An Achievable Rate Region for a Three-Phase Relay Channel

The goal is to transmit a messagew₁from node 1 to node 2 andw₂from node 2 to node 1 using the medium between the two nodes and the relay a total ofn∈^Ntimes. We start the discussion by adapting the definitions to this setup.

We assume three phases where 1 ≥ α, β, γ ≥ 0, α+β+ γ = 1 indicate timesharing be-tween the phases: In the first phase, node 1 transmits the codewordX₁ⁿ¹ of lengthn₁to the relay and node 2 using a channel p₁(yR,1,y_2,1|x₁) n₁ := n₁(n) ∈ ^N times with ⁿ_n¹ → α as n → ∞. These nodes will receive the signals Y_R,1ⁿ¹ and Y_2,1ⁿ¹ respectively. In the second phase, node 2 transmits the codewordX₂ⁿ² of lengthn₂to the relay and node 1 using a channel p₁(yR,2,y_1,2|x₂) n₂ := n₂(n) ∈ ^Ntimes with ⁿ_n² → β asn → ∞. The received signals areY_R,2ⁿ² andY_1,2ⁿ² respec-tively. In the third phase the relay node transmits X_Rⁿ³ to node 1 and node 2 using a channel p₂(y_1,3,y_2,3|x_R)n₃ := n₃(n) ∈^Ntimes with ⁿ_n³ → γ asn → ∞. The terminal nodes receive the

signalsY_1,3ⁿ³ andY_2,3ⁿ³ respectively. All channels are assumed to be memoryless and the channels in the three phases are assumed to be independent. Therefore we have a joint probability dis-tribution p(yR,1,y_2,1,y_R,2,y_1,2,y_1,3,y_2,3|x₁,x₂,x_R) = p₁(yR,1,y_2,1|x₁)p2(yR,2,y_1,2|x₂)pR(y1,3,y_1,3|x_R) which defines the considered relay channel as follows:

Definition 2.6. A discrete memoryless three-phase two-way relay channel is defined by a family np⁽ⁿ⁾ :Xⁿ₁¹ × Xⁿ₂² × Xⁿ_R³ → Yⁿ_R,1¹ × Yⁿ_2,1¹ × Yⁿ_R,2² × Yⁿ_1,2² × Yⁿ_1,3³ × Yⁿ_2,3³ o

n1∈^N,n2∈^N,n3∈^N

withn₁+n₂+n₃= n. The family consists of probability transition functions given by p⁽ⁿ⁾(yⁿ_R,1¹ ,yⁿ_2,1¹,yⁿ_R,2² ,yⁿ_1,2² ,yⁿ_1,3³ ,yⁿ_2,3³ |xⁿ₁¹,xⁿ₂²,xⁿ_R³) :=

n1

Y

i=1

p₁(y_R,1,(i),y_2,1,(i)|x_1,(i))

n2

Y

i=1

p₂(y_R,2,(i),y_1,2,(i)|x_2,(i))

n3

Y

i=1

p_R(y_1,3,(i),y_2,3,(i)|x_R,(i)) for probability functionsp₁ :X₁ → Y_R,1×Y_2,1, p₂:X₂→ Y_R,2×Y_1,2andp_R :X_R → Y_1,3×Y_2,3. Definition 2.7. A (M⁽ⁿ⁾₁ ,M₂⁽ⁿ⁾,n₁,n₂,n₃)-code for the three-phase two-way relay channel under a decode-and-forward protocol consists of an encoder at node one

xⁿ₁¹ :W₁→ Xⁿ₁¹ withW₁ =[1,2, . . . ,M₁⁽ⁿ⁾], an encoder at node two

xⁿ₂² :W₂→ Xⁿ₂² withW₂ =[1,2, . . . ,M₂⁽ⁿ⁾], an encoder at the relay node

xⁿ_R³ :W₁× W₂ → Xⁿ_R³, a decoder at node one and node two

g₁ :Yⁿ_1,2² × Yⁿ_1,3³ × W₁ → W₂ g₂ :Yⁿ_2,1¹ × Yⁿ_2,3³ × W₂ → W₁ and a decoder at the relay node

g_R :Yⁿ_R,1¹ × Yⁿ_R,2² → W₁× W₂.

Definition 2.8. When w := w(w₁,w₂) = [w₁,w₂] ∈ W := W₁ × W₂ is the message pair transmitted by the two terminal nodes, the message w₂ is decoded in error ifg₁(yⁿ_1,2² ,yⁿ_1,3³,w₁), w₂or ifg_R(yⁿ_R,1¹ ,yⁿ_R,2² ),( ˜w₁,w₂) for some ˜w₁ ∈ W₁. The probability of this error event is denoted

by

λ₁(w) := Prh

g₁(Y_1,2ⁿ²,Y_1,3ⁿ³,w₁),w₂∨g_R(yⁿ_R,1¹ ,yⁿ_R,2² ),( ˜w₁,w₂)|w(w₁,w₂) has been senti . Accordingly the corresponding error event for the messagew₁is denoted by

λ₂(w) := Prh

g₂(Y_2,1ⁿ¹,Y_2,3ⁿ³,w₂),w₁∨g_R(yⁿ_R,1¹ ,yⁿ_R,2² ),(w₁,w˜₂)|w(w₁,w₂) has been senti . Note that the definition for the error is with respect to the messages rather than with respect to the decoder. The reason for this is that we need to capture the constraint of decoding at the relay in the definition of achievable rates.

Definition 2.9. The average probability of decoding error is given by µ⁽ⁿ⁾₁ := 1

|W|

X

w∈W

λ₁(w) for messagew₂and

µ⁽ⁿ⁾₂ := 1

|W|

X

w∈W

λ₂(w) for messagew₁.

Definition 2.10. Letµ⁽ⁿ⁾₁ andµ⁽ⁿ⁾₂ be the average probabilities of decoding error for messagew₂ andw₁, respectively. The rate pair [R₁,R₂] is said to be achievable for the three-phase two-way relay channel under a decode-and-forward protocol if there exists a sequence of M₁⁽ⁿ⁾,M⁽ⁿ⁾₂ ,n₁, n2,n3-codes with ^logM_n¹⁽ⁿ⁾ →R1and ^log_n^M2(n) →R2such thatµ⁽ⁿ⁾₁ , µ⁽ⁿ⁾₂ →0 asn→ ∞.

Theorem 2.7. An achievable rate region for the three-phase two-way relay channel using a decode-and-forward protocol is the set of all rate pairs [R1,R₂] satisfying

R₁< minαI(X₁;Y_R,1);αI(X₁;Y_2,1)+γI(X_R;Y_2,3)

R₂< minβI(X2;Y_R,2);βI(X2;Y_1,2)+γI(XR;Y_1,3) (2.12) for some joint probability distribution p1(yR,1,y2,1|x1)p2(yR,2,y1,2|x2)pR(y1,3,y2,3|xR)p(x1,x2,xR) and someα, β, γ≥ 0 withα+β+γ= 1.

Remark 2.9. Due to the factorization of the channel, there is no rate loss if we restrict the probability distribution of the input p(x1,x₂,x_R) such thatX₁,X₂ andX_R are independent.

Remark 2.10 (Necessity of the feedback constraint). The restriction of the nodes not to use any feedback mechanism may not seem necessary for the considered setup. To see that it is indeed necessary, consider a toy setup: Suppose that there is no channel from node 1 to the relay, while all other channels are error and interference free and offer some rate to transmit.

For this setupI(X₁;Y_R,1)=0 and thereforeR₁is zero as well. Without the restriction it would of

course be possible to transmit some data from node 1 to node 2 in the first phase. In the second phase node 2 could forward the data received from node 1 to the relay. Therefore the decode-and-forward requirement would be satisfied. If we do not allow such cooperation between the nodes, than using a cutset bound [30] slightly adapted to the considered setup the given region can be shown to be optimal. The adaption is needed, as the proof in [30] assumes, that the messages in the network are independent, while here we transmit the same message to the relay and to the terminal node. Furthermore the encoders in [30] may use all the received signals for the encoding.

Remark 2.11(Convexity of the rate region). Looking at the rate region it is not immediately clear whether the region is convex or not. For fixed timesharing parameters the region is ob-viously convex. It remains to show that there exist parameters and probability distributions such that if [R⁽¹⁾₁ ;R⁽¹⁾₂ ] and [R⁽²⁾₁ ;R⁽²⁾₂ ] are achievable with possibly different timesharing pa-rameters, then also a convex combination of both is within the achievable rate region. Indeed the proof is not that difficult, so we will only sketch it for the given region once. It turns out, that the the weighted addition can be encapsulated in some auxiliary variable Q. Note that all terms in the above rate region can be conditioned on Q, if we change p(x1,x₂,x_R) to p(x₁,x₂,x_R|q)p(q). This will not change the region. Furthermore using the observation in Re-mark 2.9, we can add three variables Q₁,Q₂, Q₃ to the expressions and change p(x₁,x₂,x_R) to p(x₁|q₁)p(x₂|q₂)p(x₃|q₃)p(q₁)p(q₂)p(q₃). The minimum operation can now be split up into two inequalities. We receive forR^∗₁being a convex combination ofR⁽¹⁾₁ andR⁽¹⁾₁

R^∗₁= aR⁽¹⁾₁ +(1−a)R⁽¹⁾₁ the resulting inequalities

R^∗₁ ≤aα⁽¹⁾I(X⁽¹⁾₁ ;Y_R,1⁽¹⁾|Q⁽¹⁾₁ )+(1−a)α⁽²⁾I(X⁽²⁾₁ ;Y_R,1⁽²⁾|Q⁽²⁾₁ ) and

R^∗₁ ≤aα⁽¹⁾I(X₁⁽¹⁾;Y_2,1⁽¹⁾|Q⁽¹⁾₁ )+aγ⁽¹⁾I(X_R⁽¹⁾;Y_2,3⁽¹⁾|Q⁽¹⁾₃ )+

(1−a)α⁽²⁾I(X₁⁽²⁾;Y_2,1⁽²⁾|Q⁽²⁾₁ )+(1−a)γ⁽²⁾I(X_R⁽²⁾;Y_2,3⁽²⁾|Q⁽²⁾₃ ).

Now

aα⁽¹⁾I(·|Q⁽¹⁾₁ )+(1−a)α⁽²⁾I(·|Q⁽²⁾₁ )= (aα⁽¹⁾+(1−a)α⁽²⁾)I(·|Q˜₁),

where ˜Q₁with|Q˜₁|= |Q⁽¹⁾₁ |+|Q⁽¹⁾₁ |is used to include the weighted sum in the expectation over a appropriately constructed probability distribution. Note that we do not change the channel if we switch to ˜Q₁, but only the input distribution to the channel. The above steps can be performed for the other inequalities and for R₂ where the same variables ˜Q₁, ˜Q₂, ˜Q₃ can be used. We used three auxiliary variables, as the needed random variable might be different for the different

phases. We see that the resulting vector [R₁,R₂] is in the region for the timesharing parameters α^∗ = (aα⁽¹⁾ +(1 −a)α⁽²⁾), β^∗ = (aβ⁽¹⁾ +(1− a)β⁽²⁾), and γ^∗ = (aγ⁽¹⁾ +(1− a)γ⁽²⁾). Similar arguments can be used for the proof of the convexity of other regions in this thesis.

Proof. Forγ = 0 the result follows immediately by interpreting marginal channels of the two broadcast channels as a compound channel [60, 29] usedn₁andn₂times respectively. Therefore in what follows we assumeγ >0. LetR₁,R₂, p(x₁,x₂,xR),α, β, γ≥ 0 withα+β+γ =1 be given such that the inequalities in (2.12) are strict. The achievability of the closure is a consequence of the definition of achievability and will not be repeated here. It can be proved analogous to the arguments in the proof of Theorem 2.2 in Section 2.1.3.

Random Codebook Generation Letn₁ = ⌊αn⌋,n₂ = ⌊βn⌋andn₃ = n−n₁−n₂ ≤ ⌈γn⌉+1.

We generate M₁⁽ⁿ⁾ = 2^⌊nR1⌋ independent codewords X₁ⁿ¹(w₁) of length n₁ drawn according to Qn1

i=1p(x_1,(i)) where p(x₁) = P

x2∈X2

P

xR∈XR p(x₁,x₂,xR) is the marginal probability distribution.

Similarly, we generate M₂⁽ⁿ⁾ = 2^⌊nR2⌋ independent codewords Xⁿ₂²(w2) of length n₂ drawn ac-cording toQn2

i=1p(x2,(i)) andM⁽ⁿ⁾₁ M₂⁽ⁿ⁾independent codewordsX_Rⁿ³(w),w= [w1,w₂] of lengthn₃ drawn according toQn3

i=1p(xR,(i)). The random code is revealed to both terminal nodes and the relay.

Encoding Depending on the message to transmitw₁ node 1 sends the corresponding code-word xⁿ¹(w₁) using the channeln₁ times. Node 2 sendsxⁿ²(w₂) to transmit the messagew₂. To send the decoded pairw=[w₁,w₂] withw_k ∈ W_k,k∈ {1,2}, the relay sends the corresponding codeword xⁿ_R³(w).

Decoding The receiving nodes will use typical set decoding. For a strict definition of the decoding sets we choose parameter for the typical sets as ǫ₁ < ^αI(X1,Y_3α^R,1)−R1, ǫ₂ < ^βI(X2,Y_3β^R,2)−R2, ǫ₃ < ^{γI(XR,Y2,3)+αI(X}_6α ^1,Y2,1)−R1, and ǫ₄ < ^{γI(XR,Y2,3)+αI(X}_6γ ^1,Y2,1)−R1. For the first two phases the relay decides that w₁ andw₂ are transmitted, if xⁿ₁¹(w₁) and xⁿ₂²(w₂) are the only codewords jointly typical with the received signals yⁿ_R,1¹ andyⁿ_R,2² respectively, i.e.

xⁿ₁¹(w₁),yⁿ_R,1¹

∈ T_ǫ⁽ⁿ₁¹⁾(X₁,Y_R,1) and

xⁿ₂²(w₂),yⁿ_R,2²

∈ T_ǫ⁽ⁿ₂²⁾(X₂,Y_R,2). Knowingw₂, the decoder at node 2 decides that w₁ was transmitted, if this is the uniquew₁such that

xⁿ₁¹(w₁),yⁿ_2,1¹

∈ T_ǫ⁽ⁿ₃¹⁾(X₁,Y_2,1) and simultaneously xⁿ_R³(w1,w₂),yⁿ_2,3³

∈ T_ǫ⁽ⁿ₄³⁾(XR,Y_2,3). Decoding at receiver 1 works in an analogous way. To keep the definition of the decoder consistent the decoders map to the default message w₁ = 1 and w₂ =1 if no or more than one codewords is found in their respective decoding sets.

Analysis of the Probability of Error Now we bound the probability of error for the decod-ing. We give the proof for the transmission of message w₁ to receiver 2. The proof for the messagew₂ follows from analogous arguments. We have

µ⁽ⁿ⁾₂ = P⁽¹⁾_e,2P˜⁽²⁾_e,2+P⁽¹⁾_e,2(1−P˜⁽²⁾_e,2)+(1−P⁽¹⁾_e,2) ˜P⁽²⁾_e,2

where ˜P⁽²⁾_e,2is the average probability of the event, that the decoding at the terminal node fails and P⁽¹⁾_e,2is the average probability for a decoding error in the decoding ofw₁at the relay. Therefore we can boundµ⁽ⁿ⁾₂ form above as

µ⁽ⁿ⁾₂ ≤P⁽¹⁾_e,2+P⁽²⁾_e,2

whereP⁽²⁾_e,2is the average probability for a decoding error at node 2 given that the relay decoded without error. We can split the analysis of the overall error probability into the analysis of the error in decoding at the relay and of the error in the decoding at the terminal node. As we use the random coding argument, in the analysis, we will average over the random codebook and consider^E_xⁿ¹

1 ,xⁿ₂²,xⁿ_R³

nµ⁽ⁿ⁾_k o

≤ ^E_xⁿ1 1 ,xⁿ₂²,xⁿ_R³

nP⁽¹⁾_e,k+P⁽²⁾_e,ko

. We show that we have^E_xⁿ¹

1 ,xⁿ₂²,xⁿ_R³

nµ⁽ⁿ⁾₁ o

→ 0.

We will then conclude, that there exists at least one codebook with small average probability of error for the decoding ofw₁andw₂.

Decoding at the relay First consider the decoding at the relay. The relay is in error if either xⁿ₁¹(w₁),yⁿ_R,1¹

<T_ǫ⁽ⁿ₁¹⁾(X₁,Y_R,1) or if there exists a ˆw₁ , w₁with

xⁿ₁¹( ˆw₁),yⁿ_R,1¹

∈ T_ǫ⁽ⁿ₁¹⁾(X₁,Y_R,1).

Using the union bound it is sufficient to show that both these error events occur with arbitrarily small probability asn→ ∞.

By the law of large numbers the probability that

xⁿ₁¹(w1),yⁿ_R,1¹

< T_ǫ⁽ⁿ₁¹⁾(X1,YR,1) for se-quences

xⁿ₁¹(w1),yⁿ_R,1¹

drawn according to a joint probability distribution can be made arbitrarily small by choosingn(and as a consequencen₁) big.

The probability of

xⁿ₁¹( ˆw₁),yⁿ_R,1¹

∈ T_ǫ⁽ⁿ₁¹⁾(X1,Y_R,1) for ˆw₁ , w₁ averaged over all codewords and the random codebook can be bounded as follows:

X

yⁿ_R,1¹∈Yⁿ_R,1¹

Exⁿ₁¹













 p

yⁿ_R,1¹ |xⁿ₁¹(w1)

|W1|

X

ˆ w1=1 ˆ w1,w1

χ_T⁽ⁿ1)

ǫ1 (X1,YR,1)(xⁿ₁¹( ˆw1),yⁿ_R,1¹ )















=(|W₁| −1) X

yⁿ_R,1¹∈Yⁿ_R,1¹

X

xⁿ₁¹∈Xⁿ₁¹

p(xⁿ₁¹)p(yⁿ_R,1¹ )χ_T⁽ⁿ1)

ǫ1 (X1,YR,1)(xⁿ₁¹,yⁿ_R,1¹ )

≤ 2^{n(R1+3αǫ1−αI(X1,YR,1))+I(X1,YR,1)}. The last step follows from the properties of the typical set analogous to the procedure in the proof of Theorem 2.2 and the choice ofαn−1≤ n₁ ≤αn. Now forn→ ∞

2^{n(R1+3αǫ1−αI(X1,YR,1))+I(X1,YR,1)} →0 as we choose ǫ₁ < ^αI(X1,Y_3α^R,1)−R1. We conclude that^Exⁿ_R

nP⁽¹⁾_e,ko

can be made arbitrarily small forn large.

Decoding at the terminal node For the calculation of the probability^E{P⁽²⁾_e,2}we assume that the relay has decoded w₁ andw₂ without error. Furthermore node 2 received someyⁿ_2,1¹ drawn

according to p

yⁿ_2,1¹ |xⁿ₁¹(w₁) .

The error that may occur at node 2 is characterized by several possible events:

• E₁: xⁿ₁¹(w₁) is not jointly typical withyⁿ_2,1¹ ,

• E2: xⁿ_R³(w1,w2) is not jointly typical withyⁿ_2,3³ ,

• E₃: xⁿ₂¹( ˜w₁) is jointly typical withyⁿ_2,1¹ for some ˜w₁ , w₁, or

• E₄: xⁿ_R³( ˜w₁,w₂) is jointly typical withyⁿ_2,3³ for some ˜w₁, w₁.

For these events we can calculate the probabilityP_Ei where we take into account the random codebook generation as well as the joint randomness in the system. We can bound the average probability of error for the decoding from above by

E{P⁽²⁾_e,2} ≤ P_E1 +P_E2 + M₁P_E3P_E4,

where the average is over the random codebook as well as over the transmitted symbols. The last term follows form the observation that an error occurs if E₃andE₄happen simultaneously.

The factor M₁attributes the fact, that this may happen for each wrong message ˜w₁ , w₁. ClearlyP_E1 → 0 ifn₁ → ∞and P_E2 → 0 if n₃ → ∞which follows from the definition of strong typicality and the law of large numbers. Analogous to the proceeding above we have

P_E3 ≤2^{n(3αǫ3−αI(X1,Y2,1))+I(X1,Y2,1)} and

P_E4 ≤2^{n(3γǫ4−γI(XR,Y2,3))+2ǫ4} and therefore

M₂P_E3P_E4 ≤2^{n(R1−αI(X1,Y2,1)+3αǫ3−γI(XR,Y2,3)−3γǫ4)+I(X1,Y2,1)+2ǫ4}. We can now conclude that^En

P⁽²⁾_e,2o

→0 asn→ ∞by the choice ofǫ₃ < ^{γI(XR,Y2,3)+αI(X}_6α ^1,Y2,1)−R1 andǫ₄< ^{γI(XR,Y2,3)+αI(X}_6γ ^1,Y2,1)−R1.

This proves that^E_xⁿ¹

1 ,xⁿ₂²,xⁿ_R³

nµ⁽ⁿ⁾₁ o

→ 0 forn → ∞. Similarly^E_xⁿ¹

1 ,xⁿ₂²,xⁿ_R³

nµ⁽ⁿ⁾₂ o

→0. Therefore

Exⁿ₁¹,xⁿ₂²,xⁿ_R³

nµ⁽ⁿ⁾₁ +µ⁽ⁿ⁾₂ o

→0 and we can conclude that there is at least one codebook such thatµ⁽ⁿ⁾₁ and µ⁽ⁿ⁾₂ can be made arbitrarily small by choosingnlarge enough. The closure of the region is achievable using similar arguments as in the proof of Theorem 2.2. Therefore the claim is

proved.

Im Dokument Achievable Rates and Coding Strategies for the Two-Way Relay Channel (Seite 63-69)