On the Compound MIMO Broadcast Channels with Confidential Messages

On the Compound MIMO Broadcast Channels with Confidential Messages Mari K obayas hi SUPELEC Gif-sur-Yvette, France Y i ngbin Liang University of Hawaii Honolulu , HI, USA Shlomo Shamai (Sh itz) T echnion -Israel Institute of T e chnolog y Haifa, Israel M ´ erouane Debbah SUPELEC Gif-sur-Yvette, France Abstract — W e study the compound multi-in put multi -output (MIMO) broadcast channel with confidential messages (BCC), where one transmitt er sends a common message to two receive rs and two confid ential messages respectiv ely to each receiv er . The channel state may take one of a finite set of states, and the transmitter knows the state set but does not kn ow t he realization of the state. W e study achievable rates with perfect secrecy in the high SNR re gime by characterizing an achievable secrecy degree of freedom (s.d. o.f.) region for two models, the Gaussian MIMO-BCC and th e ergodic fadi ng mu lti-inpu t single-output (MISO)-BCC without a commo n message. W e show that by exploiting an additional tempora l dimension due to state v ariation in the ergodic fading model, t he achi ev able s.d. o.f. region can be significantly impro ved compared to the Gaussian model with a constant state, although at the p rice of a larger d elay . I . I N T RO D U C T I O N In most pr actical scenarios, perfec t cha nnel state infor ma- tion at transmitter (CSIT) may not be av ailable due to time- varying nature of wireless channels ( in p articular f or fast fading ch annels) and limited resourc es f or channe l estimation. Howe ver , many wireless app lications m ust gu arantee secur e and reliable commu nication in the presence of the chan nel uncertainty . In this p aper, we con sider such a scenario in the con text of the multi-inpu t m ulti-outpu t (MIM O) broadca st channel, in which a transmitter e quipped with m ulti-antenn as wishes to send one comm on message to two receivers an d two co nfidential m essages respectively to th e two receivers. The channel unce rtainty at the transmitter is mod eled as a compou nd channe l, i.e., th e chan nel to two receivers may take one state from a finite set o f states. The transmitter knows the state set, but do es no t know the r ealization of the ch annel state. The transmitter need s to send all messages reliably while keeping each co nfidential me ssage perfectly secret fro m the non-in tended receiver , no m atter wh ich c hannel state occ urs. W e note that the co mpoun d MIMO br oadcast chann el with confidential messages (BCC) is not yet fully understood. This can b e expected fr om two spe cial cases studied in [1] and [2]. One the o ne h and, it is well known th at withou t secrecy constraints the capacity region of the MIMO-BC under general CSIT is unk nown. Mor eover , ev en the d.o .f. r egion of the com pound MI MO-BC is not fully known despite the recent prog ress [1]. On th e other hand, althou gh the secrecy capacity region of th e two-user MISO-BCC has recen tly been characterized [ 2], the secrecy capacity o f a gen eral MIMO- BCC remains open. In this paper, we study achie vable secrecy degree of freedom (s.d.o.f .) regions of the MIMO-BCC, wh ich c haracterize the behavior of an achiev able secrecy rate region in the high signal-to-n oise (SNR) regime. W e consid er two compo und MIMO-BCC mo dels. The first model is th e Gaussian co m- pound MI MO-BCC, in which th e chan nel remain s in the sam e state durin g the entire tran smission. W e assume that each terminal is equipped with multiple antennas and the transmitter sends on e commo n message as we ll as two confiden tial messages to two receivers. W e pr opose a beamfor ming scheme to o btain an ac hiev able s.d.o. f., and char acterize the impact of the number of an tennas an d th e number of c hannel states on this region. W e show that with M tra nsmit antennas, N k receive anten nas and J k states f or k = 1 , 2 , a positiv e s.d.o .f. is ensure d to both r eceiv ers o nly if the nu mber o f transmit antennas is suf ficiently large, i.e. M > max( J 1 N 1 , J 2 N 2 ) . The secon d model we study is the ergodic fading comp ound multi-inpu t sing le-output (MISO)-BCC, where the channel remains in o ne state for a block du ration and then chan ges indepen dently fro m one block to another . W e mod el th e channel state at each block as a set of ran dom variables u ni- formly distributed over a finite set. Applying the variable-rate transmission strategy proposed fo r th e ergo dic fadin g wiretap channel with partial CSIT [3], we cha racterize an achiev able s.d.o.f. region. It is shown that time v ariation of th e channel (which introduces an a dditional temporal dimensio n) enables to imp rove the s.d.o .f. region com pared to the Gaussian m odel with constant chan nel state, although the secon d model applies only to d elay-toleran t ap plications. W e note that th e compo und MIMO-BCC yields a num ber of previously stud ied models as sp ecial cases. For the special case of perfect CSIT , th e secrecy capacity region o f the two-user MISO-BCC has bee n rec ently c haracterized in [ 2]. A more general two-user MIMO-BCC is con sidered in [4], wh ere the secrecy capac ity region o f the MIMO-BCC with one c ommon message and one co nfidential message is characterized . For the frequ ency-selective BCC mo deled as a special T oeplitz structure o f the MIMO -BCC, the s.d.o.f. region is analyzed in [5] . All ab ove studies d o not address the compo und natur e of the channel. For the special case of only one confidential message, the capacity o f the degra ded MI MO co mpoun d wiretap c hannel is character ized and an ach iev able s.d.o.f. of the MIMO co mpoun d wiretap ch annel is deriv ed in [ 6]. The s.d. o.f. o f th e co mpoun d wiretap parallel channels is 1 considered in [ 7], [8]. The paper is organized as fo llows. I n Sectio ns II and III, we study the Gaussian MIMO-BCC and the ergodic fading MISO-BCC, respectiv ely . Sectio n IV co ncludes th e paper . In th is paper, we ado pt the following notation s. W e let [ x ] + = max { 0 , x } and C ( x ) = log(1 + x ) . W e use x n to denote the sequ ence ( x 1 , . . . , x n ) , an d use u, v , w , x , y to de- note the r ealization of the rand om variables U, V , W, X , Y . W e use | A | , A H , tr ( A ) to deno te the determin ant, the hermitian transpose, and th e trace o f a ma trix A , r espectiv ely . I I . G AU S S I A N C O M P O U N D M I M O - B C C A. Mod el a nd Definitions W e con sider the MIMO-BCC, where the transmitter sen ds the confiden tial messages W 1 , W 2 to rece i vers 1 and 2 as well as a common message W 0 to both r eceiv ers. The transmitter , and receivers 1 a nd 2 are equ ipped with M , N 1 , N 2 antennas, respectively . The transmitter k nows a discrete set of possible channel states, and each receiver has perfect CSI. The chann el output of r eceiv er k in state j at each chan nel use is given by y k,j = H j k x + ν j k , j = 1 , . . . , J k , k = 1 , 2 (1) where J k denotes the number of p ossible chann el states o f receiver k , H j k ∈ C N k × M is the channel matrix of user k in state j , ν j k ∼ N C ( 0 , I ) is an ad ditiv e white Gaussian noise (A WGN) and is ind ependen t and id entically d istributed (i.i.d .) over k and j , and the covariance S x of the in put vecto r x satisfies the p ower constraint tr ( S x ) ≤ P . For the chann el matrices, we hav e the fo llowing assum ption. Assumption 2.1: Any M rows taken fro m th e matrices H 1 1 , . . . , H J 1 1 , H 1 2 , . . . , H J 2 2 has rank M . Definition 1: A (2 nR 0 , 2 nR 1 , 2 nR 2 , n ) block code fo r the Gaussian compoun d MIMO-BCC consists of fo llowing: • Three message sets : W i = { 1 , . . . , 2 nR i } and W i is unifor mly distributed over W i for i = 0 , 1 , 2 . • A stochastic encoder that ma ps a message set ( w 0 , w 1 , w 2 ) ∈ ( W 0 , W 1 , W 2 ) into a co dew ord x n . • T w o decode rs : decod er k maps a r eceiv ed sequ ence y n k,j into ( ˆ w ( k,j ) 0 , ˆ w k ( j ) ) ∈ ( W 0 , W k ) for j = 1 , . . . , J k and k = 1 , 2 . A rate tuple ( R 0 , R 1 , R 2 ) is a chievable if for any ǫ > 0 , there exists a (2 nR 0 , 2 nR 1 , 2 nR 2 , n ) block code such that the a vera ge error probability of r eceiv ers 1 an d 2 at state ( j, l ) satisfy P ( n ) e, 1 ,j ≤ ǫ, P ( n ) e, 2 ,l ≤ ǫ, and nR 1 − H ( W 1 | Y n 2 ,l ) ≤ nǫ, nR 2 − H ( W 2 | Y n 1 ,j ) ≤ nǫ (2) for any j = 1 , . . . , J 1 , l = 1 , . . . , J 2 . No te that (2) require s perfect secrecy for the con fidential messages at th e n on- intended receiv er . W e further define the d egree of fr eedom (d.o. f.) of the common message and the secrecy degree o f f reedom of the confidential messages as r 0 = lim P →∞ R 0 ( P ) log( P ) , r k = lim P →∞ R k ( P ) log( P ) , k = 1 , 2 . B. S ecr ecy Degr ee of F reedom Region An achiev able secrecy rate region for th e d iscrete m emory- less broad cast ch annel with one com mon and two con fidential messages was given in [9]. W e can extend this r esult to the correspo nding comp ound chan nel stud ied in this paper and obtain a n achiev able secrecy r ate region given by 0 ≤ R 0 ≤ min k,j I ( U ; Y k,j ) (3) 0 ≤ R 1 ≤ min j,l [ I ( V 1 ; Y 1 ,j | U ) − I ( V 1 ; Y 2 ,l , V 2 | U )] 0 ≤ R 2 ≤ min j,l [ I ( V 2 ; Y 2 ,l | U ) − I ( V 2 ; Y 1 ,j , V 1 | U )] over all possible jo int distributions of ( U, V 1 , V 2 , X ) satisfying U → ( V 1 , V 2 ) → X → ( Y 1 ,j , Y 2 ,l ) , ∀ j, l . (4) Based on the p receding region, we obtain the following theorem on a n ach iev able s.d.o. f. region. . Theor em 1: Consid er the Ga ussian co mpoun d MIMO-BCC with M tr ansmit antennas, N k receive antennas and J k channel states at receiver k fo r k = 1 , 2 . If J 1 N 1 < M an d J 2 N 2 < M , an a chiev able s.d.o.f. region is a unio n o f ( r 0 , r 1 , r 2 ) that satisfies r 1 ≤ min( N 1 , M − J 2 N 2 ) r 2 ≤ min( N 2 , M − J 1 N 1 ) r 0 + r 1 ≤ N 1 r 0 + r 2 ≤ N 2 (5) Pr o of: (Ou tline) W e apply a simple linear beamform ing strategy to p rovide an achievable s .d.o .f. region. W e fir st prove a useful lemma. Lemma 1: For 0 ≤ r 1 ≤ min( N 1 , M − J 2 N 2 ) and 0 ≤ r 2 ≤ min( N 2 , M − J 1 N 1 ) , th ere exist v 1 k , . . . , v r k k for k = 1 , 2 , each with dimension M tha t form a matrix V k = [ v 1 k · · · v r k k ] , such that H j k V k ′ = 0 fo r k ′ 6 = k , j = 1 , . . . , J k (6) and rank ( H j k V k ) = r k for j = 1 , . . . , J k . The pr oof of L emma 1 is omitted due to space limitations. Based on V 1 and V 2 giv en in Lemma 1, for the given 0 ≤ r 1 ≤ min( N 1 , M − J 2 N 2 ) and 0 ≤ r 2 ≤ min ( N 2 , M − J 1 N 1 ) , we let v 1 0 , . . . , v K 0 be orthno rmal vector s in the null space o f [ V 1 , V 2 ] , where K = M − rank [ V 1 V 2 ] . Hence, if we let V 0 = [ v 1 0 , . . . , v K 0 ] , then V H 0 [ V 1 , V 2 ] = 0 . W e fo rm the tra nsmit vector at each channel use by Gaussian superpo sition co ding x = V 0 u 0 + V 1 u 1 + V 2 u 2 (7) where u 0 , u 1 , u 2 are mutually independ ent with i.i.d . e ntries u k,i ∼ N C (0 , p k,i ) f or any k , i with u k,i denoting the i -th element of u k . From (6), th e receiv ed sign als are given by y j 1 = H j 1 ( V 0 u 0 + V 1 u 1 ) + n j 1 , j = 1 , . . . , J 1 (8) y l 2 = H l 2 ( V 0 u 0 + V 2 u 2 ) + n l 2 , l = 1 , . . . , J 2 (9) 2 By letting U = V 0 u 0 , V k = U + V k u k , X = V 1 + V 2 , we obtain I ( U ; Y k,j ) = log | I + H j k ( V 0 diag ( p 0 ) V H 0 + V k diag ( p k ) V H k ) H j k H | | I + H j k V k diag ( p k ) V H k H j k H | (10) I ( V k ; Y k,j | U ) = log | I + H j k V k diag ( p k ) V H k H j k H | (11) In order to find the s.d.o.f., we con sider eq ual po wer allocation over all beamfor ming directions. we first notice that the pre- log factor of log | I + P A | is determined b y rank ( A ) as P → ∞ . From Lemma 1, we o btain rank ( H j 1 V 1 V H 1 H j 1 H ) = rank ( H j 1 V 1 ) = r 1 rank ( H j 2 V 2 V H 2 H j 2 H ) = r 2 . W e th en obtain r 0 = rank “ H j 1 ( V 0 V H 0 + V 1 V H 1 ) H j 1 H ” − rank ( H j 1 V 1 V H 1 H j 1 H ) = N 1 − r 1 and similarly , r 0 = N 2 − r 2 , which concludes the pr oof. By using the beam formin g sch eme similar to that fo r Theorem 1, we o btain the f ollowing corollaries. Cor olla ry 2.1: For the Gaussian compoun d MIMO-BCC with J 1 N 1 < M an d J 2 N 2 ≥ M , an achievable s.d.o.f. r egion includes ( r 0 , r 1 , r 2 ) that satisfies r 1 = 0 , r 2 ≤ min( N 2 , M − J 1 N 1 ) , and r 0 ≤ min( N 1 , N 2 − r 2 ) . Cor olla ry 2.2: For the Gaussian compoun d MIMO-BCC with J 1 N 1 ≥ M an d J 2 N 2 ≥ M , an achievable s.d.o.f. r egion includes ( r 0 , 0 , 0) with r 0 satisfying r 0 ≤ min( M , N 1 , N 2 ) . T o gain in sight into these results, we con sider some special cases. For the c ase of perf ect CSIT ( J 1 = J 2 = 1 ), th e optimal strategy in the high SNR regime is to tran smit th e confid ential message k in th e null space o f the channe l ma trix of the othe r k ′ . This yields the s.d .o.f. r 1 ≤ min( N 1 , M − N 2 ) , r 2 ≤ min( N 2 , M − N 1 ) for M > ma x( N 1 , N 2 ) . Clearly , the s.d.o .f. of u ser k cor respond s to the s.d .o.f. o f the MIMO wiretap channel [10] wher e the transmitter sends on e confidential message to receiver k in the p resence of an eavesdropper (user k ′ 6 = k ). In addition, Theorem 1 states th at if J 1 = J 2 = 1 and the total numb er of rece i ve a ntennas is large, i.e. , N 1 + N 2 > M , we can achieve the sum d.o .f. M . This is certainly op timal since th e M IMO-BC achieves th e sum d .o.f. of min( M , N 1 + N 2 ) = M . For the case with a single r eceiv e anten na at eac h receiver , i.e., N 1 = N 2 = 1 , and withou t common message, i.e., r 0 = 0 , we have r 1 ≤ min(1 , M − J 2 ) and r 2 ≤ min (1 , M − J 1 ) . This resu lt c an be com pared to the d .o.f. of th e com pound MIMO-BC [1]. A po siti ve s.d.o .f. tup le (1 , 1) is ach iev able if J 1 < M and J 2 < M . If th e cha nnel un certainty of on e user increa ses, for examp le, J 2 = M , the s.d.o.f. of user 1 collapses. Moreover , the s.d .o.f. of both users become s zero if J 1 ≥ M , J 2 ≥ M . W e remark that secr ecy constraints significantly reduce d. o.f. and s ometimes may yield pessimistic results with r espect to [ 1]. I I I . E R G O D I C F A D I N G C O M P O U N D M I S O - B C C A. Mod el a nd Definitions W e co nsider the MISO-BCC, where th e transmitter with M antennas send s th e c onfidential m essages W 1 , W 2 respectively to tw o recei vers, each equipp ed with single anten na. W e consider th e ergod ic block fading model, in which the cha nnel remains in one state fo r a block of T ch annel uses and ch anges to ano ther chann el state indepen dently from on e block to another . W e a ssume th at the fading process is stationary and ergodic over time. Hence, the ch annel state at block t is given by the set o f ran dom variables ( A 1 [ t ] , A 2 [ t ] , H [ t ]) ∈ A , where A = { 1 , . . . , J 1 } × { 1 , . . . , J 2 } × { 1 , . . . , N } deno tes th e space of fading states and e ach r andom variable is unif ormly distributed over its set. Under non-perfec t CSIT , the tr ansmitter is a ssumed to know H [ t ] an d J 1 J 2 possible states at block t but no t the realization o f A 1 [ t ] and A 2 [ t ] , and receiver k is assumed to know both H [ t ] and A k [ t ] . At each block t , the channel of user k is expressed b y two r andom vectors h A k [ t ] k [ H [ t ]] , for which we den ote h A k [ t ] k [ t ] for the notation al simplicity . Finally , we assume tha t for each t , any M vectors taken fr om { h 1 1 [ t ] , . . . , h J 1 1 [ t ] , h 1 2 [ t ] , . . . , h J 2 2 [ t ] } has rank M . For each cha nnel use at blo ck t , th e ergod ic fading com- pound M ISO-BCC is exp ressed by y k [ t ] = h j k [ t ] H x [ t ] + ν k [ t ] , w .p. P ( A k [ t ] = j | H [ t ]) = 1 J K , ∀ j for k = 1 , 2 and t = 1 , . . . , m , wher e w .p . de notes with probab ility , ν k [ t ] ∼ N C (0 , 1) is an A WGN and i.i.d. over k , t , and th e input covariance S x [ t ] of x [ t ] satisfies the long-term power constrain t 1 m P m t =1 tr ( S x [ t ]) ≤ P . W e let n = mT denote the total numb er o f symb ols over m blocks. The definition for th e s.d.o.f. is the same as that in Section II-A. B. V ariable-Rate T ransmission W e first n ote that as m → ∞ , T → ∞ , the e r godic achiev able secrecy rate region is given by the union of all ( R 1 , R 2 ) such th at [2], 0 ≤ R 1 ≤ E [ I ( V 1 ; Y 1 )] − E [ I ( V 1 ; Y 2 , V 2 )] 0 ≤ R 2 ≤ E [ I ( V 2 ; Y 2 )] − E [ I ( V 2 ; Y 1 , V 1 )] (12) where the exp ectation is with regard to th e fading space A an d the un ion is over all possible d istributions V 1 , V 2 , X satisfy ing ( V 1 , V 2 ) → X → ( Y 1 , Y 2 ) . (13) It ca n be seen that the ergodic secrecy ra te of user k can be expressed by R k ≤ E [ I ( V k ; Y k )] − E [ I ( V k ; V k ′ )] − E [ I ( V k ; Y k ′ | V k ′ )] (14) where the fir st two term s can be interpr eted as the ergodic Marton bro adcast rate without secrecy constraint, and th e last term E [ I ( V k ; Y k ′ | V k ′ )] represents the information accumulated at the n on-inten ded re cei ver k ′ . W e next ad apt the variable-rate transmission pr oposed in [3, Th eorem 2 ] to the com pound MISO-BCC. W e fo cus on the zer o-forc ing beamformin g to provid e an achievable s.d.o.f. 3 region. At each channel use of block t , th e transmitter forms the codeword x [ t ] = x 1 [ t ] + x 2 [ t ] = v 1 [ t ] u 1 [ t ] + v 2 [ t ] u 2 [ t ] ( 15) where v k [ t ] deno tes a u nit-norm beamfo rming vector of user k (to be sp ecified below) and u k [ t ] ∼ N C (0 , p k [ t ]) is sym bol of user k , an d u 1 [ t ] , u 2 [ t ] are m utually independ ent. Clearly , the Markov ch ain (13) is satisfied by letting V k = v k [ t ] u k [ t ] and X = V 1 + V 2 at e ach t . Following [3], we assume tha t th e transmitter sen ds the codeword x k [ t ] to user k at r ate given by R k, tx [ t ] = I ( u k [ t ]; y k [ t ]) − I ( u k [ t ]; u k ′ [ t ]) (16) (a) = I ( u k [ t ]; y k [ t ]) = J k X j =1 P ( A k [ t ] = j | H [ t ]) I ( u k [ t ]; y k [ t ] | A k [ t ] = j ) where (a ) follows from the indepen dency between u 1 [ t ] an d u 2 [ t ] . This variable-rate strategy enab les to lim it th e leaked informa tion at the n on-inten ded receiver k ′ at each block t such that I ( u k [ t ]; y k ′ [ t ] | u k ′ [ t ]) = J k ′ X j =1 P ( A k ′ [ t ] = j | H [ t ]) I ( u k [ t ]; y k ′ [ t ] | u k ′ [ t ] , A k ′ [ t ] = j ) ≤ R k, tx [ t ] (17) for k ′ 6 = k and k = 1 , 2 . By co mbining (1 6) and ( 17), the av eraged secrecy rate of user k over m blocks is g iv en by R m k = 1 m m X t =1 R k, tx [ t ] − 1 m m X t =1 I ( u k [ t ]; y k ′ [ t ] | u k ′ [ t ]) = 1 m m X t =1 R k [ t ] (18) where the secr ecy rate of user k at b lock t is g i ven by R k [ t ] = [ R k, tx [ t ] − I ( u k [ t ]; y k ′ [ t ] | u k ′ [ t ])] + (19) W e rem ark th at similar to [3], the variable rate strategy av oids the non- intended r eceiv er k ′ to accumulate th e in formation on symbol k over m b locks, whenever the cha nnel condition is better than the transmission rate of user k . C. S ecr ecy Degr e e of F reedom R e gion In the f ollowing, we provide the s.d.o.f. analysis for dif ferent cases of ( J 1 , J 2 ) . Theor em 2: Th e two-user ergo dic fading co mpoun d MISO- BCC with J 1 < M , J 2 < M achiev es th e s.d.o .f. region { ( r 1 , r 2 ) : r 1 ≤ 1 , r 2 ≤ 1 } . Pr o of: A t each block t , the tra nsmitter for ms x [ t ] given in (15) by choo sing v 1 [ t ] o rthogon al to h 1 2 [ t ] , . . . , h J 2 2 [ t ] and v 2 [ t ] orthogo nal to h 1 1 [ t ] , . . . , h J 1 1 [ t ] . This yields the receiv ed signals for k = 1 , 2 given by y k [ t ] = φ j k,k [ t ] u k [ t ] + ν k [ t ] , w . p. P ( A k [ t ] = j | H [ t ]) = 1 J k , ∀ j where φ j k,i [ t ] = h j k [ t ] H v i [ t ] . I t can be shown that v 1 [ t ] an d v 2 [ t ] can be ch osen such that φ j k,k [ t ] 6 = 0 . Since th e ZF cr eates two par allel ch annels fo r any p air ( A 1 [ t ] , A 2 [ t ]) , the averaged secrecy rate of user k over m blocks is re adily given by R m k ≤ 1 mJ k m X t =1 J k X j =1 C ( p k [ t ] | φ j k,k [ t ] | 2 ) As m → ∞ , th e corner point (1 , 0 ) , (0 , 1) is achieved by allocating p 1 [ t ] = P , ∀ t , p 2 [ t ] = P , ∀ t , respectively , and the rate point (1 , 1) is achieved by eq ual power allo cation p 1 [ t ] = p 2 [ t ] = P / 2 at each t . Time-sharing between three p oints yields the region. Theor em 3: Th e two-user ergo dic fading comp ound MISO- BCC with J 1 < M , J 2 ≥ M achieves the s.d.o.f. region (see Fig.1) th at includes ( r 1 , r 2 ) satisfying r 1 ≤ M − 1 J 2 ,  J 2 M − 1 − 1  r 1 + r 2 ≤ 1 (20) Pr o of: At each b lock t , the transmitter chooses v 1 [ t ] orthog onal to the first M − 1 states 1 h 1 2 [ t ] , . . . , h M − 1 2 [ t ] and v 2 [ t ] orthogon al to h 1 1 [ t ] , . . . , h J 1 1 [ t ] to for m the codew ord (15) at each t . T his yield s the rec ei ve sign als y 1 [ t ] = φ j 1 , 1 [ t ] u 1 [ t ] + ν 1 [ t ] , w .p. P ( A 1 [ t ] = j | H [ t ]) = 1 J 1 , ∀ j y 2 [ t ] = 8 > > < > > : φ j 2 , 2 [ t ] u 2 [ t ] + ν 2 [ t ] , w .p. P ( A 2 [ t ] = j | H [ t ]) = 1 J 2 for j ≤ M − 1 φ j 2 , 1 [ t ] u 1 [ t ] + φ j 2 , 2 [ t ] u 2 [ t ] + ν 2 [ t ] w .p. P ( A 2 [ t ] = j | H [ t ]) = 1 J 2 for M ≤ j ≤ J 2 W e re mark that the incr eased ch annel unc ertainty at user 2 ( J 2 ≥ M ) incu rs two effects. First, it decr eases the transmission rate of user 2 due to interferen ce from u ser 1. Second, it dec reases the secrecy r ate of user 1 since u ser 2 observes u 1 [ t ] with p robability J 2 − M +1 J 2 , if A k [ t ] is between M and J 2 . W e obtain the secrecy r ates at b lock t given by R 1 [ t ] ≤ 2 4 1 J 1 J 1 X j =1 C ( p 1 [ t ] | φ j 1 , 1 [ t ] | 2 ) − 1 J 2 J 2 X j = M C ( p 1 [ t ] | φ j 2 , 1 [ t ] | 2 ) 3 5 + R 2 [ t ] ≤ 1 J 2 M − 1 X j =1 C ( p 2 [ t ] | φ j 2 , 2 [ t ] | 2 ) + 1 J 2 J 2 X j = M C p 2 [ t ] | φ j 2 , 2 [ t ] | 2 1 + p 1 [ t ] | φ j 2 , 1 [ t ] | 2 ! Plugging these exp ressions into (1 8) an d letting m → ∞ , the corner po int (0 , 1) , ( M − 1 J 2 , 0) is achiev ed by letting p 2 [ t ] = P and p 1 [ t ] = P for all t . Under equ al p ower allocation p 1 [ t ] = p 2 [ t ] = P 2 for all t ,  M − 1 J 2 , M − 1 J 2  is achiev ed. Time-sharing of these three p oints yields the region . Theor em 4: Consid er the two-user e rgodic compo und MISO-BCC with J 1 ≥ M , J 2 ≥ M . W e define the f unction f ( J 1 , J 2 ) = M − 1 J 1 + M − 1 J 2 − 1 − M − 1 J 1 + J 2 . If f ( J 1 , J 2 ) ≤ 0 , an ach iev able region is given by the time-sh aring between ( M − 1 J 2 , 0) a nd (0 , M − 1 J 1 ) . If f ( J 1 , J 2 ) > 0 , an achiev able region (see Fig. 2) is time -sharing b etween these two points and ( r s , r s ) with r s = M − 1 J 1 + M − 1 J 2 − 1 . Pr o of: W ithout loss of gen erality , the tr ansmitter chooses v 1 [ t ] orth ogona l to h 1 2 [ t ] , . . . , h M − 1 2 [ t ] and v 2 [ t ] orthog onal 1 The same result holds for any M − 1 set tak en from { h 1 2 [ t ] , . . . , h J 2 2 [ t ] } . 4 J 1 0 J 2 C M−1 J 1 M−1 J 2 M−1 J 1 M−1 Fig. 2. s.d. o.f. regio n for J 1 ≥ M , J 2 ≥ M to h 1 1 [ t ] , . . . , h M − 1 1 [ t ] to form the codeword given in (15) at block t . T his yields th e receiv e signals y k [ t ] = 8 > > > < > > > : φ j k,k [ t ] u k [ t ] + ν k [ t ] , w .p. P ( A k [ t ] = j | H [ t ]) = 1 J k for j ≤ M − 1 φ j k,k [ t ] u k [ t ] + φ j k,k ′ [ t ] u k ′ [ t ] + ν k [ t ] , w .p. P ( A k [ t ] = j | H [ t ]) = 1 J k for M ≤ j ≤ J k for k = 1 , 2 . By taking into accoun t the two effects caused by the increased ch annel u ncertainty mentioned above, we o btain the secrecy rate of user k at b lock t is given by R k [ t ] = 2 4 1 J k M − 1 X j =1 C ( p k [ t ] | φ j k,k [ t ] | 2 ) − 1 J k ′ J k ′ X j = M C ( p k [ t ] | φ j k ′ ,k [ t ] | 2 ) + 1 J k J k X j = M C 0 @ p k [ t ] | φ j k,k [ t ] | 2 1 + p k ′ [ t ] | φ j k,k ′ [ t ] | 2 1 A 3 5 + for k = 1 , 2 . Pluggin g the above expression into (18) and letting m → ∞ , th e corn er point A =  M − 1 J 2 , 0  , B =  0 , M − 1 J 1  is achieved by letting p 1 [ t ] = P , p 2 [ t ] = P, ∀ t , respectively . Und er equ al power allocatio n p 1 [ t ] = p 2 [ t ] = P / 2 , ∀ t , we h av e two different be haviors accord ing to the value of f ( J 1 , J 2 ) . Interestingly , if f ( J 1 , J 2 ) > 0 , the s.d.o.f. point C = ( r s , r s ) which dominates the line segment A B is achieved, as shown in Fig.2. On the c ontrary , if f ( J 1 , J 2 ) ≤ 0 , the p oint ( r s , r s ) is below the line segment A B. This can be easily verified b y comparing r s and th e inter section betwe en the line segment A B and r 2 = r 1 . W e remark that an achie vable s.d .o.f. with the er godic model gradua lly decreases as the u ncertainty increa ses. Moreover, the tim e variation of the channel state creates an additio nal temporal d imension, an d significantly improves the s.d. o.f. with respect to the Gaussian model with constant channel s tate. W e provide a simp le example to illustrate the difference be- tween two models. Consider the co mpoun d M ISO-BCC with M = 7 , J 1 = J 2 = 8 . The ergod ic m odel achiev es (1 / 2 , 1 / 2 ) which domin ates the time-sharin g between the corn er points (3 / 4 , 0) an d (0 , 3 / 4) . Th e Gaussian model yields zer o s.d.o .f. for both u sers. This r adical difference is be cause the nu mber of chann el states over which pe rfect secrecy must b e kept for the G aussian m odel equals th e nu mber o f wir etappers, w hich is not the ca se for th e ergod ic mod el. I V . C O N C L U S I O N S W e have studied th e two-user co mpoun d M IMO-BCC, fo r which we have fo und that time variation of th e chann el state provides an addition al tempo ral dimension for the ergodic model, wh ich im proves an ach iev able s.d.o.f . region com pared to the Gau ssian mod el with a con stant fading state, alth ough at the price o f a larger delay . W e n ote also that in con trast to the compou nd MI MO-BC [1], the gain by multiletter ap proache s (i.e. combin ing se veral time instance s) is no t expected h ere. Finally , we conjectu re that an achiev able s.d.o.f. region pr o- vided in the pap er is indeed the s.d.o.f. region an d the proof remains as a f uture investigation. A C K N O W L E D G M E N T This work is p artially supp orted by the Eu ropean Commis- sion in th e framework of the FP7 Network of Ex cellence in W ireless Com munication s NEWCOM++. R E F E R E N C E S [1] H. W eingart en, S. Shamai, and G. Kramer , “On the compound MIMO broadca st channel, ” in P r oceedi ngs of Annual Information Theory and Applicat ions W orkshop (IT A) , San Diego, CA, 2007. [2] R. Liu and H. V . Poor , “Secrec y capacity region of a multi-ante nna Gaussian broadcast channel with confident ial messages, ” IEEE T rans. on Inform. Theory , vol. 55, no. 3, pp. 1235–1249, March 2008. [3] P . Gopal a, L. Lai, and H. El Gamal, “On the secrecy cap acity of fadin g channe ls, ” IEE E Tr ans. on Inform. Theory , vol . 54, no. 10, October 2008. [4] H. D. Ly , T . Liu, and Y . Liang, “MIMO Broadcasting with Common, Pri va te, and Confidential Messages, ” in Proc . Inter . Symp. Inform. Theory and its App. (ISIT A) , New Zealand, December 2008. [5] M. Kobaya shi, M. Debbah, and S. Shamai (Shitz ), “Secured communica- tions ov er freque ncy-sele cti ve fa ding channels: A practi cal V ander monde precodi ng, ” to appear in E urasip, Special Issue on W ir eless Physical Securit y . [6] Y . Liang, G. Kramer , H. V . Poor , and S. Shamai (Shitz ), “Compound wire-ta p channel s, ” in Proc . 45th Annu. Allert on Conf . Communication , Contr ol and Computing , Montice llo, IL, USA, 2007. [7] ——, “Recent results on compound wire-tap channels, ” in Pro c. IEEE Internati onal Symposium on P ersonal , Indoor and Mobile Radio Com- municati ons (P IMRC) , Cannes, France, 2008. [8] T . L iu, V . Prabhakaran, and S. V ishwa nath, “The secrecy capacity of a class of parallel Gaussian compound wiretap channels, ” in Proc . IEEE Internat ional Symposium on Information Theory (ISIT) , T oronto, Ontario, Canada, 2008, pp. 116–120. [9] J. Xu, Y . Cao, and B. Chen, “Capacity Bounds for Broadcast Channels with Confidential Messages, ” submitted to ”IE EE T rans. on Inform. Theory”, May 2008, also available Ar xiv preprint arXiv:0805.4374 , 2008. [10] A. Khisti, G. W ornell, A. Wie sel, and Y . Eldar , “On the Gaussian MIMO wiretap channe l, ” in Proc. IEEE International Symposium on Informatio n Theory (ISIT) , Nice, France, 2007. 5 receiver 1 Y 1 n W 0 W 1 receiver 2 Z 1 n W 0 W 0 W 1 transmitter T( h ) T( g ) X n encoder decoder1 decoder2 AWGN AWGN