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On the achievable rates of a secondary link coexisting with a primary multiple access network
EURASIP Journal on Wireless Communications and Networking volume 2014, Article number: 203 (2014)
Abstract
An achievable rate region for a primary multiple access network coexisting with a secondary link of one transmitter and a corresponding receiver is analyzed. The rate region depicts the sum primary rate versus the secondary rate and is established assuming that the secondary link performs rate splitting. The achievable rate region is the union of two types of rate regions. The first type is a rate region established assuming that the secondary receiver cannot decode any primary signal, whereas the second is established assuming that the secondary receiver can decode the signal of one primary link. The achievable rate region is determined first assuming discrete memoryless channel (DMC), then the results are applied to a Gaussian channel. In the Gaussian channel, the performance of rate splitting is characterized for the two types of rate regions. Moreover, a necessary and sufficient condition to determine which primary signal the secondary receiver can decode without degrading the range of primary achievable sum rates is provided. When this condition is satisfied by a certain primary user, the secondary receiver can decode its signal and achieve larger rates without reducing the sum of the primary achievable rates as compared to the case in which it does not decode any primary signal. It is also shown that the probability of having at least one primary user satisfying this condition grows with the primary signaltonoise ratio.
1 Introduction
A potential benefit of allowing secondary users to share primary bands is the enhancement of the spectrum utilization. As introduced in [1, 2], cognitive radios, or secondary users, are frequencyagile devices that can utilize unused spectrum bands through dynamic spectrum access. In dynamic spectrum access, secondary users should sense the spectrum and identify unused bands or spectrum holes. If a band is sensed and found to be in low use by primary users, i.e., underutilized, a secondary user may opportunistically access this band by adjusting its transmit parameters to fully utilize this band without causing excessive interference on the primary users. However, a secondary user has to leave this band and switch to another if the demand by primary users increases.
The notion of dynamic spectrum access has opened research in different problems regarding the new functionalities that a secondary user should perform, e.g., spectrum sensing, spectrum sharing, spectrum mobility, and spectrum management [2, 3]. Moreover, information theoretic bounds on potential achievable rates by cognitive radio networks are being investigated. In most of those works, cooperation between primary and secondary transmitters is considered. In [4], an achievable rate region of primary versus secondary users’ rates is introduced when a cognitive transmitter has full knowledge of the primary message in a twotransmitter tworeceiver interference channel and the primary user cooperates with the secondary link through rate splitting introduced in [5]. In [6, 7], the notion of conferencing is introduced for the interference channel where the cognitive link is assumed to know part or all of the message of the primary transmitter.
In this paper, we consider a primary multiple access channel (MAC) that consists of two transmitters and a common receiver shared by a secondary link comprising a single transmitter and a corresponding receiver. The secondary transmitter is assumed to employ rate splitting by dividing its signal into two parts: one part is decodable by the secondary receiver and treated as noise by the primary receiver, whereas the other part is decodable at both receivers. Such rate splitting scheme has also been suggested in [8] for a partially connected interference multiple access channel, with all users belonging to the same class of quality of service (QoS). The scheme has been shown to achieve the semideterministic capacity of the addressed setup to within a quantifiable gap. In [9], interference mitigation for a similar setup of interfering MAC has been considered. Authors have shown that signal scale alignment can be achieved through layered lattice codes, which potentially reduces interference by a factor of half for linear deterministic channels.
While we conduct our analysis for the discrete memoryless channel (DMC), we will give particular focus on the Gaussian setup, which is in essence similar to that discussed in [10, 11], with a primary multiple access network and a secondary transmitterreceiver pair. We investigate and characterize necessary and sufficient conditions under which interference cancellation (IC) at either primary or secondary users can strictly improve the performance of the achievable rates. Namely, we determine the case when the primary is able to cancel the interference of the secondary while not deteriorating the QoS for the secondary network. We also determine the case when the secondary can completely decode and cancel the interference of at least one primary transmitter while not hurting the primary achievable rates. In particular, we

State the achievable rate region {\mathcal{R}}^{o} in the DMC assuming that all of the primary signals are treated as noise at the secondary receiver

State the achievable rate region {\mathcal{R}}_{i}^{r}, where the signal of primary transmitter i is to be fully decodable at the secondary receiver besides being decodable at the primary receiver

Show that there exists a case in which {\mathcal{R}}_{i}^{r} contains {\mathcal{R}}^{o}

Analyze the effect of rate splitting in a Gaussian setup where a necessary and sufficient condition is determined so that the union of the above regions is obtained without rate splitting

Derive a necessary and sufficient condition so that the secondary receiver can decode the signal of a primary user without affecting the range of achievable primary sum rates, but only enhances the range of achievable secondary rates. We call this condition primary decodability condition for Gaussian (PDCG) channel

Show, numerically, that the probability of having at least one primary user satisfying PDCG monotonically increases with the signaltonoise ratio of the primary users
We conduct our analysis assuming a Gaussian communication channel as in [10], but for general channel gains, and adoption of rate splitting techniques. Some of the results in this paper have been presented in [11]. The introduced network model of a MAC primary network shared by secondary operations has been addressed in some resource allocation frameworks without rate splitting by secondary users [12–16]. Rate splitting by a secondary link, however, has been introduced in [17] where the secondary user is assumed to know the codebook of a primary transmitter and opportunistically splits its rate into two parts and decodes it in the following way. It decodes the first part treating both the primary signal and the second part as noise, decodes and cancels the primary signal, and then decodes the second part. This scheme is generalized in this paper as we consider the cases when the signal of one primary transmitter is decodable at the secondary receiver and when all the primary signals are treated as noise.
The rest of this paper is organized as follows. In Section 2, the DMC models are defined. In Section 3, the achievable rate regions are established for the defined DMC models. Then, obtained results are applied in a Gaussian channel setup in Section 4, and the paper is concluded in Section 5.
2 Channel model
In our formulation, we denote random variables by X, Y, ⋯ with realizations x, y, ⋯ from sets , , ⋯, respectively. The communication channel is considered to be discrete and memoryless.
2.1 Basic channel model
We consider a basic channel C_{ B } defined by a tuple \left({\mathcal{X}}_{1},{\mathcal{X}}_{2},{\mathcal{X}}_{s},\omega ,{\mathcal{Y}}_{p},{\mathcal{Y}}_{s}\right), where {\mathcal{X}}_{1}, {\mathcal{X}}_{2} are two finite input alphabet sets of the primary transmitters and {\mathcal{X}}_{s} is a finite input alphabet set of the secondary transmitter. Sets {\mathcal{Y}}_{p} and {\mathcal{Y}}_{s} are two finite output alphabet sets at the primary and secondary receivers, respectively, and ω is a collection of conditional channel probabilities ω(y_{ p }y_{ s }x_{1}x_{2}x_{ s }) of \left({y}_{p},{y}_{s}\right)\in {\mathcal{Y}}_{p}\times {\mathcal{Y}}_{s} given \left({x}_{1},{x}_{2},{x}_{s}\right)\in {\mathcal{X}}_{1}\times {\mathcal{X}}_{2}\times {\mathcal{X}}_{s}, with marginal conditional distributions:
Since the channel is memoryless, the conditional probability ω^{n}(y_{ p }y_{ s }x_{1}x_{2}x_{ s }) is given by
where
The same also holds for the marginal conditional distributions {\omega}_{p}^{n}\left({\mathbf{\text{y}}}_{p}{\mathbf{\text{x}}}_{1}{\mathbf{\text{x}}}_{2}{\mathbf{\text{x}}}_{s}\right) and {\omega}_{s}^{n}\left({\mathbf{\text{y}}}_{s}{\mathbf{\text{x}}}_{1}{\mathbf{\text{x}}}_{2}{\mathbf{\text{x}}}_{s}\right). Let {\mathcal{\mathcal{M}}}_{1}=\{1,\cdots \phantom{\rule{0.3em}{0ex}},{M}_{1}\}, {\mathcal{\mathcal{M}}}_{2}=\{1,\cdots \phantom{\rule{0.3em}{0ex}},{M}_{2}\} be message sets for primary transmitters 1 and 2, respectively, and {\mathcal{\mathcal{M}}}_{s}=\{1,\cdots \phantom{\rule{0.3em}{0ex}},{M}_{s}\} be a message set for the secondary transmitter. A code (n,M_{1},M_{2},M_{ s },ε) is a collection of M_{1}, M_{2}, and M_{ s } codewords such that

1.
Sender a, a=1,2,s, has an encoding function ϕ _{ a }:i→x _{ ai }, i\in {\mathcal{\mathcal{M}}}_{a} and {\mathbf{\text{x}}}_{\mathit{\text{ai}}}\in {\mathcal{X}}^{n}

2.
The primary receiver has M _{1} M _{2} disjoint decoding sets {\mathcal{D}}_{\mathit{\text{pij}}}\subseteq {\mathcal{Y}}_{p}^{n}, \mathit{\text{ij}}\in {\mathcal{\mathcal{M}}}_{1}\times {\mathcal{\mathcal{M}}}_{2} and a decoding function ψ _{ p }:y _{ p }→i j if {\mathbf{\text{y}}}_{p}\in {\mathcal{D}}_{\mathit{\text{pij}}}, where \mathit{\text{ij}}\in {\mathcal{\mathcal{M}}}_{1}\times {\mathcal{\mathcal{M}}}_{2}

3.
The secondary receiver has M _{ s } disjoint decoding sets {\mathcal{D}}_{\mathit{\text{sk}}}\subseteq {\mathcal{Y}}_{s}^{n}, k\in {\mathcal{\mathcal{M}}}_{s} and a decoding function ψ _{ s }:y _{ s }→k if {\mathbf{\text{y}}}_{s}\in {\mathcal{D}}_{\mathit{\text{sk}}}, where k\in {\mathcal{\mathcal{M}}}_{s} (see Figure 1)
\begin{array}{l}{\mathit{\text{Pe}}}_{p}=\frac{1}{{M}_{1}{M}_{2}{M}_{s}}\sum _{i,j,k}{\omega}_{p}^{n}\left({\mathbf{\text{y}}}_{p}\notin {\mathcal{D}}_{\mathit{\text{pij}}}{\mathbf{\text{x}}}_{1i}{\mathbf{\text{x}}}_{2j}{\mathbf{\text{x}}}_{\mathit{\text{sk}}}\right),\end{array}(1)

4.
Probability of error for the primary network and the secondary link is less than ε, that is, P e _{ p }≤ε and P e _{ s }≤ε, respectively, where
A rate tuple (R_{1},R_{2},R_{ s }) of nonnegative real values is achievable if for any η>0, 0<ε<1 there exists a code such that
with sufficiently large n.
2.2 Rate splitting channel
Rate splitting channel, C_{ RS }, is a modified version of the basic channel C_{ B }, where C_{ RS } is defined by a tuple ({\mathcal{X}}_{1},{\mathcal{X}}_{2},{\mathcal{X}}_{s},\omega ,{\mathcal{Y}}_{p},{\mathcal{Y}}_{s}) with its elements are as defined in C_{ B }. Moreover, the input message sets for the primary transmitters are also {\mathcal{\mathcal{M}}}_{1} and {\mathcal{\mathcal{M}}}_{2} exactly as in C_{ B }. However, the secondary user is assumed to have two finite message sets {\mathcal{\mathcal{L}}}_{s}=\{1,\cdots \phantom{\rule{0.3em}{0ex}},{L}_{s}\}, {\mathcal{N}}_{s}=\{1,\cdots \phantom{\rule{0.3em}{0ex}},{N}_{s}\}. Hence, a code (n,M_{1},M_{2},L_{ s },N_{ s },ε) over the channel C_{ RS } is a collection of M_{1}, M_{2}, L_{ s }N_{ s } codewords such that

1.
Primary transmitter a, a = 1,2, has an encoding function ϕ _{ a }:i→x _{ ai }, i\in {\mathcal{\mathcal{M}}}_{a}, {\mathbf{\text{x}}}_{\mathit{\text{ai}}}\in {\mathcal{X}}_{a}^{n}

2.
The secondary transmitter has an encoding function ϕ _{ s }: k l → x _{ skl }, \mathit{\text{kl}}\in {\mathcal{\mathcal{L}}}_{s}\times {\mathcal{N}}_{s}, {\mathbf{\text{x}}}_{\mathit{\text{skl}}}\in {\mathcal{X}}_{s}^{n}

3.
The primary receiver has M _{1} M _{2} N _{ s } disjoint decoding sets {\mathcal{D}}_{\mathit{\text{pijl}}}\subseteq {\mathcal{Y}}_{p}^{n}, \mathit{\text{ijl}}\in {\mathcal{\mathcal{M}}}_{1}\times {\mathcal{\mathcal{M}}}_{2}\times {\mathcal{N}}_{s} and a decoding function ψ _{ p }:y _{ p }→i j l if {\mathbf{\text{y}}}_{p}\in {\mathcal{D}}_{\mathit{\text{pijl}}}, where \mathit{\text{ijl}}\in {\mathcal{\mathcal{M}}}_{1}\times {\mathcal{\mathcal{M}}}_{2}\times {\mathcal{N}}_{s}

4.
The secondary receiver has L _{ s } N _{ s } disjoint decoding sets {\mathcal{D}}_{\mathit{\text{skl}}}\subseteq {\mathcal{Y}}_{s}^{n}, \mathit{\text{kl}}\in {\mathcal{\mathcal{L}}}_{s}\times {\mathcal{N}}_{s} and a decoding function ψ _{ s }:y _{ p }→k l if {\mathbf{\text{y}}}_{p}\in {\mathcal{D}}_{\mathit{\text{skl}}}, where \mathit{\text{kl}}\in {\mathcal{\mathcal{L}}}_{s}\times {\mathcal{N}}_{s} (see Figure 2)

5.
Probability of error for primary network and secondary link is less than ε, that is, P{e}_{p}^{o}\le \epsilon and P{e}_{s}^{o}\le \epsilon, respectively, where
\begin{array}{lll}P{e}_{p}^{o}& =& \frac{1}{{M}_{1}{M}_{2}{L}_{s}{N}_{s}}\sum _{i,j,k,l}\phantom{\rule{0.3em}{0ex}}{\omega}_{p}^{n}\phantom{\rule{0.3em}{0ex}}\left({\mathbf{\text{y}}}_{p}\phantom{\rule{0.3em}{0ex}}\notin \phantom{\rule{0.3em}{0ex}}{\mathcal{D}}_{\mathit{\text{pijl}}}{\mathbf{\text{x}}}_{1i}{\mathbf{\text{x}}}_{2j}{\mathbf{\text{x}}}_{\mathit{\text{skl}}}\right)\phantom{\rule{0.3em}{0ex}},\end{array}(4)
A rate tuple (R_{1},R_{2},S,T) of nonnegative real values is achievable over the channel C_{ RS } if there exists a code (n,M_{1},M_{2},L_{ s },N_{ s },ε) such that for any arbitrary 0<ε<1 and η>0
with sufficiently large n.
Lemma 1.
If a rate tuple (R_{1},R_{2},S,T) is achievable for C_{ RS }, then a rate tuple (R_{1},R_{2},R_{ s }) where R_{ s }=S+T is achievable for C_{ B }.
Proof.
It is sufficient to show that if (n,M_{1},M_{2},L_{ s },N_{ s },ε) is a code for C_{ RS }, then (n,M_{1},M_{2},L_{ s }N_{ s },ε) is a code for C_{ B }. To do so, let {\mathcal{D}}_{\mathit{\text{pij}}}={\cup}_{l=1}^{{N}_{s}}{\mathcal{D}}_{\mathit{\text{pijl}}}. Then,
So, if (n,M_{1},M_{2},L_{ s },N_{ s },ε) is a code for C_{ RS }, then P{e}_{p}^{o}\le \epsilon and P{e}_{s}^{o}\le \epsilon; hence, from (9), P e_{ p }≤ε and P e_{ s }≤ε when k and M_{ s } of (1) are replaced with kl and L_{ s }N_{ s }, respectively, meaning that (n,M_{1},M_{2},L_{ s }N_{ s },ε) is a code for C_{ B }.
2.3 Rate splitting channel with decodable primary signal at the secondary receiver
We introduce another channel, {C}_{\mathit{\text{RS}}}^{p}, in which the secondary user splits its set of messages into two sets, exactly as the case of C_{ RS }. However, we assume that the signal of one primary transmitter is decodable at the secondary receiver. Without loss of generality, assume this is the first primary transmitter. Thus, {C}_{\mathit{\text{RS}}}^{p} is defined by a tuple ({\mathcal{X}}_{1},{\mathcal{X}}_{2},{\mathcal{X}}_{s},\omega ,{\mathcal{Y}}_{p},{\mathcal{Y}}_{s}) with its elements defined as in C_{ B } and C_{ RS }. A code for {C}_{\mathit{\text{RS}}}^{p} is the same as in C_{ RS }, except that conditions 4 and 5 are replaced by

4.
Secondary receiver has M _{1} L _{ s } N _{ s } disjoint decoding sets {\mathcal{D}}_{\mathit{\text{sikl}}}\subseteq {\mathcal{Y}}_{s}^{n} and a decoding function ψ _{ s }:y _{ s }→i k l if {\mathbf{\text{y}}}_{s}\in {\mathcal{D}}_{\mathit{\text{sikl}}}, where \mathit{\text{ikl}}\in {\mathcal{\mathcal{M}}}_{1}\times {\mathcal{\mathcal{L}}}_{s}\times {\mathcal{N}}_{s}

5.
Probability of error for the primary network and the secondary link is less than ε, that is, P{e}_{p}^{r}\le \epsilon and P{e}_{s}^{r}\le \epsilon, respectively, where
\begin{array}{l}P{e}_{p}^{r}=\frac{1}{{M}_{1}{M}_{2}{L}_{s}{N}_{s}}\sum _{i,j,k,l}\phantom{\rule{0.3em}{0ex}}{\omega}_{p}^{n}\phantom{\rule{0.3em}{0ex}}\left({\mathbf{\text{y}}}_{p}\notin {\mathcal{D}}_{\mathit{\text{pijl}}}{\mathbf{\text{x}}}_{1i}{\mathbf{\text{x}}}_{2j}{\mathbf{\text{x}}}_{\mathit{\text{skl}}}\right)\phantom{\rule{3em}{0ex}}\end{array}(10)
A rate tuple (R_{1},R_{2},S,T) of nonnegative real values is achievable over the channel {C}_{\mathit{\text{RS}}}^{p} if for any arbitrary η>0 and 0<ε<1, the inequalities (6) to (8) are satisfied for sufficiently large n.
Lemma 2.
If a rate tuple (R_{1},R_{2},S,T) is achievable for {C}_{\mathit{\text{RS}}}^{p}, then a rate tuple (R_{1},R_{2},R_{ s }) where R_{ s }=S+T is achievable for C_{ B }.
Proof
The proof follows exactly as the proof of Lemma 1 noting that if {\mathcal{D}}_{\mathit{\text{skl}}}={\cup}_{i=1}^{{M}_{1}}{\mathcal{D}}_{\mathit{\text{sikl}}}, then
At the end of this section, it is worth noting that C_{ B } furnishes a general structure for the communication setup of the system and does not explicitly pose any restrictions on the communication strategy used or limits the ability of certain receivers to decode the signals of noncorresponding transmitters. Yet, based on the primarysecondary nature of communication, we explicitly study special instances of C_{ B }, in particular C_{ RS } and {C}_{\mathit{\text{RS}}}^{p}, in which the secondary user is capable of employing rate splitting, and potentially decode the signal of one primary user. Hence, it follows clearly that achievable rates for C_{ RS } and {C}_{\mathit{\text{RS}}}^{p} are also achievable for C_{ B } as established in Lemmas 1 and 2.
3 Achievable rate region
In this section we investigate an achievable rate region for C_{ B }. We first analyze two achievable rate regions, one for C_{ RS } and another for {C}_{\mathit{\text{RS}}}^{p}, and then state the overall achievable rate region. The random variables U, W, and Q are defined over the finite sets , , and , respectively, where Q is a timesharing parameter. Let the set {\mathcal{P}}^{\ast} contain all Z = Q U W X_{1}X_{2}X_{ s }Y_{ p }Y_{ s } such that

X_{1}, X_{2}, U, and W are conditionally independent given Q

X_{ s }= f(U WQ)
Since X_{ s }= f(U WQ), then and can be considered as input sets to the channels C_{ RS } and {C}_{\mathit{\text{RS}}}^{p}.
3.1 Achievable rate region for C_{ RS }
Theorem 1.
For anyZ\in {\mathcal{P}}^{\ast}, δ^{o}(Z) is the set of achievable rate tuples (R_{1},R_{2},S,T) for C_{ RS } if the following inequalities are satisfied:
Proof
Please refer to Appendix 1.
Corollary 1.
For {\delta}^{o}={\cup}_{Z\in {\mathcal{P}}^{\ast}}{\delta}^{o}\left(Z\right) , any rate tuple of δ ^{o} is achievable.
We focus on the achievable rates by the primary network R_{ p }=R_{1}+R_{2} and the secondary link R_{ s }=S+T. Let {\mathcal{R}}^{o}\left(Z\right) be the set of all rate tuples (R_{ s },R_{ p }) having (R_{1},R_{2},S,T) satisfy (13) to (22) for all Z\in {\mathcal{P}}^{\ast}, then the following theorem describes {\mathcal{R}}^{o}\left(Z\right).
Theorem 2.
For anyZ\in {\mathcal{P}}^{\ast}, the achievable rate region{\mathcal{R}}^{o}\left(Z\right)of the defined channel C_{ RS } consists of all rate pairs (R_{ s },R_{ p }) that satisfy
where
and
Proof.
The proof can follow systematically using the FourierMotzkin elimination scheme, yet we use a different approach that determines the rate tuples (R_{ s },R_{ p }) of the corner points of {\mathcal{R}}^{o}\left(Z\right), which will essentially be utilized in the proofs of other statements in the rest of this work. To that end, we refer to Figure 3.

Point A:
{R}_{s}^{A}=0, i.e., S^{A}= T^{A}= 0. Thus, the maximum rate at which the primary network can operate is determined from (16) as

Point B:
At this point, we find the maximum possible rate at which the secondary user can transmit when the primary rate is {R}_{p}^{B}={\rho}_{p}^{o}. In this case, the relations of (13) to (22) are reduced to
Since T is irrelevant in (32), then S can be set to
Hence, using chain rule in (30) and (33), the maximum value for T would be
and {R}_{s}^{B}={S}^{B}+{T}^{B}.

Point D:
{R}_{1}^{D}={R}_{2}^{D}={R}_{p}^{D}=0, then (13) to (22) are reduced to
Since T is irrelevant in (37), S can be set to
Then,
and {R}_{s}^{D}={S}^{D}+{T}^{D}={\rho}_{s}^{o}.

Point C:
At {R}_{s}^{C}={\rho}_{s}^{o}, the maximum possible primary rate R_{ p }=R_{1}+R_{2} has to satisfy
Using chain rule, (43) can be rewritten as
Thus, if I(Y_{ p };WQ)σ^{∗}>0, then (44) will be dominated by (42). Otherwise, (44) dominates (42). So, {R}_{p}^{C} will be given by
where [ x] ^{+} = max {0,x}. The following is to show that both points \left({R}_{s}^{B},{R}_{p}^{B}\right) and \left({R}_{s}^{C},{R}_{p}^{C}\right) lie on the line {R}_{s}+{R}_{p}={\rho}_{\mathit{\text{sp}}}^{o}:
For Point B, using direct substitution with
and
it is clear that {R}_{s}^{B}+{R}_{p}^{B}={\rho}_{\mathit{\text{sp}}}^{o}.
For Point C, we consider the following two possibilities:

σ^{∗}≥I(Y_{ p };WQ):
Here min{I(Y_{ s };WQ),I(Y_{ p },WQ)}=I(Y_{ p };WQ). Consequently,
and

σ^{∗}<I(Y_{ p };WQ):
Since I(Y_{ p };WX_{1}X_{2}Q)≥I(Y_{ p };WQ), therefore I(Y_{ s };WQ)<I(Y_{ p };WQ). Consequently,
and
Therefore, both rate tuples \left({R}_{s}^{B},{R}_{p}^{B}\right) and \left({R}_{s}^{C},{R}_{p}^{C}\right) lie on the line {R}_{s}+{R}_{p}={\rho}_{\mathit{\text{sp}}}^{o}.
Note that, in the appendix of Han and Kobayashi [5], they argued that part of the achievable rate region by their introduced scheme was bounded by lines of slopes 0.5 and 2. Although from (13) to (22) reducing T by a value of r may result in increase of R_{ p } by 2r, the proof that point \left({R}_{s}^{C},{R}_{p}^{C}\right) lies on the line {R}_{s}+{R}_{p}={\rho}_{\mathit{\text{sp}}}^{o} means that a bound of slope 2 does not exist for {\mathcal{R}}^{o}\left(Z\right).
Corollary 2.
Any rate tuple (R_{ s },R_{ p }) of the region
is achievable.
3.2 Achievable rate region for {C}_{\mathit{\text{RS}}}^{p}
Since in {C}_{\mathit{\text{RS}}}^{p} the signal of one primary user has to be decodable at the secondary receiver, the model of {C}_{\mathit{\text{RS}}}^{p} can be considered as the modified interference channel model, C_{ m }, introduced in [5]. The signals of the two primary users can be treated as if they are produced from a single source, splitting its signal into two parts and encoding each part separately such that one part is decodable at both receivers while the other is decodable only at the primary receiver. For this channel, we define the set {\delta}_{i}^{r}\left(Z\right) as the set of all achievable rate tuples (R_{1},R_{2},S,T) when the signal of primary transmitter i, i ∈ {1,2}, is decodable by the secondary receiver. Without loss of generality, we assume that i = 1. Hence, the achievable rate region for {C}_{\mathit{\text{RS}}}^{p} takes the following form.
Theorem 3.
For anyZ\in {\mathcal{P}}^{\ast}, {\delta}_{1}^{r}\left(Z\right)is the set of achievable rate tuples (R_{1},R_{2},S,T) over the channel {C}_{\mathit{\text{RS}}}^{p}if the following inequalities are satisfied:
Proof
The proof follows exactly the proof of Theorem 3.1 in [5].
Corollary 3.
For{\delta}_{1}^{r}={\cup}_{Z\in {\mathcal{P}}^{\ast}}{\delta}_{1}^{r}\left(Z\right), any rate tuple of {\delta}_{1}^{r}is achievable.
For {C}_{\mathit{\text{RS}}}^{p}, the region {\mathcal{R}}_{i}^{r}\left(Z\right) is the set of rate tuples (R_{ s },R_{ p }) where R_{ s }= S + T, R_{ p }= R_{1} + R_{2}, and (R_{1},R_{2},S,T) is an element of {\delta}_{i}^{r}\left(Z\right) for any Z\in {\mathcal{P}}^{\ast}, i ∈ {1,2}.
Theorem 4.
For anyZ\in {\mathcal{P}}^{\ast}, the achievable rate region{\mathcal{R}}_{1}^{r}\left(Z\right)for the channel{C}_{\mathit{\text{RS}}}^{p}consists of all rate pairs (R_{ s },R_{ p }) that satisfy
where
and
as shown in Figure 4.
Proof
From the similarity between {C}_{\mathit{\text{RS}}}^{p} and the modified interference channel of Han and Kobayashi [5], the derivation of the achievable rate region can be found in the appendix of [5]. The analysis goes as that done for {\mathcal{R}}^{o}\left(Z\right) in C_{ RS }. Hence, the corner points of the {\mathcal{R}}_{1}^{r}\left(Z\right) are shown in Figure 4 and are given as follows.

Point A:
\begin{array}{lll}{R}_{s}^{A}& =& 0,\end{array}(69)\begin{array}{lll}{R}_{p}^{A}& =& {\rho}_{p}^{r}=I({Y}_{p};{X}_{2}{X}_{1}\mathit{\text{WQ}})+{\sigma}_{p}^{\ast}.\end{array}(70) 
Point B:
\begin{array}{lll}{R}_{s}^{B}& =& I\left({Y}_{s};U{\mathit{\text{WX}}}_{1}Q\right){\left[{\sigma}_{p}^{\ast}I\left({Y}_{s};{X}_{1}\mathit{\text{WQ}}\right)\right]}^{+}\\ +\text{min}\left\{I\left({Y}_{p};W{X}_{1}Q\right),I\left({Y}_{p};{\mathit{\text{WX}}}_{1}Q\right)\underset{p}{\overset{\ast}{\sigma}},\right.\\ I\left({Y}_{s};WQ\right)+{\left[I\left({Y}_{s};{X}_{1}\mathit{\text{WQ}}\right){\sigma}_{p}^{\ast}\right]}^{+},\\ \left(\right)close="\}">I\left({Y}_{s};W{X}_{1}Q\right)& ,\end{array}\n(71)\begin{array}{lll}{R}_{p}^{B}& =& {\rho}_{p}^{r}=I({Y}_{p};{X}_{2}{X}_{1}\mathit{\text{WQ}})+{\sigma}_{p}^{\ast}.\end{array}(72) 
Point C:
\begin{array}{lll}{R}_{s}^{C}& =& 2{\rho}_{\mathit{\text{sp}}}^{r}{\rho}_{s2}^{r},\end{array}(73)\begin{array}{lll}{R}_{p}^{C}& =& {\rho}_{s2}^{r}{\rho}_{\mathit{\text{sp}}}^{r}.\end{array}(74) 
Point D:
\begin{array}{lll}{R}_{s}^{D}& =& {\rho}_{2p}^{r}{\rho}_{\mathit{\text{sp}}}^{r},\end{array}(75)\begin{array}{lll}{R}_{p}^{D}& =& 2{\rho}_{\mathit{\text{sp}}}^{r}{\rho}_{\mathit{\text{sp}}}^{r}.\end{array}(76) 
Point E:
\begin{array}{lll}{R}_{s}^{E}& =& I({Y}_{s};U{\mathit{\text{WX}}}_{1}Q)+{\sigma}_{s}^{\ast},\end{array}(77)\begin{array}{lll}{R}_{p}^{E}& =& I({Y}_{p};{X}_{2}{\mathit{\text{WX}}}_{1}Q){[{\sigma}_{s}^{\ast}I({Y}_{p};W\left{X}_{1}Q\right)]}^{+}\\ +\text{min}\left\{I({Y}_{s};{X}_{1}\mathit{\text{WQ}}),I({Y}_{s};{\mathit{\text{WX}}}_{1}\leftQ\right){\sigma}_{s}^{\ast},\right.\\ I({Y}_{p};{X}_{1}Q)+{\left[I({Y}_{p};W{X}_{1}Q){\sigma}_{s}^{\ast}\right]}^{+},\\ \left(\right)close="\}">I({Y}_{p};{X}_{1}\mathit{\text{WQ}})& .\end{array}\n(78) 
Point F:
\begin{array}{lll}{R}_{s}^{\mathit{\text{rF}}}& =& {\rho}_{s}^{r}=I\left({Y}_{s};U{\mathit{\text{WX}}}_{1}Q\right)+{\sigma}_{s}^{\ast},\end{array}(79)\begin{array}{lll}{R}_{p}^{F}& =& 0.\end{array}(80)
Corollary 4.
Any rate tuple ( R_{ s },R_{ p }) of the regions
is achievable.
Constraining the signal of one primary user to be decodable at the secondary receiver might result in a degradation in the achievable primary rate especially when the secondary rate is small. In general, {\mathcal{R}}^{o} and {\mathcal{R}}_{i}^{r} do not necessarily contain one another; however, there exists a case in which {\mathcal{R}}^{o}\left(Z\right)\subseteq {\mathcal{R}}_{i}^{r}\left(Z\right). The following theorem characterizes that case.
Theorem 5.
For a givenZ\in {\mathcal{P}}^{\ast}, {\mathcal{R}}^{o}\left(Z\right)\subseteq {\mathcal{R}}_{i}^{r}\left(Z\right)if and only if
Proof
Please refer to Appendix 2.
Corollary 5.
If for allZ\in {\mathcal{P}}^{\ast}condition (82) is satisfied, then{\mathcal{R}}^{o}\subseteq {\mathcal{R}}_{i}^{r}, where{\mathcal{R}}_{i}^{r}={\cup}_{Z\in {\mathcal{P}}^{\ast}}{\mathcal{R}}_{i}^{r}\left(Z\right).
Theorem 5 shows that when a primary user encodes its messages at a rate decodable at both receivers, the primary network may achieve the same rate range when none of the signals of its users is decodable at the secondary receiver. Moreover, at every primary rate, the secondary rate is enhanced (see Figure 5).
Consequently, if for any Z\in {\mathcal{P}}^{\ast} condition (82) is satisfied, then allowing the secondary receiver to decode the signal of primary user i at this Z enhances the range of the secondary achievable rates without reducing the range of the achievable primary sum rates.
We call Corollary 5 the primary decodability condition (PDC).
3.3 Achievable rate region for the channel C_{ B }
From C_{ RS } and {C}_{\mathit{\text{RS}}}^{p}, we define
and
Hence, an achievable rate region for the channel C_{ B }
or equivalently,
At this point, it is worth reflecting the resulting achievable rate region on the HanKobayashi region derived for the 2×2 interference channel, denoted {\mathcal{R}}_{\mathit{\text{HK}}}, especially with the adopted channel model C_{ B } which is well related to the interference channel C in [5]. In the light of the considered communication setup and adopted rate splitting communication scheme, we can note that the two primary transmitters of our setup C_{ B } can be viewed as a common transmitter in C splitting its signal into X_{1}, X_{2}.
However, since the transmitters are sending independent messages and having no control over the codebook of each other, their transmit strategies adopted in can be considered as only three realizations of the possible rate splitting strategies for the common transmitter in C. Thus, we can note that \mathcal{R}\subseteq {\mathcal{R}}_{\mathit{\text{HK}}}. In particular, for the two primary transmitters of C_{ B } behaving as a common transmitter in C, rate region spans only the rate splitting strategies of such common transmitter when the secondary receiver (1) cannot decode any primary signal, (2) can only decode the whole signal of user 1, and (3) can only decode the whole signal of user 2.
Note that, inequalities (15) and (49) used in δ^{o}(Z) and {\delta}_{1}^{r}\left(Z\right), respectively, to limit the error in decoding the public part of the secondary signal at the primary receiver while the primary signals are decoded successfully. In fact, the primary receiver may not be interested in limiting the probability of such error event. Similarly, inequality (56) in {\delta}_{1}^{r}\left(Z\right) may not be relevant as the secondary receiver is not interested in limiting the probability of error in decoding the primary signal when the two parts of its signal are decoded successfully. However, removing (15) from the definition of δ^{o}(Z) and (49) and (56) from the definition of {\delta}_{1}^{r}\left(Z\right) does not enhance the achievable rate region .
To demonstrate this fact, we define δ^{′ o}(Z) exactly as δ^{o}(Z) but without the constraint of (15), and δ 1 ′ r (Z) exactly as {\delta}_{1}^{r}\left(Z\right) but without the constraints (49) and (56). Let {\mathcal{R}}^{\mathrm{\prime o}}\left(Z\right) and {\mathcal{R}}_{1}^{\mathrm{\prime r}}\left(Z\right) be two sets of rate tuples (R_{ s },R_{ p }) such that R_{ s }= S + T and R_{ p }= R_{1}+ R_{2}, and the rate tuple (R_{1},R_{2},S,T) is an element of δ^{′ o}(Z) and δ 1 ′ r (Z), respectively. Also, we define
Theorem 6.
If{\mathcal{R}}_{1}^{\prime}=\bigcup _{Z\in {\mathcal{P}}^{\ast}}{\mathcal{R}}_{1}^{\prime}\left(Z\right), then{\mathcal{R}}_{1}^{\prime}={\mathcal{R}}_{1}.
Proof
Please refer to Appendix 3.
Corollary 6.
For
then
4 Gaussian channel
In this section, we quantify the obtained achievable rate regions in a Gaussian channel model. A memoryless Gaussian channel of the introduced system is defined by a tuple ({\mathcal{X}}_{1},{\mathcal{X}}_{2},{\mathcal{X}}_{s},\omega ,{\mathcal{Y}}_{p},{\mathcal{Y}}_{s}) with {\mathcal{X}}_{1}={\mathcal{X}}_{2}={\mathcal{X}}_{s}={\mathcal{Y}}_{p}={\mathcal{Y}}_{s}=\Re (the field of real numbers), and a channel probability ω specified by
for {x}_{1}\in {\mathcal{X}}_{1}, {x}_{2}\in {\mathcal{X}}_{2}, {x}_{s}\in {\mathcal{X}}_{s}, {y}_{p}\in {\mathcal{Y}}_{p}, and {y}_{s}\in {\mathcal{Y}}_{s}, where n_{ p } and n_{ s } are independent Gaussian additive noise samples with zero mean and variance N_{0}, and {g}_{1}^{p}, {g}_{2}^{p}, {g}_{s}^{p}, {g}_{1}^{s}, {g}_{2}^{s}, and {g}_{s}^{s} are the channel power gains. Power constraints are imposed on codewords x_{1}(i), x_{2}(j), x_{ s }(k) (i\in {\mathcal{\mathcal{M}}}_{1}, j\in {\mathcal{\mathcal{M}}}_{2}, k\in {\mathcal{\mathcal{M}}}_{s}):
Also, we define a subclass \mathcal{G}({P}_{1},{P}_{2},{P}_{s}) of {\mathcal{P}}^{\ast} as follows: Z=\varphi {\mathit{\text{UWX}}}_{1}{X}_{2}{X}_{s}{Y}_{p}{Y}_{s}\in \mathcal{G}({P}_{1},{P}_{2},{P}_{s}) if and only if Z\in {\mathcal{P}}^{\ast}, σ^{2}(X_{1})= P_{1}, σ^{2}(X_{2})= P_{2}, and σ^{2}(X_{ s }) = P_{ s }, with X_{1}, X_{2}, U, and W being zero mean Gaussian and X_{ s }= U + W. Hence, we have the following rate regions achievable:
Assume the secondary user splits its power into λ P_{ s } and \stackrel{\u0304}{\lambda}{P}_{s} such that 0≤λ≤1 and \lambda +\stackrel{\u0304}{\lambda}=1. The part of secondary signal decodable at the primary and secondary receivers is encoded with power \stackrel{\u0304}{\lambda}{P}_{s} where the other part is encoded with power λ P_{ s }. Let τ(x)=0.5 log2(1+x), and the relevant quantities in Theorems 2 and 4 will be given by
4.1 Performance of rate splitting
We study the effect of rate splitting by the secondary link on the achievable rate regions {\mathcal{R}}_{g}^{o} and {\mathcal{R}}_{\mathit{\text{ig}}}^{r}, i ∈ {1,2} and hence {\mathcal{R}}_{\mathit{\text{ig}}}. For each region, there exists a case for which no rate splitting determines the overall region, i.e., each achievable rate region is obtained at λ = 0 or λ = 1. We say that rate splitting does not affect an achievable rate region if \mathcal{A}\left(Z\right) coincides on at λ = 0 or λ = 1, Z\in \mathcal{G}({P}_{1},{P}_{2},{P}_{s}), where \mathcal{A}=\bigcup _{Z\in \mathcal{G}({P}_{1},{P}_{2},{P}_{s})}\mathcal{A}\left(Z\right), meaning that either decoding the whole secondary signal at the primary receiver or not decoding it at all determines .
4.1.1 For {\mathcal{R}}_{g}^{o}
The region {\mathcal{R}}_{g}^{o} is obtained when the secondary receiver is assumed to treat the primary interference as noise. The following theorem determines the effect of rate splitting on {\mathcal{R}}_{g}^{o}.
Theorem 7.
For Z\in \mathcal{G}({P}_{1},{P}_{2},{P}_{s}) , an achievable rate region {\mathcal{R}}^{o}\left(Z\right) can only coincide on {\mathcal{R}}_{g}^{o} at λ = 0, if and only if
or equivalently,
Proof
Please refer to Appendix 4.
Theorem 7 shows that rate splitting does not affect the achievable rate region {\mathcal{R}}_{g}^{o} when inequality (97) is satisfied. Hence, a primary receiver decoding all the secondary signal is preferable at this case. Figure 6 depicts this case for different values of λ. It is clear that {\mathcal{R}}^{o}\left(Z\right) at smaller λ contains {\mathcal{R}}^{o}\left(Z\right) at larger λ. This figure was obtained at {g}_{1}^{p}=2.5664, {g}_{2}^{p}=3.7653, {g}_{1}^{s}=0.1812, {g}_{2}^{s}=0.1784, {g}_{s}^{p}=2.3620, and {g}_{s}^{s}=8.6065 and at the following power setup. The noise variance N_{0} = 1 unit power and \frac{{P}_{1}}{{N}_{0}}=\frac{{P}_{2}}{{N}_{0}}={\text{SNR}}_{p}=10 dB and \frac{{P}_{s}}{{N}_{0}}={\text{SNR}}_{s}=10 dB. Note that in this case, the maximum secondary throughput does not depend on λ, so the best performance from the primary rate point of view is to decode all the secondary signal by setting λ = 0.
Moreover, when inequality (97) is not satisfied, rate splitting affects {\mathcal{R}}_{g}^{o} as for any two different values of λ the corresponding {\mathcal{R}}^{o}\left(Z\right)s do not contain one another. Hence, {\mathcal{R}}_{g}^{o} is obtained by varying λ from 0 to 1. Figure 7 represents the case when (97) is not satisfied for the following parameters. {g}_{1}^{p}=1.5066, {g}_{2}^{p}=0.8290, {g}_{1}^{s}=0.1902, {g}_{2}^{s}=0.0122, {g}_{s}^{p}=1.1953, and {g}_{s}^{s}=10.3229 with the same power setup of Figure 6.
Also, it is shown in [11] that when (97) is not satisfied, the sum throughput of the whole network, i.e., R_{ s }+R_{ p }, increases with λ. That is, as λ increases, the primary sum rate decreases but the secondary rate gains an increase larger than the decrease in rate encountered by the primary network. Figure 8 depicts R_{ s }+R_{ p } for the same simulation parameters of Figure 7. It is clear that the increase in the total sum rate, R_{ s }+R_{ p }, is accompanied by a decrease in the sum primary rate R_{ p }. Hence, the sum primary rate has to be protected above a minimum limit.
4.1.2 For {\mathcal{R}}_{\mathit{\text{ig}}}^{r}, i∈ {1,2}
The region {\mathcal{R}}_{\mathit{\text{ig}}}^{r} is obtained when the secondary receiver can decode the signal of primary user i. The rate splitting effect on this region is determined in the following theorem.
Theorem 8.
ForZ\in \mathcal{G}({P}_{1},{P}_{2},{P}_{s})and i ∈ {1,2}, an achievable rate region{\mathcal{R}}_{i}^{r}\left(Z\right) can only coincide on {\mathcal{R}}_{\mathit{\text{ig}}}^{r}at λ = 0 if and only if
or equivalently,
Proof
Please refer to Appendix 5.
Hence, if inequality (99) is satisfied, {\mathcal{R}}_{\mathit{\text{ig}}}^{r} is obtained without rate splitting, specifically, when λ = 0.
Figures 9 and 10 show the performance of rate splitting under same power setup used with Figures 6 and 7 where it is assumed that the secondary receiver can decode the signal of primary user 1. In Figure 9, the achievable rate region {\mathcal{R}}_{1g}^{r} coincides on {\mathcal{R}}_{1}^{r}\left(Z\right) when inequality (99) is satisfied. The parameters for this scenario are {g}_{1}^{p}=5.5303, {g}_{2}^{p}=4.2865, {g}_{1}^{s}=0.6542, {g}_{2}^{s}=0.8121, {g}_{s}^{p}=3.9334, and {g}_{s}^{s}=8.1575.
In Figure 10, the opposite scenario is considered where inequality (99) is not satisfied. It is obvious that the overall rate region {\mathcal{R}}_{1g}^{r} is obtained by varying λ from 0 to 1 as a consequence of the fact that rate regions corresponding to different values of λ do not include one another if inequality (99) is not satisfied. The channel gains for Figure 10 are {g}_{1}^{p}=9.566, {g}_{2}^{p}=14.5045, {g}_{1}^{s}=0.0808, {g}_{2}^{s}=0.2894, {g}_{s}^{p}=0.7032, and {g}_{s}^{s}=16.6226.
Consequently, the achievable rate region {\mathcal{R}}_{\mathit{\text{ig}}} coincides on {\mathcal{R}}_{\mathit{\text{ig}}}\left(Z\right) at λ = 0 if and only if (99) is satisfied.
4.2 Decoding one primary signal
In Section 3.2, we have discussed the achievable rate region in the DMC case assuming that the signal of one primary transmitter has to be reliably decoded by the secondary receiver. Although this may impose a constraint on the range of achievable sum rates by the primary network, we showed in Theorem 5 and Corollary 5 that there exists a condition for which this constraint only enhances the achievable rates for the secondary link without degrading the range of achievable rates by the primary network. This condition is called PDC. When applying this condition to the given Gaussian channel, the PDC would be as follows: If for all Z\in \mathcal{G}({P}_{1},{P}_{2},{P}_{s})I({Y}_{p};{X}_{i}W)\le I({Y}_{s};{X}_{i}\left\mathit{\text{UW}}\right), then {\mathcal{R}}_{g}^{o}\subseteq {\mathcal{R}}_{\mathit{\text{ig}}}^{r}. Equivalently, the following inequality must hold:
But since I(Y_{ s };X_{ i }U W) does not depend on λ, then a necessary and sufficient condition to have (100) satisfied is
We call inequality (101) primary decodability condition for Gaussian channel (PDCG).
Figure 11 shows a scenario for which three rate regions are obtained: {\mathcal{R}}_{g}^{o}, {\mathcal{R}}_{1g}^{r}, and {\mathcal{R}}_{2g}^{r}. It is clear that {\mathcal{R}}_{g}^{o}\subseteq {\mathcal{R}}_{1g}^{r} meaning that primary user 1 satisfies the PDCG described in (101), whereas primary user 2 does not. By decoding the signal of primary user 1 at the secondary receiver, the range of achievable primary rates in {\mathcal{R}}_{g}^{o} remains the same for {\mathcal{R}}_{1g}^{r} while the secondary link can achieve higher rate at a given primary rate in {\mathcal{R}}_{1g}^{r} than in {\mathcal{R}}_{g}^{o}. The power setup used to produce this figure is the same as that of Figure 6, and the channel gains are {g}_{1}^{p}=0.3413, {g}_{2}^{p}=10.2047, {g}_{1}^{s}=0.2821, {g}_{2}^{s}=0.3782, {g}_{s}^{p}=0.2495, and {g}_{s}^{s}=6.3337.
Note that a primary user that satisfies PDCG does not always exist, so we evaluate the probability of PDCG as the probability of finding at least one primary user satisfying (101). We assume N_{0} = 1 unit power and {g}_{1}^{s} and {g}_{2}^{s} are i.i.d. exponentially distributed with mean μ_{ s }, whereas {g}_{1}^{p} and {g}_{2}^{p} are i.i.d. exponentially distributed with mean μ_{ p }, where {g}_{1}^{s}, {g}_{2}^{s}, {g}_{1}^{p}, and {g}_{2}^{p} are mutually independent. A closed form formula for the probability of PDCG is difficult to obtain, so we evaluate it numerically by generating 10^{7} different values for each channel gain element and calculating the average number of times at which neither primary user satisfies (101) at a given P_{1} and P_{2}; then by subtracting it from 1, we get a numerical estimate for the probability of PDCG. A simulation has been done in which we assume that \frac{{P}_{1}}{{N}_{0}}=\frac{{P}_{2}}{{N}_{0}}={\text{SNR}}_{p}. We vary {\text{SNR}}_{p} and evaluate the corresponding probability of PDCG. This simulation is done for the following pairs of (μ_{ p },μ_{ s }): (1,1), (1,5), (5,1), and (5,5). The result is shown in Figure 12, where it is obvious that the probability of PDCG increases with {\text{SNR}}_{p} and that the increase in μ_{ s } yields more increase in probability of PDCG. The monotonic increase of such probability with {\text{SNR}}_{p} can be seen by explicitly expressing the probability of event (101) as P\left(\frac{{g}_{j}^{s}{\text{SNR}}_{p}+1}{{g}_{j}^{p}{\text{SNR}}_{p}+1}\le \frac{{g}_{i}^{s}}{{g}_{i}^{p}},\text{for some}\phantom{\rule{2.77626pt}{0ex}}i,j\in \{1,2\}\right), which is essentially monotonically increasing in {\text{SNR}}_{p} and approaches 1 as {\text{SNR}}_{p} goes to ∞. While it is hard to mathematically show the dependence of the probability of PDCG on μ_{ s } and μ_{ p }, we can justify the increase of such probability with μ_{ s } relative to μ_{ p } because it statistically implies higher quality of the channel to the secondary receiver than that to the primary, hence more chances of (101).
5 Conclusions
In this work, we have analyzed an achievable rate region for a primary multiple access network coexisting with a secondary link that comprises one transmitter and a corresponding receiver. The achievable rate regions depict the sum primary rate versus the secondary rate. We have considered DMC where the secondary link employs rate splitting and investigated two types of achievable rate regions: the first is when the secondary receiver treats the primary signal as noise, whereas the second is when the secondary is able to decode the signal of only one primary transmitter. An overall achievable rate region is the union of those two regions. Moreover, we have shown that there exists a case for which allowing the secondary receiver to decode a primary signal results in an achievable rate region that includes the achievable rate region obtained when the secondary receiver does not decode the primary signal. Subsequently, we have investigated the performance of rate splitting in th e Gaussian channel where it has been found that rate splitting by the secondary user is useless when the channel between the secondary transmitter and the primary receiver supports larger rate than the channel between the two secondary nodes. Furthermore, on decoding the signal of a primary transmitter at the secondary receiver, a necessary and sufficient condition has been provided to allow the secondary user to decode the primary signal without reducing the range of achievable primary sum rates; in fact, it can only increase the range of achievable secondary rates. Finally, we have shown numerically that the probability of finding at least one primary user that satisfies this condition increases with the signaltonoise ratio of the primary users.
Appendix 1  Proof of Theorem 1
It is sufficient to show that there exists at least one code for which if the rate tuple (R_{1},R_{2},S,T) satisfies (13) to (22), then the rate tuple is achievable. We use the following random code.
Random code generation
A random code is generated as follows. Let q = (q^{(1)},⋯,q^{(n)}) be a random i.i.d sequence of {\mathcal{Q}}^{n}, {\mathbf{\text{u}}}_{k}=\left({u}_{k}^{\left(1\right)},\cdots \phantom{\rule{0.3em}{0ex}},{u}_{k}^{\left(n\right)}\right), k\in {\mathcal{\mathcal{L}}}_{s} a sequence of random variables of {\mathcal{U}}^{n} that are i.i.d given q. Moreover, u_{ k } and {\mathbf{\text{u}}}_{{k}^{\prime}} are independent ∀ k ≠ k^{′}, k,{k}^{\prime}\in {\mathcal{\mathcal{L}}}_{s}. Similarly, generate w_{ l }, l\in {\mathcal{N}}_{s}, x_{1i}, i\in {\mathcal{\mathcal{M}}}_{1} and x_{2j}, j\in {\mathcal{\mathcal{M}}}_{2}, which are respectively i.i.d. given q.
Encoding
For primary user 1 to send a message i\in {\mathcal{\mathcal{M}}}_{1}, it sends x_{1i}. Similarly, for primary user 2 to send a message j\in {\mathcal{\mathcal{M}}}_{2}, it sends x_{2j}. For the secondary user to send a message \mathit{\text{kl}}\in {\mathcal{\mathcal{L}}}_{s}\times {\mathcal{N}}_{s}, it sends {f}^{n}\left({\mathbf{\text{u}}}_{k}{\mathbf{\text{w}}}_{l}\mathbf{\text{q}}\right)=\left({f}^{\left(1\right)}\left({u}_{k}^{\left(1\right)}{w}_{l}^{\left(1\right)}{q}^{\left(1\right)}\right),\cdots \phantom{\rule{0.3em}{0ex}},{f}^{\left(n\right)}\left({u}_{k}^{\left(n\right)}{w}_{l}^{\left(n\right)}{q}^{\left(n\right)}\right)\right), where q is known at the transmitters.
Decoding: jointly typical decoding
We use the concept of jointly typical sequences and the properties of typical sets introduced in Chapter 15 of [18] to implement the decoding functions. Let {A}_{\epsilon}^{\left(n\right)} denote the set of typical (q,x_{1},x_{2},w_{ l },y_{ p }) sequences, then the primary receiver decides ijl if \left(\mathbf{\text{q}},{\mathbf{\text{x}}}_{1i},{\mathbf{\text{x}}}_{2j},{\mathbf{\text{w}}}_{l},{\mathbf{\text{y}}}_{p}\right)\in {A}_{\epsilon}^{\left(n\right)}. Also, let {B}_{\epsilon}^{\left(n\right)} denote the set of typical (q,u,w,y_{ s }) sequences, then secondary receiver decides kl if \left(\mathbf{\text{q}},{\mathbf{\text{u}}}_{k},{\mathbf{\text{w}}}_{l},{\mathbf{\text{y}}}_{s}\right)\in {B}_{\epsilon}^{\left(n\right)}.
Probability of error analysis
By the symmetry of the random code generation, the conditional probability of error does not depend on the transmitted messages. Hence, the conditional probability of error is the same as the average probability of error. So, let ijkl = 1111 be sent. An error occurs if the transmitted codewords are not typical with the received sequences.
For the primary receiver
Let the event
hence, the probability of error averaged over the random code is
where {E}_{p}^{c}\left(111\right) denotes the complement of E_{ p }(111). Using union bound, we have
From the properties of jointly typical sequences [18], P\phantom{\rule{0.3em}{0ex}}\left({E}_{p}^{c}\left(111\right)\right)\to 0 as n → ∞, and
where the last equality holds from the assumption that X_{1}, X_{2}, U, and W are conditionally independent given Q. Similarly, for other E_{ p }(ijl ≠ 111) and applying Equations (6) to (8), we get
Thus, if (13) to (19) are satisfied, \stackrel{\u0304}{P}{e}_{p}^{o}\to \epsilon as n → ∞.
For the secondary receiver
Let the event
hence, the probability of decoding error averaged over the random code is
where {E}_{s}^{c}\left(11\right) denotes the complement of E_{ s }(11). Using union bound, we have
Since P\left({E}_{s}^{c}\right(11\left)\right)\to \epsilon as n → ∞, then
So, if (20) to (22) are satisfied, \stackrel{\u0304}{P}{e}_{s}^{o}\to \epsilon as n → ∞.
This concludes the proof.
Appendix 2  Proof of Theorem 5
Sufficiency part
Suppose (82) is satisfied, we use Figure 5 to prove that {\mathcal{R}}^{o}\left(Z\right)\subseteq {\mathcal{R}}_{i}^{r}\left(Z\right). It is sufficient to show that {R}_{p}^{{A}^{o}}={R}_{p}^{{A}^{r}}, {R}_{s}^{{B}^{o}}\le {R}_{s}^{{B}^{r}}, {R}_{s}^{{D}^{o}}\le {R}_{s}^{{F}^{r}} and that lines 2{R}_{s}+{R}_{p}={\rho}_{2p}^{r} and {R}_{s}+{R}_{p}