IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 2, NO. 5, OCTOBER 2013
531
MIMO Precoding and Relay Selection for the Decode-and-Forward Relay Networks Duckdong Hwang, Chang-Yeong Oh, and Tae-Jin Lee
Abstract—We consider the joint relay and pre-coder selection in the decode-and-forward (DaF) relay network, where multiple relays are employed with multiple antennas. Depending on the channel information, the destination informs the relay and precoder selection information to the source and relays with finite bits of information. The source and the selected relay use their own pre-coders to transmit the signal using orthogonal channels. We present a selection criterion based on an upper bound of the pair-wise symbol error probability and provide the conditions for the code-books to achieve the full diversity. For a single antenna destination, we show that the proposed criterion guarantees the diversity order of Ms + KMr , where Ms , Mr are the numbers of antennas at the source and the relay respectively and K is the number of relays in the network. Index Terms—Decode-and-forward relay network, pre-coder, relay selection, limited feedback.
I. I NTRODUCTION OOPERATIVE relays [1] enhance the reliability and throughput of wireless transmissions. Relays happen to be positioned in geographically appropriate locations and offer us an opportunity to benefit from them or can be placed in a network (like cellular system) according to the network planning to widen the coverage of the network service. Most popular relaying techniques in practice are the amplify-andfrward (AaF) and the decode-and-forward (DaF) protocols [1]. The techniques jointly taking advantage of the cooperative relay and the multiple input multiple output (MIMO) antennas have drawn much of the research interest recently [2]–[6]. The authors of [2] considered MIMO beam-forming for the AaF network with perfect channel status information (CSI) assumption. In a closed loop MIMO system, the CSI feedback relies on narrow feedback pipe and thus the trade-off between efficient representation of the CSI and the performance degradation is an inherent challenge. Similar to the pre-coder selection in the point-to-point MIMO channel [7], finite feedback based mode selections in the AaF network are considered in [4]. As far as DaF protocol is concerned, the MIMO techniques in the DaF network are considered in [3], [5]. A closed loop
C
Manuscript received April 12, 2013. The associate editor coordinating the review of this letter and approving it for publication was L. Lampe. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF-2010-0020210), and by the MSIP(Ministry of Science, ICT & Future Planning), Korea, under the ITRC (Information Technology Research Center) support program supervised by the NIPA (National IT Industry Promotion Agency)(NIPA-2013-(H0301-13-1005)). The authors are with the College of Information and Communication Engineering, Sungkyunkwan University, Suwon, Korea (e-mail:
[email protected]). T.-J. Lee is the corresponding author (e-mail:
[email protected]). Digital Object Identifier 10.1109/WCL.2013.070113.130266
control of the relay link with finite feedback bits and precoding at the source and the relay, is considered in [5] while a criterion to select a relay among multiple MIMO relays is presented in [3] for general MIMO signaling modes. We, in this letter, focus on the finite feedback based pre-coding with relay selection in the relay network when there are multiple DaF relays available and aim to exploit the benefit from the multiple relays and the MIMO at the same time. Working with the maximum likelihood decoding [8], [9], where all the combinations of error types at the relays are considered is very hard. The authors of [3] elaborate on steps of upper bounding the pair-wise symbol error probability and end up with a simple and handy error rate expression. Since the result of [3] is for general space time signaling, we need a modification of the expression for the pre-coding signaling. A criterion to jointly select a relay and pre-coders at the source and at the selected relay is presented and its diversity performance is analyzed. When there are K relays with Mr antennas each and the source has Ms antennas and the destination has a single antenna, the fully available diversity order in the network is Ms + KMr . We show that the proposed criterion guarantees the full diversity order when the code-books of the source and the relay meet some rank conditions. Though the criterion is developed in the symbol error probability sense, we show that the same diversity gain is achieved in the outage probability sense as well with the same code-book conditions. Therefore, the proposed technique exploits the benefits from all the multiple antennas in the network into the diversity performance of the transmission. The letter is organized as follows. In Section II, we present the system model and the selection criteria appears and is analyzed in Section III. The simulation results are presented in Section IV. Notations: The bold lower case letter represents a vector and the bold upper case letter represents a matrix. The notations A† and diag(a1 , . . . , aJ ) are the hermitian transpose of matrix A and the diagonal matrix with a1 , . . . , aJ as its diagonal elements respectively. The notation Aa,b represents the element of A at the a-th row and b-th column position. AF denotes the Frobenius norm of A. II. S YSTEM M ODEL In Fig. 1, the relay network with K decode-and-forward relays is depicted. The source is equipped with Ms antennas, each of the relays has Mr antennas while the destination has only one receive antenna. The channel between the source and the destination is denoted by the 1 × Ms vector h0 , the channel from the source to the k-th relay is denoted by the Mr × Ms matrix Hk and the channel from the k-th relay
c 2013 IEEE 2162-2337/13$31.00
532
IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 2, NO. 5, OCTOBER 2013
the maximum ratio combining to yd (1) and yd (2) to get the final demodulated symbol xˆ. For the simplicity of analysis, we normalize the distance from the source to the destination as one and assume that all the relays are d1 (d1 ≤ 1) apart from the source and d2 (d2 ≤ 1) apart from the destination. Therefore, the elements of h0 have unit variance; the elements of Hk have variance of dα 1 and the elements of gk have the variance of dα 2 , where α is the path loss exponent. III. R ELAY AND P RE - CODER S ELECTION
Fig. 1.
The MIMO relay network with K relays.
to the destination is denoted by the 1 × Mr vector gk . The elements of these channel matrices (vectors) are independent and identically distributed (i.i.d.) complex Gaussian random variables with zero mean and the channels are stable for a certain amount of time such that the closed loop feedback control of this letter operates. When the numbers of code-book elements at the source and at the relays are Bs and Br respectively, the source has a code-book F = {f1 , . . . , fBs } and all the relays use a common code-book M = {m1 , . . . , mBr }. The code-word vectors fi and mj have dimensions Ms × 1 and Mr × 1 respectively. All the relays operate in the half-duplex mode. Therefore, in the first phase the source pre-codes a modulated message symbol x with the pre-coder fi and sends the product signal fi x through the Ms antennas. When the source transmit power is Ps , the received signals at the destination and at the k-th relay are given as √ (1) yd (1) = Ps h0 fi x + nd (1), √ yk = Ps Hk fi x + nk , (2) where the scalar nd (1) and the Mr × 1 vector nk are the complex white Gaussian noise with zero mean and variance of one. We assume that the k-th relay knows the channel Hk and the destination knows all the channel information (h0 , H1 , . . . , HK , g1 , . . . , gK ). From the channel information, the destination selects a relay among K relays. The destination also decides the pre-coder indices i at the source and j at the selected relay and informs them to the other entities in the network along with the relay selection information. The amount of this feedback information is log2 (Bs )+log2 (Br )+log2 (K) bits per channel realization. If the destination selects the k-th relay, then the k-th relay applies the linear receiver to get fi† H†k yk , where the receiver can demodulate the sent signal and denote the estimated symbol as x˜. In the second phase, the relay pre-codes the estimated symbol x˜ with the pre-coder mj and sends the product signal mj x˜ through the Mr antennas. When the relay transmit power is Pr , the received signal at the destination is given as yd (2) = Pr gk mj x ˜ + nd (2), (3) where the scalar nd (2) is the complex white Gaussian noise with zero mean and variance one. The destination applies
An upper bound of the pair-wise symbol error probability when multiple MIMO relays cooperate using orthogonal spectrum resources is presented in Eq. (20) of [3]. A modification of the equation is necessary in our case since we use the finite feedback based joint selection of the pre-coders and the relay. Let L-ary modulation be the symbol modulation for x. Then, the space time modulated signal matrices XS (x) and XR (x) in [3] become fi x and mj x, respectively, when the i-th and the j-th pre-coders are selected at the source and at the relay. min min , where dmin Also, it is easy to see that Xmin Ki = XFi = d is the smallest constellation distance in the modulation chosen. Substituting these results into the Eq. (21) in [3] and merging it into the argument of the second exponential function of the Eq. (20) in [3] gives the upper bound of the probability that the receiver confuses x with x when the source pre-coder is fi and the k-th relay is selected with the pre-coder mj is given in high SNR as 1 P (x → x ) ≤ α exp − Ps |h0 fi |2 d1 · 4 Ps † † 1 |fi Hk Hk fi |, Pr |gk mj |2 dmin , exp − min 4 2 (4) where α = 2(L+1) and d1 = |x−x |. Note that the maximum pair-wise probability of error happens when d1 = dmin . Taking the arguments of the exponential functions gives the selection criterion for the pre-coders and relay as (i, k, j) = arg
max
fi ∈F ,k∈{1,2,...,K},mj ∈M
+ min
{Ps |h0 fi |2
Ps † † |fi Hk Hk fi |, Pr |gk mj |2 }. 2
(5)
Let F be the Ms ×Bs matrix with f1 , . . . , fBs as its columns and M be Mr × Br matrix with m1 , . . . , mBr as its columns. The following Proposition states the condition on the codebooks for the criterion (5) to achieve the full diversity order. Proposition 1: The criterion (5) achieves the full diversity order of Ms + KMr in the symbol error rate sense when the following two conditions are met. condition 1) Bs , the number of rank one pre-coders (fi ∈ F ) at the source, is greater than or equal to Ms (i.e., Bs ≥ Ms ) and the rank of the matrix F is Ms . condition 2) Br , the number of rank one pre-coders (mi ∈ M) at the relay, is greater than or equal to Mr (i.e., Br ≥ Mr ) and the rank of the matrix M is Mr . Proof: Let F = V† ΛU and M = P† ΣW be the singular value decompositions (SVD) of F and M respectively, where V = [v1 , . . . , vMs ], Λ = diag(λ1 , . . . , λMs ), U =
HWANG et al.: MIMO PRECODING AND RELAY SELECTION FOR THE DECODE-AND-FORWARD RELAY NETWORKS
[u1 , . . . , uBs ] and P = [p1 , . . . , pMr ], Σ = diag(σ1 , . . . , σMr ), W = [w1 , . . . , wBr ]. The singular values are numbered in the descending order of their magnitudes. The following property [7] is useful in the proof. max |h0 fi |2
d =
q∈{1,...,Bs }
max Hk fi 2
d =
p∈{1,...,Bs }
max |gk mj |2
d =
∈{1,...,Br }
fi ∈F
fi ∈F
mj ∈M
max
|h0 Λuq |2
max
Hk Λup 2
max
|gk Σw |2 ,
(6)
d denotes the equivalence in distribution. Then, the = metric in Eq. (5) has the same distribution as the left most side of the inequality in (7). With few steps, we can find the lower bound of the metric further as in (7). where
max {Ps |h0 Λuq |2 Ps 2 2 Hk Λuq , Pr |gk Σw | } + min 2 1 {Ps h0 Λ2 ≥ max k≤K max(Ms , Mr ) Ps 2 2 Hk ΛF , Pr gk Σ } + min 2 2 min(λ2Ms , σM ) r { max Ps |hp0 |2 ≥ max k≤K max(Ms , Mr ) p≤Ms Ps q,p 2 q 2 |Hk | , max Pr |gk | }. + min max p≤Ms ,q≤Mr 2 q≤Mr q≤Ms ,k≤K,≤Mr
the criterion (5) is developed from the symbol error rate sense, it has the same diversity performance in the outage probability sense as in the symbol error rate sense. The following Proposition 2 proves this property. Proposition 2: The criterion (5) achieves the full diversity order of Ms + KMr in the outage probability sense when the same rank conditions as in the Proposition 1 are met. Proof: In the DaF relay network with a single relay, the outage event occurs in two cases. The first case occurs when all the links from the source (to the destination and to the relay) are in outage at the same time. The last case is the combined cooperative channel to the destination (from the source and from the relay) is in outage while the source to the relay channel is not in outage. The information rate of the channel from the source to the kth relay when the source uses the pre-coder fi is given (taking the factor of 1/2 by considering the half-duplex relaying into account) as ISRk (fi ) =
1 log2 (1 + Ps |fi† H†k Hk fi |). 2
(9)
Therefore, the outage probability from the source to the k-th relay channel with the data rate R is P (OutSRk (fi )) = P (|fi† H†k Hk fi |
