Rate-Distortion-Based Physical Layer Secrecy with Applications to Multimode Fiber
Eva C. Song, Emina Soljanin, Paul Cuff, H. Vincent Poor, Kyle Guan
aa r X i v : . [ c s . CR ] D ec IEEE TRANSACTIONS ON COMMUNICATIONS 1
Rate-Distortion-Based Physical Layer Secrecywith Applications to Multimode Fiber
Eva C. Song,
Student Member, IEEE,
Emina Soljanin,
Fellow, IEEE,
Paul Cuff,
Member, IEEE,
H. Vincent Poor,
Fellow, IEEE, and Kyle Guan,
Member, IEEE
Abstract
Optical networks are vulnerable to physical layer attacks; wiretappers can improperly receivemessages intended for legitimate recipients. Multimode fiber (MMF) transmission can be modeled viaa broadcast channel in which both the legitimate receiver’s and wiretapper’s channels are multiple-input-multiple-output complex Gaussian channels. Our work considers the theoretical aspect of thissecurity problem in the domain of a broadcast channel. Source-channel coding analyses based on theuse of distortion as the metric for secrecy are developed. Alice has a source sequence to be encoded andtransmitted over this broadcast channel so that the legitimate user Bob can reliably decode while forcingthe distortion of the wiretapper, or eavesdropper, Eve’s estimate as high as possible. Tradeoffs betweentransmission rate and distortion under two extreme scenarios are examined: the best case where Eve hasonly her channel output and the worst case where she also knows the past realization of the source. Itis shown that under the best case, an operationally separate source-channel coding scheme guaranteesmaximum distortion at the same rate as needed for reliable transmission. Theoretical bounds are given,and particularized for MMF. Numerical results showing the rate distortion tradeoff are presented andcompared with corresponding results for the perfect secrecy case.
E. C. Song, P. Cuff and H. V. Poor are with the Department of Electrical Engineering, Princeton University, Princeton, NJ08544, USA e-mail: { csong,cuff, poor } @princeton.edu.E. Soljanin and K. Guan are with Bell Labs, Alcatel-Lucent, Murray Hill, NJ 07974, USA e-mail: { emina,kyle.guan } @alcatel-lucent.com October 9, 2018 DRAFT IEEE TRANSACTIONS ON COMMUNICATIONS
Index Terms rate-distortion, MIMO, optical fiber communication, source-channel coding, secrecy, SDM, MMF
I. I
NTRODUCTION
Single mode fiber systems are believed to have reached their capacity limits. In particular,techniques such as wavelength-division multiplexing (WDM) and polarization-division multi-plexing (PDM) have been heavily exploited in the past few years, leaving little room for furtherimprovement in capacity [1]. Space-division multiplexing (SDM) is a promising solution formeeting the growing capacity demands of optical communication networks. One way of realizingSDM is via the use of multimode fiber (MMF). While multimode transmission provides greatercapacity, the security of such systems can be an issue because a wiretapper can eavesdropupon MMF communication by simply bending the fiber [2] . MMF is a multiple-input-multiple-output (MIMO) system [1] that captures the charateristics of crosstalk among different modes.The secrecy capacity of a Gaussian MIMO broadcast channel was studied in [3], but the resultcannot be applied directly to MMF because the channel is not the same. The secrecy capacityof this channel was studied in [2] where it is shown that the channel conditions required forperfect secrecy are quite demanding.The concepts of “perfect secrecy”, “partial secrecy”, “strong secrecy”, “weak secrecy”, “equiv-ocation” and “distortion” will be applied repeatedly in this paper. We shall now briefly summarizethe relationships among those terms. Please note that, even though “perfect secrecy” is a non-asymptotic concept, here for convenience, we refer to “perfect secrecy” in the asymptotic sense,i.e. the information leakage is arbitrarily small as the blocklength goes to infinity. Informationtheoretic secrecy typically considers one of the two regimes, perfect secrecy or partial secrecy.Perfect secrecy essentially requires no information leakage to the eavesdropper. In the regime ofpartial secrecy, one must quantify the degree of secrecy obtained. Equivocation rate is a metricfound in the literature which measures how much of the signal is leaked to the eavesdropper
DRAFT October 9, 2018UBMITTED PAPER 3 regardless of whether the eavesdropper can use the leaked information in a constructive way.Another approach, taken in this work, is to use distortion to measure the difference between theoriginal content and eavesdropper’s estimate of it.The notions of strong and weak secrecy are both referring to the perfect secrecy regime, andequivocation is usually involved in the analysis. Under either strong or weak secrecy, distortionis at the maximum, because negligible information is leaked to the eavesdropper that wouldallow her to make a better estimate. However, implication in the other direction does not holdin general. In order to keep the distortion at a maximum, neither strong or weak secrecy isnecessarily required.This work focuses on the partial secrecy regime. It should be noted that equivocation anddistortion are not two independent measures. As pointed out in recent work [4], equivocationbecomes a special case of distortion when causal information is revealed to the eavesdropper andthe distortion is measured by log loss. It can be seen from our results herein that high distortioncan be achieved even if all the past information of the source is given to the eavesdropper.Furthermore, partial secrecy comes at a much lower cost than perfect secrecy.Distortion was also used in [5] and [6] as a metric for secrecy in the context of a noiselessnetwork with secret key sharing. In this work, we are concerned with physical layer secrecy inMMF systems. This prompts us to formulate the problem as a source-channel coding problemalong the lines studied in a general setting in [7], some results of which can be directly appliedto MMF systems.The rest of the paper is structured as follows. In Section II, we introduce the system model intwo ways: the general source-channel coding model for theoretical derivation; and the particularMMF channel model for application. In Section III, we provide theoretical bounds with source-channel coding for general broadcast channels. This is our main theoretical contribution of thepaper. In Section IV, we apply the general results from Section III to the MMF model whenthe channel is time-invariant and discuss the secrecy outage in the case of a random channel
October 9, 2018 DRAFT IEEE TRANSACTIONS ON COMMUNICATIONS in which the channel state information (CSI) is not available to the transmitter. In Section V,we provide numerical evaluation to the MMF source-channel model under Hamming distortion.Finally, in Section VI, we conclude the paper and discuss open problems from this work.II. S
YSTEM M ODEL
We first introduce some notation that will be used throughout this paper. A sequence X , ..., X n is denoted by X n . Limit taken with respect to “ n → ∞ ” is abbreviated as “ → n ”. In the casethat X is a random variable, x is used to denote a realization and X is used to denote thesupport of that random variable. R and C are reserved to denote the real field and complex field,respectively. A complex Gaussian distribution is denoted by CN ( µ, Σ , C ) , where µ is the mean, Σ is the covariance matrix, and C is the relation matrix. A Markov relation is denoted by thesymbol − − . The total variation distance between two distributions P and Q are denoted by || P − Q || T V . For a distortion measure d : S × T 7→ R + , the distortion between two sequencesis defined to be the per-letter average distortion d ( s k , t k ) = k P ki =1 d ( s i , t i ) . The maximumdistortion ∆ , the average distortion achieved by guesses based only on the prior distribution ofthe source, is defined as ∆ , min t E [ d ( S, t )] . (1) A. Source-Channel Coding Model for General Broadcast Channel
A source node (Alice) has an independent and identically distributed (i.i.d.) sequence S k that she intends to transmit over a memoryless broadcast channel P Y Z | X such that a legitimateuser (Bob) can reliably decode the source sequence, while keeping the distortion between aneavesdropper (Eve) and Alice as high as possible. The source sequence S k is mapped to thechannel input sequence X n through a source-channel encoder. Upon receiving Y n , Bob makes anestimate ˆ S k of the original source sequence S k . Let f k,n : S k
7→ X n be a source-channel encoderand g k,n : Y n
7→ S k be the corresponding decoder. For almost lossless reconstruction, we require DRAFT October 9, 2018UBMITTED PAPER 5 the probability that Bob’s reconstruction differs from the original goes to zero asymptoticallywith the source blocklength. That is, P h S k = ˆ S k i → k . Similarly, Eve also makes an estimate T k of S k upon receiving Z n and some other sideinformation. We will examine two extreme cases based on the amount of side information Evehas. Fig. 1: Source-channel coding model with an i.i.d. source and broadcast channel.
1) No causal information available to Eve:
The case in which Eve has only her own chan-nel output but no side information about the source corresponds to the best scenario for thelegitimate users of the network, Alice and Bob. With the channel output alone, Eve has verylimited resources in hand to make the estimate. Let t k be Eve’s estimate of the original sourcesequence s k . The system model is shown in Fig. 1; however, the dashed line represents additionalinformation that is not available to the eavesdropper in this first scenario. We use the lower case t k ( z n ) to denote Eve’s deterministic estimation functions of her observation z n and the capitalletter T k = t k ( Z n ) to denote the function of the random sequence Z n . The following definitionsin this section are for time-invariant channels. Definition 1.
For a given distortion function d ( s, t ) , a rate distortion pair ( R, D ) is achievableif there exists a sequence of encoder/decoder pairs f k,n and g k,n such that kn = R, October 9, 2018 DRAFT IEEE TRANSACTIONS ON COMMUNICATIONS lim n →∞ P h S k = ˆ S k i = 0 , and lim inf n →∞ min t k ( z n ) E (cid:2) d ( S k , t k ( Z n )) (cid:3) ≥ D. Note that the rate-distortion pair ( R, D ) captures the tradeoff between Bob’s rate for reliabletransmission and Eve’s distortion, which is different from rate-distortion theory in the traditionalsense.
2) With causal information available to Eve:
On the other hand, we are also interested inthe case in which, at each time instance j , Eve gets to see the past realization of the sourcesequence S j − . This would be the worst scenario for the legitimate users. The definition for anachievable rate distortion pair ( R, D ) is similar to Definition 1 given in the previous subsectionexcept the last condition is replaced by lim inf n →∞ min { t j ( z n ,s j − ) } kj =1 E " k k X j =1 d ( S j , t j ( Z n , S j − ) ≥ D. The system model is shown in Fig. 1 with the dashed line representing the availability of thecausal information.
B. MMF Channel Model
Now we particularize the general broadcast channel described above to an MMF broadcastchannel as shown in Fig. 2. An M -mode MMF is modeled as a memoryless MIMO channelwith input X an M -dimensional complex vector. Here M is a positive integer.Unlike wireless MIMO which has a total power constraint, MMF channels have the followingper mode power constraint averaged over n uses of the channel: n n X i =1 (cid:12)(cid:12)(cid:12) X ( m ) i (cid:12)(cid:12)(cid:12) ≤ for all modes m ∈ [1 : M ] . (2)More generally (as in [3]), we will consider a power constraint of the form n n X i =1 X i X † i (cid:22) Q, (3) DRAFT October 9, 2018UBMITTED PAPER 7
Fig. 2: MMF channel model where Q ∈ { A ∈ H M × M : A (cid:23) , A ii = 1 } and H denotes the set of Hermitian matrices. Oneelement in this set is the identity matrix I (constraint (2)). We will focus on the case that Q = I for simplicity. A detailed discussion of the MMF channel model can be found in [1].
1) The Legitimate User Communications Model:
The channel between Alice and Bob P Y | X is complex, Gaussian, MIMO, with input X ∈ C M as described above, and output Y ∈ C M given by Y = HX + N, (4)where N ∼ CN (0 , σ N I, is M -dimensional, uncorrelated, zero-mean, complex, Gaussian noiseand H is an M × M complex matrix. Bob’s channel matrix H is of the form H = p E L Ψ , (5)where Ψ ∈ C M × M is unitary and E L is a constant scalar that measures the average power ofthe channel. We refer to E L/σ N as the SNR of the channel. Matrix Ψ , the unitary factor ofthe channel H , describes the modal crosstalk [1].
2) The Eavesdropper Communications Model:
The channel between Alice and Eve P Z | X isalso complex, Gaussian, MIMO, with input X ∈ C M as described above, and output Z ∈ C M given by Z = H e X + N e , (6) October 9, 2018 DRAFT IEEE TRANSACTIONS ON COMMUNICATIONS where N e ∼ CN (0 , σ N e I, is M -dimensional uncorrelated, zero-mean, complex, Gaussiannoise, and H e is an M × M complex matrix. Eve’s channel matrix H e is of the form H e = p E L e √ ΦΨ e , (7)where Ψ e ∈ C M × M is unitary, Φ is diagonal with positive entries, and E L e is the averagepower of Eve’s channel. Note that Eve has a different signal to noise ratio SNR e = E L e /σ N e .The diagonal component Φ of the channel matrix H e corresponds to the mode-dependent loss(MDL) as introduced in [1]. III. T HEORETICAL B OUNDS
In this section, we focus on the general broadcast channel introduced in Section II-A only. Wefirst make some general observations about the communication between Alice and Bob, as well asthe communication between Alice and Eve. If Eve is not present, Alice and Bob can communicatelosslessly at any rate lower than R , max X I ( X ; Y ) H ( S ) because separate source-channel coding isoptimal for point-to-point communication. Ideally, we want to force maximum distortion ∆ uponEve. But higher distortion to Eve may come at the price of a lower communication rate to Bob.The technical content of this section is organized as follows: the rate-distortion region for the“no causal information” case is first given in Theorem 1; to prepare for the achievability proofof Theorem 1, an operational separation scheme is discussed; also, an achievable rate-distortionregion is given in Theorem 5 for the causal case under Hamming distortion; and finally, anexample with a binary symmetric channel and Hamming distortion is provided for illustration.Before starting the new results, we shall provide a recap of what have been done in the literatureregarding this problem and what our main advances are in this work. For noiseless channels, thesource coding problems of both the no-causal-information and the causal-information cases weresolved in [8] and [6], respectively. There, secrecy was obtained by using a secret key sharedbetween Alice and Bob. As for physical layer secrecy of a memoryless broadcast channel, theresult for transmitting two messages, one confidential and one public, from Csisz´ar and K¨orner [9] DRAFT October 9, 2018UBMITTED PAPER 9 have been known for many decades. In their work, weak secrecy were considered. This result wasstrengthened in [10] by considering strong secrecy. The same rate region was obtained in [10],however the metric for secrecy is stronger. In our work, the source-channel coding schemes wepropose operationally separate source and channel coding that require dividing the bit sequenceproduced by source coding into two messages which are then processed by the channel coding.The channel coding part functions in a way that is similar to [9] or [10], except that the publicmessage in their work is not required to be decoded in our case, and we refer to that messageas the “non-confidential” message. This type of channel coding setting was also used in [7]for the causal-information case and it is shown that only weak secrecy is required to combinethe source and channel coding. In this work, we will connect the source coding (from [8]) andchannel coding for the no-causal-information case. Unlike the causal-information case, strongsecrecy from the channel is needed. This will require modifying some of the settings from [10].We also extend the result for the causal-information case from [7] to include the rate-distortiontradeoff.We now state the rate-distortion result for general source-channel coding with an i.i.d. sourcesequence and a discrete memoryless broadcast channel P Y Z | X when no causal information isavailable to Eve. In the following theorem, we will see that the source sequence can be deliveredalmost losslessly to Bob at a rate arbitarily close to R while the distortion to Eve is kept at ∆ ,as long as the secrecy capacity is positive. Theorem 1.
For an i.i.d. source sequence S k and memoryless broadcast channel P Y Z | X , if thereexists W − − X − − Y Z such that I ( W ; Y ) − I ( W ; Z ) > , then ( R, D ) is achievable if andonly if R < max X I ( X ; Y ) H ( S ) , (8) D ≤ ∆ , (9) where ∆ was defined in (1) . October 9, 2018 DRAFT0 IEEE TRANSACTIONS ON COMMUNICATIONS
Remark : The requirement I ( W ; Y ) − I ( W ; Z ) > implies the existence of a secure channelwith a positive rate, i.e. the eavesdropper’s channel is not less noisy than the intended receiver’schannel. So instead of demanding a high secure transmission rate with perfect secrecy toaccommodate the description of the source, we need only to ensure the existence of a securechannel with positive rate. This will suffice to ensure the eavesdropper’s distortion is maximal.The converse is straightforward. Each of the inequalities (8) and (9) is true individually for anychannel and source, (8) by channel capacity coupled by optimality of source-channel separation,and (9) by definition.The idea for achievability is to operationally separate the source and channel coding (see Fig.3). The source encoder compresses the source and splits the resulting message into a confidentialmessage and a non-confidential message. A channel encoder is concatenated digitally with thesource encoder so that the channel delivers both the confidential and non-confidential messagesreliably to Bob and keeps the confidential message secret from Eve, as in [9]. The overall source-channel coding rate will have the following form: R = kn = k log |M| · log |M| n = R ch R src , where |M| is the total cardinality of the confidential and the non-confidential messages; R ch and R src arethe channel coding and source coding rates, respectively.Let us look at two models in the following subsections that will help us establish the platformfor showing the achievability of Theorem 1. Fig. 3: Operational separate source-channel coding: the confidential and non-confidential messages satisfy M s ∈ [1 : 2 kR ′ s = 2 nR s ] and M p ∈ [1 : 2 kR ′ p = 2 nR p ] DRAFT October 9, 2018UBMITTED PAPER 11
A. Channel Coding and Strong Secrecy
Consider a memoryless broadcast channel P Y Z | X and a communication system with a confi-dential message M s and a non-confidential message M p that must allow the intended receiverto decode both M s and M p while keeping the eavesdropper from learning anything about M s .Problems like this were first studied by Csisz´ar and K¨orner [9] in 1978, as an extension ofWyner’s work in [11]. However, their model and our model differ in that the second receiver intheir setting is required to decode the public message M p . The mathematical formulation andresult of our channel model is stated below. We focus on the message pairs ( M s , M p ) whosedistribution satisfies the following: P M s | M p = m p ( m s ) = 2 − nR s (10)for all ( m s , m p ) . Later we will show a source encoder can always prepare the input messagesto the channel of this form. Definition 2. A ( R s , R p , n ) channel code consists of a channel encoder F c (possibly stochastic)and a channel decoder g c such that F c : M s × M p
7→ X n and g c : Y n
7→ M s × M p where |M s | = 2 nR s and |M p | = 2 nR p . Definition 3.
The rate pair ( R s , R p ) is achievable under weak secrecy if for all ( M s , M p ) satisfying (10) , there exists a sequence of ( R s , R p , n ) channel codes such that lim n →∞ P h ( M s , M p ) = ( ˆ M s , ˆ M p ) i = 0 and lim n →∞ n I ( M s ; Z n | M p ) = 0 . October 9, 2018 DRAFT2 IEEE TRANSACTIONS ON COMMUNICATIONS
Note that because the eavesdropper may completely or partially decode M p , the secrecy re-quirement is modified accordingly to consider I ( M s ; Z n | M p ) instead of I ( M s ; Z n ) . To guaranteetrue secrecy of M s , we want to make sure that even if M p is given to the eavesdropper, thereis no information leakage of M s , because I ( M s ; Z n | M p ) = I ( M s ; Z n , M p ) if M s and M p areindependent. Theorem 2 (Theorem 3 in [7]) . A rate pair ( R s , R p ) is achievable under weak secrecy if R s ≤ I ( W ; Y | V ) − I ( W ; Z | V ) , (11) R p ≤ I ( V ; Y ) (12) for some V − − W − − X − − Y Z . The proof can be found in [7]. Let us denote the above region as R . We now strengthenthe result by considering strong secrecy introduced in [12]. Later we will use strong secrecy toconnect the operationally separate source and channel encoders. Definition 4.
The rate pair ( R s , R p ) is achievable under strong secrecy if for all ( M s , M p ) satisfying (10) , there exists a sequence of ( R s , R p , n ) channel codes such that lim n →∞ P [( M p , M s ) = ( ˆ M s , ˆ M p )] = 0 and lim n →∞ I ( M s ; Z n | M p ) = 0 . In general, weak secrecy does not necessarily imply that strong secrecy is also achievable;however, in this particular setting we have the following claim:
Theorem 3.
A rate pair ( R s , R p ) achievable under weak secrecy is also achievable under strongsecrecy. DRAFT October 9, 2018UBMITTED PAPER 13
The following two lemmas will assist the proof of Theorem 3 by providing a sufficientcondition for satisfying the secrecy constraint lim n →∞ I ( M s ; Z n | M p ) = 0 . Lemma 1. If || P Z n | M p = m p P M s | M p = m p − P Z n M s | M p = m p || T V ≤ ǫ ≤ , then I ( M s ; Z n | M p = m p ) ≤ − ǫ log ǫ |M s | . Proof:
Let ǫ z n = (cid:12)(cid:12)(cid:12)(cid:12) P M s | M p = m p − P M s | Z n = z n ,M p = m p (cid:12)(cid:12)(cid:12)(cid:12) T V . Therefore, E P Zn | Mp = mp [ ǫ z n ] = (cid:12)(cid:12)(cid:12)(cid:12) P Z n | M p = m p P M s | M p = m p − P Z n M s | M p = m p (cid:12)(cid:12)(cid:12)(cid:12) T V ≤ ǫ. By Lemma 2.7 [13], | H ( M s | M p = m p ) − H ( M s | Z n = z n , M p = m p ) | ≤ − ǫ z n log ǫ zn |M s | . Notethat f ( x ) , − x log x is concave. And by applying Jensen’s inequality twice, we have I ( M s ; Z n | M p = m p ) = (cid:12)(cid:12)(cid:12) E P Zn | Mp = mp [ H ( M s | M p = m p ) − H ( M s | Z n = z n , M p = m p )] (cid:12)(cid:12)(cid:12) ≤ E P Zn | Mp = mp [ | H ( M s | M p = m p ) − H ( M s | Z n = z n , M p = m p ) | ] ≤ E P Zn | Mp = mp (cid:20) − ǫ z n log ǫ z n |M s | (cid:21) ≤ − ǫ log ǫ |M s | . Lemma 2.
If for every ( m s , m p ) , there exists a measure θ m p on Z n such that || P Z n | M p = m p ,M s = m s − θ m p || T V ≤ ǫ n then lim n →∞ I ( M s ; Z n | M p ) = 0 where ǫ n = 2 − nβ for some β > . A proof of Lemma 2 is given in Appendix A.If there exist channel codes such that P h ( M s , M p ) = ( ˆ M s , ˆ M p ) i → n and measure θ m p forall ( m s , m p ) such that || P Z n | M p = m p ,M s = m s − θ m p || T V ≤ ǫ n , then Theorem 3 follows immediately. October 9, 2018 DRAFT4 IEEE TRANSACTIONS ON COMMUNICATIONS
The existence of such a code and measure is assured by the same codebook construction andchoice of measure as in [10].
B. Source Coding
Recall from our problem setup in Section II that the sender Alice has an i.i.d. source sequence S k . A source encoder is needed to prepare S k by encoding it into a pair of messages ( M s , M p ) that satisfies P M s | M p = m p ( m s ) = 2 − kR ′ s = 2 − nR s so that it forms a legitmate input to the channelmodel in Section III-A. Definition 5. An ( R ′ s , R ′ p , k ) source code consists of an encoder f s and a decoder g s such that f s : S k
7→ M s × M p g s : M s × M p
7→ S k where |M s | = 2 kR ′ s and |M p | = 2 kR ′ p . Definition 6.
A rate distortion triple ( R ′ s , R ′ p , D ) is achievable under a given distortion measure d ( s, t ) if there exists a sequence of ( R ′ s , R ′ p , k ) source codes such that lim k →∞ P (cid:2) S k = g s ( f s ( S k )) (cid:3) = 0 and the message pair generated by the source encoder satisfies P M s | M p = m p ( m s ) = 2 − kR ′ s andfor all P Z n | M s M p such that I ( M s ; Z n | M p ) → n k →∞ min t k ( z n ) E (cid:2) d k ( S k , t k ( Z n )) (cid:3) ≥ D. Theorem 4. ( R ′ s , R ′ p , D ) is achievable if R ′ s > ,R ′ s + R ′ p > H ( S ) , DRAFT October 9, 2018UBMITTED PAPER 15 and D ≤ ∆ . The general idea for achievability is to consider the ǫ -typical S k sequences and partition theminto bins of equal size so that each bin contains sequences of the same type. The identity M p ofthe bin is revealed to all parties, but the identity M s of each sequence inside a bin is perfectlyprotected. Each of such partitions is treated as a codebook. It was shown in [8] that, for thenoiseless case in which Eve is given m p instead of z n , the distortion averaged over all suchcodebooks achieves the maximum distortion ∆ as k → ∞ and therefore there must exist onepartition that achieves ∆ . In order to transition from the result in [8] to our claim in Theorem4, we only need to show min t k ( z n ) E (cid:2) d k ( S k , t k ( Z n )) (cid:3) ≥ min t k ( m p ) E (cid:2) d k ( S k , t k ( M p )) (cid:3) . Proof:
First, observe that min t k ( · ) E (cid:2) d k ( S k , t k ( · )) (cid:3) = 1 k k X i =1 min t ( i, · ) E [ d ( S i , t ( i, · ))] (13)Next, we claim the channel output sequence z n does not provide Eve anything more than m p and therefore min t ( i,z n ) E " k k X i =1 d ( S i , t ( i, Z n )) ≥ min t ( i,m p ) E " k k X i =1 d ( S i , t ( i, M p )) − δ ′ ( ǫ ) (14)The analysis is similar to that in [7], but for the sake of clarity, we present the complete proofof (14) in Appendix B. Here strong secrecy comes into play. This is also pointed out within theproof in Appendix B that I ( M s ; Z n | M p ) → n is needed.Finally, combining (14) with (13) give us the desired result. Strictly speaking, the source encoder may violate the condition (10) on ( k + 1) |S| number of bins, because ( k + 1) |S| is anupper bound on the number of types of sequence with length k . However, this is just a very small (polynomial in k ) numberof bins compared with the total number (roughly kH ( S ) ) of bins. Therefore, for this small portion of “bad” bins that violates (10) , we can just let the source encoder declare an error on the confidential message M s and constructs a dummy M s uniformlygiven the bin index m p . This will contribute only an ǫ factor to the error probability. October 9, 2018 DRAFT6 IEEE TRANSACTIONS ON COMMUNICATIONS
C. Achievability of Theorem 1
With all the elements from Section III-A and III-B, we are now ready to harvest the achiev-ability proof of Theorem 1 using Theorems 2 and 4 by concatenating the channel encoder withthe source encoder.
Proof:
Fix ν ≥ ǫ > . Fix P S . Let R s ′ = 2 ν , R p ′ = H ( S ) − ν and R ′ = R s ′ + R p ′ . Weapply the same codebook construction and encoding scheme as in Section III-B by partioningthe ǫ -typical S k sequences into kR p ′ bins and inside each bin we have kR s ′ sequences so that P [ S k = g s ( f s ( S k ))] ≤ ǫ . Recall that all the sequences inside one bin are of the same type, so itis guaranteed that P M s | M p = m p ( m s ) = 1 |M s | = 12 kR ′ s for all m p , m s , which implies I ( M s ; M p ) = 0 .Let R s and R p be the channel rates. R p is seen as a function of R s on the boundary ofthe region given in Theorem 2 and this is denoted by R p ( R s ) . Suppose max ( R s ,R p ) ∈R R s > ,i.e. there exists W − − X − − Y Z such that I ( W ; Y ) − I ( W ; Z ) > (justified in AppendixC). R p ( R s ) is continuous and non-increasing. Thus, R p achieves the maximum at R s = 0 ,which would be the channel capacity max X I ( X ; Y ) of P Y | X for reliable transmission. By thecontinuity of R p ( R s ) , ( R s , R p ) = (2 ν kn , R p (0) − δ ( ν )) is achievable under strong secrecy, i.e. P [( M s , M p ) = ( ˆ M s , ˆ M p )] ≤ ǫ and I ( M s ; Z n | M p ) ≤ ǫ , where δ ( ν ) → as ν → .From the above good channel code under strong secrecy we have P Z n | M s M p such that I ( M s ; Z n | M p ) → n . Therefore, we can apply Theorem 4 to achieve lim inf k →∞ min t k ( z n ) E (cid:2) d k ( S k , t k ( Z n )) (cid:3) = D. The error probability is bounded by the sum of the error probabilities from the source codingand channel coding parts i.e. P h S k = ˆ S k i < ǫ . Finally, we verify the total transmission rate to DRAFT October 9, 2018UBMITTED PAPER 17 complete the proof: R = kn = R s + R p R ′ s + R ′ p = R p (0) − δ ( ν ) + 2 RνH ( S ) + ν ≥ R p (0) − δ ( ν ) H ( S ) + ν ν → −→ max X I ( X ; Y ) H ( S ) . We next state the rate-distortion result for source-channel coding with an i.i.d. source sequenceand discrete memoryless broadcast channel P Y Z | X when causal information is available to Eve.The result comes from the rate matching of [7]. Theorem 5.
For an i.i.d. source sequence S k and a memoryless broadcast channel P Y Z | X , arate distortion pair ( R, D ) is achievable if R ≤ min (cid:18) I ( V ; Y ) I ( S ; U ) , I ( W ; Y | V ) − I ( W ; Z | V ) H ( S | U ) (cid:19) ,D ≤ αR · ∆ + (cid:16) − αR (cid:17) · min t ( u ) E [ d ( S, t ( U ))] for some distribution P S P U | S P V P W | V P X | W P Y Z | X , where α = [ I ( V ; Y ) − I ( V ; Z )] + I ( S ; U ) .Example: binary symmetric broadcast channel (BSBCC) and binary source with Hammingdistortion To visualize Theorem 1 and Theorem 5, we will illustrate the results with a BSBCC andbinary source under Hamming distortion, defined as d H ( s, t ) = , s = t, , otherwise.With the above setting, suppose S i ∼ Bern ( p ) , and the broadcast channel is binary symmetricwith crossover probabilities to the intended receiver and the eavesdropper p and p , respectively. October 9, 2018 DRAFT8 IEEE TRANSACTIONS ON COMMUNICATIONS
Assume p ≤ . and p < p < . . It is well known that this can be treated as a physicallydegraded channel in capacity calculation. Let us make the following definitions: f ( x ) is the linear interpolation of the points (cid:18) log n, n − n (cid:19) , n = 1 , , , ... (15) d ( x ) , min( f ( x ) , − max s P S ( s )) , (16) h ( x ) , x log 1 x + (1 − x ) log 11 − x is the binary entropy function, (17) x ∗ x , x ∗ (1 − x ) + (1 − x ) ∗ x is the binary convolution, (18)where P S ( · ) is the probability mass function of the random variable S . The corresponding rate-distortion regions for the no-causal-information and causal-information cases are given in thefollowing corollaries. Corollary 1.
For an i.i.d. Bern ( p ) source sequence S k and BSBCC with crossover probabilities p and p , when no causal information is available, ( R, D ) is achievable if and only if R < − h ( p ) h ( p ) ,D ≤ p. Corollary 2.
For an i.i.d. Bern ( p ) source sequence S k and BSBCC with crossover probabilities p and p , when causal information is available, ( R, D ) is achievable if R ≤ h ( p ) − h ( p ) h ( p ) ,D ≤ p or h ( p ) − h ( p ) h ( p ) < R ≤ − h ( p ) h ( p ) ,D ≤ α ′ p + (1 − α ′ ) d (cid:18) h ( γ ∗ p ) − h ( γ ∗ p ) − h ( p ) + h ( p ) R (cid:19) where γ ∈ [0 , . solves h ( γ ∗ p ) = 1 − h ( p ) + h ( p ) − Rh ( p ) and α ′ = h ( γ ∗ p ) − h ( γ ∗ p )1 − h ( γ ∗ p ) . DRAFT October 9, 2018UBMITTED PAPER 19
These corollaries result directly from applying Theorem 1 and Theorem 5, respectively. Theregion given in Corollary 2 is calculated in a similar fashion as the region given by Theorem 7 of[7]. An numerical example with p = 0 . , p = 0 . and p = 0 . is plotted in Fig.4. Interpretationof the plot is deferred until the end of Section V. R=k/n A ve r a g e D i s t o r t i on no−causal−informationcausal−informationperfect secrecy ratereliable transmission rate Fig. 4: Achievable distortion-rate curves. On the horizontal axis is the symbol/channel use source-channel codingrate and on the vertical axis is the average Hamming distortion.
IV. MMF M
AIN R ESULTS
A. Fixed MMF Channel
We now apply the above result to the MMF model introduced in Section II-B by finding therate distortion regions for the MMF model defined in (4) and (6) under the two scenarios. Inthis section, as before, we assume the channels are time-invariant. First of all, we will give theachievable rate region under strong secrecy (therefore also under weak secrecy).
Theorem 6.
The following rate region for one confidential and one non-confidential message is
October 9, 2018 DRAFT0 IEEE TRANSACTIONS ON COMMUNICATIONS achievable under strong secrecy for a complex Gaussian channel: R s ≤ log | HKH † + σ N I || σ N I | − log | H e KH e † + σ N e I || σ N e I | (19) R p ≤ log | HQH † + σ N I || HKH † + σ N I | (20) for some K and Q , where (cid:22) K (cid:22) Q , K ∈ H M × M , Q satisfies the power constraint in (3) ,and H and H e are the channel gain matrices.Proof: According to Theorem 2 and 3, R s ≤ I ( W ; Y | V ) − I ( W ; Z | V ) (21) R p ≤ I ( V ; Y ) (22)for some V − − W − − X − − Y Z and E [ XX † ] (cid:22) Q , is an achievable rate pair.We restrict the channel input X to be a circularly symmetric complex Gaussian vector. Let V ∼ CN (0 , Q − K, , B ∼ CN (0 , K, such that B and V are independent, and W = X = V + B . Therefore, X ∼ CN (0 , Q, satisfies the power constraint. Similar to results in [3], therate pair ( R s , R p ) satisfying inequalities (19) and (20) can be achieved.An immediate corollary follows directly from the above theorem. Corollary 3.
The following rate pairs are achievable under strong secrecy for MMF with channelgains defined in (5) and (7) and equal full power allocation Q = I : R s ≤ log | SNR K + I || SNR e Ψ e K Ψ e † Φ + I | (23) R p ≤ log | ( SNR + 1) I || SNR K + I | (24) for some K where (cid:22) K (cid:22) I , K ∈ H M × M , SNR = E L/σ N and SNR e = E L e /σ N e . With the secrecy capacity region of MMF, we can evaluate its rate distortion region ( R, D ) under the two extreme cases, without and with causal information at Eve’s decoder respectively.For the best case scenario (no causal information), we will give a sufficient condition to force DRAFT October 9, 2018UBMITTED PAPER 21 maximum distortion ∆ between Alice and Eve. For the worst case scenario (with causal in-formation), we will give an achievable rate-distortion region and look at the particular case ofHamming distortion. Theorem 7.
For an i.i.d source sequence S k , if min j ∈{ ,...,M } ¯ φ j < SNRSNR e (25) where ¯ φ j ’s are the diagonal entries of Φ , then the following rate distortion pair ( R, D ) isachievable with no causal information at the eavesdropper: R < M log(
SNR + 1) H ( S ) (26) D ≤ ∆ . (27)Theorem 7 follows from Theorem 1 and Corollary 3. Note that (25) is a sufficient conditionfor the existence of a secure channel with strictly positive rate from Alice to Bob. A discussionof this condition is provided in Appendix D. Theorem 8.
For an i.i.d. source sequence S k and Hamming distortion, the following distortionrate curve D ( R ) is in the achievable region with causal information at the eavesdropper: D = d ( H ( S )) , if R ≤ R ∗ s H ( S ) (28) D = ¯ α ( K )∆ + (1 − ¯ α ( K )) d (cid:18) R s ( K ) R (cid:19) , if R ∗ s H ( S ) < R ≤ R ∗ p H ( S ) (29) where d ( · ) is as defined in (16) ; K , { K ∈ H M × M , (cid:22) K (cid:22) I } , R ∗ s = max K ′ ∈K log | SNR K ′ + I || SNR e √ ΦΨ e K ′ Ψ e † √ Φ + I | ,R ∗ p = M log( SNR + 1) ,R s ( K ) = log | SNR K + I || SNR e √ ΦΨ e K Ψ e † √ Φ + I | , ¯ α ( K ) = ¯ β ( K ) − ¯ γ ( K )¯ β ( K ) , October 9, 2018 DRAFT2 IEEE TRANSACTIONS ON COMMUNICATIONS ¯ β ( K ) = log | ( SNR + 1) I || SNR K + I | , ¯ γ ( K ) = log | SNR e Φ + I || SNR e √ ΦΨ e K Ψ e † √ Φ + I | . The result given in Theorem 8 can be derived directly from Theorem 5 and Corollary 3.
B. Secrecy Outage under Random MMF Channel
All the results we have seen thus far were derived for a time-invariant channel, which meansthat both the transmitter and the receivers are informed about the channel state. However, inMMF, the channels H and H e vary with time and the CSI is not available at the transmitterdue to the long round-trip delay over the large distances common in optical transmission, eventhough it has a long coherence time, i.e. H and H e are essentially constant over n channel uses.In Corollary 2, Theorem 7 and Theorem 8, we have chosen Q = I as the channel input powerfor simplicity. This power allocation strategy also tends to minimize the outage probability forperfect secrecy [14].The randomness of H = √ E L Ψ and H e = √ E L e √ ΦΨ e comes from the unitary component Ψ , Ψ e and the diagonal component Φ . The random matrices Ψ and Ψ e are uniformly distributedin ⊖ , where ⊖ is the set of all M × M unitary matrices [1]. The diagonal matrix Φ = diag { ¯ φ , ..., ¯ φ M } , where ¯ φ i = M φ i P Mj =1 φ j . Here φ = φ min and φ max , and φ i ∼ U nif [ φ min , φ max ] for i = 3 , ..., M .In this situation of no CSI at the transmitter with long coherence time, performance istypically measured by outage probability. The capacity C = max Q log | I + HQH † | and secrecycapacity C s = max Q (cid:2) log | I + HQH † | − log | I + H e QH e † | (cid:3) for a deterministic MIMO Gaussianbroadcast channel were given in [15] and [16], respectively. For the case of no causal informationat the eavesdropper, the CSI does not really affect the performance much. The channel capacitybetween Alice and Bob M log(1 + SNR ) does not depend on the channel realization due to theunitary component of the channel. Hence, the encoder can choose the source-channel coding rate DRAFT October 9, 2018UBMITTED PAPER 23 to be just below M log( SNR +1) H ( S ) , and maximum distortion can be achieved if the channel satisfies thecondition in Theorem 7. For the case where Eve has causal source information, we consider onlythe input power Q = I and Hamming distortion. Without knowledge of CSI, the source-channelencoder picks a source-channel coding rate ¯ R , a pair of source coding rates ( ¯ R ′ s , ¯ R ′ p ) on the linesegment R ′ s + R ′ p = H ( S ) , R ′ s , R ′ p ≥ and a real value α ∈ [0 , . Let ¯ R s , ¯ R ′ s ¯ R and ¯ R p , ¯ R ′ p ¯ R. We define the outage probability of such a choice of parameters to be P out ( I, ¯ R ′ s , ¯ R, α ) = 1 − X K ∈K P ΦΨ e Ψ (cid:2) ( ¯ R s , ¯ R p ) ∈ R ΦΨ e Ψ ( I ) and ¯ α ( K ) ≥ α (cid:3) , where R ΦΨ e Ψ ( I ) denotes the region given in Corollary 3, K = { K : 0 (cid:22) K (cid:22) I, ¯ R s =log | SNR K + I || SNR e Ψ e K Ψ e † Φ+ I | and ¯ R p = log | ( SNR +1) I || SNR K + I | } ,and ¯ α ( K ) is defined as in Theorem 8. Note that R ΦΨ e Ψ ( Q ) is a random variable because the channels P Y Z | X are now random. Under this setof parameters ( ¯ R ′ s , ¯ R, α ) , we can achieve distortion α ∆ + (1 − α ) d ( ¯ R ′ s ) with probability − P out ( I, ¯ R ′ s , ¯ R, α ) , where d ( · ) was defined in (16) . Proving the existence of a good codebook inthis case is an important information theoretic problem, most recently addressed in [17].V. N UMERICAL R ESULTS
In this section, we present numerical results illustrating achievable rate distortion regionsof an MMF under the two information models with a time-invariant channel. Let us considermeasuring the eavesdropper’s distortion using Hamming distortion and a Bern( p ) i.i.d. sourcesequence. Fig. 3 shows numerical results corresponding to Theorem 7 and Theorem 8 underequal power allocation. The channels are simulated as a − mode MMF with SNR = 20 dB ,SNR e = 10 dB , and MDL = 20 dB .In each plot, the vertical line on the right is the maximum reliable transmission rate betweenAlice and Bob and the vertical line on the left is the maximum perfect secrecy transmission ratethat can be obtained with separate source-channel coding. The horizontal line is the maximum October 9, 2018 DRAFT4 IEEE TRANSACTIONS ON COMMUNICATIONS distortion which is also the rate distortion curve from Theorem 7 with no causal information atEve. The curve obtained from Theorem 8 shows the tradeoff between the transmission ratebetween Alice and Bob and the distortion forced on Eve with causal information. We seein Fig. 5(a), p = 0 . , that with our source-channel coding analysis, we gain a free regionfor maximum distortion, as if under perfect secrecy, (from the left vertical line to the kink)because we effectively use the redundancy of the source. In Fig. 5(b) with p = 0 . , sincethere is no redundancy in the source, the distortion curve drops immediately after the maximumperfect secrecy rate. Note that the transmission rates are not considered beyond the right verticallines because they are above the maximum reliable transmission rates and Bob cannot lossleslyreconstruct the source sequences. R=k/n A ve r a g e D i s t o r t i on (a) maximum distortionachievable distortionreliable transmission rateperfect secrecy rate R=k/n A ve r a g e D i s t o r t i on (b) maximum distortionachievable distortionreliable transmission rateperfect secrecy rate Fig. 5: Achievable distortion-rate curves. On the left is the Bern(0.3) i.i.d. source case and on the right is theBern(0.5) i.i.d. source case. On the horizontal axes are the symbol/channel use source-channel coding rate and onthe vertical axes are the average Hamming distortions.
VI. C
ONCLUSION
In this work, we have examined the rate-distortion-based secrecy performance of an insecureMMF communication system. The sender is assumed to have an i.i.d. source sequence whichthe intended receiver and the eavesdropper both try to reconstruct. Two source-channel coding
DRAFT October 9, 2018UBMITTED PAPER 25 models with different information availability at the eavesdropper have been considered. Wehave shown that, when no causal source information is disclosed to the eavesdropper, under ageneral broadcast channel and any distortion measure, it is possible to send the source at themaximum rate that guarantees lossless reconstruction at the intended receiver while keeping thedistortion at the eavesdropper as high as if it only has the source prior distribution. When thepast source realization is causally disclosed to the eavesdropper, we have applied the theoreticalresults in [7] to the particular case of an MMF channel. Numerical results for an i.i.d. Bernoullisource and Hamming distortion have been provided.Only the theoretical formulation is given to calculate the secrecy outage probability for randomMMF under equal full power allocation Q = I . The optimality of this power strategy andthe statistics for different sets of parameters given in Section IV-B remain an open problem.Moreover, in our model, it is required that the intended receiver reconstruct the source losslessly.In a more general setting, one can allow lossy reconstruction of the source at the intended receiver,which is an interesting problem for further research. October 9, 2018 DRAFT6 IEEE TRANSACTIONS ON COMMUNICATIONS A PPENDIX AP ROOF OF L EMMA ( m s , m p ) , suppose there exists θ m p such that || P Z n | M p = m p ,M s = m s − θ m p || T V ≤ ǫ n (30)where ǫ n = 2 − nβ for some β > . Then we have the following: || P Z n | M p = m p − θ m p || T V = X z n (cid:12)(cid:12) P Z n | M p = m p ( z n ) − θ m p ( z n ) (cid:12)(cid:12) (31) = X z n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X m s P M s | M p = m p ( m s ) P Z n | M p = m p ,M s = m s ( z n ) − X m s P M s | M p = m p ( m s ) θ m p ( z n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X z n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X m s |M s | P Z n | M p = m p ,M s = m s ( z n ) − X m s |M s | θ m p ( z n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X z n X m s |M s | (cid:12)(cid:12) P Z n | M p = m p ,M s = m s ( z n ) − θ m p ( z n ) (cid:12)(cid:12) (32) = X m s |M s | X z n (cid:12)(cid:12) P Z n | M p = m p ,M s = m s ( z n ) − θ m p ( z n ) (cid:12)(cid:12) ≤ X m s |M s | ǫ n (33) = ǫ n (34)where (32) follows from triangle inequality and (33) follows from (30) . DRAFT October 9, 2018UBMITTED PAPER 27 || P Z n | M p = m p P M s | M p = m p − P Z n M s | M p = m p || T V = X z n X m s (cid:12)(cid:12) P Z n | M p = m p ( z n ) P M s | M p = m p ( m s ) − P Z n | M p = m p ,M s = m s ( z n ) P M s | M p = m p ( m s ) (cid:12)(cid:12) = 1 |M s | X z n X m s (cid:12)(cid:12) P z n | M p = m p ( z n ) − P Z n | M p = m p ,M s = m s ( z n ) (cid:12)(cid:12) = 1 |M s | X z n X m s (cid:12)(cid:12) P Z n | M p = m p ( z n ) − θ m p ( z n ) + θ m p ( z n ) − P Z n | M p = m p ,M s = m s ( z n ) (cid:12)(cid:12) ≤ |M s | X z n X m s ( (cid:12)(cid:12) P Z n | M p = m p ( z n ) − θ m p ( z n ) (cid:12)(cid:12) + (cid:12)(cid:12) P Z n | M p = m p ,M s = m s ( z n ) − θ m p ( z n ) (cid:12)(cid:12) )= 1 |M s | X m s ( X z n (cid:12)(cid:12) P Z n | M p = m p ( z n ) − θ m p ( z n ) (cid:12)(cid:12) + X z n (cid:12)(cid:12) P Z n | M p = m p ,M s = m s ( z n ) − θ m p ( z n ) (cid:12)(cid:12) ) ≤ |M s | X m s ( ǫ n + ǫ n )= 2 ǫ n By applying Lemma 1, we have I ( M s ; Z n | M p ) = X m p P M p ( m p ) I ( M s ; Z n | M p = m p ) ≤ X m p P M p ( m p )( − ǫ n log 2 ǫ n |M s | ) ≤ · − nβ ( nR s ) (35)where (35) goes to as n → ∞ . A PPENDIX BP ROOF OF (14)
For each i , we have I ( S i ; Z n | M p ) ≤ I ( M s S i ; Z n | M p )= I ( M s ; Z n | M p ) + I ( S i ; Z n | M s M p ) ≤ ǫ (36) October 9, 2018 DRAFT8 IEEE TRANSACTIONS ON COMMUNICATIONS for large enough n . (36) follows from strong secrecy of the channel and Fano’s inequality. Notethat weak secrecy is not sufficient to give us the desired result in our proof. We now define P i , P S i Z n M p ¯ P i , P M p P S i | M p P Z n | M p i.e. ¯ P i is the Markov chain S i − − M p − − Z n . By Pinsker’s inequality, || P i − ¯ P i || T V ≤ √ D ( P i || ¯ P i ) = 1 √ I ( S i ; Z n | M p ) ≤ r ǫ (37) min t ( i,z n ) E [ d ( S i , t ( i, Z n ))] ≥ min t ( i,z n ,m p ) E [ d ( S i , t ( i, Z n , M p ))] ≥ min t ( i,z n ,m p ) E ¯ P i [ d ( S i , t ( i, Z n , M p ))] − δ ′ ( ǫ ) (38) = min t ( i,m p ) E ¯ P i [ d ( S i , t ( i, M p ))] − δ ′ ( ǫ ) (39) ≥ min t ( i,m p ) E [ d ( S i , t ( i, M p ))] − δ ′ ( ǫ ) (40)where (38) and (40) use the fact that P i and ¯ P i are close in total variation from (37) ; (39) usesthe Markov relation S i − − M p − − Z n of distribution ¯ P i . The technical details can be found inLemma 2 and 3 from [7]. Averaging over k , we obtain (14) .A PPENDIX CJ USTIFICATION OF THE CONDITION max ( R s ,R p ) ∈R R s > From Theorem 2 or 3, we have max ( R s ,R p ) ∈R R s > is equivalent to I ( W ; Y | V ) − I ( W ; Z | V ) > (41) DRAFT October 9, 2018UBMITTED PAPER 29 for some V − − W − − X − − Y Z . We claim this can be simplified to I ( W ; Y ) − I ( W ; Z ) > (42)for some W − − X − − Y Z .To see (42) ⇒ (41) , we can simply let V = ø . To see (41) ⇒ (42) , observe that if thereexists V − − W − − X − − Y Z such that (41) holds, then there has to exist at least one value v such that I ( W ; Y | V = v ) − I ( W ; Z | V = v ) > . We can redefine the distribution as P ¯ W ¯ X ¯ Y ¯ Z , P W XY Z | V = v . It can be verified that the Markovity ¯ W − − ¯ X − − ¯ Y ¯ Z holds and P ¯ Y ¯ Z | ¯ X = P Y Z | X .A PPENDIX DS UFFICIENT CONDITION ON T HEOREM max K ∈H M × M , (cid:22) K (cid:22) I | SNR K + I || SNR e Ψ e K Ψ e † Φ + I | > . (43)However, (43) is computationally heavy to verify. If we restrict K to be of the form K = Ψ e † ΛΨ e where Λ is diagonal with diagonal entries λ i ∈ [0 , , then (43) has a much simpler form: Q Mi =1 (1 + SNR λ i ) Q Mi =1 (1 + SNR e λ i ¯ φ i ) > . (44)Therefore, if there exists a j ∈ { , ..., M } such that ¯ φ j < SNRSNR e , we can choose λ j = 1 and λ i = 0 for i = j to satisfy (44) . A CKNOWLEDGMENT
The authors would like to thank Dr. Peter Winzer from Bell Labs, Alcatel Lucent, Dr. MatthieuBloch and Dr. Rafael Schaefer for fruitful discussions and great support on this project.
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