the approximately universal [12] and thus D-MG optimal over the two-product channel, perfect codes (see Appendix 6), and after considering the code's ...
D-MG Optimality and Explicit Coding for Wireless Relay Networks With Reduced Channel Knowledge Petros Elia and P. Vijay Kumar Dept. of Electrical Engineering University of Southern California Los Angeles, CA 90089-2565 {elia,vijayk}@usc.edu
Abstract We propose an efficient variant of the linear-processing wireless relay network scheme presented in [1], which exhibits a half-duplex constrained, diversity-multiplexing gain (D-MG) performance that is equal to that of other cooperative-diversity increasing schemes such as non-dynamic amplify-and-forward or non-dynamic decodeand-forward, but with the extra advantages that the intermediate relays do not need channel information and do not perform time and energy consuming decoding. We also explicitly construct a network encoding scheme, based on perfect space-time codes, and a coding methodology for reducing the effect of the half-duplex constraint and which potentially allows for asymptotic full-duplex D-MG optimality, even when the half-duplex constraint is present. Several extra implementation advantages are pointed out, making the proposed coding scheme an ideal candidate for encoding over such relay networks. This is substantiated by simulations. Finally we will present a scheme based on vectorized perfect codes which achieves the dynamic amplify-and-forward D-MG performance, presented in [8], for any number of relays.
1 1.1
Introduction: Distributed space-time codes in wireless relay networks and the equivalent channel Relay scheme and channel model
In this relay network cooperative-diversity scheme, a set of independent nodes cooperates for improving communication over a slow fading channel. Slightly deviating from [1], during the first stage, the original transmitter’s single antenna sequentially transmits a vector θf = θ · f1 f2 · · · fn of n signals, where each fi is from an information set A, and where θ is the normalization factor such that E[kθfi k2 ] = SNR. Each of the n − 1 intermediate relay i, i = 2, 3, · · · , n then receives the n-length vector ri = θgi f + v i ,
(1)
where each gi , i = 2, 3, · · · , n, is the complex Gaussian CN (0, 1) fading coefficient corresponding to the channel from the transmitter to the ith intermediate relay, and where v i = vi (0) vi (1) · · · vi (n − 1) is the additive noise vector with each element vi (τ ) corresponding to the CN (0, 1) Gaussian random variable representing the additive noise affecting relay i at time τ . In the second stage, each intermediate relay i, independently
performs linear-processing (time-averaging based on space-time codes) on ri and transmits xi = ri Ai where each n × n matrix Ai is unitary. (in [1] each Ai is chosen randomly). The signal at the final destination receiver-node is then of the form y =
n X
hi xi + w
(2)
i=1
where each hi , i = 1, ..., n is the CN (0, 1) fading coefficient corresponding to the channel from the ith intermediate relay to the final receiver-node, and w(τ ) the CN (0, 1) Gaussian random variables representing the additive noise at the receiver-node at time τ . Remark 1 Setting A1 = In , translates into considering the original transmitting node as an intermediate relay, with g1 = 1 known to the receiver and the transmitter. Compared to [1], this results in savings of one intermediate relay. For uniformity we will assume that the transmitter does not know g1 . As we proceed, it will become apparent that this does not affect our analysis. Consequently, we will henceforth consider gi , hi , i = 1, · · · , n to be i.i.d, zero-mean, complex Gaussian random variables.
1.2
Equivalent point-to-point channel model
All results assume temporally disjoint first and second stages. The authors in [1] introduce the concept of a distributed space-time code by having the linear-processing at the intermediate relays be performed by unitary linear-dispersion matrices [14], where each such matrix uniquely defines a row of the codematrices from a space-time code. The rationale behind this becomes clearer after rewriting (2) as: P P P y = ni=1 hi (θgi f + v i )Ai + w = θ ni=1 hi gi f Ai + ni=1 hi v i Ai + w which easily transforms into the familiar channel model, y = θHX + W, H = g1 h1 g2 h2 · · ·
g n hn , W =
n X i=1
f A1 hi v i Ai + w, X = ... f An
(3)
It becomes apparent from the above setup that matrix X can be considered as an n × n codematrix and matrix H as the 1 × n channel fading coefficient matrix. The authors in [1] show that due to the required unitary nature of the Ai ’s and due to the fact that the hi ’s are known to the receiver, it is the case that the effective additive noise elements W |hi are spatially and temporally white, zero mean, Gaussian random variables, with P variance V ar(W |hi ) = (1 + ni=1 |hi |2 )In . P n−1 e−h Considering that the probability density function of h = ni=1 |hi |2 is p(h) = h(n−1)! , it is not difficult to see that for unitary Ai ’s, in the SNR scale of interest, the W |hi can be considered, without loss of generality, to be CN (0, 1) random variables, conditioned on considering the gi , hi fading coefficients to be i.i.d. CN (0, SNR0 ) random variables. We then proceed with the following definition. Definition 1 The ‘two-product channel’ is the n × 1 channel, modelled as in (3), with the channel matrix H as in (3) representing fading coefficients which are products of two i.i.d. CN (0, SNR0 ) random variables. Furthermore, the effective additive noise consists of spatially and temporally white CN (0, 1) random variables.
1.2.1
Existing performance bounds and the need for diversity-multiplexing gain analysis
In the high-SNR regime, the PEP diversity bound of the second stage of the network scheme, as given in [1], is a function only of SNR and of the minimum eigenvalue of any difference of any two codematrices. It is easy to see that random coding allows for no guarantees on the finite duration D-MG performance of the network. Furthermore, directly applying the existing PEP bound through the union bound as in [15, 4], on the approximately universal [12] and thus D-MG optimal over the two-product channel, perfect codes (see Appendix 6), and after considering the code’s eigenvalue bounds from [4], we conclude that the optimal D-MG tradeoff in the equivalent two-product channel (only the second stage), is lower bounded as: deq (r) ≥ n(1 − 2r), req,max ≥ 21 , implying a maximum diversity d(0) = n.
2
Exact optimal diversity-multiplexing gain tradeoff of the two-product channel
Our task here is to find the optimal D-MG tradeoff of the equivalent two-product channel. To do so, we utilize the fact that perfect codes are both information lossless (unitary linear-dispersion matrices [13, 14]) as well as approximately universal. Due to the approximate universality of cyclic-division-algebra (CDA)/perfect space-time codes, the ‘error contribution channel region’ in which perfect codes decode erroneously, coincides with the outage region of the two-product channel, directly given by the following lemma. P Lemma 1 For λn = HH † = ni=1 khi k2 kgi k2 := SNR−µ being the only non-zero eigenvalue of the two-product channel, then the corresponding channel outage region which equals the error contribution region of the perfect codes, is given by: B = {µ ≥ 1 − r}.
(4)
Proof: Directly from the mismatched eigenvalue theorem presented in [4], we know that the minimum Euclidean distance between any two codematrices after the action of the channel, is lower bounded as d2E = kθH∆Xk2F ≥ θ2 λn ln where ln corresponds r to the smallest code eigenvalue of ∆X∆X † . From [4] we have that θ2 = SNR1− n , r −n (n−1) ˙ ˙ SNR1− nr SNR− nr (n−1) SNR−µ := SNRc , where as , and as a result, d2E (µ) ≥ ln ≥SNR . ˙ ˙ denoting exponential equality and inequalities respectively. in [6], we have =, ≥ and ≤ Due to its double-exponentially decreasing term, the entire probability of error, given a channel µ, is upper bounded as: c
˙ | · P EP (µ) = SNRnr e−SNR = ˙ SNR−∞ = 0, P (err|µ)≤|X
∀c > 0, as SNR → ∞ ¤
To find the volume of the outage region, we present in the following theorem the pdf of the eigenvalues in the region of interest B. Theorem 2 For hi , gi , i = 1, 2, · · · P , n being CN (0, SNR0 ) random variables, then the ˙ n−1 probability density function of λn = ni=1 khi k2 kgi k2 is upper bounded as fλn (λn )≤λ n −µ and for λn = SNR −µn ˙ fµ (µ)≤SNR . (5) 0 Proof: Consider h1 , g1 being two independent identically distributed CN (0, k), k =SNR ˙ , 2 2 random variables and thus x = kh1 k , y = kg1 k , two i.i.d. exponential random variz ables with fx (x) = e−kx , fy (y) = e−ky , fx,y (x, y) = e−kx e−ky and fx,y (x, xz ) = e−kx e−k x , x, y, z ∈ R+ . As a result, from [10, Eq 3.471.9], we have that
fkh1 k2 kg1 k2 (z) =
R∞ 0
z 1 −kw −k w e e dw w
=
R∞ 0
z 1 −k(w+ w ) e dw w
√ = 2K0 (2k z)
where K0 (·) corresponds to the modified Bessel function of the second kind. From [11, 0 (x) Section 6.6], we observe that limx→0 −Kln(x) → 1 and using the fact that for x ≥ 0.55 then π −x ˜ √ e < 1 is always finite (and decreasing in a double-exponential rate), we K0 (x)≤ 2πx ˙ get that for z ≤1, Rz R Rz √ √ √ ˙ − k z ln( z)dz = K0 ( z)dz ≤ ˙ − ln( z)dz. 0
0
0
Furthermore,
˙ z(1 − ln(z)) P (kh1 k2 kg1 k2 ≤ z) ≤ (6) Rz √ R z since Fz (z) = ˙ − 0 ln( t)dt = ˙ − 12 0 ln(t)dt = 21 (t − t ln(t))|z0 = z − z ln(z) + 0 ln(0). From a simple geometric argument we have that ˙ [z(1 − ln(z))]n . P (λn ≤ z) ≤ [P (kh1 k2 kg1 k2 < z)]n ≤ ˙ SNR0 , we have It is not difficult to see that in the SNR range of interest and for λn ≤ n ˙ n . Differentiation, change of variables and simplifications due to the that Fλn (λn )≤λ asymptotic nature of SNR, result in the final expression of the theorem. ¤ We note that in [6, 4], pµ (µ) for the n × 1 Rayleigh fading channel, is given by −µ −µn SNR −µn ˙ p (µ)=SNR ˙ e which reduces to the two-product pdf expression of f (µ)≤SNR µ
µ
for µ ∈ B. Furthermore, from [4] we see that the error contribution region of the CDA code in the Rayleigh fading channel (outage region in [6]), coincides with the error contribution region B = {µ ≥ 1 − r} in the two-product channel. As a result, we proceed as in [4], where B and fµ (µ) completely defined the probability of error as Z Z Pe ≤ fµ (µ)dµ ≤ SNR−µn dµ = ˙ maxµ∈B {SNR−µn } = ˙ SNR−n(1−r) (7) µ∈B
µ∈B
with the first inequality being due to the double exponential nature of the probability of error in B, and with the next two equalities being due to Varadhan’s Lemma and (4). Consequently deq (r) ≥ n(1 − r), and considering that knowledge of gi at each relay i reduces the two-product channel to the Rayleigh fading channel, directly results in the following theorem: Theorem 3 The optimal diversity-multiplexing gain tradeoff of the two-product channel is given by deq (r) = n(1 − r). Surprisingly, the above expression is as in the Rayleigh fading channel [6], which is essentially the two-product channel given channel knowledge at the intermediate relays.
3
Coding schemes for half-duplex constrained, linearprocessing relay networks
As described in [8], for practical considerations such as the large ratio between the transmission and reception powers at the relay antenna, it is the case that relay communications are bound by the half-duplex constraint. This is reflected in the two-stage setup of the overall relay network scheme and is considered in most related literature [7, 8]. In this section, we start by describing the extra limitations and extra coding requirements, besides D-MG optimality and information losslessness, involved with having
the half-duplex constraint, and then proceed to involve perfect codes as stepping stones towards encoding for half-duplex constrained, linear-processing relay networks. Due to lack of information extraction at the intermediate relays, if deq (r) describes the D-MG performance of a ST code carrying mn discrete information symbols over the n-time slot second stage, then d(r) = deq (r(m + 1)) describes the corresponding halfduplex constrained D-MG performance over the entire mn + n-time slot relay network transmission. On the other hand, D-MG analysis as in [15, Section 3.1.2], indicates that the above code will only achieve req,max = m over the n × nr , nr ≥ m Rayleigh fading channel and consequently over the n × 1 two-product channel (m ≤ 1). From the above two arguments it becomes clear that optimal performance can only be achieved by codes that carry, on the average, one discrete information symbol per channel use. The following bound is immediate. Theorem 4 The optimal D-MG performance of the half-duplex constrained linear-processing relay networks is 1 dentire (r) = n(1 − 2r), Coutage,entire,max = ˙ log2 (SNR). (8) 2 From the above theorem and the results in [7], we also see that, Corollary 5 Linear-processing relay networks have the same high-SNR probability of outage as the equivalent non-dynamic selection-decode-and-forward and non-dynamic amplify-and-forward networks. We note that the same cooperative gains are here achieved without requiring that the intermediate relays know the channel or that they perform decoding. In [12, Theorem 4.1 and Prop 5.2] we see that approximate universality, in any MISO channel with i.i.d fading coefficients, can be achieved by diagonalizing space-only codes that are approximately universal over the parallel channel, i.e. codes that given a discrete alphabet, satisfy a non-vanishing product distance property in a channel whose coefficient matrix is diagonal. Sporadic instances of such codes were explicitly constructed in [12]. Based on the above sufficiency criteria, we present a new unified construction of D-MG optimal codes over the two-product channel. Explicit constructions of approximately universal codes over any n × 1 MISO channel with i.i.d. fading coefficients, for all n We consider the construction of the ‘diagonal-restricted perfect code’ Xd , given as: Xd = {diag(x) = diag(f · G), ∀f ∈ An }.
(9)
Choosing the information alphabet A to be discrete and choosing G to be the lattice generator matrix of the perfect code, provides the code with a non-vanishing productdistance (see [2]), and hence makes the code approximately universal over the channel of interest.This, together with |A| = SNRr , satisfy the related conditions in [12, Theorem 4.1]. We observe that this code achieves all the desired properties but it does not allow for Ai = In for any i and as a result (Remark 1), the corresponding entire-network D-MG performance is dentire (r) ≥ (n − 1)(1 − 2r). The task is to find a code which satisfies the same conditions on the transmission rate and the minimum determinant of the difference of codematrices and which will allow for A1 = In . This is found in the newly constructed class of integral-restricted perfect codes, given as: P k Xir = {X = n−1 k=0 fk Γ , ∀fk ∈ A}
where Γ from (10) provides the linear-dispersion matrices Ai = Γn−i , i = 1, · · · , n, whose integral nature guarantees the determinant-rate bound. This construction is closely related to the diagonal-restricted codes but exhibits better overall performance as shown in the simulations depicted in Figure 1. Furthermore, this encoding scheme has the 0
0
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−1
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−1
PCWE
PCWE
10
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10
2bpcu ir−perfect 2bpcu diagonal 4bpcu ir−perfect 4bpcu diagonal 8bpcu ir−perfect 8bpcu diagonal
−3
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50
2bpcu ir−perfect 2bpcu diagonal 4bpcu ir−perfect 4bpcu diagonal 8bpcu ir−perfect 8bpcu diagonal 0
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Figure 1: 2 × 2 IR-perfect vs diagonal-perfect in 2 × 1 Rayleigh fading (left) and twoproduct (right) channels advantage of allowing for no constellation expansion, since due to the special nature of Γ [2], and after a small modification, the transmitted signals belong in the powernormalized QAM-HEX. In addition, the sparse and integer nature of the Ai ’s, provides for fast encoding at the intermediate relays. Specifically, the scheme only requires one multiplication with a small Gaussian integer, per channel use. This compares with n high-resolution multiplications and n − 1 high-resolution additions per channel use for the case of random-coding. Another advantage is that the codes manage to translate the existence of only one receive antenna at the destination, into a reduction of the sphere-decoding complexity at the final destination receiver, from O(n2 ) to O(n). Finally the encoding allows for the ability to maintain optimal rate, ease of construction and minimum delay, all for any n, trades that become important due to the potentially high number of relays. Figure 2 shows the simulation comparison between the proposed scheme and a good alternative, the Alamouti code. The superiority of the IR-perfect code, is revealed for the relay network scenario, and is mainly due to the lack of the complex-conjucacy operation. The dominance of the n × n IR-perfect code is expected to grow as n increases due to 0
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2bpcu Alam 2bpcu ir−perfect 4bpcu Alam 4bpcu ir−perfect
P1.5 4
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rate = 2bpcu, ir−perfect rate = 2bpcu, Alam rate = 4bpcu, ir−perfect rate = 4bpcu, Alam 10
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60
Figure 2: Alamouti vs. 2 × 2 IR-perfect in 2 × 1 two-product channel(left) and in n = 2 relay network (right) the drop in the rate of the orthogonal design.
The same dominance of the IR-perfect code holds over the full perfect code, which achieves overall network performance of dentire (r) = n(1 − (n + 1)r), due to the fact that the first stage is bound to have duration of n2 time slots. For high values of n, orthogonal designs are expected to carry on the average m ≈ 12 symbols per channel use and be of dimension n×n2 . Analysis similar as that presented in the proof of Lemma 1, gives that the D-MG performance in the second stage is given by deq (r) = n(1 − mr ) ≈ n(1 − 2r). The first stage duration needs to be of length 2n2 m ≈ n2 since conjugation cannot be described as a matrix transformation, resulting in a D-MG performance over the entire relay network being dentire (r) = deq ( 2r ) ≈ deq (4r) = n(1−4r). m
3.1
Diversity increasing cooperation for base-station communication: Reducing the half-duplex effect
A practical relay-network scenario given in [7], talks of several relays with one transmit/receive antenna, cooperating in their task to communicate with a single base-station. It is logical to assume that the centrality of such a base-station will allow it the ability to utilize multiple receive antennas. This can correspond to a wireless telephony setup where each mobile user utilizes the surrounding users to increase the reliability of the transmission to the base-station. For the case where a base-station centered, l.p relay network, utilizes an n × n space-time code whose D-MG performance over the equivalent (second-stage) channel is deq (r), it is easy to see that if we let the code map on the average mn information elements and if the base station utilizes m receive antennas, m < n, then the half-duplex D-MG performance of the entire network is upper bounded m r), rentire, max ≤ m+1 implying a potential overall network outage as dentire (r) ≤ deq ( m+1 m m capacity of up to Centire,max = ˙ m+1 log2 (SNR). As a result, a relay network with intermediate relays with only one antenna, can potentially reduce the effect of the half-duplex.The above bound hints towards utilizing n × n codes that map mn information elements and maintain sufficiently good eigenvalue bounds for increasing spectral efficiency. For this we turn again to the general family of CDA/perfect codes © ¢¢ªthe ‘m-layered ¡ and¡ consider Pm−1 j n × n perfect code’ Xm , given by: Xm = X = j=0 Γ diag f j · G , which maps the mn information symbols from {f 0 , f 1 , · · · , f m−1 }. One could then accept an increase in decoding complexity and equipment, both only at the base station, for the benefit of having the intermediate relays transmit reliably, with power approaching the square root of the previously required power (m = 1).
4
Optimal construction for the dynamic receive-andforward scheme
In [8], a cooperation scheme has been proposed for an arbitrary number of relays, with the special property that the original transmitter transmits continuously, during every time slot. In this proposed scheme, during every other time slot (even time index), the intermediate relays take turns in relaying the signal transmitted during the previous time slot by the transmitter. For the case of a single intermediate relay, the equations that describe the scheme are y1 = g1 x1 + v1 , y2 = g1 x2 + h2 [g2 x1 + w1 ] + v2 y3 = g1 x3 + v3 , y4 = g1 x4 + h2 [g2 x3 + w3 ] + v4 . In the above equations, g1 , g2 , h2 respectively describe the fading from source-to-destination, source-to-relay and relay-to-destination. Variables wi , vi respectively describe the receiver
noise at the intermediate relay and final destination at time i. Finally, xi , yi respectively describe the source transmitted signal and the destination received signal, at time i. As a result, Theorem 6 Letting x1 , x2 , x3 , x4 come from a column-by-column vectorization of a 2 × 2 D-MG optimal CDA space-time code, will allow for the above dynamic relaying scheme to achieve the optimal D-MG performance, as described in [8], of d(r) = (1 − r) + (1 − 2r)+ . Proof: We representation Y = HX + W of ¸the above scheme i.e. ¸ use · the equivalent ¸· ¸ · · y1 y3 g1 0 x1 x3 v1 v3 = + and modify to reflect h2 g2 g1 x2 x4 h2 w1 + v2 h2 w3 + v4 y2 y4 ½· ¸ ¾ · ¸ £ ∗ ∗ ∗ ¤ v1 1 0 ∗ v1 h2 w1 + v2 noise whitening by Σ = E = to the h2 w1 + v2 0 |h2 |2 + 1 equivalent whitened model · ¸ · ¸· ¸ · ¸ y y g 0 x x v v 1 3 1 1 3 1 3 Σ−1 = Σ−1 + Σ−1 . y2 y4 h2 g2 g1 x2 x4 h w + v2 h2 w3 + v4 · ¸ · ¸ · 2 1 ¸ ¸ · g1 0 v1 v3 u1 u3 g1 0 −1 −1 For Σ := , Σ = g1 h2 g2 h2 w1 + v2 h2 w3 + v4 u2 u4 h2 g2 g1 2 2 1+|h | 1+|h 2 2| ¸ · ¸ · ¸ · ¸· ¸ · z1 z3 y1 y3 g1 0 x1 x3 u1 u3 we also let := Σ−1 = + z2 z4 y2 y4 Ah g g A x x u2 u4 · 2 2 1 ¸ 2 4 g1 0 and proceed to find the where A = 1+|h1 2 |2 . We now define Heff = Ah2 g2 g1 A outage probability by calculating · ¸ 1 + ρ|g1 |2 ρg1 (Ah2 g2 )∗ † det(I + ρHeff Heff ) = det ρ(Ag2 h2 )g1∗ 1 + ρ{|Ah2 g2 |2 + |g1 A|2 } = 1 + ρ|g1 |2 + A2 {ρ|g2 h2 |2 + ρ|g1 |2 + ρ2 |g1 |4 }. This gives that Pout (r) = P r {log [1 + ρ|g1 |2 + A2 {ρ|g2 h2 |2 + ρ|g1 |2 + ρ2 |g1 |4 }] < 2r log(ρ)} 2 2 2 and R then for G1 = |g1 | , G2 = |g2 | H2 = |h2 | , the probability of outage is given by exp(−[G1 + G2 + H2 ])dG1 dG2 dH2 where {G1 ,G2 ,H2 }∈O © ª {G1 , G2 , H2 } ∈ O ⇒ log{1 + ρG1 + A2 ρG2 H2 + ρG1 + ρ2 G21 } ≤ 2r log(ρ) i.e., {G1 , G2 , H2 } ∈ O ⇒ 1 + ρG1 + A2 {ρG2 H2 + ρG1 + ρ2 G21 } ≤ ρ2r . At this point, we ˙ ρ0 ([8]), we can approximate outage as {G1 , G2 , H2 } ∈ O ⇒ can see that since G1 , G2 , H2 ≤ 1 + ρG1 + ρG2 H2 + ρG1 + ρ2 G21 ≤ ρ2r and can thus write Z Pout = exp(−[G1 + G2 + H2 ])dG1 dG2 dH2 {G1 ,G2 ,H2 }∈O
where {G1 , G2 , H2 } ∈ O ⇒ log{1 + ρG1 + ρG2 H2 + ρG1 + ρ2 G21 } ≤ 2r log(ρ). From this, it follows that the effective (whitened) channel Heff has the same outage as does the original channel H, and this outage curve is met by the use of the approximately universal CDA code. In a similar fashion, we present in the following theorem the D-MG optimal solution for the dynamic AAF scheme for any number n of relays. Theorem 7 We consider the scheme where the source transmits continuously and where each intermediate relay Ri , i = 1, · · · , n − 1 transmits at time t = (1 + i) + (n − 1)2 k, k = 0, 1, · · · from a set x1 , x2 , · · · coming from a column-by-column vectorization of a (n−1)2 × (n − 1)2 D-MG optimal CDA space-time code. This n − 1 intermediate relay dynamic AAF scheme achieves the optimal D-MG performance, as described in [8], of d(r) = (1 − r) + (n − 1)(1 − 2r)+ .
5
Conclusion
We considered an efficient variant of the wireless relay-network model proposed in [1], where the intermediate relays perform space-time code related averaging on the received signal without knowledge of the fading channel and without time consuming decoding. Our contribution begun with the surprising result that the optimal D-MG tradeoff of the entire relay network is as though the intermediate relays had knowledge of the channel and used it to decode the signals from the original source and then again used it to re-encode and transmit. Furthermore, improving upon the existing random-coding methodology, we have explicitly constructed coding schemes, based on suitable variants of perfect space-time codes [3, 2], satisfying approximate universality, information losslessness and decoding ease, which along with other positive trades, make the proposed coding schemes ideal for encoding over such relay networks. This claim was substantiated by simulations. It is worth noting that the usually high number of relays in a network and the accompanying requirement of having unitary linear-dispersion matrices, provided for a meaningful raison d’etre for high-dimensional perfect codes, both for reasons of encoding as well as for outage analysis of unknown high-dimensional channels. Furthermore, a base-station version of the network was presented together with an explicit coding technique that can potentially allow for the asymptotic elimination of the half-duplex rate-reduction effect. Finally we presented a scheme based on vectorized perfect codes and proceeded to show that it achieves the dynamic amplify-and-forward D-MG bound, as described in [8], for any number of relays.
6
Appendix: Distributed perfect space-time codes
As previously shown, D-MG optimal codes for the two-product channel can be found in the general category of cyclic division algebra (CDA) space-time codes. The extra requirement that the linear dispersion matrices be unitary, directly points towards the CDA information lossless sub-category of perfect-codes, first introduced in [3] for n = 2, 3, 4, 6 and recently generalized in [2] for all nt , nr and all T ≥ nt . As is shown in more detail in [2], for G corresponding to an orthogonal signallinglattice generator matrix and for Γ to the unitary basis of the division algebra over this lattice, it is the case that the CDA code-matrix form is µ ¶ ¡ ¢ Pn−1 j P X = j=0 Γ diag f j · G = nu=1 f Au with f j = [fj,0 fj,1 · · · fj,n−1 ] being QAM n-tuples. For the useful case of the integralrestriction perfect code, it is the case that the linear dispersion matrices
Au = Γn−u u = 1, 2, · · · , n,
0 0 ··· 1 0 ··· Γ = 0 1 ···
.. .
γ 0 0
(10)
The perfect code requirement that the lattice generator matrix G and the power sharing matrices Γj , j = 0, 1, · · · , n − 1 in (10), be unitary matrices guarantees that A†u Au = I, essential for the code’s information losslessness and for the spatial and temporal whiteness of the effective noise at the final receiver in the network. The same readily holds also for the case of full-perfect distributed codes.
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