Noncoherent Receiver for Amplify-and-Forward Relaying with M-FSK ...

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2011 proceedings.

Noncoherent Receiver for Amplify-and-Forward Relaying with M -FSK Modulation Ha X. Nguyen and Ha H. Nguyen Department of Electrical & Computer Engineering University of Saskatchewan 57 Campus Dr., Saskatoon, SK, Canada S7N5A9 [email protected], [email protected]

Abstract—This paper develops a detection scheme based on implicit pilot-symbol-assisted architecture for non-coherent amplifyand-forward (AF) relay networks. The networks use M -ary frequency-shift-keying (FSK) modulation and multiple relays to assist communication from a source to a destination. Built on the fact that every transmitted symbol from the source to the destination can be considered as a pilot symbol, the destination combines the signals from the source and the relays to perform detection by using the estimated channels obtained from the implicit pilot symbols. A tight upper bound on the bit-errorrate (BER) of the developed scheme is obtained and used to show that the proposed scheme achieves a full diversity order. Moreover, simulation results reveal that the proposed scheme can significantly improve the BER performance when compared to previously proposed schemes in a temporally-correlated fading environment.

I. I NTRODUCTION The last decade has witnessed an explosive interest in wireless relay communications, from both industry and research community. This is due mainly to its ability to provide spatial diversity (or cooperative diversity) in mobile units that cannot be equipped with multiple antennas. Specifically, a relay communication network can be designed so that one mobile in the network sends (or receives) its signal to (or from) the base station via other mobiles or base station in the network. With such a configuration, a virtual multiple antenna system is created although the multiple antennas are not collocated, hence providing spatial diversity [1]–[3]. The cooperative diversity transmission is typically divided into two phases. In the first phase, the source broadcasts its signal to all the relays and destination. In the second phase, the relays either amplify-and-forward the received signal, or decode, re-encode, and forward the re-encoded signal. The former process is commonly referred to as amplify-andforward (AF), whereas the latter is known as decode-andforward (DF) [1]–[3]. The amplify-and-forward (AF) protocol is quite attractive since it puts less processing burden on the relays. The AF protocol is further categorized as variable-gain or fixed-gain relaying based on the availability of channel state information (CSI) at the relays. The variable-gain AF relaying scheme requires the instantaneous CSI of the source-relay link at the corresponding relay to maintain a fixed transmit power at all time. On the other hand, the fixed-gain AF relaying scheme

Tho Le-Ngoc Department of Electrical & Computer Engineering McGill University 3480 University St., Montreal, QC, Canada H3A2A7 [email protected]

does not need the instantaneous CSI, but the average signalto-noise ratio of the source-relay link in order to maintain a fixed average transmit power at each relay [1]–[4]. A majority of previous research works in wireless relay communications assumes that all the receivers in the network (i.e., at relays and destination) have the perfect knowledge of CSI of all the transmission links propagated by their received signals in order to perform a coherent detection. However, such an assumption is unrealistic or implies a high cost for a network with multiple relay transmission links, especially in a fast fading environment. To alleviate the need of requiring CSI at the receivers, differential or noncoherent modulation/demodulation techniques can be used. In particular, references [5]–[8] focus on the differential phase-shift keying (DPSK) for both AF and DF protocols. In [9], the maximum-likelihood (ML) receiver and a suboptimal noncoherent AF receiver have been studied for on-off keying (OOK) and binary frequency-shift-keying (BFSK). The paper shows that a full diversity order is achieved for BFSK but not for OOK. However, the ML receiver involves integrals and hence is very complicated for implementation. Reference [10] shows that there is no closed-form ML detector for a non-coherent AF network. Reference [11] proposes a noncoherent detection scheme based on the generalized likelihood ratio test (GLRT) method, in which the likelihood function of each hypothesis is evaluated using the maximum-likelihood estimation of the channel gain. For the case of BFSK and single-relay transmission, the paper shows that the GLRT receiver achieves near full diversity order. More recently, the maximum energy selection (MES) receiver was developed in [12] by selecting the maximum output from the square-law detectors of all branches, and was shown to achieve the full spatial diversity. The GLRT receiver in [11] needs to know only the noise power at the relays and destination, while the MES receiver does not need any CSI nor noise information [12]. These two receivers therefore have low complexity. It should be pointed out that references [9], [11], [12] assume that there is no temporal correlation in any wireless channel of the network. As such, the receivers developed in these references might not work well when temporal channel correlation exists. This observation motivates our study of yet another detection scheme for non-coherent AF relay networks

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that makes use of the implicit pilot-symbol-assisted technique. Such a technique was originally developed in [13] for pointto-point communications with multiple antennas. Although the main concept is inspired by the work in [13], there are significant differences between our development and analysis of the technique for AF relay networks and those reported in [13] for point-to-point multiple-antenna systems. II. P ROPOSED F RAMEWORK FOR N ONCOHERENT AF R ELAY S YSTEMS A. System Model Consider a wireless relay network with one source, K relays, and one destination. The K relays retransmit signals to the destination over orthogonal channels. For convenience, the source, relays, and destination are denoted and indexed by node 0, node i, i = 1, . . . , K, and node K + 1, respectively. Every node is equipped with one antenna and operates in a half-duplex mode, i.e., a node cannot transmit and receive simultaneously. Fixed-gain AF protocol is employed at the relays. Without loss of generality, assume that orthogonal channels are made available by means of time-division multiplexing. This implies that signal transmission from the source to destination is completed in (K + 1) time slots, i.e., the duration is (K + 1)T where T is the symbol duration (or time slot duration). In the following discussion, we adopt the convention that epoch k starts at t = k(K + 1)T and ends at (k + 1)(K + 1)T . In the first time slot at epoch k, the source broadcasts a M -FSK signal and the received signal at node i, i = 1, . . . , K + 1, is written as y0,i (t) = E0 h0,i [k]xm (t) + n0,i (t), t ∈ Tk,0 (1) where Tk,j = [k(K +1)T +jT, k(K +1)T +(j+1)T ) denotes the interval of time slot (j + 1) at epoch k, h0,i [k] denotes the channel fading coefficient between node 0 and node i at epoch k, which is assumed to be fixed during time slot Tk,0 , and n0,i (t) is zero-mean additive white Gaussian noise (AWGN) at node i whose two-sided power spectral density (PSD) is N0 /2. The quantity E0 is the average transmitted symbol energy of the source. The transmitted waveform xm (t), chosen from an M -ary FSK constellation, is given in complex baseband as jπt 1 (2m − M − 1) , m = 1, . . . , M xm (t) = √ exp T T (2) The channel between any two nodes is assumed to be constant over (K + 1) time slots, but varies dependently every (K +1) time slots. In particular, the channel gains are modeled as circularly symmetric complex Gaussian random variables (i.e., the magnitudes of the channel gains have the flat Rayleigh fading property) with the following Jake’s autocorrelation function: 2 J0 (2πfi,j l) (3) φi,j [l] = σi,j In the above equation, indices i, j refer to transmit and receive nodes, respectively; fi,j is the maximum Doppler frequency experienced by the transmission between node i and node j;

2 is the average signal strength of the i–j link; and J0 (x) σi,j is the 0th order Bessel function of the first kind. For AF with fixed-gain relaying, the ith relay node amplifies and retransmits the received signal with a fixed scaling factor given by Ei Ei = (4) βi = 2 +N E{|y0,i (t)|2 } E0 σ0,i 0

where Ei is the average transmitted symbol energy of relay i. The received signal at the destination via the ith relay node at epoch k, i.e., during the time interval t ∈ Tk,i is written as yi,K+1 (t) = βi E0 h0,i,K+1 [k]xm (t−iT )+w0,i,K+1 (t) (5) where h0,i,K+1 [k] = h0,i [k]hi,K+1 [k] denotes the ith overall relay channel between the source and destination at epoch k, and w0,i,K+1 (t) = βi hi,K+1 [k]n0,i (t − iT ) + ni,K+1 (t) is the total additive noise corrupting the received signal. The noise ni,K+1 (t) is also a zero-mean AWGN with two-sided PSD of N0 /2. Similar to [13], it shall be shown in the next section that every transmitted symbol from the source to the destination can be considered as an implicit pilot symbol. Hence the first task at the destination is to estimate the overall channels of all the links from the source to destination using these implicit pilot symbols. Then based on the estimated overall channels, the destination performs a MRC detection as if the overall channels are perfectly known. B. Channel Estimation Exploiting the orthogonal property of FSK signalling [13], [14], the received signals at the destination are correlated with the following sum waveform, r(t), in order to estimate the overall channels: r(t) =

M

xm (t).

(6)

m=1

The outputs of the correlators are1 g0,K+1 [k] = E0 h0,K+1 [k] + v0,K+1 [k], (7) g0,i,K+1 [k] = βi E0 h0,i,K+1 [k] + u0,i,K+1 [k], (8) where u0,i,K+1 [k] = βi hi,K+1 [k]v0,i [k] + vi,K+1 [k], i = 1, . . . , K and all the noise terms vi,K+1 [k], i = 0, . . . , K, and v0,i [k], i = 1, . . . , K, are zero-mean complex Gaussian random variables with variance M N0 . Observe from (7) and (8) that g0,K+1 [k] and g0,i,K+1 [k] are independent of the transmitted waveform xm (t). This implies that one can estimate the overall channels h0,K+1 [k] and h0,i,K+1 [k] without explicit pilot symbols from the source. By employing the LMMSE estimation algorithm [15], [16], the estimation of h0,K+1 [k] and h0,i,K+1 [k], denoted by 1 Due to space limitation, detailed derivations are omitted and only the final expressions are given.


ˆ 0,i,K+1 [k] respectively, can be obtained as ˆ 0,K+1 [k] and h h follows: −1 ˆ 0,i [k] = Ψ(0) Φ(0) g0,K+1 [k], (9) h gg hg −1 ˆ 0,i,K+1 [k] = Ψ(i) Φ(i) g0,i,K+1 [k], i = 1, . . . , K h gg hg (10) where g0,K+1 [k] and g0,i,K+1 [k] are the 2P × 1 vectors formed by stacking 2P consecutive values of g0,K+1 [k + l] and g0,i,K+1 [k + l], l = −P, . . . , P , respectively. That is, g0,K+1 [k] = (g0,K+1 [k − P ] g0,K+1 [k − P + 1] t

g0,K+1 [k + P ]) , (11)

...

g0,i,K+1 [k] = (g0,i,K+1 [k − P ] g0,i,K+1 [k − P + 1] ...

t

g0,i,K+1 [k + P ]) ,

i = 1, . . . , K,

(12)

t

where (·) stands for the transpose operation. Further(i) more, Ψhg is the correlation vector between h0,K+1 [k] and g0,K+1 [k] for i = 0 or between h0,i,K+1 [k] and g0,i,K+1 [k] (i) for i = 1, . . . , K. The matrix Φgg is 2P × 2P auto-correlation matrix of g0,K+1 [k] for i = 0 or g0,i,K+1 [k] for i = 1, . . . , K. In essence, the estimations in (9) and (10) use the P closest data symbols from the past and P closet data symbols from the future to estimate the channel at a given symbol time. It should be noted that there is a trade-off between complexity and performance. Additional pilot symbols may be used to improve the performance but that will increase the complexity. With the system model and channel properties described in (i) the previous section, Ψhg can be computed as (0) Ψhg = E0 φ0,K+1 [k − P ] E0 φ0,K+1 [k − P + 1] ... E0 φ0,K+1 [k + P ] (13) (i) Ψhg = βi E0 φ0,i,K+1 [k − P ] βi E0 φ0,i,K+1 [k − P + 1] . . . βi E0 φ0,i,K+1 [k + P ] , i = 1, . . . , K, (14) where φ0,i,K+1 [l] is the auto-correlation function of the overall channel h0,i,K+1 . Assuming a channel model based on fixed relays as in [15], one has φ0,i,K+1 [l] = φ0,i,K+1 [l]φ0,i,K+1 [l] 2 2 σi,K+1 J0 (2πf0,i l)J0 (2πfi,K+1 l) = σ0,i

(15)

(i)

The (n, m)-th element of Φgg can be computed as Φ(i) (n, m) = ⎧gg ⎪ ⎨E0 φ0,K+1 [(n − m)] + M N0 δ[n − m], i = 0, 2 M N0 δ[n − m] (16) βi2 E0 φ0,i,K+1 [(n − m)] + βi2 σi,K+1 ⎪ ⎩ +M N0 δ[n − m], i = 1, . . . , K. where δ[·] is the discrete-time Dirac delta function. Let the estimation errors of channels h0,K+1 [k] and ˆ 0,K+1 [k] and h0,i,K+1 [k] be e0,K+1 [k] = h0,K+1 [k] − h

ˆ 0,i,K+1 [k], respectively. From e0,i,K+1 [k] = h0,i,K+1 [k] − h the LMMSE property [17], the estimation errors e0,K+1 [k] and e0,i,K+1 [k] are zero-mean complex random variables with variances given, respectively, as −1 H (0) (0) 2 2 Ψ = σ0,K+1 − Ψhg Φ(0) (17) σ ˜0,K+1 gg hg −1 H (i) (i) 2 2 Ψhg = σ0,i,K+1 − Ψhg Φ(i) , (18) σ ˜0,i,K+1 gg 2 2 2 = σ0,i σi,K+1 , i = 1, . . . , K denotes the where σ0,i,K+1 overall channel variance.

C. Data Detection For the purpose of data detection, the received signals at the destination in (1) and (5) are rewritten as ˆ 0,K+1 [k]xm (t) + E0 e0,K+1 [k]xm (t) y0,K+1 (t) = E0 h + n0,K+1 (t), (19) ˆ 0,i,N +1 [k]xm (t − iT ) yi,K+1 (t) = βi E0 h + βi E0 e0,i,N +1 [k]xm (t − iT ) + w0,i,K+1 (t)

(20)

First, the destination correlates the received signals in (19) and (20) with the following vector of basis waveforms in M FSK: t

(21) x(t) = x∗1 (t) x∗2 (t) . . . x∗M (t) where (·)∗ stands for the conjugate operation. The outputs of the correlators can be shown to be: 0,K+1 [k]xm [k] y0,K+1 [k] = E0 H + E0 E0,K+1 [k]xm [k] + n0,K+1 [k], (22) 0,i,N +1 [k]xm [k] yi,K+1 [k] = βi E0 H + βi E0 E0,i,N +1 [k]xm [k] + w0,i,K+1 [k],

(23)

where the M × 1 vector xm [k], m = 1, . . . , M , represents the transmitted symbol at epoch k. Note that xm [k] has 1 as its mth element and 0 as its other elements. The estimated channel matrix (or channel error matrix) between node i and i,j (or Ei,j ), is an M × M matrix containing node j, H ˆ i,j estimated channel gains (or channel estimation errors) h (or ei,j ) on its main diagonal and 0 at its other elements. Similarly, the estimated overall channel matrix (or overall 0,i,K+1 (or E0,i,K+1 ), is also an channel error matrix), H M × M matrix containing overall channel gains (or overall ˆ 0,i,K+1 (or e0,i,K+1 ) on its main channel estimation errors) h diagonal and 0 at its other elements. The elements of M × 1 noise vectors n0,K+1 [k] and w0,i,K+1 [k] are i.i.d. zero-mean 2 + 1 N0 , random variables with variance N0 and βi2 σi,K+1 respectively. The destination then combines the received signals in (22) and (23) to detect the transmitted information. Using the


0,K+1 [k] and H 0,i,K+1 [k] as they are the channel estimates H true channels, the output of the maximum ratio combiner is K H H εi H r[k] = ε0 H 0,K+1 [k]y0,K+1 [k] + 0,i,K+1 [k]yi,K+1 [k], i=1

−1 (24) 2 where the combining weights ε0 = E0 σ 0,K+1 + N0 and −1 2 2 εi = βi2 E0 σ 0,i,K+1 + βi2 σi,K+1 N0 + N 0 are used to maximize the SNR of the combiner’s output. The destination subsequently decides the transmitted symbol as m ˆ = arg

max

m=1,...,M

Re(rm [k])

(25)

where rm [k] is the mth element of the M × 1 vector r[k]. Re(x) takes the real part of a complex number x. III. U PPER B OUND ON BER P ERFORMANCE AND D IVERSITY O RDER The exact BER performance analysis of the proposed detection scheme appears difficult due to the non-Gaussian nature of the noise in (23). Instead, this section derives an upper bound on the BER by assuming that the noise is Gaussian. It follows from (24) that the instantaneous SNR at the combiner’s output can be written as γ=

K

γi

(26)

i=0

where

ˆ 0,i,K+1 |2 βi2 E0 |h . 2 2 βi2 E0 σ 0,i,K+1 + βi2 σi,K+1 N0 + N 0

The upper-bound on BER can be evaluated using the moment-generating function (MGF) method [18]. To simplify the analysis, it is assumed that BFSK is employed2 , i.e., M = 2 . The average BER can be upper bounded as π2 π2 K 1 g g 1 dθ = dθ Mγ M Pe ≤ γi π 0 π 0 i=0 sin2 θ sin2 θ (27) where g = 12 for BFSK. The MGF of γi , Mγi (s), can be obtained by performing ˆ 0,i,K+1 |2 ˆ 0,K+1 |2 for i = 0, or |h integration over the pdf of |h for i = 1, . . . , K. One can verify that [15] Mγ0 (s) = (1 + sγ 0 )−1 , and

1 exp Mγi (s) = γis

where γ0 =

1 γis

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2 2 −σ 0,i,K+1 ) βi2 E0 (σ0,i,K+1 . 2 2 2 2 βi E0 σ 0,i,K+1 + βi σi,K+1 N0 + N0

By substituting (28) and (29) into (27), an upper bound on the BER can be obtained by performing a single integration. Although such an upper bound can be calculated numerically, it does not readily tell us about the diversity order achieved by the proposed detection method. To analyze the diversity order, a looser upper bound on the BER is presented next that does not involve any integration. By using the following inequality [19] 1 1 < exp(x)E1 (x) ≤ , ∀x > 0, 0 < η < 1 (30) 1+x η+x one has π2 K 1 1 1 Pe ≤ dθ (31) g π 1 + sγ 1 + ηγ s 0

(28) 1 γis

(29)

2 2 −σ 0,K+1 ) E0 (σ0,K+1 2 E0 σ 0,K+1 + N0

2 It should be noted that for M > 2 the pairwise error probability can (M −FSK) (BFSK) be upper bounded as Pe < M P [13]. It implies that if the 2 e network can achieve a full diversity order with M = 2, it also achieves a full diversity order with arbitrary values of M .

0 i=1

i

s= sin2 θ

Then the Chernoff bound can be used to further upper bound the BER by setting θ = π/2 in the above expression, which results in 1 (32) Pe ≤ K π(1 + γ 0 /2) i=1 (1 + ηγ i /2) It is now clearly seen that, when the transmitted powers at the source and relays are sufficiently large, the variances of the 2 2 ,σ 0,i,K+1 → 0), estimation errors approach zero (i.e., σ 0,K+1 E0 σ 2

ˆ 0,K+1 |2 E0 |h , γ0 = 2 E0 σ 0,K+1 + N0 γi =

γi =

β 2 E0 σ 2

0,K+1 hence γ 0 → , γ i → β 2 σi 2 0,i,K+1 . This means that N0 i i,K+1 N0 +N0 in the high signal-to-noise ratio region, a full diversity order of K + 1 is achieved by the proposed detection scheme. This result agrees with the conclusion in [16] that the presence of channel estimation errors does not affect the diversity order. The result of full diversity order shall also be confirmed by the BER curves obtained by simulation in Section IV.

IV. S IMULATION R ESULTS In all the simulations presented and discussed in this section, the transmitted powers are set to be the same for the source and all the relays, i.e., Ei = E, i = 0, . . . , K. The noise components at the relays and destination are modeled as i.i.d. CN (0, 1) random variables. The average quality of a transmission link is assumed to be a function of the relative distance between a transmitting node and a receiving node. 2 = d−ν In particular we set σi,j i,j where ν is the path loss exponent and di,j is the distance between node i and node j. All the simulation results are reported with ν = 4. Also for simplicity, we set d0,1 = d0,2 = · · · = d0,K = d1 , d1,K+1 = d2,K+1 = · · · = dK,K+1 = d2 , and d0,K+1 = d0 . Unless other stated, the parameter P (which defines the size of observations used in channel estimation – see (12)) is set to 2 and the Doppler frequencies are normalized as 10f0,i T = fi,K+1 T = f0,K+1 T = 0.01, i = 1, . . . , K. It should be emphasized again that the fading coefficients are independent among different transmission links, but they are time correlated according to the Jake’s model. Figs. 1, 2, and 3 compare the performance of the proposed detection scheme with that of three previously proposed


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schemes, namely GLRT [11], MES [12], and the “optimum” detection scheme in [9]. The comparison is done for a tworelay network with BFSK modulation and under three different scenarios: the relays are placed close to the source (Fig. 1), close to the destination (Fig. 2), and at the midpoint between the source and the destination (Fig. 3). Plots of the upper bound from (27) and performance of the coherent detection scheme (i.e., when the destination has perfect knowledge of CSI of all the transmission links) are also included in these figures. The three figures show that our proposed scheme outperforms other schemes in all three considered scenarios of relay positions (i.e., different channel conditions). The superior performance of our proposed scheme comes from the fact that it can make use of the correlation information to aid in the

estimation of the overall channels. It is also noted that the performance gap to the coherent scheme becomes smaller for the proposed scheme when the relays are close to the source or at the midpoint between the source and destination. This observation is intuitively satisfying since in such scenarios the destination is affected by less noise amplification from the relays. The tightness of the derived upper bound on the BER performance of our proposed scheme can also be confirmed from these figures. Next, Fig. 4 illustrates how the size of observations (determined by the parameter P ) used for channel estimation, affects the average BER of the proposed scheme. Investigated here is a single-relay network with BFSK modulation and when the relay is placed close to the source. It can be seen that the BER performance improves as P increases. In general, the parameter P should be chosen to tradeoff among different system requirements, such as the BER performance, implementation complexity, and decoding delay. As observed from the figure, when P increases from 0 to 1, there is a gain of about 2 dB in average transmitted power per node. However, the power gain is quite insignificant when P increases from 2 to 3. Such a behavior is also expected since when the estimation error reaches a saturation point, increasing P will not result in a significant BER improvement. However, in the high SNR region, the performance of the proposed scheme can be made to approach that of the coherent scheme by keep increasing P . The BER comparison of the proposed scheme and other schemes is shown in Fig. 5 for the case of a single-relay network and 4-FSK modulation (M = 4) and when the relay is placed at the midpoint between the source and destination. The figure again confirms that the proposed scheme outperforms other schemes for 4-FSK. A diversity order of 2 is also observed with all the schemes from the figure.


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Simulation results were presented to corroborate the analysis. Performance comparison reveals that the proposed scheme outperforms the previously proposed schemes. Finally, it should be pointed out that, although this paper focuses on a scenario where all the nodes in the network are equipped with a single antenna, the developed framework can be extended to a more generalized multi-antenna AF relay networks, i.e., when the source, relays and destination are equipped with multiple antennas.

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Average Power per Node (dB) Fig. 5. Error performance of a single-relay network by different schemes when M = 4 (4-FSK), d0 = 1, d1 = 1, d2 = 1 (the relays are placed at the midpoint between the source and the destination), and P = 2.

V. C ONCLUSIONS In this paper, we have considered amplify-and-forward multiple-relay networks in which the source transmits to the destination with the help of K relays and the channels are temporally-correlated Rayleigh flat fading. The networks are investigated when M -FSK is employed at both the source and relays to facilitate noncoherent communications. Making use of the orthogonal property of FSK signalling, the destination first estimates the overall channel coefficients based on a LMMSE approach and then detects the information symbol with a (approximate) maximum ratio combiner. An upperbound of the BER expression was also derived and used to show that the proposed scheme achieves a full diversity order.

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