The inner and outer bounds on the capacity of the 3-user GBC-SKT with unequal state variances

The “Gaussian BroadCast channel (GBC) with States non-causally Known at the Trans-mitter” (GBC-SKT) introduces Gaussian broadcast channel with additive i.i.d. Gaussian states non-causally known at the transmitter. In this model additive interferences on links are considered as states which are only non-causally known to the transmitter. Such cases are encountered in, e.g., wireless multiple-antenna multicasting scenario. Previously the approximate capacity of the K-user GBC-SKT has been studied where all states but one are assumed to have the same variance. In this work, the capacity of the 3-user GBC-SKT channel communicating only common messages is studied. All three states are assumed to have different variances. The key point is how to simultaneously pre-code the common message against three states. Two transmission strategies are proposed by combination of power-sharing and time-sharing methods and two inner bounds are derived. From ana-lytical and simulation results it is seen that one inner bound outperforms the other inner bound. Moreover, the outer bound on the capacity of this channel is driven and the conditions on the state variances for which, the capacity of the 3-user GBC-SKT channel is derived to within 1 bpcu, are extracted.


INTRODUCTION
In the 'Writing on Dirty Paper' (WDP) channel [1], a channel with output Y = X + S + Z was examined, the state S and the noise Z are Gaussian random variables. If S is known to the encoder, it does not reduce the capacity of the channel and the capacity is shown to be the capacity of a standard Gaussian channel, even though S is unknown to the receiver.
The same coding technique as the one used for the WDP channel has also been used to attain the capacity of the Gaussian BroadCast (GBC) channel [2,3] which is a degraded Broad-Cast (BC) channel. The 'GBC with States non-causally Known at the Transmitter' (GBC-SKT) is the combination of the GBC and Costa's WDP channels. The 2-user GBC-SKT is studied in [4] where inner bounds combining Marton coding [5] and Gelf'and-Pinsker binning [6] are derived. The authors of [4] also derived the capacity for the model with statistically equivalent channel outputs in which only private messages are sent over the channel. on no interference decoding fails to achieve the capacity region. The achievable rate region is derived based on multiple description coding wherein the encoder transmits a common description as well as multiple dedicated private descriptions to the many possible channel realizations of the users. It is shown that MD coding yields larger inner bounds than the single description scheme for a class of compound MISO BC.
In [11] closed-form capacity bounds for an exponential version of the dirty paper channel is presented. This channel can be used to model the non-coherent Gaussian dirty paper channel. First, a superposition modulation-like technique is used to obtain a capacity upper bound, then a closed-form capacity lower bound is obtained by using a Costa-like strategy.
In [12] we derived the approximate capacity for the 2-user GBC-SKT channel considering the states with unequal variances and unequal channel gains, communicating both common and private messages. For the inner bound we divided the user message into two parts. For sending one part, interference is considered as noise while for sending the other part, the codewords are pre-coded against the states and they will be sent by the power-sharing strategy. For the outer bound we used the same idea as [9], the capacity should be decreasing in the state variance, but for the assumption of independent states with unequal variances. We obtained the gap between inner and outer bounds at most 1.25 bpcu for all channel parameters and independent Gaussian distributed channel states.
In [13] we extended the result of [12] to obtain the approximate capacity for the K -user GBC-SKT channel in which (K − 1) of K channel outputs are statistically equivalent, that is, they have the same channel gain and the same variance for the channel states. For this model we showed the approximate capacity is achieved to within 1.5 bpcu for all channel parameters and independent Gaussian distributed channel state information.
Here, we derive inner and outer capacity bounds on the 3user GBC-SKT channel communicating common messages to three users. We assume the channel states to have different variances. We propose an outer bound and two inner bounds. For the inner bounds we propose two new approaches which use combinations of time-and power-sharing methods. The difference is about how and where interference can be considered as noise and when we can pre-coded the message against states. We obtain the conditions for which inner and outer bounds are within 1 bpcu. Finally we present some simulation results which compare the performance of proposed inner bounds and the gap between the inner and outer bounds. According to our knowledge there is no complete work on such problems previously. It is due to the complexity of the encoding strategies when we have states with more than two different variances. This problem arises when we wish to send common message to three users. For better understanding and illustration in this study we do not consider communicating private messages to users and focus only on communicating common messages.
The remainder of the paper is organized as follows: Section 2 presents the notations and the channel model. Section 3 reviews the relevant results available in the literature. Section 4 presents the inner and outer capacity bounds for the 3-user The K -user Gaussian broadcast channel with states known at the transmitter GBC-SKT channel communicating common messages which considers states with unequal variances. In Section 5 we present some simulation results. Section 6 concludes the paper.

NOTATIONS AND THE CHANNEL MODEL
Throughout we use the same notations and the channel model as those defined in [13]. For completeness we rewrite them in this section.
[1; n] for n ∈ ℤ stands for the set of integers {1, … , n}. ⌊a⌋ stands for the largest integer number which is not greater than a.
[a] + stands for max(a, 0). Random variables are denoted by capital letters, for example, X . Vectors with length N are shown with capital letters and a superscript which denotes the vector length, for example, X N stands for the sequence of letters [X 1 , X 2 , … , X N ].
The K -user (GBC-SKT is depicted in Figure 1. The channel outputs are obtained as and Z N k , an iid Gaussian random variable with zero mean and variance N , at all receivers, that is, for all k, and S N 1 , … , S N K are iid jointly Gaussian vectors with zero mean and variance Q k and S N i ⟂ S N j , i ≠ j . Note that a k 's are taken positive without loss of generality: since both the state and the noise distributions are symmetric, the sign of the channel gain can be chosen freely. Additionally, the channel input sequence X N is subject to the power constraint: Having non-causal knowledge of the channel states, the transmitter wishes to reliably communicate the common message W 0 ∈ [1; ⌊2 NR 0 ⌋] to all receivers and the private message W k ∈ [1; ⌊2 NR k ⌋] to receiver k ∈ [1; K ].

PREVIOUS RESULTS
In this section we only review previous results of the author in [13]. We have given complete review in that reference.

2-user GBC-SKT channel
The 2-user GBC-SKT with R 0 ≠ 0, R 1 ≠ 0, R 2 ≠ 0, and a 1 ≥ a 1 = 1 has been considered. The approximate capacity of this channel has been determined. The results is stated in the following theorem: for ∈ [0, 1], = 1 − , and Q = max{Q 1 , Q 2 }. The variance of noise is assumed to be equal to 1. The exact capacity is to within a gap of 1.25 bpcu from the outer bound shown in (4).

K-user GBC-SKT channel
The approximate capacity of the general K -user GBC-SKT with two different values for channel gains and two different values for the variances of independent states, that is, we consider the following assumptions,

FIGURE 2
A schematic representation of the result in Theorem 3.1

Theorem 3.2. [13, Thm. 4] The approximate capacity for a special class of the K -user GBC-SKT with independent states:
Consider a special class of the K -user GBC-SKT with independent states satisfying (5), then an outer bound to the capacity of this channel is obtained as The variance of noise is assumed to be equal to 1. The exact capacity is to within a gap of 1.5 bpcu from the outer bound in (6).
For the outer bound we use the fact that the capacity should be decreasing in states as suggested in [14] to obtain a tighter outer bound than that proposed in [7] and [15].
For the inner bound, we proved the achievability of curves AA ′ A ′′ , BB ′ B ′′ , and point C as shown in Figure 2 and then considered the total rate region as time sharing between these regions.
We allowed the "pre-coded against state" codewords to have different power depending on state powers and channel gains, which attains higher rates than simple time-sharing strategy.
The gap between curve BB ′ B ′′ and the outer bound is 1.25 bpcu, and the gap between point C and the outer bound (4c) is 2 bpcu.
For the curve AA ′ A ′′ , which is shown with the green color in Figure 2, we obtain the exact capacity. This result includes the result of [4, Theorem 2] which only considered communicating private messages.
As is seen, the difficult point to not achieve the capacity is in communicating the common message to the users, so, in this paper, to more clarify the problem, we only focus on the transmission of common messages.
In [13] we considered two different variances for channel states and we divided the power P between two codewords as [P − Q] + and min{P, Q} to send the common message. In this paper we consider three different variances for channel states and we need to do more complex power dividing between codewords as explained in the next section. Remarks: 1. We can also use Theorem 3.2 for the case (6), it means that we consider the worst case and thus the rate that is achievable for the worst case is also achievable for the other cases. For example, for 3-user GBC-SKT with Q 1 < Q 2 < Q 3 communicating only common messages, by and also considering noise variance N ≠ 1 in (6), the following rate is achievable: We use (7) for comparison in simulation section in this paper. We also use the rate obtained by the time-sharing method for comparison which is always an inner bound for such problems.

THE CAPACITY RESULT FOR THE 3-USER GBC-SKT
In this section we derive two inner bounds and one outer bound on the capacity of the 3-user GBC-SKT communicating common messages to three users, that is, R 1 = R 2 = R 3 = 0, and we assume R 0 = R in the rest of document. The channel gains are assumed to be one, that is, a 1 = a 2 = a 3 = 1. We assume Gaussian distributed independent states to have different variances. Without loss of generality, we consider Q 1 < Q 2 < Q 3 .
Theorem 4.1. The first inner bound: For the 3-user GBC-SKT with independent states with unequal variances, the following rate is achieved: Proof. The transmission strategy consists of both one timesharing and two power-sharing phases. We divide the transmission time unit into three parts with duration , ∕2, and ∕2. At duration we divide power P into two parts [P − Q 3 ] + and min{P, Q 3 }. Note that The strategy is graphically represented in Figure 3.
1. At duration , we transmit power P with two codewords X N SAN3 with power [P − Q 3 ] + , X N PAN3 with power min{P, Q 3 }. (SAN stands for 'state as noise' and PAN stands for 'pre-coded against the state sequences'). 2. At duration ∕2, we transmit power P with three codewords Note that here we again transmit X N PAN1 with power min{P, Q 2 }, no need to use Q 1 . The reason was illustrated in [12] for the 2-user GBC-SKT channel.
The codewords are defined as follows: 1) The power-sharing codeword for three users, X N SAN3 , with power [P − Q 3 ] + , carries the message W SAN3 with rate R SAN3 which treats all state sequences as additional noise. Since the state variance at the third receiver is the largest, the codeword X N SAN3 can be decoded at all users if its rate is The coding scheme I for sending common message for K = 3 limited by 2) The power-sharing and time-sharing codeword for users 1 and 2, X N SAN2 with power [min{P, Q 3 } − Q 2 ] + , carries the message W SAN2 with rate R SAN2 , transmitted in of time and treats state sequences S 1 , S 2 as additional noise. Since the state variance at the second user is larger than the state variance at the first user, the codeword X N SAN2 can be decoded at users 1, 2 if its rate is limited by, 3) The power-sharing and time-sharing codewords, X N PAS3 , X N PAS2 , and X N PAS1 , with powers min{P, Q 3 }, min{P, Q 2 }, and min{P, Q 2 }, which are pre-coded against the state sequences S N 3 , S N 2 , and S N 1 , and are transmitted on , ∕2, and ∕2 portions of the three channel uses, respectively, where = 1 − . This strategy attains the following achievable rate, According to Figure 3, it is seen that at duration , the following rate is achievable: at duration , the following rate is achievable: By increasing , (14) increases and (15) decreases, so maximum of the achievable rate at any time occurs when R 1 = R 2 which results 2 log and by solving (16), is obtained as shown in (17).

Theorem 4.2. The second inner bound:
The following rate is achievable for the 3-user GBC-SKT with independent states with unequal variances: where is as shown in (21).
Proof. The transmission strategy consists of one power-sharing and two time-sharing phases. We divide power P into three parts [P − Q 3 ] + , [min{P, Q 3 } − Q 2 ] + , and min{P, Q 3 }. We consider two divisions for the transmission time unit: ( , ) and (1∕3, 1∕3, 1∕3). The strategy is graphically represented in Figure 4. The codewords are defined as follows: 1) The power-sharing codeword, X N SAN3 with power [P − Q 3 ] + , carries the message W SAN3 with the following rate that should be decoded at three users.
2) The power-sharing and time-sharing codeword X N PAS3 0 for user 3 with power [min{P, Q 3 } − Q 2 ] + which is pre-coded against the state sequence S N 3 , and is transmitted in portion of time.
that should to be decoded at user 3.
3) The power-sharing and time-sharing codeword for users 1 and 2, X N SAN2 with power [min{P, Q 3 } − Q 2 ] + , is transmitted in portion of time with following rate that should to be decoded at users 1 and 2. 4) Three power-sharing and time-sharing codewords, X N PAS1 , X N PAS2 , and X N PAS3 , each with power min{P, Q 2 }, which are pre-coded against the state sequences S N 1 , S N 2 , and S N 3 , respectively, and they are transmitted on 1 3 portions of the transmission time unit with the following rate, According to Figure 4, It is seen that at duration , the following rate is achievable: and at duration , the following rate is achievable: Increasing increases (27) but decreases (28), so to have the maximum total rate, should be chosen such that R 1 = R 2 which results 2 log which yields (21) for .

Remarks:
1) For P < Q 2 , we have Substituting them in (9) and (20) yields the time-sharing rate, (10) and (21)  . Substituting them in (9) and (20) yields R 1 and R 2 , respectively, As we see from (32), R 2 > R 1 , that is, the second inner bound is greater than the first one for this case. 4) It is not possible to compare directly the two proposed inner bounds because of their complexities which is mostly due to the fractional expressions for parameters and . In Section 5, through some simulation results, we see that the second inner bound outperform the first inner bound. We can justify this fact by saying that for the codewords with power min{P, Q 2 } in the derivation of the second inner bound we obtain the same rate for three users which is seemingly the better strategy. It is also concluded from (32).

Theorem 4.3. The outer bound: The capacity of the 3-user GBC-SKT with independent states with unequal variances is upper bounded by
Proof. Applying Fano's inequality yields, For the positive entropy term in (34) we have, For the negative entropy term in (34) we have, In the above, (36a) follows from the fact that conditioning reduces entropy. Equation (36b) follows from the fact that the determinant of the following transformation matrix is equal to one.

Remarks:
In the following we consider two special classes of the 3-user GBC-SKT channel for which the gap between inner and outer bounds is 1 bpcu.
1. For Q 1 Q 2 ≥ 1 and (Q 1 + Q 2 ) ≥ (Q 3 + 2), the negative entropy term, (36d), simplifies to, Equations (38a) and (35d) yield the following outer bound, For P ≥ Q 3 , (39a) reduces to, For P < Q 3 , (39a) reduces to, Combining two cases together we have the following equivalent form for the outer bound (39a), This rate can be achieved by sending one codeword with power [P − Q 3 ] + which considers state as noise and the other codeword with power min(P, Q 3 ) pre-coded against each state and transmitted in 1 3 part of time. Comparing with the second inner bound, (20), it is seen that for = 0, we can achieve the capacity within 1bpcu.

For
Equations (35a) and (43) yield the following outer bound, Comparing with the second inner bound, (20), it is seen that for = 2 3 , we can achieve the capacity within 1bpcu. Note that subject to Q 3 ≥ Q 2 ≥ Q 1 , the condition ).

SIMULATION RESULTS
In this section we evaluate the proposed inner and outer bounds through simulations. We assume N = 1. At each experiment we change one of the parameters (P, Q 1 , Q 2 , Q 3 ) and fix the others. We compare proposed inner bounds, (9) and (20) with proposed outer bound, (33), and also with previous inner bounds, (7) and (8).
We consider three following scenarios: 1) P = 100, Q 1 = 1, Q 2 = 10: By changing Q 3 from 1.01Q 2 to 0.99P, we sketch the inner bounds in Figure 5a. As previously discussed when Q 3 ≈ Q 3 , two inner bounds and (7) will be nearly the same. From Figure 5a we see that as Q 3 grows the gap between these bounds increases. The time-sharing method and the second inner bound yield the lowest and highest achievable rates, respectively. In Figure 5b we depict two proposed inner bounds and the outer bound. In Figure 5c the gap between the proposed outer bound and each of the proposed inner bounds is shown. We see that the gap is obtained at most 0.5 bpcu for this experiment.

Gap (bpcu)
Gap between outer bound and inner bound II Gap between outer bound and inner bound I (c) The gap between the proposed outer and each of the inner bounds versus Q 3

FIGURE 5
The results for the 3-user GBC-SKT channel versus Q 3 2) P = 100, Q 1 = 1, Q 3 = 50: By changing Q 2 from 1.01Q 1 to 0.99Q 3 , we sketch the inner bounds in Figure 6a. As we see from this figure, the second inner bound yields the highest achievable rate. The other point is that three inner bounds yield nearly the same rate as Q 2 approaches Q 3 . On the other hand as Q 2 approaches Q 1 the difference between two proposed inner bounds decreases. We also depict the proposed outer and inner

FIGURE 6
The results for the 3-user GBC-SKT channel versus Q 2 bounds in Figure 6b and the gap between the proposed outer bound and each of the proposed inner bounds in Figure 6c. As we see, the gap is obtained at most 0.3 bpcu for this experiment.
3) Q 1 = 1, Q 2 = 10, Q 3 = 50: By changing P from 1.01 Q 3 to 10 Q 3 , we sketch the inner bounds in Figure 7a. As we see from this figure, when P grows, inner bounds increase and again the second proposed inner bound outperforms the others. We also depict the proposed outer and inner bounds in Figure 7b and the gap between the proposed outer bound and each of the proposed inner bounds in Figure 7c. As we see, the gap obtained is at most 0.3 bpcu for this experiment.

CONCLUSION
It this work we investigated the capacity of the 3-user Guassian broadcast channel with states known non-causally at the transmitter (GBC-SKT). The variances of the states are assumed to be different. We study the case where the transmitter communicates only common messages to the users. In fact the difficult point to not achieve the capacity of the GBC-SKT channel is in communicating common messages. The key point in deriving the inner bound is how to pre-code the common message against the states which do not have the same variance.
Here we derived two inner bounds and one outer bound on the capacity of the 3-user GBC-SKT. The novel point was in defining power-sharing and time-sharing codewords and arranging them such that we get to the higher achievable rates. Simulation results show that one of the proposed inner bounds always outperforms the other inner bound. We also find the conditions for which the gap between inner and outer bounds are at most 1 bpcu.