On MISO Broadcast Channel With Alternating CSIT and Secrecy Constraints arXiv:1503.06333v1 [cs.IT] 21 Mar 2015
Zohaib Hassan Awan, Abdellatif Zaidi, and Aydin Sezgin
Abstract We study the problem of secure transmission over a two-user Gaussian multi-input single-output (MISO) broadcast channel under the assumption that the state of the channel to each receiver is conveyed either perfectly (P) or with delay (D) to the transmitter. Denoting by S1 and S2 the channel state information at the transmitter (CSIT) of user 1 and user 2, respectively, the overall CSIT can then alternate between four possible states, i.e., (S1 , S2 ) ∈ {P, D}2 . We denote by λS1 S2 the fraction of time during which the state S1 S2 occurs and focus on the symmetric case such that λS1 S2 = λS2 S1 . Under these assumptions, we first consider the Gaussian MISO wiretap channel and characterize the secure degrees of freedom (SDoF). Next, we generalize this model to the two-user Gaussian MISO broadcast channel and establish bounds on the SDoF region. The analysis reveals that alternating CSIT allows synergistic gains in terms of SDoF; and shows that, by opposition to encoding separately over different states, joint encoding across the states enables strictly better secure rates.
I. Introduction In cellar networks, multiple nodes communicate with each-other over a shared wireless medium. Due to the broadcast and superposition nature of the wireless medium, simultaneous transmission of information over this channel emanates an important issue of interference in networks. As the communication network grows, and since due to scarcity of available resources for example, radio spectrum and available power, the detrimental effect of interference is unavoidable. A key resource that helps mitigating the effect of interference more efficiently is the availability of CSIT. In the literature, different multi-user networks are studied under ideal assumption of perfect CSIT in [2] (and references therein), where quality of CSIT plays a major role in aligning or canceling interference in networks. Recently, a growing body of research has attracted attention to study a wide variety of two-user CSIT models, e.g., with strictly causal (delayed) CSI in [3]–[7], no CSIT in [8], [9] and with mixed CSIT (perfect delayed CSI along with imperfect instantaneous CSI) in [10], [11], all from degrees of freedom (DoF) perspective. The DoF metric basically Zohaib Hassan Awan and Aydin Sezgin are with Institute of Digital Communication Systems, Ruhr-Universit¨at Bochum, 44780 Bochum, Germany. Email: {zohaib.awan, aydin.sezgin}@rub.de Abdellatif Zaidi is with Universit´e Paris-Est Marne-la-Vall´ee, 77454 Marne-la-Vall´ee Cedex 2, France. Email:
[email protected] The result in this work was presented in part at the IEEE International Symposium on Information Theory, Honolulu, USA, Jun.-Jul. 2014 [1].
1
measures the way how capacity scales asymptotically with logarithm of signal-to-noise ratio (SNR). In all these models, it is assumed that symmetric CSI is available at the transmitter, i.e., either perfect, delayed or no CSI is conveyed by both receivers. In [12], Tandon et al. studied a two-user broadcast channel with asymmetric CSI conveyed to the transmitter. In this model, the channel to one receiver is available instantaneously at the transmitter, while the channel to the other receiver is conveyed with some delay. The authors refer to this model as being one with partially perfect CSIT and they characterize the DoF region. Due to the random fluctuations in the wireless medium, it becomes difficult for the receivers to convey the same quality of CSI over time. In another related work [13], Tandon et al. studied the twouser broadcast channel by taking time varying nature of CSIT into account. Among other constraints, the authors assumed that the CSI conveyed by both receivers can vary over time, and each receiver is allowed to convey either perfect, delayed or no CSIT to the transmitter in an asymmetric manner. For this channel model, they characterize the full DoF region. As said before, in wireless networks, due to the broadcast nature of the medium, information exchange between two communicating parties can be overheard by other nodes of the network for free. The adversaries (eavesdroppers) listen to this communication and can try to extract some useful information from it. In his seminal work [14], Wyner introduced a basic information-theoretic model to study secrecy by taking physical layer attributes of the channel into account. Wyner’s wiretap channel consists of three nodes — a source, a legitimate receiver and an eavesdropper. The source wants to transmit a confidential message to the legitimate receiver and wishes to conceal it from an eavesdropper. For the degraded wiretap channel, in which the channel from the source-to-legitimate receiver is stronger than the one from source-to-eavesdropper, secrecy capacity is established. The wiretap channel introduced by Wyner is extended to study a variety of multi-user networks namely, the broadcast channel [15], [16], multi-access channel [17]–[21], relay channel [22]–[24], interference channel [25], [26], and multi-antenna channel [27]–[30]. The interested reader may also refer to [31], for a review of other related contributions. Characterizing the secrecy capacity region of these models fully can be very challenging in general. At high SNR regime, similar to DoF the notion of SDoF captures the asymptotic behaviour of the data rates that are allowed securely. More specifically, it shows how the secrecy capacity prelog or spatial multiplexing gain scales asymptotically with logarithm of SNR. In [27], Khisti et al. study a Gaussian multi-input multi-output (MIMO) wiretap channel in which perfect CSI of the legitimate receiver and eavesdropper is available at the transmitter; and establish the secrecy capacity as well as the SDoF. In [32], Liu et al. generalize the model in [27] to the broadcast setting and characterize the secrecy capacity region. For the two-user (2,1,1)–MISO broadcast channel the optimal sum SDoF is 2, and is obtained by zero-forcing the confidential messages at the unintended receivers. Yang et al. in [33] study the MIMO broadcast channel and show that strictly causal (delayed) CSI is still useful from SDoF perspective in the sense that it enlarges the secrecy region, in comparison with the same setting but with no CSIT. The coding scheme in [33] follows by an appropriate extension of Maddah Ali-Tse scheme [3] with additional
2
noise injection to account for security constraints. Recently, in [34], Zaidi et al. study the MIMO X-channel with asymmetric feedback and delayed CSIT and characterize the corresponding SDoF region. Despite of all these important recent advances for models with delayed CSI at transmitters, settings in which CSI from different receivers are observed with different delays, are still not fully understood. The model that we study in this work can be seen as a step towards better understanding this type of models. In this work, we consider a two-user Gaussian MISO broadcast channel in which the transmitter is equipped with two antennas, and each of the two receivers is equipped with a single antenna as shown in Figure 1. The transmitter wants to reliably transmit messages W1 and W2 to receiver 1 and receiver 2, respectively. In investigating this model we make three assumptions, namely, 1) the communication is subjected to a fast fading environment, 2) each receiver knows the perfect instantaneous CSI and also the CSI of the other receiver with a unit delay, and 3) the channel to each receiver is conveyed either instantaneously (P) or with a unit delay (D) to the transmitter. In both cases, it assumed that the CSI is perfect. Thus, the CSIT vector that is gotten at the transmitter from the two receivers can alternate among four possible states, PP, PD, DP and DD. Furthermore, the transmitter wants to conceal the message W1 that is intended to receiver 1 from receiver 2; and the message W2 that is intended to receiver 2 from receiver 1. Thus, each receiver plays two different roles, being at the same time a legitimate receiver of the message that is destined to it, and an eavesdropper of the message that is destined to the other receiver. We assume that both eavesdroppers are passive, i.e., they are not allowed to modify the communication. The model that we study can be seen as similar to the one in [13] but with imposed security constraints. We consider the case of perfect secrecy and focus on the asymptotic behavior of this model, where system performance is measured by SDoF. The main contributions of this work are summarized as follows. We first consider a (2,1,1)-MISO wiretap channel with alternating CSIT, and characterize fully the optimal SDoF for this model. The coding scheme in this case is based on an appropriate combination of schemes that we develop for fixed CSIT configurations, namely, PP, PD, DP states and the one that is developed previously for DD state in [33]. The converse proof follows by extending the proof of [33] developed in the context of wiretap channel with delayed CSIT to the case with alternating CSIT; and, also, uses some elements from the converse proof of [13] established for the broadcast model with alternating CSIT by taking imposed security constraints into account. We note that, our result for the MISO wiretap model is not restricted to symmetric case, i.e., λS1 S2 = λS2 S1 and holds in general. Next, we consider the broadcast setting with alternating CSIT that is shown in Figure 1. We establish inner and outer bounds on the SDoF region of this model. As part of the main ingredients that we employ for proof of the inner bound, we develop some elementary coding schemes that can be seen as an appropriate generalization of those in [13], tuned carefully so as to account for the imposed secrecy constraints. The proof of our outer bound follows by carefully extending our proof for the MISO wiretap model to the broadcast setting. The outer and inner bounds that we have constructed do not agree in
3
Fig. 1.
Two-user MISO broadcast channel with alternating CSIT, and security constraints.
general; however, for the special case in which perfect CSI or strictly causal CSI is conveyed by both receivers we recover the SDoF region in [32] and [33], respectively. Although non-optimal in general, the results of this paper show that, for the MISO broadcast channel that we study, alternating CSIT not only enables interesting synergistic gains in terms of degrees of freedom in the case without secrecy constraints as was shown in [13], but also if secrecy constraints are imposed on the communication. At this point, we highlight the key differences between the result in this paper and a similar work which was independently done in parallel in [35]. In [35], the authors studied a MISO broadcast channel with confidential messages in which the transmitter is allowed to alternate between two states, i.e., PD and DP, equal fractions of communication time. The authors characterize the complete secrecy capacity region. As opposed to the model in [35], the model that we study in this paper is more general, since it allows more leverage to the transmitter to choose between four possible states, i.e., PP, PD, DP, and DD. Specializing the results in this work to model studied in [35] reveals that the encoding scheme in [35] outperforms the scheme that we have developed in this work. However, the outer bound that we have established in this work is more general; and, it subsumes the outer bound in [35] and the one developed in the context of broadcast channel with delayed CSIT in [33]. Finally, in comparison to the results in this paper, it is worth noting that the work in [35] does not investigate the model with wiretap setting.
4
We structure this paper as follows. Section II provides a formal description of the channel model that we study along with some useful definitions. Section III states the SDoF of the (2, 1, 1)–MISO wiretap channel. In Section IV, we study the MISO broadcast channel, and extend our results for the wiretap model to the broadcast setting. The formal proof of the coding scheme that we use to establish the inner bound is given in Section V. Finally, in Section VI we conclude this paper by summarizing its contributions. Notations. We will use the following notations throughout this work. Boldface upper case letter X denotes matrices, boldface lower case letter x denotes vectors, and calligraphic letter X designates j
alphabets; at each time instant t, xt denotes [xt1 , . . . , xtn ]. For integers i ≤ j, Xi is used as a shorthand for (Xi , . . . , X j ), ⌈.⌉ denotes the ceiling operator, and φ denotes null set. The Gaussian distribution with mean g(n) n→∞ n
µ and variance σ2 is denoted by CN(µ, σ2 ). The term o(n) is some function g(n) such that lim
= 0.
The dot equality denotes the equality on the prelog factor, such that for some functions f (n) and g(n), f (n) g(n) implies f (n) = g(n) + o(n). Finally, throughout the paper, logarithms are taken to base 2. II. System Model and Definitions We consider a two-user multiple-input single-output (MISO) broadcast channel, as shown in Figure 1. In this setting, the transmitter is equipped with two transmit antennas and the two receivers are equipped with a single antenna each. The transmitter wants to reliably transmit message W1 ∈ W1 = {1, . . . , 2nR1 (P) } to receiver 1, and message W2 ∈ W2 = {1, . . . , 2nR2 (P) } to receiver 2. In doing so, the transmitter also wishes to conceal the message W1 that is intended to receiver 1 from receiver 2; and the message W2 that is intended to receiver 2 from receiver 1. Thus, in the considered system configuration, receiver 2 acts as an eavesdropper on the MISO channel to receiver 1; and receiver 1 acts an eavesdropper on the MISO channel to receiver 2. We assume that both eavesdroppers are passive, i.e., they are not allowed to modify the communication. We consider a fast fading channel model, and assume that each receiver knows the perfect instantaneous CSI and also the past CSI of the other receiver. The channel input-output relationship at time instant t is given by yt = ht xt + n1t zt = gt xt + n2t , t = 1, . . . , n
(1)
where x ∈ C2×1 is the channel input vector, h ∈ H ⊆ C1×2 is the channel vector connecting receiver 1 to the transmitter and g ∈ G ⊆ C1×2 is the channel vector connecting receiver 2 to the transmitter respectively; and ni is assumed to be independent and identically distributed (i.i.d.) white Gaussian noise, with ni ∼ CN(0, 1) for i = 1, 2. The channel input is subjected to block power constraints, as n X E[kxt k2 ] ≤ nP. t=1
(2)
5
For ease of exposition, we denote St =
h i ht gt
as the channel state matrix and St−1 = {S1 , . . . , St−1 } denotes
the collection of channel state matrices over the past (t − 1) symbols respectively. For convenience, we set S0 = ∅. We assume that, at each time instant t, the channel state matrix St is full rank almost surely. At each time instant t, the past states of the channel matrix St−1 are known to all terminals. However, the instantaneous state ht is known only to receiver 1, and the instantaneous state gt is known only to receiver 2. Communication over the wireless channel is particularly sensitive to the quality of CSIT. Although, there are numerous forms of CSIT, in this work we focus on two of them as follows. 1) Perfect CSIT: corresponds to those instances in which the transmitter has perfect knowledge of the instantaneous channel state information. We denote these states by ‘P’. 2) Delayed CSIT: corresponds to those instances in which at time t, the transmitter has perfect knowledge of only the past (t − 1) channel states. Also, we assume that at time instant t the current channel state is independent of the past (t − 1) channel states. We denote these states by ‘D’. Let S1 denotes the CSIT state of user 1 and S2 denotes the CSIT state of user 2. Then, based on the availability of the CSIT, the model that we study (1) belongs to any of the four states (S1 , S2 ) ∈ {P, D}2.
(3)
We denote λS1 S2 be the fraction of time state S1 S2 occurs, such that X
λS1 S2 = 1.
(4)
(S1 ,S2 )∈{P,D}2
Also, due to the symmetry of problem as reasoned in [13], in this work we assume that λPD = λDP , i.e., the fractions of time spent in state PD and DP are equal. Definition 1: A code for the Gaussian two user (2, 1, 1)–MISO broadcast channel with alternating CSIT (λS1 S2 ) consists of sequence of stochastic encoders at the transmitter, PP ⌉ {φ1t : W1 ×W2 ×St −→ X1 × X2 }⌈nλ t=1
(5)
⌈nλDD ⌉ {φ2t : W1 ×W2 ×St−1 −→ X1 × X2 }t=1
(6)
⌈nλPD ⌉ {φ3t : W1 ×W2 ×St−1 ×Ht −→ X1 × X2 }t=1
(7)
⌈nλDP ⌉ {φ4t : W1 ×W2 ×St−1 ×Gt −→ X1 × X2 }t=1
(8)
where the messages W1 and W2 are drawn uniformly over the sets W1 and W2 , respectively; and two decoding functions at the receivers, ˆ1 ψ1 : Yn ×Sn−1 ×Hn −→ W ˆ 2. ψ2 : Zn ×Sn−1 ×Gn −→ W
(9)
6
Definition 2: A rate pair (R1 (P), R2 (P)) is said to be achievable if there exists a sequence of codes such that, ˆ i , Wi |Wi } = 0, lim sup Pr{W
∀ i ∈ {1, 2}.
(10)
n→∞
Definition 3: A SDoF pair (d1 , d2 ) is said to be achievable if there exists a sequence of codes satisfying following, 1) Reliability condition: ˆ i , Wi |Wi } = 0, lim sup Pr{W
∀ i ∈ {1, 2},
(11)
n→∞
2) Perfect secrecy condition: lim sup n→∞
lim sup n→∞
I(W2 ; yn , Sn ) = 0, n I(W1 ; zn , Sn ) = 0, n
(12)
3) and communication rate condition: lim lim inf
P→∞
n→∞
log |Wi (n, P)| ≥ di , n log P
∀ i ∈ {1, 2}
(13)
at both receivers. Definition 4: We define the SDoF region, CSDoF (λS1 S2 ), of the MISO broadcast channel as the set of all achievable non-negative pairs (d1 , d2 ). III. SDoF of the MISO wiretap channel with alternating CSIT Before proceeding to state the main results, we first digress to provide a useful lemma which we will repetitively use in this work. Lemma 1: For the Gaussian MISO channel in (1), following inequalities hold n n n ˙ 2h(znPD , znDP , znDD |Sn )≥h(y PD , yDD |S ),
(14a)
˙ nDP , znDD |Sn ), 2h(ynPD, ynDP , ynDD |Sn )≥h(z
(14b)
n n n n n n ˙ h(znPD , znDP , znDD |Sn )≥h(y PD , yDD |zPD , zDP , zDD , S ),
(14c)
˙ nDP , znDD |ynPD , ynDP , ynDD , Sn ). h(ynPD , ynDP , ynDD |Sn )≥h(z
(14d)
Proof: The proof of Lemma 1 appears in Appendix I. The inequalities in Lemma 1 also hold with additional conditioning over message W. The following theorem characterizes the SDoF of the MISO wiretap channel with alternating CSIT. Theorem 1: The SDoF of the (2,1,1)–MISO wiretap channel with alternating CSIT (λS1 S2 ) is ds (λS1 S2 ) = 1 −
λDD . 3
(15)
7
Proof: The proof of Theorem 1 appears in Appendix II. Remark 1: The upper bound extends the converse proof of [33, Theorem 1] established in the context of SDoF of wiretap channel with delayed CSIT to the case with alternating CSIT. It also uses some elements from the converse proof of [13] established for the two-user broadcast channel with alternating CSIT by taking imposed security constraints into account. Note that, if delayed channel state information of both receivers is conveyed to the transmitter, i.e., λDD := 1, the outer bound recovers the SDoF of MISO wiretap channel with delayed CSI [33, Theorem 1]. We also notice that, the result established in Theorem 1 is not restricted to symmetric case, i.e., λPD = λDP and holds in general. Remark 2: The achievability proof of Theorem 1 follows by combining appropriately fixed CSIT schemes. It is interesting to note that, for a given SDoF, any fixed CSIT scheme can be fully alternated by other (remaining) fixed schemes. For example, the SDoF of
2 3
can be achieved by completely using
the state DD (λDD := 1) or by using any of PD, DP or PP state, 23 -rd fraction of communication time. IV. SDoF of the MISO broadcast channel with alternating CSIT In this section, before proceeding to state our result on the general model (1) with alternating CSIT, we first consider the (2, 1, 1)–MISO broadcast channel with partially perfect CSIT and establish a lower bound on the SDoF region. Proposition 1: A lower bound on the SDoF region of the two user (2,1,1)–MISO broadcast channel with partially perfect CSIT (PD state) is given by the set of all non-negative pairs (d1 , d2 ) satisfying d1 + d2 ≤ 1.
(16)
Proof: To establish the achievability proof in Proposition 1, it is sufficient to prove the two corner points (1, 0) and (0, 1), since the entire region can then be achieved by time sharing. The achievability of these points follows from the coding scheme that we use to establish the proof of Theorem 1; by choosing PD state point (1, 0) is achievable, where the transmitter is sending information to receiver 1 being eavesdropped by receiver 2. Similarly, the achievability of the other point follows by reversing the roles of both receivers. Figure 2 illustrates the achievable SDoF region of the (2, 1, 1)–MISO broadcast channel with partially perfect CSIT (PD state). For comparison reasons, the figure also shows the DoF region of the similar model without secrecy constraints in [12, Theorem 1]. Although, the optimality of lower bound in (16) is still to be shown, the visible gap between the SDoF and DoF regions in Figure 2 highlights the rate loss incurred due to imposed secrecy constraints. We now turn our attention to consider the MISO broadcast channel with alternating CSIT (λS1 S2 ) and state our main result.
8
1
d2
( 12 , 1)
(1, 12 )
Achievable SDoF with partially perfect CSIT (PD state) (16) DoF with partially perfect CSIT (PD state) [12] DoF with partially perfect CSIT (DP state) [12]
0 0
Fig. 2.
d1
1
SDoF and DoF regions of the (2, 1, 1)–MISO broadcast channel with partially perfect CSIT (PD or DP state) .
A. Outer Bound
The following theorem provides an outer bound on the SDoF region of the MISO broadcast channel with alternating CSIT. Theorem 2: An outer bound on the SDoF region CSDoF (λS1 S2 ) of the two user (2,1,1)–MISO broadcast channel with alternating CSIT is given by the set of all non-negative pairs (d1 , d2 ) satisfying
d1 ≤ ds
(17a)
d2 ≤ ds
(17b)
3d1 + d2 ≤ 2 + 2λPP + 2λPD
(17c)
d1 + 3d2 ≤ 2 + 2λPP + 2λPD .
(17d)
Proof: The outer bound follows by generalizing the converse that we have established in Theorem 1 in the context of the MISO wiretap channel with alternating CSIT to the broadcast setting. The proof of the upper bounds (17a) and (17b) in Theorem 2 follows along the lines of the one established in Theorem 1 and is omitted for brevity. The proof of (17c) and (17d) appears in Appendix III.
9
B. Inner Bound We now establish an inner bound on the MISO broadcast channel with alternating CSIT (λS1 S2 ). For convenience, we first define the following quantity dlow = ds − s
6λPD . 11
(18)
The following theorem provides an inner bound on the SDoF region of the MISO broadcast channel with alternating CSIT. Theorem 3: An inner bound on the SDoF region CSDoF (λS1 S2 ) of the two user (2,1,1)–MISO broadcast channel with alternating CSIT is given by the set of all non-negative pairs (d1 , d2 ) satisfying d1 ≤ ds
(19a)
d2 ≤ ds
(19b)
d2 λPP + λPD d1 + ≤1+ 2 2 dlow s d2 λPP + λPD d1 + low ≤ 1 + . 2 2 ds
(19c) (19d)
Proof: The proof of Theorem 3 appears in Appendix IV. Remark 3: The region established in Theorem 3 reduces to the DoF region of the MISO broadcast channel with alternating CSIT and no security constraints in [13, Theorem 1] by setting ds = dlow := 1 s in (19). For the special case in which instantaneous or delayed CSI is conveyed by both receivers, i.e., λPP := 1 or λDD := 1, respectively, the outer and inner bounds coincide and SDoF region is established. Figure 3 sheds light on the benefits of alternation between states and shows the SDoF regions of DD, partially perfect CSIT (PD state), PP states and the region obtained by alternation between PD and DP states. Although, the optimality of inner bounds in (16) and (19) is still to be shown, it can be easily seen from Figure 3 that alternation between PD and DP states enlarges the SDoF region in comparison to only PD state. This gain also illustrates the fact that, by joint encoding across these states higher SDoF is achievable. Alternation between the states not only provide significant gain in terms of DoF region as previously noted in [13], it also enlarges the DoF region with security constraints. In Figure 4 we compare the DoF region and achievable SDoF region with alternation between PD and DP states. The visible gap between the two regions illustrates the rate loss incurred due to imposed secrecy constraints. Remark 4 (Synergistic Gains in Asymmetric Configurations): In Theorem 3, the inner bound provides synergistic benefits of alternating CSIT under symmetric assumption of λPD = λDP . Like the model without security constraints [13], we note that this gain in terms of SDoF is not restricted to symmetric setting and is also preserved under asymmetric setting, i.e., λPD , λDP . We consider a simple example in which states PD and DP occur λPD = 1/6 and λDP = 5/6 fractions of time, respectively, such that λPD , λDP and λPD + λDP = 1. From (16), it is easy to note that by coding independently over these states, the SDoF of 1
10
1
(1, 1)
0.9 0.8
( 32 , 32 )
0.7
d2
0.6 0.5
( 21 , 21 )
0.4 0.3
Achievable SDoF with partially perfect CSIT (PD state) (16) SDoF with DD state ( [33, Theorem 3], (19))
0.2
Achievable SDoF with alternation between PD and DP states (19) SDoF/DoF with PP state (19)
0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
d1
Fig. 3.
SDoF region of (2, 1, 1)–MISO broadcast channel with alternating CSIT.
1 ( 12 , 1)
( 56 , 65 )
d2
( 23 , 23 )
(1, 12 ) Achievable SDoF with alternation between PD and DP states (19) DoF with alternation between PD and DP states [13, Theorem 1]
0
Fig. 4.
d1
SDoF and DoF regions of (2, 1, 1)–MISO broadcast channel with alternation between PD and DP states.
1
11
is achievable. However, by synergistically using these states one can still obtain higher SDoF as follows. By synergistically using states PD and DP (that gives a SDoF of 4/3 as will be specified later — S4/3 1 scheme) 1/3-rd fraction of time and using DP state in the remaining fraction of time as a separate state, we get 4 2 10 1 ≥1 SDoF = × + × (1) = 3 3 |{z} 3 9 DP |{z}
(20)
S4/3 1
which shows the benefits of alternating CSIT under asymmetric configurations. V. Coding Scheme In this section, we construct some elemental encoding schemes that provide the main building blocks to establish the inner bound of Theorem 3. For simplicity of analysis and in accordance with DoF framework, in this work we neglect the effect of additive Gaussian noise at the receivers. A. Coding scheme achieving 2-SDoF The following scheme achieves 2-SDoF. •
S2 – using PP state, (d1 , d2 ) = (1, 1) is achievable.
Due to the availability of perfect CSI of both receivers, the transmitter can zero-force the information leaked to the unintended receiver. Thus, it can be readily shown that one symbol is securely transmitted to each receiver in a single timeslot, yielding 1-SDoF at each receiver. B. Coding scheme achieving 1-SDoF The following scheme achieves 1-SDoF. •
S1 – using DD state, (d1 , d2 ) = ( 21 , 21 ) is achievable.
For the case in which delayed CSI of both receivers is conveyed to the transmitter, (d1 , d2 ) = ( 21 , 21 ) SDoF is achievable. The coding scheme in this case is established in [33] and so for the sake of completeness, we outline it briefly. In this scheme, the transmitter wants to send two symbols (v1 , v2 ) to receiver 1 and wishes to conceal them from receiver 2; and two symbols (w1 , w2 ) to receiver 2 and wishes to conceal them from receiver 1. The coding scheme consists of four phases each composed of only one timeslot each, and is concisely shown in Table 1. At t = 1, the transmitter injects artificial noise u = [u1 , u2 ] from both antennas where each receiver gets a linear combination of artificial noise denoted by L1 (u) and M1 (u), respectively. At t = 2, the transmitter sends two symbols (v1 , v2 ) intended for receiver 1 along with a linear combination of past channel output of receiver 1 (at t = 1). At the end of second timeslot, since receiver 1 already knows the CSI and also the past channel output, it subtracts out the contribution of y1 from y2 to obtain one equation with two unknowns (v1 , v2 ). Thus, receiver 1 requires one extra
12
Timeslot
1
2
3
x
u := [u1 u2 ]
[v1 v2 ]T + L1 (u)
[w1 w2 ]T + M1 (u)
4 L3 (w1 , w2 , M1 (u)) +M2 (v1 , v2 , L1 (u))
Rx1
y1 = L1 (u)
y2 = L2 (v1 , v2 , L1 (u))
y3 = L3 (w1 , w2 , M1 (u))
y4 = L3 (w1 , w2 , M1 (u)) +M2 (v1 , v2 , L1 (u))
Rx2
z1 = M1 (u)
z2 = M2 (v1 , v2 , L1 (u))
z3 = M3 (w1 , w2 , M1 (u))
z4 = L3 (w1 , w2 , M1 (u) +M2 (v1 , v2 , L1 (u))
TABLE I Yang et al. scheme using DD state [33].
equation to decode the intended symbols, being available as side information at receiver 2 (z2 ). At t = 3, the transmission scheme is similar to the one in timeslot 2 with the roles of receiver 1 and receiver 2 being reversed. In the third timeslot the transmitter sends two symbols (w1 , w2 ) intended for receiver 2 along with a linear combination of channel output of receiver 2 at t = 1. By means of past CSI, the transmitter can learn the side information available at the unintended receivers, i.e., z2 , y3 ; and at t = 4, multicasts a linear combination of them. Following security analysis as will be specified later, it can be readily shown that 2 symbols are securely transmitted to each receiver over a total of 4 time slots yielding 1/2 SDoF at each receiver.
C. Coding schemes achieving 4/3-SDoF The following schemes achieve 4/3 SDoF. 1) S14/3 – using DP, PD states for ( 21 , 21 ) fractions of time, (d1 , d2 ) = ( 32 , 23 ) SDoF is achievable. 2) S4/3 – using DD, DP, PD states for ( 13 , 31 , 31 ) fractions of time, (d1 , d2 ) = ( 32 , 32 ) SDoF is achievable. 2 1) S4/3 — Coding scheme using DP and PD states: In the coding scheme that follows, we highlight the 1 benefits of alternation between the states. We now show that by using PD and DP states, (d1 , d2 ) = ( 23 , 32 ) SDoF is achievable. In this case, transmitter wants to transmit four symbols (v1 , v2 , v3 , v4 ) to receiver 1 and wishes to conceal them from receiver 2; and four symbols (w1 , w2 , w3 , w4 ) to receiver 2 and wishes to conceal them from receiver 1. The communication takes place in six phases, each comprising of only one time slot. In this scheme, the transmitter alternates between different states and chooses DP state at t = 1, 3, 5, and PD state at t = 2, 4, 6. In the first phase the transmitter chooses DP state and injects artificial noise, u = [u1 , u2 ]T . The channel input-output relationship is given by y1 = h1 u,
(21a)
z1 = g1 u.
(21b)
13
At the end of phase 1, the past CSI of receiver 1 is conveyed to the transmitter. In the second phase, utilizing the leverage provided by alternating CSIT model, the transmitter switches from DP to PD state and sends v˜ := [v1 , v2 ]T along with a linear combination of channel output y1 of receiver 1 during the first phase. Due to the availability of past CSI of receiver 1 (h1 ) in phase 1 and since the transmitter already knows u, it can easily re-construct the channel output y1 . During this phase, the transmitter sends v1 y1 x2 = + . (22) v2 φ At the end of phase 2, the channel input-output relationship is given by y2 = h2 v˜ + h21 y1 ,
(23a)
z2 = g2 v˜ + g21 y1 . | {z }
(23b)
interference
At the end of phase 2, receiver 2 feeds back the delayed CSI to the transmitter. Since receiver 1 knows the CSI (h2 ) and also the channel output y1 from phase 1, it subtracts out the contribution of y1 from the channel output y2 , to obtain one equation with two unknowns (v˜ := [v1 , v2 ]T ). Thus, receiver 1 requires one extra equation to successfully decode the intended variables, being available as interference or side information at receiver 2. In the third phase, the transmitter switches from PD to DP state and sends w ˜ := [w1 , w2 ]T and v3 along with a linear combination of channel output z1 of receiver 2 during the first phase. The transmitter can easily re-construct z1 , since it already knows the perfect CSI (g1 ) and u. In phase 3, perfect CSI of receiver 2 (g3 ) at the transmitter is utilized in two ways, 1) it zero-forces the interference at receiver 2 being caused by symbol v3 , and in doing so 2) it also secures symbol v3 which is intended to receiver 1, being eavesdropped by receiver 2. During this phase, the transmitter sends w1 z1 x3 = + + b1 v3 , φ w2
(24)
where b1 ∈ C2×1 is the precoding vector chosen such that g3 b1 = 0. At the end of phase 3, the channel input-output relationship is given by y 3 = h3 w ˜ + h31 z1 +h3 b1 v3 , | {z }
(25a)
z3 = g3 w ˜ + g31 z1 .
(25b)
interference
At the end of phase 3, receiver 1 feeds back the delayed CSI to the transmitter. Since receiver 2 knows the CSI (g3 ) and also the channel output z1 from phase 1, it subtracts out the contribution of z1 from the channel output z3 , to obtain one equation with two unknowns (w ˜ := [w1 , w2 ]T ). Thus, it requires one extra equation to successfully decode the intended variables being available as interference or side information
14
at receiver 1. Receiver 1 gets the intended symbol v3 embedded in with some interference (h3 w ˜ + h31 z1 ) from the transmitter. If this interference can be conveyed to the receiver 1, it can then subtracts out the interference’s contribution from y3 and decodes v3 through channel inversion. At the end of phase 3, due to availability of delayed CSI (g2 , h3 ), the transmitter can learn the interference at receiver 2 in phase 2 and at receiver 1 in phase 3, respectively. In the fourth phase, the transmitter switches from DP to PD state and sends the interference (g2 v˜ + g21 y1 ) at receiver 2 during the second phase and fresh information w3 , where perfect CSI of receiver 1 (h4 ) is utilized to zero-force the interference being caused by symbol w3 at receiver 1. During this phase, the transmitter sends g v˜ + g21 y1 2 + b2 w3 , x4 = (26) φ
where b2 ∈ C2×1 is the precoding vector chosen such that h4 b2 = 0. At the end of phase 4, the channel input-output relationship is given by y4 = h41 (g2 v˜ + g21 y1 ),
(27a)
z4 = g41 (g2 v˜ + g21 y1 ) + g4 b2 w3 .
(27b)
At the end of phase 4, since receiver 1 knows the CSI and also the channel output y1 from phase 1, it subtracts out the contribution of y1 from the channel outputs (y2 , y4 ) and decodes (v1 , v2 ) through channel inversion. Similarly, since receiver 2 knows the CSI and z2 from phase 2, it first subtracts out the contribution of z2 from the channel output z4 and decodes w3 through channel inversion. In the fifth phase, the transmitter switches from PD to DP state and sends the interference (h3 w ˜ + h31 z1 ) at receiver 1 during phase 3 and fresh information v4 to receiver 1, where perfect CSI of receiver 2 (g5 ) is utilized to zero-force the interference being caused by symbol v4 at receiver 2. During this phase, the transmitter sends h3 w ˜ + h31 z1 + b3 v4 , x5 = φ
(28)
where b3 ∈ C2×1 is the precoding vector chosen such that g5 b3 = 0. At the end of phase 5, the channel input-output relationship is given by y5 = h51 (h3 w ˜ + h31 z1 ) + h5 b3 v4 ,
(29a)
z5 = g51 (h3 w ˜ + h31 z1 ).
(29b)
At the end of phase 5, since receiver 2 knows the CSI and also the channel output z1 from phase 1, it subtracts out the contribution of z1 from the channel outputs (z3 , z5 ) and decodes (w1 , w2 ) through channel inversion. Receiver 1 gets the intended symbol v4 embedded within the same interference as in phase 3. If this interference can be conveyed to the receiver 1, it can then subtracts out the interference’s contribution from y5 and decodes v4 .
15
In the sixth phase, the transmitter switches from DP to PD state and sends interference (h3 w ˜ + h31 z1 ) at receiver 1 during phase 3 with fresh information w4 for receiver 2, where perfect CSI of receiver 1 (h6 ) is utilized to zero-force the interference being caused by symbol w4 at receiver 1. During this phase the transmitter sends h3 w ˜ + h z 31 1 + b4 w4 , x6 = φ
(30)
where b4 ∈ C2×1 is the precoding vector chosen such that h6 b4 = 0. At the end of phase 6, the channel input-output relationship is given by y6 = h61 (h3 w ˜ + h31 z1 ),
(31a)
z6 = g61 (h3 w ˜ + h31 z1 ) + g6 b4 w4 .
(31b)
At the end of phase 6, since receiver 1 knows the CSI and by using y6 , subtracts out the contribution of (h3 w+h ˜ 31 z1 ) from the channel outputs (y3 , y5 ) and decodes v3 and v4 through channel inversion. Similarly, since receiver 2 knows the CSI and also z5 , it can then subtracts out the contribution of (h3 w ˜ + h31 z1 ) from channel output z6 and decodes w4 through channel inversion. Security Analysis. At the end of phase 6, the channel input-output relationship is given by h2 h41 g 2 0 y = 0 0 0
0
0
h21
0
0
h41 g21
h3 b1
0
0
0
h5 b3
0
0
0
1
0
0
0
|
g 3 g51 h3 0 z = g h 61 3 0 0 |
{z
}
H∈C6×6
0
0
g31
0
0
g51 h31
g4 b2
0
0
0
g6 b4
g61 h31
0
0
1
0
0
0
{z
G∈C6×6
0 v˜ 0 v3 1 , v 4 h51 h1 u 0 h3 w ˜ + h g u 31 1 h61 | {z } C6×1
0 w ˜ 0 w 3 g41 . w4 0 g u 1 0 g2 v˜ + g21 h1 u 1 | {z } }
(32)
C6×1
(33)
16
The information rate to receiver 1 is given by I(v1, v2 , v3 , v4 ; y|Sn ) and is evaluated as I(v1 , v2 , v3 , v4 ; y|Sn ) = I(v, ˜ v3 , v4 ; y|Sn ) = I(v, ˜ v3 , v4 , h1 u, h3 w ˜ + h31 z1 ; y|Sn ) − I(h1 u, h3 w ˜ + h31 z1 ; y|v, ˜ v3 , v4 , Sn ) (a)
= rank(H) log(P) − 2 log(P)
= 6 log(P) − 2 log(P) = 4 log(P)
(34)
where (a) follows from [33, Lemma 2]. Similarly, the information leaked to receiver 2 is given by I(v1 , v2 , v3 , v4 ; z|Sn ) and can be bounded as I(v1 , v2 , v3 , v4 ; z|Sn ) = I(v, ˜ v3 , v4 ; z|w, ˜ w3 , w4 , Sn ) ≤ I(g2 v, ˜ v3 , v4 ; z|w, ˜ w3 , w4 , Sn ) = I(g2 v, ˜ v3 , v4 , u; z|w, ˜ w3 , w4 , Sn ) − I(u; z|g2 v, ˜ w, ˜ w3 , w4 , S n ) ≤ I(g2 v˜ + g21 h1 u, v3 , v4 , g1 u; z|w, ˜ w3 , w4 , Sn ) − I(u; z|g2 v, ˜ w, ˜ w3 , w4 , Sn ) (a)
= 2 log(P) − 2 log(P)
=0
(35)
where (a) follows from [33, Lemma 2]. From the above analysis, it can be easily seen that 4 symbols are securely transmitted to receiver 1 over a total of 6 time slots, yielding d1 = 2/3 SDoF at receiver 1. Using similar reasoning, it can be readily shown that 4 symbols are transmitted securely to receiver 2 over 6 time slots, which yields d2 = 2/3 SDoF at receiver 2. 2) S4/3 — Coding scheme using DD, DP and PD states: In the previous coding scheme S4/3 , availability of 2 1 only delayed CSI of both receivers in the first two phases suffices to achieve 4/3 SDoF. Thus, by choosing DD state at t = 1, 2, DP state at t = 3, 5, and PD state at t = 4, 6; (d1 , d2 ) = ( 32 , 32 ) SDoF pair is achievable. VI. Conclusion In this paper, we studied the SDoF region of a two-user multi-input single-output broadcast channel. We assume that each receiver knows its own CSI and also the past CSI of the other receiver; and, each receiver is allowed to convey either the instantaneous or delayed CSIT. Thus, the overall CSIT vector obtained at the transmitter can alternate between four possible states. Under these assumptions, we first consider the Gaussian MISO wiretap channel and characterize the full secure SDoF. Next, we generalize this model to the two-user Gaussian MISO broadcast setting and establish bounds on the SDoF region. The coding scheme that we establish in this work is based on schemes developed previously — in the context of broadcast setting with alternating CSIT and delayed CSIT — by carefully tuning them to
17
account for secrecy constraints. The result established in this work explored the synergistic benefits of alternating CSIT in terms of SDoF; and shows that in comparison to encoding separately over different states, joint encoding across the states provides strictly better secure rates.
Appendix I Proof of Lemma 1 For convenience, we denote the channel output at each receiver as yn := (ynPP , ynPD , ynDP , ynDD ), zn := (znPP , znPD , znDP , znDD ), where ynS1 S2 (znS1 S2 ) denotes the part of channel output at receiver 1 (receiver 2), when (S1 , S2 ) ∈ {P, D}2 channel state occurs. We first introduce a property which we will use in the proof of Lemma 1. We refer to this as property of channel output symmetry [6, Lemma 4], [33], [35]. We focus our attention to states PD and DD at the receiver 2. Recall that for states PD and DD the channel input-output relationship at receiver 2 is given by zI,t = gI,t xI,t + nI,2t ,
∀ I ∈ {PD, DD}.
(36)
Next, we consider a statistically indistinguishable receiver 2˜ which has access to states PD and DD, where the channel outputs to the receiver 2˜ are 1) independent from channel outputs at the receiver 2 and 2) identically distributed as the channel outputs at receiver 2. The channel input-output relationship at receiver 2˜ at t-th time instant is given by z˜I,t = g˜ I,t xI,t + n˜ I,2t ,
∀ I ∈ {PD, DD}
(37)
where g˜ I,t and gI,t are identically distributed, and independent of each other — and all other random variables — for I ∈ {PD, DD}. The additive white Gaussian noise n˜ I,2 is assumed to be i.i.d., with n˜ I,2t ∼ CN(0, 1) for I ∈ {PD, DD} and is independent from all random variables. Let λgI,t denotes the probability distribution from which, gI,t and g˜ I,t are independent and identically drawn, for I ∈ {PD, DD}. Let Sn := {gI,t , g˜ I,t }nt=1 , for I ∈ {PD, DD}. Property 1: The channel output symmetry states that t−1 n t−1 n h(zPD,t , zDD,t |zt−1 zPD,t , z˜ DD,t |zt−1 PD , zDD , S ) = h(˜ PD , zDD , S ).
(38)
18
Proof: We begin the proof as follows. t−1 n h(zPD,t, zDD,t |zt−1 PD , zDD , S ) (a)
t−1 ˜ PD,t , g˜ DD,t , Sn \ St )] = EλgPD,t ,λgDD,t [h(zPD,t, zDD,t |zt−1 PD , zDD , gPD,t = gPD , gDD,t = gDD , g
(b)
t−1 n = EλgPD,t ,λgDD,t [h(gPD xPD,t + nPD,2t , gDD xDD,t + nDD,2t |zt−1 PD , zDD , S \ St )]
(c)
t−1 n = EλgPD,t ,λgDD,t [h(gPD xPD,t + n˜ PD,2t , gDD xDD,t + n˜ DD,2t |zt−1 PD , zDD , S \ St )]
(d)
t−1 = EλgPD,t ,λgDD,t [h(gPD xPD,t + n˜ PD,2t , gDD xDD,t + n˜ DD,2t |zt−1 ˜ PD,t = gPD , g˜ DD,t = gDD , Sn \ St )] PD , zDD , g
(e)
t−1 = EλgPD,t ,λgDD,t [h(˜zPD,t, z˜ DD,t |zt−1 ˜ PD,t = gPD , g˜ DD,t = gDD , gPD,t , gDD,t , Sn \ St )] PD , zDD , g
t−1 n = h(˜zPD,t , z˜ DD,t |zt−1 PD , zDD , S )
(39)
where (a) follows due to the definition of differential entropy, (b) follows because xI,t is independent from (gI,t , g˜ I,t ), (c) follows because nI,2t and n˜ I,2t are independent from all other random variables and have same statistics, (d) and (e) follow because since (gI,t , g˜ I,t ) belong to same distribution λgI,t and due to the independence of xI,t and (gI,t , g˜ I,t ); for I ∈ {PD, DD}. We now provide the proof of (14a) and (14c); due to the symmetry the rest of the inequalities follow straightforwardly. We begin the proof as follows. 2h(znPD , znDP , znDD |Sn ) = 2h(znPD, znDD |Sn ) + 2h(znDP |znPD , znDD , Sn ) (f)
≥ 2h(znPD, znDD |Sn ) + h(znDP |znPD , znDD , Sn ) + h(znDP |znPD , znDD , xn , Sn ) | {z } ≤no(log(P))
(g)
≥2
n X
t−1 n n n n n h(zPD,t , zDD,t |zt−1 PD , zDD , S ) + h(zDP |zPD , zDD , S )
t=1
n X (h) t−1 n t−1 n = h(zPD,t, zDD,t |zt−1 zPD,t , z˜ DD,t |zt−1 PD , zDD , S ) + h(˜ PD , zDD , S ) t=1
+ h(znDP |znPD , znDD , Sn ) ≥
n X
t−1 n n n n n h(zPD,t , zDD,t , z˜ PD,t , z˜DD,t |zt−1 PD , zDD , S ) + h(zDP |zPD , zDD , S )
=
n X
t−1 n h(zPD,t , zDD,t , z˜ PD,t , z˜DD,t , yPD,t , yDD,t |zt−1 PD , zDD , S )
t=1
t=1
− h(yPD,t , yDD,t |znPD , znDD , z˜ PD,t , z˜ DD,t , Sn ) + h(znDP |znPD , znDD , Sn ) =
n X
t−1 n h(zPD,t , zDD,t , yPD,t , yDD,t |zt−1 PD , zDD , S )
t=1
+ h(˜zPD,t , z˜ DD,t |znPD , znDD , yPD,t , yDD,t , Sn ) | {z } =no(log(P))
19
− h(yPD,t , yDD,t |znPD , znDD , z˜ PD,t , z˜ DD,t , Sn ) +h(znDP |znPD , znDD , Sn ) | {z } =no(log(P))
n (i) X t−1 t−1 t−1 n ≥ h(zPD,t , zDD,t , yPD,t , yDD,t |zt−1 PD , zDD , yPD , yDD , S ) t=1
+ h(znDP |znPD , znDD , Sn ) + no(log(P)) = h(znPD , znDD , ynPD , ynDD |Sn ) + h(znDP |znPD , znDD , Sn ) + no(log(P)) (j)
≥ h(znPD , znDD , ynPD , ynDD |Sn ) + h(znDP |znPD , znDD , ynPD , ynDD , Sn ) + no(log(P))
= h(znPD , znDP , znDD , ynPD , ynDD |Sn ) + no(log(P))
(40)
≥ h(ynPD , ynDD |Sn ) + no(log(P))
(41)
where ( f ) and ( j) follow from the fact that conditioning reduces entropy, (g) follows because given (xn , Sn ), znDP can be recovered within bounded noise distortion, (h) follows from the property of channel output symmetry (38), (i) follows from the fact that conditioning reduces entropy; and, (˜zPD,t , z˜ DD,t ) and (yPD,t , yDD,t ) can be reconstructed within bounded noise distortion form (znPD , znDD , yPD,t , yDD,t , Sn ), and (znPD , znDD , z˜PD,t , z˜ DD,t , Sn ), respectively. We can also bound the term in (40) as follows. Continuing from (40), we get 2h(znPD , znDP , znDD |Sn ) ≥ h(znPD , znDP , znDD , ynPD , ynDD |Sn ) + no(log(P)) = h(znPD , znDP , znDD |Sn ) + h(ynPD , ynDD |znPD , znDP , znDD , Sn ) + no(log(P)).
(42)
From (42), it implies that h(znPD , znDP , znDD |Sn ) ≥ (ynPD , ynDD |znPD , znDP , znDD , Sn ) + no(log(P)).
(43)
This concludes the proof. Appendix II Proof of Theorem 1 Achievability. We first construct some elemental coding schemes which form the basic building blocks to establish the achievability in Theorem 1. These schemes have some connections to the one in Section V, and, so we outline them briefly. S1 —Coding schemes achieving 1-SDoF: For PP, and DP states 1-SDoF is achievable. Due to the availability of perfect CSI of the unintended receiver (wire-taper), the transmitter can zero-force the information leaked to it. Thus, it can be readily shown that one symbol is securely transmitted to the legitimate receiver in a single timeslot, yielding 1-SDoF. For the case in which PD state occurs, the transmitter transmits one confidential message (v) along with the artificial noise (u). In this state, perfect CSI of the legitimate receiver is utilize to zero-force the
20
injected artificial noise. The communication takes place in only one time slot and the transmitter sends v x = + bu, (44) φ where b ∈ C2×1 denotes the precoding vector chosen such that h1 b = 0. The channel input-output relationship is given by y = h11 v
(45a)
z = g11 v + g1 bu.
(45b)
Receiver 1 knows the CSI (h) and easily decodes v via channel inversion. Receiver 2 gets the confidential message v embedded in artificial noise and is unable to decode it. Then, following security analysis similar to in Section V, it can be readily shown that 1 symbol is securely transmitted to the legitimate receiver over 1 timeslot yielding 1-SDoF. S2/3 —Coding scheme achieving 2/3-SDoF: For the case in which DD state occurs, 2/3 SDoF is achievable. The coding scheme in this case is similar to the one in [33, Section IV-B-2] for the wiretap channel with delayed CSIT from both receivers and is omitted for brevity. The achievable SDoF of Theorem 1 then follows by choosing PP, PD, DP and DD states λPP , λPD , λDP and λDD fractions of time, respectively, which yields ds = λPP (1) + λPD (1) + λDP (1) + λDD
2 3
λDD = λPP + λPD + λDP + λDD − 3 | {z } 1
λDD =1− . 3
(46)
Converse Proof. The converse proof extends the proof established earlier in the context of MIMO wiretap channel with delayed CSIT [33] and the one established for the MISO broadcast channel with alternating CSIT by taking imposed secrecy constraints into account [13]. We begin the proof as follows. nRe = H(W|zn ) = H(W|zn , Sn ) = H(W|Sn ) − I(W; zn |Sn ) = I(W; yn |Sn ) + H(W|yn , Sn ) − I(W; zn |Sn ) (a)
≤ I(W; yn |Sn ) − I(W; zn |Sn ) + nǫn = h(ynPP , ynPD , ynDP , ynDD |Sn ) − h(ynPP , ynPD , ynDP , ynDD |W, Sn ) − I(W; znDD |Sn ) −I(W; znPP , znPD , znDP |znDD , Sn ) + nǫn
21
≤ h(ynPP |Sn ) + h(ynPD |Sn ) + h(ynDP|Sn ) + h(ynDD |Sn ) − h(ynDD |W, Sn ) −h(ynPP , ynPD , ynDP |ynDD , W, Sn ) − I(W; znDD |Sn ) − I(W; znPP , znPD , znDP |znDD , Sn ) +nǫn | {z } ≥0
(b)
≤ h(ynPP |Sn ) + h(ynPD |Sn ) + h(ynDP|Sn ) + h(ynDD |Sn ) − h(ynDD |W, Sn ) − h(ynPP , ynPD , ynDP |ynDD , W, xn , Sn ) −I(W; znDD |Sn ) + nǫn | {z } ≥no(log(P))
(c)
≤ h(ynPP |Sn ) + h(ynPD |Sn ) + h(ynDP|Sn ) + I(W; ynDD |Sn ) − I(W; znDD |Sn ) +nǫn | {z }
(47)
:=η1
where ǫn → 0 as n → ∞; (a) follows from Fano’s inequality, (b) follows from the non-negativity of I(W; znPP , znPD , znDP |znDD , Sn ) and the fact that conditioning reduces entropy, and (c) follows because (ynPP , ynPD , ynDP ) can be obtained within bounded noise distortion form (xn ,Sn ). We now bound η1 in two ways as follows. η1 = I(W; ynDD |Sn ) − I(W; znDD |Sn ) = h(ynDD |Sn ) − h(ynDD |W, Sn ) − h(znDD |Sn ) + h(znDD |W, Sn ) 1 ≤ h(ynDD |Sn ) − h(znDD |W, Sn ) − h(znDD |Sn ) + h(znDD |W, Sn ) 2 1 = h(ynDD |Sn ) + h(znDD |W, Sn ) − h(znDD |Sn ) 2 (e) 1 n n ≤ h(yDD |S ) − h(znDD |Sn ) 2
(d)
(48)
where (d) follows from (14b) by conditioning over W and setting ynPD = ynDP = znDP := φ, and (e) follows due to the fact that conditioning reduces entropy. We can also bound η1 as follows. η1 = I(W; ynDD |Sn ) − I(W; znDD |Sn ) ≤ I(W; ynDD , znDD |Sn ) − I(W; znDD |Sn ) = I(W; ynDD |znDD , Sn ) ≤ h(ynDD |znDD , Sn ) − h(ynDD |znDD , W, xn , Sn ) | {z } ≥no(log(P))
(f)
≤ h(znDD |Sn )
(49)
where ( f ) follows from (14c) by setting znPD = znDP = ynPD := φ, and noticing that ynDD can be obtained within bounded noise distortion form (xn ,Sn ).
22
From (48) and (49), we get
1 η1 ≤ min − h(znDD |Sn ), h(znDD |Sn ) 2 α = max min h(ynDD |Sn ) − , α α 2 2 ≤ h(ynDD |Sn ) 3 h(ynDD |Sn )
(50)
where α := h(znDD |Sn ). Then, replacing (50) in (47), we get 2 nRe ≤ h(ynPP|Sn ) + h(ynPD |Sn ) + h(ynDP|Sn ) + h(ynDD |Sn ) + nǫn 3 2 ≤ λPP + λPD + λPD + λDD n log(P) + nǫn 3 1 = λPP + λPD + λPD + λDD − λDD n log(P) + nǫn 3 (g) λDD = 1− n log(P) + nǫn 3
(51)
where (g) follows from (4). Then, dividing both sides by n log(P) and taking lim P → ∞ and lim n → ∞, we get ds ≤ 1 −
λDD . 3
(52)
This concludes the proof. Appendix III Proof of Theorem 2 The outer bound follows by generalizing the proof that we establish earlier in the context of MISO wiretap channel with alternating CSIT in Theorem 1 to the broadcast setting. We now provide the proof of (17c). The proof of (17d) follows due to the symmetry. We begin the proof as follows. nR1 =H(W1 |zn ) (a)
≤I(W1 ; yn |Sn ) − I(W1 ; zn |Sn ) + nǫn =I(W1 ; ynPP , ynPD , ynDP , ynDD |Sn ) − I(W1 ; znPP , znPD , znDP , znDD |Sn ) + nǫn =I(W1 ; ynPD , ynDP , ynDD |Sn ) − I(W1 ; znPD , znDP , znDD |Sn ) + I(W1 ; ynPP|ynPD , ynDP , ynDD , Sn ) − I(W1 ; znPP |znPD , znDP , znDD , Sn ) +nǫn | {z } ≥0
(b)
≤ h(ynPD , ynDP , ynDD |Sn ) − h(ynPD , ynDP , ynDD |W1 , Sn ) − h(znPD, znDP , znDD |Sn ) + h(znPD, znDP , znDD |W1 , Sn ) | {z } :=η1
23
+ h(ynPP |ynPD , ynDP , ynDD , Sn ) − h(ynPP |ynPD , ynDP , ynDD , W1 , Sn ) + nǫn (c)
≤η1 + h(ynPP|Sn ) − h(ynPP|ynPD , ynDP , ynDD , W1 , xn , Sn ) +nǫn | {z } ≥no(log(P))
≤η1 + h(ynPP|Sn ) + nǫn
(53)
where ǫn → 0 as n → ∞; (a) follows from Fano’s inequality, (b) follows from the non-negativity of I(W; znPP |znPD , znDP , znDD , Sn ), and (c) follows due to the fact that conditioning reduces entropy. We now bound η1 as follows. η1 = h(ynPD , ynDP , ynDD |Sn ) − h(ynPD, ynDP , ynDD |W1 , Sn ) − h(znPD , znDP , znDD |Sn ) +h(znPD , znDP , znDD |W1 , Sn ) 1 ≤ h(ynPD , ynDP , ynDD |Sn ) − h(znDP , znDD |W1 , Sn ) − h(znPD , znDP , znDD |Sn ) 2
(d)
+h(znPD , znDP , znDD |W1 , Sn ) 1 = h(ynPD , ynDP , ynDD |Sn ) − h(znDP , znDD |W1 , Sn ) − h(znDP , znDD |Sn ) − h(znPD |znDP , znDD , Sn ) 2 +h(znDP , znDD |W1 , Sn ) + h(znPD |znDP , znDD , W1 , Sn ) 1 = h(ynPD , ynDP , ynDD |Sn ) + h(znDP , znDD |W1 , Sn ) − h(znDP , znDD |Sn ) − h(znPD |znDP , znDD , Sn ) 2 +h(znPD |znDP , znDD , W1 , Sn ) 1 ≤ h(ynPD , ynDP , ynDD |Sn ) + h(znDP , znDD |Sn ) − h(znDP , znDD |Sn ) − h(znPD|znDP , znDD , Sn ) 2
(e)
+h(znPD |znDP , znDD , Sn ) 1 = h(ynPD , ynDP , ynDD |Sn ) − h(znDP , znDD |Sn ) 2
(54)
where (d) follows from (14b) by conditioning over W1 , and (e) follows from the fact that conditioning reduces entropy. We can also bound R1 as follows. nR1 = H(W1 |zn , W2 ) (f)
≤ I(W1 ; yn |W2 , Sn ) − I(W1 ; zn |W2 , Sn ) + nǫn
≤ I(W1 ; ynPD , ynDP , ynDD , znPD , znDP , znDD |W2 , Sn ) − I(W1 ; znPD , znDP , znDD |W2 , Sn ) +I(W1 ; ynPP |ynPD , ynDP , ynDD , W2 , Sn ) − I(W1 ; znPP |znPD , znDP , znDD , W2 , Sn ) +nǫn | {z } ≥0
≤ I(W1 ; ynPD , ynDP , ynDD |znPD , znDP , znDD , W2 , Sn ) + h(ynPP |Sn ) − h(ynPP|ynPD , ynDP , ynDD , W2 , xn , Sn ) +nǫn | {z } ≥no(log(P))
(g)
≤ h(ynPD , ynDP , ynDD |znPD , znDP , znDD , W2 , Sn ) − h(ynPD , ynDP , ynDD |znPD , znDP , znDD , W2 , W1 , Sn )
24
+h(ynPP|Sn ) + nǫn ≤ h(ynPD , ynDP , ynDD |znPD , znDP , znDD , W2 , Sn ) − h(ynPD , ynDP , ynDD |znPD , znDP , znDD , W2 , W1 , xn , Sn ) | {z } ≥no(log(P))
+h(ynPP|Sn )
+ nǫn
(h)
≤ h(ynPD , ynDD |znPD , znDP , znDD , W2 , Sn ) + h(ynDP |ynPD , ynDD , znPD , znDP , znDD , W2 , Sn ) + h(ynPP |Sn ) +nǫn (i)
≤ h(znPD , znDP , znDD |W2 , Sn ) + h(ynDP |Sn ) + h(ynPP |Sn ) + nǫn
(55)
where ( f ) follows from Fano’s inequality, (g) and (h) follow because ynPP and (ynPD , ynDP , ynDD ) can be obtained within bounded noise distortion form (xn ,Sn ), and (i) follows from (14c) by conditioning over W2 . Next, we bound the term R2 as follows. nR2 = H(W2 ) (j)
≤ I(W2 ; zn |Sn ) + nǫn = I(W2 ; znPD , znDP , znDD |Sn ) + I(W2 ; znPP |znPD , znDP , znDD , Sn ) + nǫn = h(znPD , znDP , znDD |Sn ) − h(znPD , znDP , znDD |W2 , Sn ) + h(znPP |znPD , znDP , znDD , Sn ) −h(znPP |znPD , znDP , znDD , W2 , Sn ) + nǫn ≤ h(znDP , znDD |Sn ) + h(znPD |znDP , znDD , Sn ) − h(znPD , znDP , znDD |W2 , Sn ) + h(znPP |Sn ) − h(znPP |znPD , znDP , znDD , W2 , xn , Sn ) +nǫn | {z } ≥no(log(P))
(k)
≤ h(znDP , znDD |Sn ) + h(znPD |Sn ) + h(znPP |Sn ) − h(znPD, znDP , znDD |W2 , Sn ) + nǫn
(56)
where ǫn → 0 as n → ∞; ( j) follows from Fano’s inequality, and (k) follows because znPP can be obtained within bounded noise distortion form (xn ,Sn ) and due to the fact that conditioning reduces entropy. Then, by replacing (54) in (53) and appropriately combining (53), (55) and (56), we get n(3R1 + R2 ) ≤ 2h(ynPD , ynDP , ynDD |Sn ) − h(znDP , znDD |Sn ) + 2h(ynPP |Sn ) + h(znPD , znDP , znDD |W2 , Sn ) +h(ynDP |Sn ) + h(ynPP |Sn ) + h(znDP , znDD |Sn ) + h(znPD |Sn ) + h(znPP |Sn ) −h(znPD , znDP , znDD |W2 , Sn ) + nǫ′n = 2h(ynPD , ynDP , ynDD |Sn ) + 3h(ynPP|Sn ) + h(znPP |Sn ) + h(ynDP|Sn ) + h(znPD |Sn ) + nǫ′n ≤ 2[h(ynPP |Sn ) + h(ynPD |Sn ) + h(ynDP |Sn ) + h(ynDD |Sn )] + h(ynPP |Sn ) + h(znPP |Sn ) +h(ynDP |Sn ) + h(znPD |Sn ) + nǫ′n ≤ [2(λPP + λPD + λDP + λDD ) + 2λPP + λPD + λDP ]n log(P) (l)
= (2 + 2λPP + 2λPD )n log(P)
(57)
25
where (l) follows from (4) and due to the symmetry of states PD and DP. Then, dividing both sides of (57) by n log(P) and taking lim P → ∞ and lim n → ∞, we get 3d1 + d2 ≤ 2 + 2λPP + 2λPD .
(58)
This concludes the proof. Appendix IV Proof of Theorem 3 We now show that the region in Theorem 3 is achievable. The SDoF region in (19) is characterized by the following corner points, (ds , 0), (0, ds), (2 + λPP + λPD )dlow (2 + λPP + λPD )dlow s s , , low 2 + dlow 2 + d s s [(2 + λPP + λPD )dlow − 2ds ]+ s , and ds , dlow s [(2 + λPP + λPD )dlow − 2ds ]+ s , ds . dlow s
(59)
(60)
(61)
The achievability of the two corner points (ds , 0) and (0, ds) readily follows from the encoding scheme
that we use to establish the achievability in Theorem 1. In what follows we show that the point (59) is achievable. The scheme in this case is obtained by appropriately choosing the elemental coding schemes of Section V as shown in Table II. By straightforward algebra, the achievable SDoF for this scheme at receiver 1 is then given by, (8 + 4λ + 6λ ) 1 8 + 4λPP − dlow PP PD s d1 = λPP + low 2 2 + ds
(9 + 3λ + 4λ ) − 6λ − 4λ − 6 2 dlow PP PD PP PD s 3 2 + dlow s 4λPD 1 8 λ PD dlow = λPP + + ] − 2λ − λ [2 − PP PD s 3 3 3 2 + dlow s dlow λ 4λ PD PD s 2 − + + λ = PP 3 3 2 + dlow s 2 + 2λPD + 3
=
(2 + λPP + λPD )dlow s 2 + dlow s
.
(62)
Due to the symmetry of the point, d2 follows by similar steps. The achievability of point (60), follows along the same lines as shown above. More specifically, by appropriately choosing the encoding schemes of Section V as shown in Table III, the achievable SDoF at
26
Scheme
CSIT State
Time
(d1 , d2 )
S2
PP
1
(1, 1)
DD
1
( 12 , 12 )
(PD, DP)
( 12 , 21 )
( 23 , 23 )
(DD, PD, DP)
( 31 , 13 , 31 )
( 23 , 23 )
S1 S4/3 1 S4/3 2
Fractional Time Usage λPP 8+4λPP −dlow s (8+4λPP +6λPD ) 2+dlow s
≥0
2λPD dlow s (9+3λPP +4λPD )−6λPP −4λPD −6 2+dlow s
P
=1
TABLE II Secure encoding scheme achieving (59).
Scheme
CSIT State
S2
Time
(d1 , d2 )
PP
1
(1, 1)
S4/3 1
(PD, DP)
( 12 , 21 )
( 23 , 23 )
S2/3
DD
1
( 23 , 0)
Fractional Time Usage 3ds − 2 3
.[dlow s (4 + λPP 2dlow s
+ λPD ) − (3dlow + 2)ds ] s
3 .[ds (2 + dlow s )− 2dlow s
P
(2 + λPP + λPD )dlow s ] =1
TABLE III Secure encoding scheme achieving (60).
receiver 1 is given by 2 3 d1 = (3ds − 2) + low .[dlow (4 + λPP + λPD ) − (3dlow + 2)ds] s s 3 2ds 2 3 ) − (2 + λPP + λPD )dlow ] + low [ds(2 + dlow s s 3 2ds = (3ds − 2) + = (3ds − 2) +
low low dlow − 3dlow − 2) s (4 + λPP + λPD ) − ds (2 + λPP + λPD ) + ds (2 + ds s
dlow s 2dlow − 2ds dlow s s dlow s
= ds .
(63)
Similarly, the achievable SDoF at receiver 2 is given by 2 3 low low d2 = (3ds − 2) + low [ds (4 + λPP + λPD ) − (3ds + 2)ds] 3 2ds = =
(3ds − 2)dlow − ds (3dlow + 2) + dlow s s s (4 + λPP + λPD ) dlow s
low dlow s (4 + λPP + λPD ) − 2ds − 2ds
dlow s
27
=
dlow s (2 + λPP + λPD ) − 2ds dlow s
.
(64)
Due to the symmetry, the achievability of point (61) follows trivially. This concludes the proof.
References [1] Z. H. Awan, A. Zaidi, and A. Sezgin, “Achievable secure degrees of freedom of MISO broadcast channel with alternating CSIT,” in IEEE International Symposium on Information Theory, Honolulu, USA, June 2014, pp. 31–35. [2] S. A. Jafar, “Interference alignment — A new look at signal dimensions in a communication network,” Foundations and Trends in Communications and Information Theory, vol. 7, no. 1, pp. 1–134, 2010. [3] M. A. Maddah-Ali and D. Tse, “Completely stale transmitter channel state information is still very useful,” IEEE Trans. Inf. Theory, vol. 58, no. 7, pp. 4418–4431, Jul. 2012. [4] C. S. Vaze and M. K. Varanasi, “The degrees of freedom region of the two-user and certain three-user MIMO broadcast channel with delayed CSI,” 2011. [Online]. Available: http://arxiv.org/abs/1101.0306 [5] M. J. Abdoli, A. Ghasemi, and A. K. Khandani, “On the degrees of freedom of three-user MIMO broadcast channel with delayed CSIT,” in IEEE International Symposium on Information Theory, St. Petersburg, Russia, Aug. 2011, pp. 209–213. [6] C. S. Vaze and M. K. Varanasi, “The degrees of freedom region and interference alignment for the MIMO interference channel with delayed CSIT,” IEEE Trans. Inf. Theory, vol. 58, no. 7, pp. 4396–4417, Jul. 2012. [7] A. Ghasemi, A. S. Motahari, and A. K. Khandani, “Interference alignment for the MIMO interference channel with delayed local CSIT,” 2011. [Online]. Available: http://arxiv.org/abs/1102.5673 [8] C. Huang, S. Jafar, S. Shamai, and S. Vishwanath, “On degrees of freedom region of MIMO networks without channel state information at transmitters,” IEEE Trans. Inf. Theory, vol. 58, no. 2, pp. 849–857, Feb. 2012. [9] C. S. Vaze and M. K. Varanasi, “The degrees of freedom regions of MIMO broadcast, interference, and cognitive radio channels with no CSIT,” IEEE Trans. Inf. Theory, vol. 58, no. 8, pp. 5354–5374, Aug. 2012. [10] T. Gou and S. Jafar, “Optimal use of current and outdated channel state information: Degrees of freedom of the MISO BC with mixed CSIT,” IEEE Commun. Lett., vol. 16, no. 7, pp. 1084–1087, Jul. 2012. [11] S. Yang, M. Kobayashi, D. Gesbert, and X. Yi, “Degrees of freedom of time correlated MISO broadcast channel with delayed CSIT,” IEEE Trans. Inf. Theory, vol. 59, no. 1, pp. 315–328, Jan. 2013. [12] R. Tandon, M. A. Maddah-Ali, A. M. Tulino, H. V. Poor, and S. Shamai (Shitz), “On fading broadcast channels with partial channel state information at the transmitter,” in Int. Symp. on Wireless Commun. Sys., Paris, France, Aug. 2012, pp. 1004–1008. [13] R. Tandon, S. A. Jafar, S. Shamai (Shitz), and H. V. Poor, “On the synergistic benefits of alternating CSIT for the MISO broadcast channel,” IEEE Trans. Inf. Theory, vol. 59, no. 7, pp. 4106–4128, Jul. 2013. [14] A. D. Wyner, “The wiretap channel,” Bell System Technical Journal, vol. 54, pp. 1355–1387, Oct. 1975. [15] I. Csisz´ar and J. Korner, ¨ “Broadcast channels with confidential messages,” IEEE Trans. Inf. Theory, vol. 24, no. 3, pp. 339–348, May. 1978. [16] R. Liu, T. Liu, H. V. Poor, and S. Shamai (Shitz), “New results on multiple-input multiple-output broadcast channels with confidential messages,” IEEE Trans. Inf. Theory, vol. 59, no. 3, pp. 1346–1359, Mar. 2013. [17] E. Tekin and A. Yener, “The Gaussian multiple access wire-tap channel,” IEEE Trans. Inf. Theory, vol. 54, no. 12, pp. 5747–5755, Dec. 2008. [18] Y. Liang and H. V. Poor, “Multiple access channels with confidential messages,” IEEE Trans. Inf. Theory, vol. 54, no. 3, pp. 976–1002, Mar. 2008. [19] E. Tekin and A. Yener, “The general Gaussian multiple access and two-way wire-tap channels: Achievable rates and cooperative jamming,” IEEE Trans. Inf. Theory, vol. 54, no. 6, pp. 2735–2751, Jun. 2008. [20] Z. H. Awan, A. Zaidi, and L. Vandendorpe, “On multiaccess channel with unidirectional cooperation and security constraints,” in 50th Annual Allerton Conference Communication, Control and Computing, Monticello, IL, USA, Oct. 2012, pp. 982–987.
28
[21] ——, “Multiaccess channel with partially cooperating encoders and security constraints,” IEEE Trans. Inf. Forensics Security, vol. 8, no. 7, pp. 1243–1254, Jul. 2013. [22] L. Lai and H. E. Gamal, “The relay eavesdropper channel: Cooperation for secrecy,” IEEE Trans. Inf. Theory, vol. 54, no. 9, pp. 4005–4019, Sept. 2008. [23] Z. H. Awan, A. Zaidi, and L. Vandendorpe, “On secure transmission over parallel relay eavesdropper channel,” in 48th Annual Allerton Conference Communication, Control and Computing, Monticello, IL, USA, Sept. 2010, pp. 859–866. [24] ——, “Secure communication over parallel relay channel,” IEEE Trans. Inf. Forensics Security, vol. 7, no. 2, pp. 359–371, Apr. 2012. [25] O. O. Koyluoglu and H. E. Gamal, “Cooperative encoding for secrecy in interference channels,” IEEE Trans. Inf. Theory, vol. 57, no. 9, pp. 5682–5694, Sept. 2011. [26] Z. Li, R. Yates, and W. Trappe, “Secrecy capacity of a class of one-sided interference channel,” in IEEE International Symposium on Information Theory, Toronto, ON, Jul. 2008, pp. 379–383. [27] A. Khisti and G. W. Wornell, “Secure transmission with multiple antennas II: The MIMOME wiretap channel,” IEEE Trans. Inf. Theory, vol. 56, no. 11, pp. 5515–5532, Nov. 2010. [28] F. Oggier and B. Hassibi, “The secrecy capacity of the MIMO wiretap channel,” IEEE Trans. Inf. Theory, vol. 57, no. 8, pp. 4961–4972, Aug. 2011. [29] T. Liu and S. Shamai (Shitz), “A note on the secrecy capacity of the multi-antenna wiretap channel,” IEEE Trans. Inf. Theory, vol. 55, no. 6, pp. 2547–2553, Jun. 2009. [30] R. Bustin, R. Liu, H. V. Poor, and S. Shamai (Shitz), “MMSE approach to the secrecy capacity of the MIMO Gaussian wiretap channel,” EURASIP Journal on Wireless Commnunications and Networking. Special issue on physical layer security, Nov. 2009. [31] Y. Liang, H. V. Poor, and S. Shamai (Shitz), “Information theoretic security,” Foundations and Trends in Communications and Information Theory, vol. 5, no. 4-5, pp. 355–580, 2009. [32] R. Liu, T. Liu, H. V. Poor, and S. Shamai, “Multiple-input multiple-output Gaussian broadcast channels with confidential messages,” IEEE Trans. Inf. Theory, vol. 56, no. 9, pp. 4215–4227, 2010. [33] S. Yang, M. Kobayashi, P. Piantanida, and S. Shamai, “Secrecy degrees of freedom of MIMO broadcast channels with delayed CSIT,” IEEE Trans. Inf. Theory, vol. 59, no. 9, pp. 5244–5256, 2013. [34] A. Zaidi, Z. H. Awan, S. Shamai (Shitz), and L. Vandendorpe, “Secure degrees of freedom of MIMO X-channels with output feedback and delayed CSIT,” IEEE Trans. Inf. Forensics Security, vol. 8, no. 11, pp. 1760–1774, 2013. [35] P. Mukherjee, R. Tandon, and S. Ulukus, “MISO broadcast channels with confidential messages and alternating CSIT,” in IEEE International Symposium on Information Theory, Honolulu, USA, June 2014, pp. 216–220.
