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1 Cooperative Diversity without Temporal Diversity

1.1 System Model

Figure 1.1 Ilustration of radio signal transmit paths

1.1.1 Equivalent Channel Models

For direct transmission, our baseline for comparison, we model the channel as:

(1.1) y_d[n] = a_s, d(s)x₂[n] + z₃[n]

for say, n = 1, 2, ...N ⁄ 2, where x_s[n] is the source transmitted signal and y_d[n] is the destination received signal.

Figure 1.2 Time division channel allocations

the other terminal transmits for n = N ⁄ 2 + 1, ..., N as 1.2↑ depicts.

For cooperative diversity transmission, we model the channel during the first half of the blocks as

(1.2) y_r[n] = a_s, rx_s[n] + z_r[n]

(1.3) y_d[n] = a_s, dx_s + z_d[n]

for say, \mathchoicen = 1, ..., N ⁄ 4n = 1, ..., N ⁄ 4n = 1, ..., N ⁄ 4n = 1, ..., N ⁄ 4 where x_s[n] is the source transmitted signal and y_r[n] and y_d[n] are the relay and destination received signals, respectively. For the second half of the block, we model the received signal as:

(1.4) y_d[n] = a_r, d(s)x_r[n] + z_d[n]

for \mathchoicen = N ⁄ 4 + 1, .., N ⁄ 2n = N ⁄ 4 + 1, .., N ⁄ 2n = N ⁄ 4 + 1, .., N ⁄ 2n = N ⁄ 4 + 1, .., N ⁄ 2, where x_r[n] is the relay transmitted signal and y_d[n] is the destination received signal.

A similar setup is employed in the second half of the block, with the roles of the source and relay reversed, as 1.2↑ depicts. While again half the degrees of freedom are allocated to each source terminal for transmission to its destination, only a quarter of the degrees of freedom are available for communication to its relay.

In 1.1↑-1.4↑ a_i, j captures the effects of path-loss, shadowing, and frequency non-selective fading, and z_j[n] captures the effects of receiver noise and other forms of interference in the system, where i ∈ {s, r} and j ∈ {r, d}. Statistically, we model a_i, j as zero-mean, independent, circularly-symmetric complex gaussian random variables with variances σ²_i, j so that the magnitudes |a_i, j| are Rayleigh distributed (|a_ij|² are exponentially distributed wiht mean σ²_i, j) and phases ∠a_i, j are uniformly distributed on [0, 2π]. Furthermorre, we model z_j[n] as zeropmean mutually independent, circularly-symmetric, complex Gaussian random sequences with variance N₀

1.1.2 Parametrization

SNR≜(2P_c)/(No⋅W) = (P)/(N₀) is SNR without fading.

Spectral efficiency

R≜2r ⁄ W b ⁄ s ⁄ Hz

SNR_norm≜(SNR)/(2^R − 1) R_norm≜(R)/(log(1 + SNR_{σ²_s, d(s)}))

1.2 Cooperative Transmission Protocols

1.2.1 Fixed Protocols

1.2.1.1 Amplify-and Forward Transmission

For amplify-and-forward transmission, the appropriate channel model is 1.2↑-1.4↑. The source terminal transmits its information as x_s[n] , say, for n = 1, ..., N ⁄ 4. During this interval, the relay processes y_r[n], and relays the information by transmitting

(1.5) x_r[n] = β⋅y_r[n − N ⁄ 4]

for n = N ⁄ 4 + 1, ..., N ⁄ 2. To remain within its power constraint (with high probability), an amplify relay must use gain:

β ≤ √((P)/(|a_s, r|²P + N₀))

where we allow the amplifier gain to depend upon the fading coefficient a_s, r between the source and relay, which the relay estimates to high accuracy. This transmission scheme can be viewed as repetition coding from two separate transmitters, except that the relay transmitter amplifies its own receiver noise. The destination can decode its received signal y_d[n] for n = 1, ..., N ⁄ 2 by first appropriately combining the signals form the two subblocks using a suitably designed matched-fiter (maximum-ratio combiner).

1.2.1.2 Decode and Forward Transmission

For decode-and forward transmission, the appropriate channel model is again 1.2↑-1.4↑. the source terminal transmits its information as x_s[n] , say , for n = 0, ..., N ⁄ 4. During this interva, the relay processes y_r[n] by decoding an estimate x̂_s[n] of the source transmitted signal.

Under a repetition coded scheme, the relay transmits the signal

x_r[n] = x̂_s[n − N ⁄ 4]

for n = N ⁄ 4 + 1, ...N ⁄ 2 .

Decoding at the relay can take on variety of forms. For example, the relay might fully decode the source message by estimating the source code-word, or it might employ symbol-by-symbol decoding and allow the destination to perform full decoding. These options allow for trading off performance and complexity at the relay terminal. Beacuse the performance of symbol-bysymbol decoding varies with the choice of coding and modulatio, we focus on full decoding in the sewuel; symbol-by-symbol decoding of binary transmisssions has been treated from uncodded perspecitve in [1].

2 Appendix C: Mutual Information Calculations

2.1 Amplify-and-forward Mutual Information

In this section, we compute the maximum average mutual information for amplify and forward transmission. We write the equvalent channel 1.2↑-1.4↑, with relay processing 1.5↑, in vector form as:

y_d[n] = a_s, dx_s + z_d[n]; y_r[n] = a_s, rx_s[n] + z_r[n]; x_r[n] = β⋅y_r[n − N ⁄ 4] = β⋅a_s, rx_s[n − N ⁄ 4] + β⋅z_r[n − N ⁄ 4];

y_d[n] = a_r, d(s)x_r[n] + z_d[n]; y_d[n] = a_r, d(s)β⋅a_s, rx_s[n − N ⁄ 4] + a_r, d(s)⋅β⋅z_r[n − N ⁄ 4] + z_d[n] y_d[n + N ⁄ 4] = a_r, d(s)β⋅a_s, rx_s[n] + a_r, d(s)⋅β⋅z_r[n] + z_d[n + N ⁄ 4]

⎡⎢⎣ y_d[n] y_d[n + N ⁄ 4] ⎤⎥⎦ = ⎡⎢⎣ a_s, d(s) a_r, d(s)β⋅a_s, r ⎤⎥⎦x_s[n] + ⎡⎢⎣ 0 1 0 a_r, d(s)β 0 1 ⎤⎥⎦⋅⎡⎢⎢⎢⎣ z_r[n] z_d[n] z_d[n + N ⁄ 4] ⎤⎥⎥⎥⎦

where the source signal has power constraint E[x_s] ≤ P_s, and relay amplifier has constraint:

β ≤ √((P_r)/(|a_s, r|²P_s + N_r))

and the noise has covariance E[zz^T] = diag(N_r, N_d, N_d)

E⎡⎢⎢⎢⎣⎡⎢⎢⎢⎣ z₁ z₂ z₃ ⎤⎥⎥⎥⎦⋅[ z₁ z₂ z₃ ]⎤⎥⎥⎥⎦ = E⎡⎢⎢⎢⎣⎡⎢⎢⎢⎣ z²₁ z₁z₂ z₁z₃ z₂z₁ z²₂ z₂z₃ z₁z₃ z₂z₃ z²₃ ⎤⎥⎥⎥⎦⎤⎥⎥⎥⎦ = ⎡⎢⎢⎢⎣ Ez²₁ 0 0 0 Ez²₂ 0 0 0 Ez²₃ ⎤⎥⎥⎥⎦ = ⎡⎢⎢⎢⎣ N_r 0 0 0 N_d 0 0 0 N_d ⎤⎥⎥⎥⎦

Note that we determine the mutual information for arbitrary transmit powers and noise levels, even though we utilize the result only for the symmetric case. Since the channel is memoryless, the average mutual information satisfies

I_AF ≤ I(x_s;y_d) ≤ logdet(I + (P_sAA^T)⋅(B⋅E[z⋅z^T]⋅B^T)^− 1)

A⋅A^T = ⎡⎢⎣ a_s, d(s) a_r, d(s)β⋅a_s, r ⎤⎥⎦⋅⎡⎢⎣ a_s, d(s) a_r, d(s)β⋅a_s, r ⎤⎥⎦^† = ⎡⎢⎣ a_s, d(s)a_s, d(s) a_s, d(s)a_r, d(s)β a_s, r a_r, d(s)β a_s, ra_s, d(s) a_r, d(s)β a_s, ra_r, d(s)β a_s, r ⎤⎥⎦ = ⎡⎢⎣ |a_s, d(s)|² a_s, d(s)a_r, d(s)β a_s, r a_r, d(s)β a_s, ra_s, d(s) |a_r, d(s)β a_s, r|² ⎤⎥⎦

†-conjugate transpose

B⋅E[z⋅z^†]⋅B^† = ⎡⎢⎣ 0 1 0 a_r, d(s)β 0 1 ⎤⎥⎦⋅⎡⎢⎢⎢⎣ N_r 0 0 0 N_d 0 0 0 N_d ⎤⎥⎥⎥⎦⋅⎡⎢⎣ 0 1 0 a_r, d(s)β 0 1 ⎤⎥⎦^† = ⎡⎢⎣ N_d 0 0 a_r, d(s)β N_ra_r, d(s)β + N_d ⎤⎥⎦

det(I₂ + (P_sAA^T)⋅(B⋅E[z⋅z^T]⋅B^T)^− 1) = det⎛⎜⎝⎡⎢⎣ 1 0 0 1 ⎤⎥⎦ + P_s⎡⎢⎣ |a_s, d(s)|² a_s, d(s)a_r, d(s)β a_s, r a_r, d(s)β a_s, ra_s, d(s) |a_r, d(s)β a_s, r|² ⎤⎥⎦⋅⎡⎢⎣ N_d 0 0 a_r, d(s)β N_ra_r, d(s)β + N_r ⎤⎥⎦⎞⎟⎠

β ≤ √((P_r)/(|a_s, r|²P_s + N_r)) β² ≤ (P_r)/(|a_s, r|²P_s + N_r)

1 + (P_s⋅|a_s, d(s)|²)/(N_d) + (P_s⋅|a_r, d(s)β a_s, r|²)/(N_r|a_r, d(s)|²⋅(P_r)/(|a_s, r|²P_s + N_r) + N_d) = 1 + SNR_s, d(s)⋅|a_s, d(s)|² + (P_s⋅|a_r, d(s)β a_s, r|²)/(|a_r, d(s)|²⋅(P_r)/(|a_s, r|²P_s ⁄ N_r + 1) + N_d) = 1 + SNR_s, d(s)⋅|a_s, d(s)|² + (P_s⋅|a_r, d(s)β a_s, r|²)/(|a_r, d(s)|²⋅(P_r)/(|a_s, r|²SNR_s, r + 1) + N_d) =

f(x, y)≜(x⋅y)/(x + y)

y_d[n] = a_s, dx_s + z_d[n]; y_r[n] = a_s, rx_s[n] + z_r[n]; x_r[n] = β⋅y_r[n − N ⁄ 4] = β⋅a_s, rx_s[n − N ⁄ 4] + β⋅z_r[n − N ⁄ 4];

I(x_s;y_d) = I(X_s;Y_d’, Y_d’’) Y_d’ = a_s, dX_s + Z_d \mathchoiceY_d’’Y_d’’Y_d’’Y_d’’ = a_rdX_r + Z_d = a_rd⋅(β⋅a_srX_s + β⋅Z_r) + Z_d = \mathchoicea_rd⋅β⋅a_srX_s + a_rd⋅β⋅Z_r + Z_da_rd⋅β⋅a_srX_s + a_rd⋅β⋅Z_r + Z_da_rd⋅β⋅a_srX_s + a_rd⋅β⋅Z_r + Z_da_rd⋅β⋅a_srX_s + a_rd⋅β⋅Z_r + Z_d

Y₃’ = a₃₁X₁ + Z₃ \mathchoiceY₃’’Y₃’’Y₃’’Y₃’’ = a₃₂X₂ + Z₃ = \mathchoicea₃₂⋅β⋅a₃₁⋅X₁ + a₃₂⋅β⋅Z₂ + Z₃a₃₂⋅β⋅a₃₁⋅X₁ + a₃₂⋅β⋅Z₂ + Z₃a₃₂⋅β⋅a₃₁⋅X₁ + a₃₂⋅β⋅Z₂ + Z₃a₃₂⋅β⋅a₃₁⋅X₁ + a₃₂⋅β⋅Z₂ + Z₃ Y₂ = a₂₁X₁ + Z₂
β = √((P)/(a²₂₁P + 1)) = √((P)/(S₂₁ + 1)) =
C ≥ max_{p(x₁)p(ŷ₂|y₂)}min{I(X₁;Y₃’) − I(Y₂;Ŷ₂|X₁Y₃’) + C₀, I(X₁;Ŷ₂, Y₃’)}
C ≥ max_{p(x₁)p(ŷ₂|y₂)}min{I(X₁;Y₃’) − I(Y₂;Ŷ₂|X₁Y₃’) + C₀, I(X₁;Ŷ₂, Y₃’)}
max_{p(x₁)p(x₂)}min{I(X₁;Y₃’) + I(X₂;Y₃’’), I(X₁;Y₂, Y₃’)}
I(X₁;Y₃’) = H(Y₃’) − H(Y₃’|X₁) = (1)/(2)log(S₃₁ + 1)
I(X₂;Y₃’’) = H(Y₃’’) − H(Y₃’’|X₂) = (1)/(2)log(S₃₂ + 1)
I(X₂;Y₃’’) = H(Y₃’’) − H(Y₃’’|X₂) = (1)/(2)log(2πe)⋅(a²₃₂β²a²₃₁P + a²₃₂β² + 1) − (1)/(2)log(2πe) = (1)/(2)log(a²₃₂β²a²₃₁P + a²₃₂β² + 1)
I(X₁;Y₂, Y₃’) = H(Y₂Y₃’) − H(Y₂Y₃’|X₁) = H(Y₂) + H(Y₃’|Y₂) − H(Y₂|X₁) − H(Y₃’|Y₂X₁) =
≤ (1)/(2)log(2πe)(S₂₁ + 1) + (1)/(2)log(2πe)⎛⎝E[Y^’2₃] − (E²[Y^’₃⋅Y₂])/(E[Y²₂])⎞⎠ − (1)/(2)⋅log(2πe) − (1)/(2)⋅log(2πe) =
= (1)/(2)log(S₂₁ + 1) + (1)/(2)log⎛⎝S₃₁ + 1 − (E²[(a₃₁X₁ + Z₃)⋅(a₂₁X₁ + Z₃)])/(E[Y²₂])⎞⎠ = (1)/(2)log(S₂₁ + 1) + (1)/(2)log⎛⎝S₃₁ + 1 − (a²₃₁a²₂₁⋅E²[X²₁])/(S₂₁ + 1)⎞⎠ =
= (1)/(2)log(S₂₁ + 1) + (1)/(2)log⎛⎝S₃₁ + 1 − (S₂₁S₃₁)/(S₂₁ + 1)⎞⎠ =
max_{p(x₁)p(x₂)}min⎧⎩(1)/(2)log(S₃₁ + 1) + (1)/(2)log(a²₃₂β²a²₃₁P + a²₃₂β² + 1), (1)/(2)log(S₂₁ + 1) + (1)/(2)log⎛⎝S₃₁ + 1 − (S₂₁S₃₁)/(S₂₁ + 1)⎞⎠⎫⎭
(1)/(2)log(S₃₁ + 1)⋅(a²₃₂β²a²₃₁P + a²₃₂β² + 1) = (1)/(2)log(S₂₁ + 1)⋅⎛⎝S₃₁ + 1 − (S₂₁S₃₁)/(S₂₁ + 1)⎞⎠ (1)/(2)log(S₃₁ + 1)⋅⎛⎝a²₃₂⋅(P)/(S₂₁ + 1)⋅a²₃₁P + a²₃₂⋅(P)/(S₂₁ + 1) + 1⎞⎠ = (1)/(2)log(S₂₁ + 1)⋅⎛⎝S₃₁ + 1 − (S₂₁S₃₁)/(S₂₁ + 1)⎞⎠
min⎧⎩(1)/(2)log(S₃₁ + 1)⋅⎛⎝(S₃₂⋅S₃₁)/(S₂₁ + 1) + (S₃₂)/(S₂₁ + 1) + 1⎞⎠, (1)/(2)log(S₂₁ + 1)⋅⎛⎝S₃₁ + 1 − (S₂₁S₃₁)/(S₂₁ + 1)⎞⎠⎫⎭
(S₂₁ + 1)⋅⎛⎝S₃₁ + 1 − (S₂₁S₃₁)/(S₂₁ + 1)⎞⎠ = ((S₃₁ + 1)(S₂₁ + 1) − S₂₁S₃₁) = S₃₁S₂₁ + S₂₁ + S₃₁ + 1 − S₂₁S₃₁ = 1 + S₃₁ + S₂₁
Expand[(S₃₁ + 1)(S₂₁ + 1)] = S₃₁S₂₁ + S₂₁ + S₃₁ + 1
(S₃₁ + 1)⋅⎛⎝(S₃₂⋅S₃₁)/(S₂₁ + 1) + (S₃₂)/(S₂₁ + 1) + 1⎞⎠ = (S₃₁ + 1)⋅⎛⎝(S₃₂⋅S₃₁ + S₃₂ + S₂₁ + 1)/(S₂₁ + 1)⎞⎠ = 1 + S₃₁ + (S₃₂(S₃₁ + 1)²)/(S₂₁ + 1)

C ≤ max_{p(x₁)p(x₂)}min{I(X₁;Y₃’) + I(X₂;Y₃’’), I(X₁;Y₂, Y₃’)}

y₃ = g₃₁x₁ + g₃₂x₂ + z₃ y₂ = g₂₁x₁ + z₂ z₃ ~ N(0, 1) z₂ ~ N(0, 1)

x₂ = √((P)/(S₂₁ + 1))y₂ = g₂₁√((P₂)/(S₂₁ + 1))x₁ + √((P₂)/(g²₂₁P₁ + 1))z₂ y₃ = g₃₁x₁ + g₃₂x₂ + z₃ = g₃₁x₁ + g₃₂⋅g₂₁⋅√((P₂)/(g²₂₁P₁ + 1))x₁ + g₃₂⋅√((P₂)/(g²₂₁P₁ + 1))z₂ + z₃ =

I(X₁;Y₃) = h(Y₃) − h(Y₃|X₁) = h(Y) − h(g₃₁x₁ + g₃₂⋅g₂₁⋅√((P₂)/(g²₂₁P₁ + 1))x₁ + g₃₂⋅√((P₂)/(g²₂₁P₁ + 1))z₂ + z₃|X₁) =

= h(g₃₁x₁ + g₃₂⋅g₂₁⋅√((P₂)/(g²₂₁P₁ + 1))x₁ + g₃₂⋅√((P₂)/(g²₂₁P₁ + 1))z₂ + z₃) − h(g₃₂√((P₂)/(g²₂₁P₁ + 1))z₂ + z₃) =

= (1)/(2)log2πe⎛⎝g²₃₁P + g²₃₂⋅g²₂₁⋅(P₂)/(g²₂₁P₁ + 1)⋅P₁ + g²₃₂⋅(P₂)/(g²₂₁P₁ + 1) + 1⎞⎠ − (1)/(2)log2πe⋅⎛⎝g²₃₂(P₂)/(g²₂₁P₁ + 1) + 1⎞⎠ =

= (1)/(2)log(⎛⎝g²₃₁P + g²₃₂⋅g²₂₁⋅(P₂)/(g²₂₁P₁ + 1)⋅P₁ + \mathchoiceg²₃₂⋅(P₂)/(g²₂₁P₁ + 1) + 1g²₃₂⋅(P₂)/(g²₂₁P₁ + 1) + 1g²₃₂⋅(P₂)/(g²₂₁P₁ + 1) + 1g²₃₂⋅(P₂)/(g²₂₁P₁ + 1) + 1⎞⎠)/(⎛⎝g²₃₂(P₂)/(g²₂₁P₁ + 1) + 1⎞⎠) = (1)/(2)log⎛⎜⎜⎝1 + (g²₃₁P + g²₃₂⋅g²₂₁⋅(P₂)/(g²₂₁P₁ + 1)⋅P₁)/(⎛⎝g²₃₂(P₂)/(g²₂₁P₁ + 1) + 1⎞⎠)⎞⎟⎟⎠

(1)/(2)log⎛⎜⎜⎝1 + (S₃₁ + (S₃₂⋅S₂₁)/(S₂₁ + 1))/(⎛⎝(S₃₂)/(S₂₁ + 1) + 1⎞⎠)⎞⎟⎟⎠ = \mathchoice(1)/(2)log⎛⎝1 + (S₃₁(S₂₁ + 1) + S₃₂⋅S₂₁)/(S₂₁ + S₃₁ + 1)⎞⎠(1)/(2)log⎛⎝1 + (S₃₁(S₂₁ + 1) + S₃₂⋅S₂₁)/(S₂₁ + S₃₁ + 1)⎞⎠(1)/(2)log⎛⎝1 + (S₃₁(S₂₁ + 1) + S₃₂⋅S₂₁)/(S₂₁ + S₃₁ + 1)⎞⎠(1)/(2)log⎛⎝1 + (S₃₁(S₂₁ + 1) + S₃₂⋅S₂₁)/(S₂₁ + S₃₁ + 1)⎞⎠ = \mathchoice(1)/(2)log⎛⎝1 + ((S₃₁ + S₃₂)S₂₁ + S₃₁)/(S₂₁ + S₃₁ + 1)⎞⎠(1)/(2)log⎛⎝1 + ((S₃₁ + S₃₂)S₂₁ + S₃₁)/(S₂₁ + S₃₁ + 1)⎞⎠(1)/(2)log⎛⎝1 + ((S₃₁ + S₃₂)S₂₁ + S₃₁)/(S₂₁ + S₃₁ + 1)⎞⎠(1)/(2)log⎛⎝1 + ((S₃₁ + S₃₂)S₂₁ + S₃₁)/(S₂₁ + S₃₁ + 1)⎞⎠

(S₃₁S₂₁ + S₃₂ + S₃₂S₂₁)/(S₂₁ + S₃₁ + 1) = (S₃₁S₂₁ + ⎛⎝\cancelto0(S₃₂)/(S₂₁) + S₃₂⎞⎠⋅S₂₁)/(S₂₁ + S₃₁ + 1) = (S₃₁S₂₁ + S₃₂⋅S₂₁)/(S₂₁ + S₃₁ + 1) = ((S₃₁ + S₃₂)⋅S₂₁)/(S₂₁ + S₃₁ + 1)

1. Прочитај го чланакот [1] имаат прикажано како се прави maximum-ratio combiner.

2. Помини го Appendix A од докторската на Laneman. Површно го поминав. Истите концепти се од EIT и NIT но може Laneman додал нешто интересно. Обично има многу добри објаснувања.

Bibliography

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[2] Thomas M. Cover and Cyril S.K. Leung. An achievable rate region for the multiple- access channel with feedback. IEEE Trans. Inform. Theory, 27(3):292–298, May 1981.

[3] Frans M.J. Willems and Edward C. van der Meulen. The discrete memoryless multiple- access channel with cribbing encoders. IEEE Trans. Inform. Theory, 31(3):313–327, May 1985.