The Proposed Two-Way Distributed Space-Time Coding Protocol

Although the protocol presented in Sec. 4.3 allows cooperation in two phases, it has mainly two disadvantages: a computationally demanding ML decoder at the relays and the impossibility to use the direct link between the two terminals. In [113], a three-phase PDF-II protocol (3-phase PDF) was proposed to extend 2-phase PDF and mitigate its drawbacks.

In 3-phase PDF, as shown in Fig. 2.12, first _T1 transmits while T2 remains

silent, then _T2 transmits while T1 remains silent, and finally, the relays encode

and broadcast the signal to the terminals as in the second phase of 2-phase PDF. However, it was shown in [113] that two-phase protocols perform better than three-phase protocols for a large constellation size due to the higher symbol rate.

4.4 The Proposed Two-Way Distributed Space-Time Coding Protocol

In this chapter, we propose a modification to the three-phase protocol that re- sults in a symbol rate equivalent to that of the two-phase protocol. The proposed strategy achieves better performance and low decoding complexity. Moreover, the proposed strategy is also able to use the direct link between the two destination terminals.

Let us assume that T _{≥ 2 is an even number. In the first phase, from time slot} 1 to T /2,T1 andT2 transmit simultaneously

p

P_T1xT1 and

p

P_T2xT2, respectively,

where x_T1 and xT2 are two T /2× 1 vectors of transmitted signals, [xT1]i ∈ XT1,

[x_T2]i ∈ XT2, |XT1| = |ST1|

2 _{and |X}

T2| = |ST2|

2_{. Here,}

x_T1 =GT1(sT1), (4.1)

where GT1(·) is a function that maps the entries [sT1]2i−1 and [sT1]2i onto [xT1]i

resulting in a constellation with the square of the size of ST1. The symbol [xT2]i

is the respective combination of the symbols [s_T2]_2i−1 and [s_T2]2i of T2 with the

function _GT2(·). The received signal at the rth relay is given by:

y_R1,r =pP_T1frxT1 +

p

P_T2grxT2 + nR1,r (4.2)

where n_R1,r is the noise vector at the rth relay in the first phase. In the second phase, from time slot T /2 + 1 to T , _T1 and T2 transmit simultaneously

p P_T1xT1

and ₋pP_T2xT2, respectively. The only difference with the first phase is that T2

transmits the same symbols multiplied by _{−1. The received signal at the rth} relay during the second phase is given by:

y_R2,r =pP_T1frxT1 −

p

P_T2grxT2 + nR2,r (4.3)

where n_R2,r is the noise vector at the rth relay in the second phase. We assume that the entries of all noise vectors can be modeled as independent and identically distributed Gaussian random variables with zero mean and variance σ2 _{= 1.}

Chapter 4: Distributed Space-Time Block Coding in Decode and Forward Relay Networks

4.4.1

Decoding Procedure at the Relays

Using (4.2) and (4.3), the rth relay can decode the symbols as: ˆ x_T1,r = arg min x_T1 yR1,r + yR2,r− 2 p P_T1frxT1 , (4.4) ˆ x_T2,r = arg min x_T2 yR1,r − yR2,r − 2 p P_T2grxT2 . (4.5)

Let us explain in detail the decoding procedure at the relays. The rth relay receives the signals given by (4.2) and (4.3) during the first two phases. Using the received signals, the rth relay finds ˜x_T1,r and ˜xT2,r as follows:

˜ x_T1,r = round (y_R1,r + y_R2,r)f∗ r 2pP_T1|fr|2 ! , (4.6) ˜ x_T2,r = round (y_{R1,r − yR2,r})g∗ r 2pP_T2|gr|2 ! (4.7)

where round(·) rounds the argument to the nearest constellation point. In case of an integer constellation, this can be carried out with just a hard decision in the real and imaginary parts. Using ˆx_T1,r and ˆxT2,r, the rth relay can obtain ˜sT1,r

and ˜s_T2,r by performing the inverse operation of GT1(·) and GT2(·), i.e.,

˜s_T1,r =G_T−1₁ (ˆxT1,r), (4.8)

˜s_T2,r =G_T−12 (ˆxT2,r). (4.9)

The GTt and G_T−1t operations can be implemented by merging and splitting

the bits that represent each symbol, respectively. Note that the relay de- coding, given by (4.6) and (4.7), rounds simply the real and imaginary parts to the nearest constellation point in case of an integer constellation. Fur- thermore, the use of _GTt and G_T−1t in the proposed scheme has a negligible

complexity. On the other hand, the relay decoding complexity of 2-phase PDF explained in Sec. 4.3 [113] has the order of _|ST1||ST2|, since its system is

4.4 The Proposed Two-Way Distributed Space-Time Coding Protocol

if s_T1 and sT2 are drawn from M -QAM, then the ML decoder of 2-phase PDF

has the order of M2 _{in this case. Hence, the relay decoding complexity of our}

proposed method is much lower than that of the 2-phase PDF which is quadratic.

An equivalent transmission to the previous two phases can be performed if_T1

transmits p2P_T1xT1 whileT2 remains silent from 1 to T /2, and thenT2 transmits

p

2P_T2xT2 while T1 remains silent from T /2 + 1 to T . In this case, the direct link

can be used as the terminal that remains silent can listen to the transmission of the other one. Since one terminal is off at a certain time slot, the other terminal uses the double power. After decoding the received symbols, the relay combines the decoded symbols from the two terminals into one symbol as

s_R,r =_F(˜sT1,r, ˜sT2,r) (4.10)

where s_R,r is a T _{× 1 vector. The selection of F(·, ·) for our protocol is discussed} in the next section. The third and final phase of the proposed protocol, from time slot T + 1 to 2T , is exactly the same as the second phase of the protocol described in Sec. 4.3. The relay precodes the symbol vector or its conjugate and scales it before broadcasting it to the terminals. The received signals at T1 and

T2 are given by:

y_T1 = R X r=1 p P_RrfrAr˘sR,r+ nT1, (4.11) y_T2 = R X r=1 p P_RrgrAr˘sR,r+ nT2, (4.12)

respectively, where either ˘s_R,r = s_R,r or ˘s_R,r = s∗

R,r, and nT1 and nT2 are the

noise vectors at _T1 and T2, respectively. The selection of ˘sR,r and Ar on each

relay depends on the used DSTC scheme as explained in Sec. 2.4.1. Any linear space-time codes can be applied to the proposed technique directly. For example, in case of the Alamouti code with two relay nodes used in the simulation part,

Chapter 4: Distributed Space-Time Block Coding in Decode and Forward Relay Networks

˘s_R,r and Ar are chosen as follows:

A1 = " 1 0 0 1 # , A2 = " 0 1 −1 0 # , ˘s_R,1 = s_R,1, ˘s_R,2 = s∗_R,2.

4.4.2

Decoding Procedure at the Communicating Termi-

nals without Direct Link

The ML decoding problem at_T2 without considering the direct link between the

communicating terminals can be expressed as: arg min S∈S kyT2 − S gk 2 (4.13) where g = [pP_R1g1,· · · , p P_RRgR] T_{, S = [˜s} R,1, ˜sR,2, . . . , ˜sR,R] ∈ S, S is the

set containing the LT possible combinations of the symbol matrices and LT =

(|ST1||ST2|)

RT_{. As it can be observed from (4.13), the ML decoding is quite com-}

plex even for moderate numbers of relays and small constellation sizes. Therefore, the ML decoder is approximated by considering an error free decoding at the re- lays. This also means that s_R,1 = _{· · · = sR,R} = s_R and, therefore, the decoder can be expressed as:

ˆs_R,T2 = arg min sR y_T2 − R X r=1 p P_RrgrAr˘sR 2 . (4.14)

To find ˆs_T1 at T2, the decoder uses the inverse of F(·, ·) (denoted as F−1(·, ·))

with the knowledge of s_T2, i.e.,

ˆs_T1 =F−1(ˆsR,T2, sT2). (4.15)