 . and p1|0 = ρ−L1|0 , respectively, the maximum achievable diversity order is min{Ls + L0|1 , R · min{L1|0 , L}} with the correlator-based decoder. Proof: We first divide all summation terms (i.e., all possible relay status) in (24) into three different classes and then within each class we find the dominant terms which contribute to the diversity order. ˆ K−1 ∈ Uij , Consider the first class that corresponds to S 0 where Uij is defined in Lemma 4. For this class, m =i nSm = ˆ K−1 |Ci ) ≤ ρ−(L0|1 +Ls ) from (25). Using the nSj ≥ 1 and P(S 0 result in Lemma 4, we can deduce that the dominant term in ˆ K−1 ∈ Uij has an exponent L0|1 + Ls , (24) within the class S 0 which occurs when nSj = 1 and n0 = 0.. Consider the second class that corresponds to the absence of any harmful relay, i.e., m =i nSm = 0. For this class, we can directly use the result presented in Section III-B that P(Ci → . −k1 L . −k2 min{Ls ,L1|0 } ˆ K−1 ) = ˆ K−1 |Ci ) = Cj |Ci , S ρ and P(S ρ , 0 0 where k1 and k2 are the number of useful relays and silent relays, respectively, with k1 + k2 = R in this class. Consider the third class that corresponds to all other relay status. Since there is at least one harmful relay, we ˆ K−1 |Ci )P(Ci → ˆ K−1 |Ci ) ≤ ρ−(L0|1 +Ls ) and P(S have P(S 0 0 ˆ K−1 ) ≤ ρ−(L0|1 +Ls ) from (25), which implies an exCj |Ci , S 0 ponent in ρ no less than L0|1 + Ls . Taking into account all summation terms in (24) and using Lemma 1, we can express the maximum achievable diversity order as
(38). k1 L + k2 · min{L1|0 , Ls } . min Ls + L0|1 , min k1 +k2 =R. Solving the minimization completes the proof. Proposition 4 suggests that when Ls + L0|1 is dominantly small, the system diversity does not scale with the number of relays R when the destination node adopts the correlator-based decoder. Furthermore, different from the result presented in Section III-B with perfect relay censoring, here, with imperfect relay censoring, the suboptimal correlator-based decoder may not attain the diversity of order R · min{Ls , L} when L0|1 or L1|0 is small. Note that the derivation of the system diversity with the noncoherent ML decoder is difficult since there exists no closed-form ML decision rule, as shown in (21), (23), and (24). V. S IMULATION R ESULTS Here, we present the decoding block error rate (BLER) performance results at the destination. We set Ls = 2, L = 2, and N = 16. The channel gains are complex Gaussian and are independent across different channel paths, node pairs, and symbol blocks (time instants). The SNR is defined as the total transmit power from the R relays per subchannel, i.e., SNR = ρR/N with the noise power normalized to one. The transmit power of the source node is ρ. The relay decoding error rate . is ps = ρ−Ls . As commented in Remark 1, the codebook used in Slot II transmission is constructed by following a similar procedure as in [19] and is summarized in Table I. A. Perfect Relay Censoring We first present the results for perfect relay censoring, i.e., p1|0 = 0 and p0|1 = 0..
(39) CHIU et al.: DIVERSITY OF NONCOHERENT DISTRIBUTED SFC RELAY SYSTEMS WITH RELAY CENSORING. 3499. Fig. 2. The BLER performance for the noncoherent partial-knowledge ML decoder in (5) and coherent partial-knowledge ML decoder [28], [29], both with perfect relay censoring, for different configurations of R and K.. 1) Noncoherent Receiver With Partial Information: The BLER performance results with partial information of channel statistics and relay decoding status at the destination are presented in Fig. 2 for the four combinations of R = 2, 4 (the number of relays) and K = 8, 16 (the codebook size). The chan2 2 = σr,1 = nel power delay profile (PDP) is uniform, i.e., σr,0 0.5, ∀r in the simulation. The results show that a diversity of order 4 and 8 is achieved for R = 2 and R = 4, respectively, confirming that the potential diversity of distributed relays is identical to that of conventional MIMO-OFDM systems with noncoherent SFC [19]. The difference between the two curves with R = 2 (and respectively R = 4) corresponds to the coding gain as a result of the different code rates [19]. In Fig. 2, we also show the BLER performance of the coherent receiver with known relay decoding status for R = 2, K = 8 and R = 4, K = 8. The same space–frequency code C presented in Table I is used for encoding in the coherent case, since this codebook C also satisfies the full diversity condition specified in [28], [29] for coherent space–frequency coded systems. As can be seen, both coherent and noncoherent SFC cooperative relaying systems with the ML decoding achieve full diversity, which confirms that the noncoherent SFC MIMOOFDM system requiring no instantaneous CSI at the receiver can achieve a full diversity as its coherent counterpart [19]. The coherent case, as expected, has a coding gain advantage over the noncoherent case, because instantaneous CSI is known at the coherent receiver while only statistical CSI is available to the noncoherent receiver for the decoding. Fig. 3 plots the BLER results for different PDPs, with R = 2. As can be seen, all curves exhibit the same diversity of order 4. Furthermore, the BLER performance degrades as the channel has a more asymmetricPDP. This can be explained by (7). 2 When the total power L−1 l=0 σr,l is a constant of unity, the ˆ 2 Q), which can be approximated by magnitude of λ−1 (SΣ R 2 −1 2 2 −1 σr,1 ) , increases as the PDP becomes λ (Σ ) = r=1 (σr,0 more asymmetric due to the arithmetic-geometric inequality. This agrees with the intuition that, when the PDP is more asymmetric, more power is needed to make the weaker channel contribute to the maximum diversity. As shown in Fig. 3, the 2 2 σr,1 ] = [0.99 0.01] does not reach diversity order for PDP [σr,0 4 until the SNR rises above 20 dB or so.. Fig. 3. The BLER performance for the noncoherent partial-knowledge ML decoder in (5) with perfect relay censoring, for different PDPs.. Fig. 4. The BLER performance for the suboptimal correlator-based decoder in (12) and the noncoherent partial-knowledge ML decoder in (5) with perfect 2 σ 2 ] = [0.99 0.01] and relay censoring, for two different PDPs, i.e., [σr,0 r,1 2 2 [σr,0 σr,1 ] = [0.9 0.1].. 2) Completely Noncoherent Receiver: Fig. 4 shows the BLER performance for the completely noncoherent suboptimal correlator-based decoder in (12) in comparison with the noncoherent partial-knowledge ML decoder in (5), under perfect relay censoring and different channel PDPs. We observe that the two decoders yield almost identical performance even though the completely noncoherent correlator-based decoder does not have partial knowledge. There is a minor performance gap in low SNR regimes, particularly when the PDP is more asymmetric, since correlator-based decoding assumes a uniform PDP. Nevertheless, in high SNR regimes, the performance overlaps. This confirms the near-optimal performance and the diversityachieving capability of a completely noncoherent suboptimal correlator-based decoder under perfect relay censoring. B. Imperfect Relay Censoring Here, we present the results for imperfect relay censoring and examine the effect of p1|0 and p0|1 = 0 on the BLER.
(40) 3500. Fig. 5. The BLER performance for the ML decoder in (21) with imperfect relay censoring, for different configurations of p1|0 and R, with p0|1 = 0.. Fig. 6. The BLER performance for the completely noncoherent suboptimal correlator-based decoder and the ML decoder in (21) with imperfect relay censoring, for two cases of the relay decoding status with R = 4. Case I: 2 useful, 1 useless, and 1 harmful relays. Case II: 3 useful and 1 harmful relays.. performance. Fig. 5 depicts the performance of ML decoding in (21) with p0|1 = 0 and different settings of p1|0 . As can be seen, an error floor occurs for all constant values of p1|0 (even with p0|1 = 0), which justifies Proposition 2. A smaller p1|0 reduces the likelihood that all relays are useless relays and therefore reduces the BLER where the error floor occurs. Also, an increased number of relays reduces the BLER where the error floor occurs, which confirms the analytical result in (27) that the BLER where the error floor occurs approaches (p1|0 )R as the SNR goes to infinity. Fig. 6 compares the performance of the suboptimal correlator-based decoder and the optimal ML decoder in (21) for two cases of the relay decoding status with R = 4, namely, Case I where there are 2 useful, 1 useless, and 1 harmful relays, and Case II where there are 3 useful and 1 harmful relays. The two cases are constructed to illustrate a result of Proposition 4 that the suboptimal correlator-based decoder does not scale with the number of relays if L0|1 is small (i.e., there is a nonnegligible probability of harmful relays). This is verified in Fig. 6, where the correlator-based decoder suffers from an error floor for both cases with one harmful relay. In contrast, the ML decoder does not suffer from an error floor. Also, as. IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 10, OCTOBER 2014. Fig. 7. The BLER performance for the completely noncoherent suboptimal correlator-based decoder and the ML decoder in (21) with imperfect relay censoring (p1|0 = 0 and p0|1 = 1), as well as the noncoherent partial-knowledge ML decoder with perfect relay censoring.. shown in Fig. 6, Case II outperforms Case I with the same total number of relays, indicating that the relay conditions may affect the diversity order of the system with a completely noncoherent ML decoder. Fig. 7 shows the performance of the suboptimal correlatorbased decoder and the optimal ML decoder in (21) when all relays are active, i.e., p1|0 = 0 and p0|1 = 1. Several observations can be made. First, the diversity order is 2 for the suboptimal decoder for both R = 2 and R = 4. This agrees well with the analytical result in Proposition 4 with parameters . . Ls = 2, L = 2, p0|1 = 1 = ρ−0 , and p1|0 = 0 = ρ−∞ . In this case, increasing the number of relays does not increase the diversity of the system with correlator-based decoding. Second, the ML decoder achieves a diversity of order 2 and 4 for R = 2 and R = 4, respectively. This shows that the ML decoder can yield the maximal achievable diversity of order R2 Ls as stated in Proposition 3. Third, by comparing the performance for decoders with perfect and imperfect relay censoring for R = 2, it is seen that the imperfection in censoring decreases the diversity gain. VI. C ONCLUSION We have studied the diversity of a noncoherent distributed SFC wireless two-hop relay system with DF relaying. We approached the problem from a previously unexamined perspective that takes into account both perfect and imperfect relay censoring to examine the potential of the system under practical operation. The main findings are summarized as follows: • Perfect relay censoring: We show that a diversity of order R · min{Ls , L} can be achieved with the optimal ML decoder with or without partial information of channel statistics and relay decoding status at the noncoherent receiver. The same diversity order was shown also achievable with a suboptimal correlator-based decoder, which was proposed as a convenient but motivated scheme for analytical development. Our results suggest that with perfect relay censoring the diversity of the noncoherent distributed SFC.
(41) CHIU et al.: DIVERSITY OF NONCOHERENT DISTRIBUTED SFC RELAY SYSTEMS WITH RELAY CENSORING. system is on average identical to that of the conventional SFC MIMO-OFDM system. • Imperfect relay censoring: We characterize the relationship between censoring error probabilities p1|0 (which corresponds to useless relays) and p0|1 (which corresponds to harmful relays) and the diversity with a completely noncoherent receiver. If p1|0 is a constant that does not scale with the transmit power of the relay (ρ), the diversity order is zero with the ML decoder. If p0|1 is a constant that does not scale with the transmit power of the relay, the diversity order is upper bounded by R2 Ls with the . ML decoder. If p0|1 and p1|0 scale with ρ as p0|1 = ρ−L0|1 . −L1|0 and p1|0 = ρ , respectively, the diversity order is given by min{Ls + L0|1 , R · min{L1|0 , L}} with the correlatorbased decoder. The results suggest that a carefully implemented relay censoring scheme is crucial in achieving the full diversity potential of a distributed relay system. A PPENDIX A P ROOF OF L EMMA 2 Given transmitted si in Slot I, the relays can be grouped into three disjoint sets: 1) the set of useful relays Ru|i , 2) the set of silent relays Rs|i , and 3) the set of harmful relays Rh|i . The probabilities that a relay is in each set are characterized as follows. The probability that the jth relay Rj is in Ru|i is.
(42) P Rj ∈ Ru|i = P (Si,j = 1|s = si )
(43). = P Rj active, ˆsj = si |s = si.
(44) = 1 − pjs 1 − pj1|0 . The probability that the kth relay Rk is in Rs|i is.
(45).
(46) P Rk ∈ Rs|i = P Rk silent, ˆsk = s|s = si
(47). + P Rk silent, ˆsk = s|s = si.
(48). = 1 − pks pk1|0 + 1 − pk0|1 pks . The probability that the lth relay Rl is in Rh|i and has erroneously decoded si as ˆsl = sm (m = i) is P(Rl ∈ Rh|i , ˆsl = sm ) = P(Rl active, ˆsl = sm |s = si ). Furthermore, we have. where the second diversity equivalence follows from the condi. tion P(Rj active, ˆsj = sp |s = si ) = P(Rj active, ˆsj = sq |s = . si ). Thus, P(Rl ∈ Rh|i , ˆsl = sm ) = P(Rl active, ˆsl = s) = p0|1 ps . Combining the results for the three cases completes the proof. A PPENDIX B P ROOF OF L EMMA 3 Taking the logarithm of the right-hand-side of (23) yields .
(49)
(50)
(51) nSi log (1 − ps ) 1 − p1|0 + nSm log p0|1 ps m =i.
(52). +n0 log (1 − ps )p1|0 + ps (1 − p0|1 ) .. K−1 .
(53) P Rl active, ˆsl = s|s = si P(s = si ). i=0 K−1. .
(54) = P Rl active, ˆsl = s|s = si i=0. =. K−1 i=0 j =i.
(55) P Rl active, ˆsl = sj |s = si. .
(56) = P Rl active, ˆsl = sj |s = si. (28). We examine the scaling behaviors of the first and the third term . in (28). Since ps = ρ−Ls , (1 − ps )(1 − p1|0 ) must be a constant as ρ → ∞ regardless of the scaling laws of p1|0 . For the third term in (28), if p0|1 is a constant that does not scale with ρ, . we must have (1 − ps )p1|0 + ps (1 − p0|1 ) = p1|0 + β · ps for some constant β, which immediately leads to (1 − ps )p1|0 + . ps (1 − p0|1 ) = p1|0 + ps since multiplying by a constant does . not affect the scaling laws. If p0|1 scales with ρ as p0|1 = ρ−L0|1 , we see that p0|1 vanishes rapidly in 1 − p0|1 when . ρ → ∞, which also leads to (1 − ps )p1|0 + ps (1 − p0|1 ) = p1|0 + ps . Substituting these results into (23) yields the following and completes the proof: K−1 1 ˆ log P S |Ci 0 log ρ ⎛ ⎞ .
(57)
(58) 1 ⎝ → nSm log p0|1 ps + n0 log p1|0 + ps ⎠ log ρ m =i. as ρ → ∞.. A PPENDIX C P ROOF OF L EMMA 4 ˆ K−1 ∈ Uij is The received signal at the destination when S 0 given by. P(Rl active, ˆsl = s) =. 3501. rd =. √. ˆ i h + √ρEj S ˆ j h + n. ρEi S. Then, the conditional PEP is. ˆ K−1 ∈ Uij P Ci → Cj |Ci , S 0
(59). H H H = P rH d Ej Ej rd − rd Ei Ei rd > 0|Ci , Uij . 1
(60) H H H r d Ej EH > 0|C r − r E E r , U =P i i d i ij j d d ρ Δ. = P (wρ > 0|Ci , Uij ) ,.
(61) 3502. IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 10, OCTOBER 2014. Δ. H H rd − ¯rH rd with ¯rd = √rdρ = where wρ = ¯rH d Ej Ej ¯ d Ei Ei ¯ ˆ i h + Ej S ˆ j h + √n . Note that showing the lemma is Ei S ρ equivalent to showing limρ→∞ P(wρ > 0|Ci , Uij ) > 0. Δ. H H rd − ˜rH rd , where ˜rd = Define w∞ = ˜rH d Ej Ej ˜ d Ei Ei ˜ ˆ ˆ Ei S i h + Ej S j h. Clearly, wρ → w∞ almost surely as ρ → ∞. Thus, we only need to show P(w∞ > 0|Ci , Uij ) > 0, since convergence almost surely implies convergence in distribution. After simple manipulations we have.
(62). ˆ j I − EH Ei EH Ej S ˆ jh w∞ = h H S j i
(63). Δ ˆ i I − EH Ej EH Ei S ˆ ih = −hH S X − Y, i j Δ Δ Hˆ ˆ j h and Y = ˆ j (IRL − EH Ei EH Ej )S h Si where X = hH S j i H H H ˆ (IRL − EH E E E ) S h. Since I − E E E E is posj j i i RL i i j i j itive definite provided that Ei = Ej , and X and Y are independent generalized chi-square random variables with support [0, ∞), we have. P (w∞ > 0|Ci , Uij ) = P(X − Y > 0|Ci , Uij ) ≥ P ({X > 1} ∩ {Y < 1} |Ci , Uij ) = P (X > 1|Ci , Uij ) P(Y < 1|Ci , Uij ) > 0. This completes the proof. R EFERENCES [1] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity— Part I: System description,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1927–1938, Nov. 2003. [2] J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004. [3] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space–time codes for high data rate wireless communication: Performance criterion and code construction,” IEEE Trans. Inf. Theory, vol. 44, no. 2, pp. 744–765, Mar. 1998. [4] J. Laneman and G. Wornell, “Distributed space–time-coded protocols for exploiting cooperative diversity in wireless network,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2415–2425, Oct. 2003. [5] Y. Jing and B. Hassibi, “Distributed space–time coding in wireless relay networks,” IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3524–3536, Dec. 2006. [6] B. Sirkeci-Mergen and A. Scaglione, “Randomized space–time coding for distributed cooperative communication,” IEEE Trans. Signal Process., vol. 55, no. 10, pp. 5003–5017, Oct. 2007. [7] Y. Li, W. Zhang, and X. Xia, “Distributive high-rate space–frequency codes achieving full cooperative and multipath diversities for asynchronous cooperative communications,” IEEE Trans. Veh. Technol., vol. 58, no. 1, pp. 207–217, Jan. 2009. [8] K. G. Seddik and K. J. R. Liu, “Distributed space–frequency coding over broadband relay channels,” IEEE Trans. Wireless Commun., vol. 7, no. 11, pp. 4747–4759, Nov. 2008. [9] J. Boyer, D. D. Falconer, and H. Yanikomeroglu, “Multihop diversity in wireless relaying channels,” IEEE Trans. Commun., vol. 52, no. 10, pp. 1820–1830, Oct. 2004. [10] D. Chen and J. N. Laneman, “Modulation and demodulation for cooperative diversity in wireless systems,” IEEE Trans. Wireless Commun., vol. 5, no. 7, pp. 1785–1794, Jul. 2006.. [11] F. A. Onat et al., “Threshold selection for SNR-based selective digital relaying in cooperative wireless networks,” IEEE Trans. Wireless Commun., vol. 7, no. 11, pp. 4226–4237, Nov. 2008. [12] F. A. Onat, Y. Fan, H. Yanikomeroglu, and J. S. Thompson, “Asymptotic BER analysis of threshold digital relaying schemes in cooperative wireless systems,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 4938–4947, Dec. 2008. [13] T. Wang, A. Cano, G. B. Giannakis, and J. N. Laneman, “Highperformance cooperative demodulation with decode-and-forward relays,” IEEE Trans. Commun., vol. 55, no. 7, pp. 1427–1438, Jul. 2007. [14] T. Wang, G. B. Giannakis, and R. Wang, “Smart regenerative relays for link-adaptive cooperative communications,” IEEE Trans. Commun., vol. 56, no. 11, pp. 1950–1960, Nov. 2008. [15] T. Hunter and A. Nosratinia, “Diversity through coded cooperation,” IEEE Trans. Wireless Commun., vol. 5, no. 2, pp. 283–289, Feb. 2006. [16] G. Al-Habian, A. Ghrayeb, M. Hasna, and A. Abu-Dayya, “Thresholdbased relaying in coded cooperative networks,” IEEE Trans. Veh. Technol., vol. 60, no. 1, pp. 123–135, Jan. 2011. [17] B. M. Hochwald and T. L. Marzetta, “Unitary space–time modulation for multiple-antenna communications in Rayleigh flat fading,” IEEE Trans. Inf. Theory, vol. 46, no. 2, pp. 543–564, Mar. 2000. [18] V. Tarokh and I.-M. Kim, “Existence and construction of noncoherent unitary space–time codes,” IEEE Trans. Inf. Theory, vol. 48, no. 12, pp. 3112–3117, Dec. 2002. [19] M. Borgmann and H. Bolcskei, “Noncoherent space–frequency coded MIMO-OFDM,” IEEE J. Sel. Areas Commun., vol. 23, no. 9, pp. 1799– 1810, Sep. 2005. [20] M. R. Souryal, “Non-coherent amplify-and-forward generalized likelihood ratio test receiver,” IEEE Trans. Wireless Commun., vol. 9, no. 7, pp. 2320–2327, Jul. 2010. [21] P. Lin, S. Gazor, I.-M. Kim, and D. I. Kim, “Noncoherent amplify-andforward cooperative networks: Robust detection and performance analysis,” IEEE Trans. Commun., vol. 61, no. 9, pp. 3644–3659, Sep. 2013. [22] G. Farhadi and N. C. Beaulieu, “A low complexity receiver for noncoherent amplify-and-forward cooperative systems,” IEEE Trans. Commun., vol. 58, no. 9, pp. 2499–2504, Sep. 2010. [23] T.-H. Chang, W.-K. Ma, C.-Y. Huang, and C.-Y. Chi, “Noncoherent OSTBC-OFDM for MIMO and cooperative communications: Perfect channel identifiability and achievable diversity order,” IEEE Trans. Signal Process., vol. 60, no. 9, pp. 4849–4863, Sep. 2012. [24] M. C. Valenti, D. Torrieri, and T. Ferrett, “Noncoherent physical-layer network coding with FSK modulation: Relay receiver deign issues,” IEEE Trans. Commun., vol. 59, no. 9, pp. 2595–2604, Sep. 2011. [25] P. Herhold, E. Zimmermann, and G. Fettweis, “A simple cooperative extension to wireless relaying,” in Proc. Int. Zurich Seminar Commun., Mar. 2004, pp. 36–39. [26] G. L. Turin, “The characteristic function of Hermitian quadratic forms in complex normal variables,” Biometrika, vol. 47, no. 1/2, pp. 199–201, Jun. 1960. [27] R. A. Horn and C. R. Johnson, Matrix Analysis. New York, NY, USA: Cambridge Univ. Press, 1985. [28] H. Bolcskei and A. J. Paulraj, “Space–frequency coded broadband OFDM systems,” in Proc. IEEE WCNC, 2000, pp. 1–6. [29] H. Bolcskei and A. J. Paulraj, “Space–frequency codes for broadband fading channels,” in Proc. IEEE ISIT, 2001, pp. 219–219.. Sung-En Chiu received the B.S. and M.S. degrees in electronics engineering from National Chiao Tung University, Hsinchu, Taiwan, in 2008 and 2010, respectively. He is currently working toward the Ph.D. degree in electrical and computer engineering at the University of California, San Diego, CA, USA. From 2011 to 2013, he served as a 3GPP RAN1 delegate for the Industrial Technology Research Institute, Hsinchu, Taiwan. His current research interests include statistical signal processing, dynamic resource allocation, large-scale MIMO, and next generation communication systems..
(64) CHIU et al.: DIVERSITY OF NONCOHERENT DISTRIBUTED SFC RELAY SYSTEMS WITH RELAY CENSORING. Feng-Tsun Chien (S’02–M’05) received the B.S. degree from National Tsing Hua University, Hsinchu, Taiwan, in 1995, the M.S. degree from National Taiwan University, Taipei, Taiwan, in 1997, and the Ph.D. degree from the University of Southern California, Los Angeles, CA, USA, in 2004, all in electrical engineering. He joined the Department of Electronics Engineering, National Chiao Tung University, Hsinchu, in July 2005 as an Assistant Professor. His current research interests include wireless communications, statistical signal processing, game theoretic resource allocation, and network information theory. He has been serving as a Technical Program Committee Member in ICC (2009–2015), GLOBECOM (2009–2014), WCNC (2010–2015), and PIMRC (2011–2014) and is currently the treasurer of the IEEE Vehicular Technology Society, Taipei Chapter.. 3503. Ronald Y. Chang (M’12) received the B.S. degree in electrical engineering from National Tsing Hua University, Hsinchu, Taiwan, in 2000, the M.S. degree in electronics engineering from National Chiao Tung University, Hsinchu, in 2002, and the Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, CA, USA, in 2008. From 2002 to 2003, he was with the Industrial Technology Research Institute, Hsinchu. In 2008, he was a Research Intern at the Mitsubishi Electric Research Laboratories, Cambridge, MA, USA. In 2009, he worked on NASA Small Business Innovation Research projects. Since 2010, he has been with the Research Center for Information Technology Innovation, Academia Sinica, Taipei, Taiwan, where he is currently an Assistant Research Fellow. He has published about 50 papers and has four granted U.S. patents. His research interests include wireless communications and networking. He was an Exemplary Reviewer for the IEEE Communications Letters in 2012. He has contributed to various conferences as a Technical Program Committee Member, including the IEEE International Conference on Communications 2012, 2013, and 2015. He was a recipient of the Best Paper Award from the IEEE Wireless Communications and Networking Conference 2012..
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