On the capacity of statedependent Gaussian cognitive interference channel
 Shahab GhasemiGoojani^{1} and
 Hamid Behroozi^{1}Email author
https://doi.org/10.1186/168714992014196
© GhasemiGoojani and Behroozi; licensee Springer. 2014
Received: 30 June 2014
Accepted: 27 October 2014
Published: 22 November 2014
Abstract
A Gaussian cognitive interference channel with state (GCICS) is studied. In this paper, we focus on the twosender, tworeceiver case and consider the communication situation in which two senders transmit a common message to two receivers. Transmitter 1 knows only message W_{1}, and transmitter 2, referred to as the cognitive user, knows both messages W_{1} and W_{2} and also the channel’s states sequence noncausally. Receiver 1 needs to decode only W_{1} while receiver 2 needs to decode both messages. In this paper, we investigate the weak and moderate interference case where we assume that the channel gain a satisfies a≤1. In addition, inner and outer bounds on the capacity region are derived in the regime of high state power, i.e., the channel state sequence has unbounded variance. First, we show that the achievable rate by GelfandPinsker coding vanishes in the high state power regime under a condition over the channel gain. In contrast, we propose a transmission scheme (based on lattice codes) that can achieve positive rates, independent of the interference. Our transmission scheme can achieve the capacity region in a high signaltonoise ratio (SNR) regime. Also, regardless of all channel parameters, the gap between the achievable rate region and the outer bound is at most 0.5 bits.
1 Introduction
In the exchange of information among many nodes, the interference between different transmitter and receiver pairs is unavoidable. In the classical interference channel (IC), this interference exists between two different transmitters and receivers. In [1], Carleial, using superposition coding, obtains general bounds on the capacity region of discrete memoryless interference channels. By using rate splitting at transmitters and sequential decoding at destinations, Han and Kobayashi establish the best achievable rate region known to date [2]. Unfortunately, the problem of characterizing the capacity region of a general IC has been open for more than 30 years. Except for very strong Gaussian IC, strong Gaussian IC, the sum capacity of the degraded Gaussian IC and very weak interference, characterizing the capacity region of a Gaussian IC is still an open problem [3–6]. Etkin et al., by deriving new outer bounds, show that an explicit HanKobayashi version scheme can achieve capacity region within 1 bit for all channel parameters [7].
The cognitive interference channel, where one user has full noncausal knowledge of the other user’s message, is studied in [8–10]. This setup is also referred to as the interference channels with degraded message sets.
Recently, interference channels with state have received considerable attention. In general, channels with random states can model a timevarying wireless channel as well as interfering signals. The twouser statedependent Gaussian interference channel where the state information is noncausally known at both encoders is studied in [11]. By proposing an active interferencecancellation mechanism, which is a generalized dirtypaper coding (DPC) [12] technique, some achievable rate regions for this channel are obtained. A Gaussian IC with the same state at both links which is scaled differently at two receivers is studied in [13]. For the very strong interference regime, as well as for the weak regime, the sum capacity is obtained under certain conditions on channel parameters [13]. In [14], a statedependent Gaussian Zinterference channel model in the regime of high state power is investigated. By utilizing a layered coding scheme, inner and outer bounds on the capacity region are derived.
In [15], a model of cognitive statedependent interference channels is studied, in which one of the transmitters knows both messages and also the states of the channel in a noncausal manner while the other transmitter knows only one of the messages and does not know the channel states. Each of the two decoders try to decode only its intended message. By using a generalized binning principle, inner and outer bounds on the capacity region are established.
The main result of this paper is designing a novel transmission scheme for the Gaussian interference channel with state where we aim to recover a common message at two decoders. To reach this goal, we treat this channel as two statedependent Gaussian multipleaccess channels (MACs) and try to simultaneously recover the common message at both decoders. Prior to this work, different types of the statedependent twouser MAC are investigated in the literature (See e.g., [17–24]). In [17], a twouser statedependent multiaccess channel in which the state is known only at the encoder (that sends both messages) is investigated. By generalizing the GelfandPinsker model, the capacity region for both noncausal and causal state information is characterized. If the state information is noncausally known only at the encoder that sends the common message, then the capacity region for the Gaussian scenario in some cases is characterized in [18]. In [19–21], the statedependent twouser multiaccess channel in which the states of the channel are known noncausally at one of the encoders and only strictly causally at the other encoder is considered. By generalizing the framework of [21], the capacity region of this model is fully characterized in [19], and the optimal schemes for achieving the capacity region are also studied. In [22–24], the twouser multipleaccess channel with state is considered in which the states are known causally or strictly causally at both encoders or only at one encoder. For the causal state, it is shown that the capacity region is fully achievable. If the state is known strictly causally at both the encoders or only at one encoder, then the capacity region at some cases is characterized.
The capacity region of relay channel with state is investigated in [25–32]. The relay channel and the cooperative relay broadcast channel controlled by random parameters are studied in [25]. It is shown that when the state is noncausally known to the transmitter and intermediate nodes, the decodeandforward can achieve the capacity region under some cases. The relay channel with the state known noncausally at the relay is investigated in [26] and [27]. Using GelfandPinsker coding, rate splitting, and decodeandforward, a lower bound on channel capacity is obtained for this channel, and it is shown that for the degraded Gaussian channels, the lower bound meets the upper bound and thus the capacity region is achievable. The relay channel when the state is available only at the source is studied in [28–30]. By obtaining lower and upper bounds, it is shown that in a number of special cases the capacity region is achievable. A partially cooperative relay broadcast channel (PCRBC) with state is studied in [31] where two situations including the availability of the state noncausally at both the source and the relay and only at the source are analyzed. The relay interference channel with a cognitive source where only the source knows (noncausally) the interference from the interferer is considered in [32], and some achievable rate regions are obtained.
All achievable rate regions in [16] are based on random coding. In this paper, we use the latticebased coding scheme (especially lattice alignment) to establish capacity regions for this channel. A comprehensive study on the performance of lattices is presented in [33]. Performance of lattice codes over the additive white Gaussian noise (AWGN) channel is studied in [34]. A dirty paper AWGN channel in which the interference is known noncausally or causally at the transmitter is investigated in [35]. In [36], it is shown that the lattice coding strategy may outperform the DPC in a doubly dirty MAC. In [37], we also show that if the noise’s variance satisfies some constraints, then the capacity region of an additive statedependent Gaussian interference channel with two independent channel states is achieved when the state power goes to infinity. In [38], a Gaussian relay channel with a state is considered in which the additive state is either added at the destination and known noncausally at the source or experienced at the relay and known at the destination. It is shown that a scheme based on nested lattice codes can achieve the capacity region within 0.5 bits. In [39], by using nested lattice codes, the generalized degrees of freedom for the twouser cognitive interference channel are characterized where one of the transmitters has knowledge of a linear combination of the two information messages. Using lattice codes for the statedependent Gaussian Zinterference channel, some rate regions are established in [40].
Here, we evaluate the performance of latticebased coding schemes on obtaining achievable rate regions for the GCICS. Similar to [14, 36], we assume that the channel state has unbounded variance. This is referred to as a high state power regime. In addition, we consider the weak and moderate interference cases, i.e., the channel gain is smaller than one; a≤1. First, we show that the achievable rate region by random coding vanishes in a high state power regime under a condition over the channel gain. Then, by using a latticebased coding scheme, we obtain an achievable rate region for the GCICS. As Figure 1 shows, we can see that the GCICS can be treated as two statedependent MACs with a common message: one from encoders 1 and 2 to decoder 1, and the other from encoders 1 and 2 to decoder 2. For both these MACs, the capacity region is completely characterized in [19]. However in the GCICS, we need to decode the common message simultaneously at both decoders. Since these two MACs are different, we cannot apply the proposed scheme in [19] for this channel.
The main challenge of this paper is designing a scheme that can achieve a rate region close to the outer bound for the statedependent Gaussian interference channel with a common message (set W_{2}=0 in Figure 1). Although this channel can be treated as two statedependent MACs with a common message, these two MACs are different, and since the common message should be recovered simultaneously at both decoders, the known schemes in the literature cannot be directly applied. To solve this problem, we use lattice codes and obtain a linear combination of the common message, sent by two transmitters, at the decoders. Note that lattice codes are among the best codes in finding the linear combination of messages [41]. As we will show, at high signaltonoise ratios (SNRs), the achievable rate region meets the outer bound, and regardless all channel parameters, the achievable rate region is within 0.5 bits.
The paper is organized as follows: We present the channel model in Section 2. The achievable rate region by random coding is presented at Section 3. Section 4, by using lattice codes, establishes an achievable rate region for the GCICS. Using numerical examples, achievable rate regions of our proposed scheme and random coding are compared in Section 5. Section 6 concludes the paper.
2 System model
Throughout the paper, random variables and their realizations are denoted by capital and small letters, respectively. x stands for a vector of length n, (x_{1},x_{2},…,x_{ n }). Also, ∥.∥ denotes the Euclidean norm, and all logarithms are with respect to base 2.
A rate pair (R_{1},R_{2}) of nonnegative real values is achievable if there exists a sequence of $\left({2}^{{\mathit{\text{nR}}}_{1}},{2}^{{\mathit{\text{nR}}}_{2}},n\right)$ code such that ${lim}_{n\to \infty}\phantom{\rule{0.3em}{0ex}}{P}_{e}^{\left(n\right)}\to 0.$ The capacity region is defined as the convex closure of the set of all achievable rate pairs (R_{1},R_{2}).
3 Achievable rate region by random coding
In this Section, we evaluate achievable rate regions by random coding for the GCICS in the regime of high state power. In [16], by using random coding, two inner bounds for the GCICS are provided when a≤1. By evaluating the inner bound 1 of Proposition 3 in [16] (and replacing $\mathit{S}\to a\mathit{S},\phantom{\rule{1em}{0ex}}b\to a,\phantom{\rule{1em}{0ex}}c\to \frac{1}{a}$), we can see that this inner bound when the channel gain tends to zero vanishes, and thus, we cannot achieve any positive rate region by such scheme. The following theorem presents the second inner bound. To achieve this region, the GelfandPinsker coding and rate splitting in transmitter 2 is used.
Lemma 1
Proof
See Proposition 4 in [16].
4 Lattice alignment
4.1 Lattice definitions
Here, we provide some necessary definitions on lattices and nested lattice codes. The reader can find more details in [34, 41, 44].
Definition 1
Definition 2.
Definition 3.
Definition 4.
where $V=\underset{\mathcal{V}\left(\Lambda \right)}{\int}d\mathit{x}$ is the Voronoi region volume, i.e., $V=\text{Vol}\left(\mathcal{V}\right)$.
Definition 5.
that maps x into a point in the fundamental Voronoi region.
Definition 6.
The sequence is indexed by the lattice dimension n. The existence of such lattices is shown in [45, 46].
Definition 7.
and, for a fixed volumetonoise ratio greater than 2π e, $\text{Pr}\left\{\mathit{Z}\notin {\mathcal{V}}^{n}\right\}$ decays exponentially in n.
Poltyrev showed that sequences of such lattices exist [47]. The existence of a sequence of lattices Λ^{(n)} which is good in both senses (i.e., simultaneously are Poltyrevgood and Rogersgood) has been shown in [46].
Definition 8.
(Nested lattices): A lattice Λ is said to be nested in lattice Λ_{1} if Λ⊆Λ_{1}. Λ is referred to as the coarse lattice and Λ_{1} as the fine lattice.
Note that if $a\in \mathbb{Z}$, then always a Λ⊆Λ.
Definition 9.
Definition 10.
In the following, we present a key property of dithered nested lattice codes.
Lemma 2.
The Crypto Lemma [34, 48]. Let V be a random vector with an arbitrary distribution over ${\mathbb{R}}^{n}$. If D is independent of V and uniformly distributed over , then (V+D) mod Λ is also independent of V and uniformly distributed over .
Proof.
See Lemma 2 in [48].
Before presentation of our proposed scheme, we prove the following lemma that plays an important role in the proof of achievable rate region by lattice codes.
Lemma 3.
Proof.
where (11) is based on the fact that Λ⊆Λ_{1}. Now, by comparing (10) and (12), the proof of the lemma is complete.
4.2 Our proposed scheme
In this section, we obtain an achievable rate region using lattice codes for the GCICS. If we use the common encoding and decoding as it is explained in [34], then similar to random coding, we cannot achieve the capacity region within a constant gap. Thus, we require to introduce a new scheme for this channel. For presenting this scheme, we use two modulo operations at the decoder. Then using Lemma 3, we interchange modulo operations. As we will see, this scheme can achieve the capacity region at high SNRs and within 0.5 bits regardless of all channel parameters. In the following, we present our scheme in more detail.
A method to obtain a rate region is achieving two corner points of that region. Then, by time sharing between two corner points, we can achieve a rate region. Suppose that V_{1} and V_{2} are two lattice codewords that carry the information for user 1 and 2, respectively. We use DPC or a lattice scheme to decode V_{2} at decoder 2 and a scheme which estimates linear combination of the common message at both decoders to decode V_{1} for both users. In the following, we explain both schemes in more details.

Sending the private message, V_{2} (for decoder 2):

Encoding the common message, V_{1}:
where D_{1} and D_{2} are two independent dithers which are uniformly distributed on the Voronoi region . Note that by the crypto lemma, we know that the power constraint is satisfied. Now, we explain decoding at decoder 1 and 2.

Decoding the common message, V_{1}, at decoder 1
Note that to decode [(1+a) q V_{1}]mod Λ, since we have used a quantizer associated with lattice q Λ_{1}, we may map some points of lattice q Λ_{1} to [(1+a) q V_{1}]mod Λ. That’s by finding some points of q Λ_{1}, since we used a onetoone mapping, we can recover [(1+a)q V_{1}]mod Λ.
where (25) follows from (24), and (26) is based on Rogers goodness of Λ.
Now, we have ${\mathit{V}}_{1}^{\prime}\stackrel{\u25b3}{=}\left[\left(1+a\right)q{\mathit{V}}_{1}\right]\phantom{\rule{1em}{0ex}}\text{mod}\phantom{\rule{1em}{0ex}}\Lambda $ and must try to decode V_{1}. In the following lemma, we show that it is possible to decode it correctly.
Lemma 4.
if Λ≠a Λ_{1}.
Proof.

Decoding the common message, V_{1}, at decoder 2:
Remark 1
By comparing (31) with (15), we can see that at encoder 1, instead of sending V_{1}, we transmit −V_{1}. At the decoder, instead of decoding ${\mathit{V}}_{1}^{\prime}=\left[\left(q+p\right){\mathit{V}}_{1}\right]\phantom{\rule{1em}{0ex}}\text{mod}\phantom{\rule{1em}{0ex}}\Lambda $, we find ${\mathit{V}}_{1}^{\prime}=\left[\left(q+p\right){\mathit{V}}_{1}\right]\phantom{\rule{1em}{0ex}}\text{mod}\phantom{\rule{1em}{0ex}}\Lambda $. But since p≥0 and q≤0, thus p−q≥−q which enables us to estimate V_{1} correctly. Note that for the case −1≤a≤0, if we estimate [(q+p)V_{1}] mod Λ, since p+q≤−q, we cannot find the desired lattice point correctly.
4.2.1 Rateregion outer bound
For comparison, an outer bound on the capacity region of the GCICS is provided. This outer bound is similar to the bound provided in [16] obtained using a different approach.
Lemma 5.
where the union is taken over all parameters 0≤ρ_{21} and ρ_{2s}≤1 such that ${\rho}_{21}^{2}+{\rho}_{2s}^{2}\le 1$.
Proof.
4.3 Capacity results
By comparing the outer region (33) and the achievable region in (29), we conclude that the outer region is indeed tight at high SNRs for the weak and moderate interference case in the high state power regime. Thus, we have the following Corollary
Corollary 1.
where o(1)→0 as $\frac{P}{N}\to \infty $.
where (38) is based on the fact that the maximum gap occurs at $\frac{1}{{a}^{2}+1}+\frac{P}{N}=1$ for i=1,2. Thus, we have the following result.
Theorem 1
The capacity region of the statedependent Gaussian cognitive interference channel for the weak and moderate interference case in the high state power regime is achievable within 0.5 bits.
5 Numerical results
6 Conclusions
In this paper, the statedependent Gaussian cognitive interference channel in the weak and moderate interference case and in the high state power regime is studied. First, we showed that the achievable rate by random coding, which is based on GelfandPinsker coding, vanishes under a condition over the channel gain. Then, we showed that a scheme that is based on lattice codes can achieve the capacity region at high SNRs and within 0.5 bits of the outer bound for all channel parameters.
Authors’ information
The authors are members of IEEE.
Declarations
Acknowledgments
This work has been supported in part by the Iran NSF under Grant No. 88114.46. The material in this paper was presented in part at the 2nd Iran Workshop on Communication and Information Theory (IWCIT 2014), Tehran, Iran.
Authors’ Affiliations
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