Duality Between the Improved Rate Regions of Gaussian Multiple Access and Broadcast Channel for Discrete Constellations

Abstract

In this paper, we first look at the achievable rate regions of the Gaussian Multiple Access Channel (G-MAC) and Gaussian Broadcast Channel (G-BC) when discrete constellations are transmitted. Focusing on the two-user case and assuming uncoded Pulse Amplitude Modulation (PAM), we show that the rate regions of G-MAC and G-BC can be improved by power conservation at the transmitter, a factor ignored in the earlier derived rate regions. We then further investigate duality between the G-MAC and the G-BC and show that a rate pair achieved in the G-MAC can be translated to a rate pair in the dual G-BC, such that the equal sum power constraint be satisfied. Due to the similarity of the rate expressions to Shannon’s capacity formula, for an appropriate choice of signal to noise ratio (SNR) gap, we show that when finite constellations are used for transmission, rate regions of these two channels also have a dual relationship (the known for Gaussian alphabets). The rate region of the G-BC can therefore be characterized from the rate region of the dual G-MAC and vice versa.

Appendix

We can rewrite the rate equations in G-MAC and G-BC as

$$ {R_i}^{mac} = \frac{1}{2}log_2 \left (1+ \frac{h_i{P_i}^{mac}}{A_i} \right), {R_i}^{bc} = \frac{1}{2}log_2 \left(1+ \frac{h_i{P_i}^{bc}}{B_i}\right)$$

(38)

We show that if

$$ {{P_i}^{mac}}{B_i}={{P_i}^{bc}}{A_i}, i=1, ..., K $$

(39)

for all i, where

$$A_i={\tilde{\Gamma }_{i,mac}}\left( {\sum _{j=i+1}^{K}h_j{P_j} ^{mac}}+ N_{0}\right)$$

(40)

and

$$ B_i={\tilde{\Gamma }_{i,bc}}\left( h_i{\sum _{j=1}^{i-1}{P_j}^{bc}}+ N_{0}\right)$$

(41)

then $\sum _{j=1}^{K}{P_j}^{mac}=\sum _{j=1}^{K}{P_j}^{bc}$. We do this by inductively showing that

$$\sum _{j=1}^{i}{P_j}^{bc}= \frac{{{\tilde{\Gamma }_{1,bc}} N_{0}}\sum _{j=1}^{i}{P_j}^{mac}}{A_i}, $$

(42)

For i = 1,

$${{P_1}^{bc}}=\frac{{P_1}^{mac}{B_1}}{A_1}, $$

(43)

From (41)

$$ B_1={\tilde{\Gamma }_{1,bc}} N_{0} $$

(44)

So (43) becomes

$${{P_1}^{bc}}=\frac{{{\tilde{\Gamma }_{1,bc}} N_{0}}{P_1}^{mac}}{A_1}, $$

(45)

We can generalize (45) for i users as shown in (46).

$$ \sum _{j=1}^{i}{P_j}^{bc}= \frac{{{\tilde{\Gamma }_{1,bc}} N_{0}}\sum _{j=1}^{i}{P_j}^{mac}}{A_i}, $$

(46)

Assume that (46) holds for i users so for i + 1 case we have,

$$\begin{aligned} \sum _{j=1}^{i+1}{P_j}^{bc}= \sum _{j=1}^{i}{P_j}^{bc} + \frac{{B_{i+1}}{P_{i+1}}^{mac}}{A_{i+1}} \end{aligned}$$

(47)

From (41), we can get the equation for $B_{i+1}$ as

$$\begin{aligned} B_{i+1}={\tilde{\Gamma }_{i+1,bc}}\left( h_{i+1}{\sum _{j=1}^{i} {P_j}^{bc}}+ N_{0}\right) \end{aligned}$$

(48)

Substituting (48) in (47), we get

$$\begin{aligned} \sum _{j=1}^{i+1}{P_j}^{bc}&= \sum _{j=1}^{i}{P_j}^{bc} + \frac{{{\tilde{\Gamma }_{i+1,bc}}\left( h_{i+1}{\sum _{j=1}^{i} {P_j}^{bc}}+ N_{0}\right) }{P_{i+1}}^{mac}}{A_{i+1}} \end{aligned}$$

(49)

$$\begin{aligned} \sum _{j=1}^{i+1}{P_j}^{bc}&=\frac{{\tilde{\Gamma }_{i+1,bc}}N_{0}{P_{i+1}}^{mac}+ \left( P_{i+1} ^{mac}h_{i+1}\tilde{\Gamma }_{i+1,bc}+A_{i+1}\right) \sum _{j=1}^{i}{P_j}^{bc}}{A_{i+1}} \end{aligned}$$

(50)

For all $i>1$, $\tilde{\Gamma }_{i,bc}$ is equal to one in high SNR regime. Thus $\tilde{\Gamma }_{i+1,bc}$ will also equal to one, for $i\ge 1$.

We can write $A_i=P_{i+1} ^{mac}h_{i+1}\tilde{\Gamma }_{i+1,mac}+A_{i+1}$ from (40). But we know that the additional gap in the G-MAC is equal to one, for all $i<K$. Till when $i<K$, $\tilde{\Gamma }_{i+1,mac}$ will be equal to one. Thus $A_i$ could be written as $A_i=h_{i+1}P_{i+1} ^{mac}+A_{i+1}$ then (50) becomes

$$\begin{aligned} \sum _{j=1}^{i+1}{P_j}^{bc}=\frac{N_{0}{P_{i+1}}^{mac}+A_i \sum _{j=1}^{i}{P_j}^{bc}}{A_{i+1}} \end{aligned}$$

(51)

We can replace ${A_i} \sum _{j=1}^{i}{P_j}^{bc}$ from (42) as

$$\begin{aligned} {A_i} \sum _{j=1}^{i}{P_j}^{bc}= {\tilde{\Gamma }_{1,bc}}N_{0}\sum _{j=1}^{i}{P_j}^{mac}, \end{aligned}$$

(52)

Substituting (52) in (51) we get

$$\begin{aligned} \sum _{j=1}^{i+1}{P_j}^{bc}&=\frac{N_{0}{P_{i+1}}^{mac}+ {{\tilde{\Gamma }_{1,bc}} N_{0}}\sum _{j=1}^{i}{P_j}^{mac}}{A_{i+1}} \end{aligned}$$

(53)

$$\begin{aligned} \sum _{j=1}^{i+1}{P_j}^{bc}&=\frac{\tilde{\Gamma }_{1,bc} N_{0}\sum _{j=1}^{i+1}{P_j}^{mac}}{A_{i+1}} \end{aligned}$$

(54)

$$\begin{aligned} \sum _{j=1}^{i+1}{P_j}^{bc}&=\frac{\tilde{\Gamma }_{1,bc} N_{0}\sum _{j=1}^{i+1}{P_j}^{mac}}{A_{i+1}} \end{aligned}$$

(55)

By using (42) for $i=K$

$$\begin{aligned} \sum _{j=1}^{K}{P_j}^{bc}= \frac{{{\tilde{\Gamma }_{1,bc}} N_{0}}\sum _{j=1}^{K}{P_j}^{mac}}{A_K}, \end{aligned}$$

(56)

from (40)

$$\begin{aligned} A_K=\tilde{\Gamma }_{K,mac}N_{0} \end{aligned}$$

(57)

Substituting (57) into (56)

$$\begin{aligned} \sum _{j=1}^{K}{P_j}^{bc}= \frac{{{\tilde{\Gamma }_{1,bc}} N_{0}}\sum _{j=1}^{K}{P_j}^{mac}}{\tilde{\Gamma }_{K,mac}N_{0}}, \end{aligned}$$

(58)

The single user gap values for a given probability of error vary slightly with modulation scheme [3]. So, $\tilde{\Gamma }_{1,bc}\sim \tilde{\Gamma }_{K,mac}$.

$$\begin{aligned} \sum _{j=1}^{K}{P_j}^{bc}= \sum _{j=1}^{K}{P_j}^{mac} \end{aligned}$$

(59)