The Streaming-DMT of Fading Channels

Transcription

1 The Streaming-DMT of Fading Channes Ashish Khisti Member, IEEE, and Star C. Draper Member, IEEE arxiv:30.80v3 cs.it] Aug 04 Abstract We consider the sequentia transmission of a stream of messages over a boc-fading muti-input-muti-output MIMO channe. A new message arrives at the beginning of each coherence boc, and the decoder is required to output each message sequentiay, after a deay of T coherence bocs. In the specia case when T =, the setup reduces to the quasi-static fading channe. We estabish the optima diversity-mutipexing tradeoff DMT in the high signa-to-noise-ratio SNR regime, and show that it equas T times the DMT of the quasi-static channe. The converse is based on utiizing the deay constraint to ampify a oca outage event associated with a message, gobay across a the coherence bocs. This approach appears to be new. We propose two coding schemes that achieve the optima DMT. The first scheme invoves intereaving of messages, such that each message is transmitted across T consecutive coherence bocs. This scheme requires the nowedge of the deay constraint at both the encoder and decoder. Our second coding scheme invoves a sequentia tree code and is deay-universa i.e., the nowedge of the decoding deay is not required by the encoder. However, in this scheme we require the coherence boc-ength to increase as og SNR, in order to attain the optima DMT. Finay, we discuss the case when mutipe messages arrive at uniform intervas within each coherence period. Through a simpe exampe we exhibit the sub-optimaity of intereaving, and propose another scheme that achieves the optima DMT. Index Terms Rea-Time Streaming Communication, Diversity-Mutipexing Tradeoff, Boc-Fading, Tree Codes, Intereaving. I. INTRODUCTION Mutimedia appications require rea-time encoding of a source stream and a sequentia reconstruction of each sourcepacet by its paybac deadine. Both the fundamenta imits and optima communication techniques for such streaming systems can be very different from cassica communication systems. In recent years there has been a growing interest in characterizing information theoretic imits for deayconstrained communication over wireess channes. When the transmitter has channe state information CSI, a notion of deay-imited capacity can be defined ]. For sow fading channes, the deay-imited capacity is achieved using channe inversion at the transmitter 3]. In absence of transmitter CSI, an outage capacity can be defined 4], 5]. Unfortunatey the characterization of the outage capacity is in genera a chaenging probem, even in point-to-point settings 6]. A somewhat coarse metric for studying the outage capacity is the diversity-mutipexing tradeoff DMT, first introduced in 7]. The authors propose diversity order and mutipexing gain as two fundamenta metrics for communication over a wireess Part of this wor was presented at the Internationa Symposium on Information Theory, St. Petersburg, Russia, 0 ]. A. Khisti s wor was supported by an NSERC Natura Sciences Engineering Research Counci Discovery Grant and an Ontario Eary Researcher Award. S. Draper s wor was supported by the Nationa Science Foundation under CAREER grant CCF and by an NSERC Discovery Grant. channe, and estabish a tradeoff between these for quasistatic, muti-input-muti-output MIMO fading channes, in the high signa-to-noise-ratio SNR regime. A significant body of iterature on DMT now exists, see e.g., 5, Chapter 9]. Of particuar interest in this wor is the case of T independent parae MIMO fading channes where the optima DMT equas T times the DMT of the quasi-static MIMO fading channe, with a suitaby normaized mutipexing gain 7]. Practica code constructions for parae fading channes have been proposed in 8] 0]. Interestingy, when the parae channes are correated, the DMT anaysis is far more intricate and ony specia cases are nown ], ]. In the present paper we consider the probem of rea-time streaming over a MIMO boc-fading wireess channe. We assume that the transmitter observes a sequence of independent messages. One message arrives per coherence boc, right at the start of the boc. The input into the channe can depend on a the past messages, but not on any future messages. The decoder is required to output each message with a maximum deay of T coherence bocs. When T =, each message sees ony one fading reaization and the setup reduces to the quasi-static fading channe mode. In genera, each message experiences T independent fading bocs; however it must be mutipexed together with messages arriving in other coherence bocs. We decare a message to be in outage if it cannot be decoded by its deadine. We estabish the optima DMT in this streaming setting and show that it equas T times the DMT of the quasi-static MIMO fading channe. The optima DMT can be achieved by a simpe intereaving of messages across coherence bocs and transmitting each message over T parae, MIMO fading channes. We aso propose an aternative tree-code that attains the optima DMT. In this scheme the deay constraint ony needs to be reveaed to the decoder, and not to the encoder, and thus it is suitabe for appications where a common source stream must be transmitted to mutipe receivers with different decoding deays. However in order to achieve the optima DMT, the coherence boc ength for tree-codes must increase with og SNR and thus this scheme appears to require ong coherence periods. Tree codes for streaming communication over a discrete memoryess channes DMC have been studied previousy in 3] 7]. These wors, however, consider maximum ieihood and universa decoders. In contrast, our anaysis of the tree code is based on a very different outage anaysis paired with a decision directed decoder. We express our error probabiity as a sum of two terms one term decreases exponentiay in ogsnr whie the other decreases exponentiay in the coherence boc-ength. By suitaby baancing the two exponents we estabish that our proposed scheme attains the optima DMT. Another recent wor, reference 8], studies a reated setup when the transmitter sequentiay observes a

2 stream of messages, but assumes that a the messages have a common deadine. A variety of coding techniques such as adaptive joint encoding, memoryess transmission, time sharing and superposition transmission are compared in different deay and SNR regimes. Another foowup wor 9] considers the case when a the messages are avaiabe to the encoder, but have different paybac deadine at the receiver. In contrast our proposed setup requires that each incoming message must be reconstructed after a fixed decoding deadine, which is reevant in appications such as rea-time voice and video streaming. Finay in yet another reated wor 0], ], the authors study the transmission of bursty and deay-sensitive data source over a constant-rate MIMO fading channe and estabish an optima operating point on the DMT that baances the channe outage and deay-vioation probabiities. However the resuts are vaid ony for asymptoticay arge decoding deays. Furthermore it appears that the coding techniques considered in these wors do not retransmit the same information bits across mutipe coherence bocs, a ey idea expoited in the present paper. In the rest of the paper we describe the system mode in Section II, and the main resut, that characterizes the streaming- DMT in Section III. We provide the proof of the converse in Section IV. The coding schemes based on intereaving and tree-codes are presented in Section V. We discuss extension to the case of mutipe messages in Section VI and provide concusions in Section VII. Throughout the paper we wi use the foowing notation. Upper case bod-font wi be reserved for matrices e.g., H whereas ower case bod-font e.g., x wi be used for vectors. Scaar symbos wi be denoted using ower case non-bod fonts. We wi use the sans serif font for random variabes e.g., x. A sequence of symbos x i, x i+,..., x j wi be denoted using the notation x j i. Throughout the paper the symbo. = wi be reserved to denote equaity in the exponentia sense i.e., we express, fρ. = ρ b, if im ρ og fρ og ρ = b hods. The symbos and wi be defined in a simiar fashion. II. MODEL We consider an independent identicay distributed i.i.d. boc fading channe mode with a coherence period of M: Y = H X + Z, where = 0,,..., denotes the index of the coherence boc of the fading channe. The matrix H C Nr Nt denotes the channe transfer matrix in coherence period. We assume that the transmitter has N t transmit antennas and the receiver has N r receive antennas. X = x... x M] C Nt M is a matrix whose j-th coumn, x j, denotes the vector transmitted in time-sot j in the coherence boc and simiary Y C Nr M is a matrix whose j-th coumn, y j denotes the vectors received in time-sot j in boc. The additive noise matrix is Z C Nr M. Thus can aso be expressed as, y j = H x j + z j, j =,..., M. We assume that a entries of H are samped independenty from the compex Gaussian distribution with zero-mean and unit-variance i.e., CN 0,. The channe remains constant during each coherence boc and is samped independenty across bocs. A entries of the additive noise matrix Z are aso samped i.i.d. CN 0,. Finay the reaization of the channe matrices H is reveaed to the decoder, but not to the encoder. We assume an average short-term power constraint E M i= x i ] Mρ. Note that ρ denotes the transmit power which wi serve as our SNR parameter. A deayconstrained streaming code is defined as foows: Definition Streaming Code: A rate R streaming code with deay T, CR, T, consists of. A sequence of messages {w } 0 each distributed uniformy over the set I M = {,,..., MR }.. A sequence of encoding functions F : I + M CNt M, X = F w 0,..., w, = 0,,... 3 that maps the input message sequence to the channe input matrix X C Nt M. 3. A sequence of decoding functions G : C M+T I M that outputs message estimate ŵ based on the first +T observations, i.e., ŵ = G Y 0,..., Y +T, = 0,,... 4 Fig. iustrates such a setup for the case when T =. One message w arrives at the start of each coherence boc. The codeword transmitted in boc, X w 0 can depend on a the past messages, but not on any future messages. Since T =, the receiver must decode message w at the end of coherence boc + i.e., ŵ = G Y 0,..., Y +. We now define the diversity-mutipexing tradeoff DMT associated with the streaming code CR, T. The error probabiity for the -th message is Prw ŵ ] where ŵ is the decoder output 4 and the error probabiity is averaged over the random channe gains. In the DMT both error probabiity and rate are studied as a function of the SNR parameter ρ. Let e max ρ = sup 0 Prŵ w ] denote the worst-case error probabiity of a rate-rρ code. A DMT tradeoff 7] of r, d is said to be achievabe with deay T if there exists a sequence of codeboos CRρ, T achieving e max ρ such that Rρ r = im ρ og ρ, d = im ρ og e max ρ. 5 og ρ Of interest, is the optima diversity-mutipexing tradeoff, denoted by d T r. One cass of wireess systems that motivates this mode is frequency-hopping orthogona frequency-division mutipe Whie we ony focus on the Rayeigh channe mode, our resuts easiy extend to other channe modes. We caution the reader that in the above discussion e maxρ is not the maximum error probabiity with respect to a singe reaization of the fading state sequence. This ater quantity is ceary as in any sufficienty ong reaization there wi eventuay be a boc fade that induces an outage. In our definition we fix an index,, and find the error probabiity Prŵ w ] averaged over the channe gains. We subsequenty search for the index with the maximum error probabiity. E.g., for time-invariant coding schemes operating in the steady-state regime, due to symmetry, Prŵ w ] wi not depend on.

3 w 0 w w 3 w 4 w 5 x 0 w 0 x w 0,w x w 0,.. x 3 w 0,.. w 3 x 4 w 0,.. w 4 xw 0,.. w 5 T= ŵ 0 ŵ ŵ ŵ 3 ŵ 4 Fig.. Proposed Streaming Mode. One new message arrives at the start of each coherence boc. The message stream is encoded sequentiay and each message needs to be output at the receiver after T coherence bocs. In the above figure T =. access OFDMA systems. Here the frequency bands or sub-channe aocated to a user are changed at reguar intervas and in a randomized fashion. In frequency-seective channes the randomized sub-channe aocation means that the channes the the user experiences pre- and post-hop are approximatey independent, as ong as the expected recurrence time of a particuar sub-channe exceeds the coherence time of the channe. In this setting we can characterize the paybac deadine in terms of the number of hops T unti the message w must be estimated by the receiver. This can be transated into a number of channe uses, T M, where M denotes the number of symbos transmitted in each hop, and hence into time. In practica systems the vaue of M may be fixed and thus not under the contro of the appication. III. MAIN RESULT The optima tradeoff between diversity and mutipexing DMT for the quasi-static fading channe was characterized in 7]. We reproduce the resut beow for the convenience of the reader. Theorem : Zheng and Tse, 7] For the quasi-static fading channe yt = H xt + zt 6 where the entries of H C Nr Nt are samped i.i.d. CN 0,, the optima DMT tradeoff d r is a piecewise inear function connecting the points, d for = 0,,..., minn r, N t where d = N r N t. 7 In our anaysis the foowing generaization of the quasistatic DMT to L parae channes, see 7, Coroary 8] 8], 9] is usefu. Coroary : Consider a coection of L parae quasi-static fading channe y t = H x t + z t, =,..., L 8 where the entries of H C Nr Nt are a samped i.i.d. CN 0,. The DMT tradeoff is given by d L r = L d r L for any r 0, L minn r, N t. Our main resut estabishes the optima DMT for a boc fading channe mode with a deay constraint of T coherence bocs. Theorem : The optima DMT tradeoff for a streaming code cf. Definition with a deay of T coherence bocs is given by d T r = T d r, where d r is the optima DMT of the underying quasi-static fading channe 7. Comparing the resuts of Theorem with that of Coroary, we observe that the DMT of a streaming source under a deay constraint of T coherence bocs is identica to the DMT of a system with T independent and parae MIMO channes if the rate of the atter system is suitaby normaized. Indeed, one of our achievabiity schemes expoits this connection. We show that the DMT can be achieved by intereaving messages in a suitabe manner to reduce the system to a parae channe setup. However, the converse does not foow from earier resuts since the ength-t paybac deadines of successive messages are ony partiay overapping. We present a new approach that addresses the overapping character of the paybac deadines. The technique is specific to the streaming setup and appears nove. Remar : In our system mode, we assumed that the coherence boc-ength M can be arbitrariy arge. It is we nown 7] that for quasi-static channe, as we as its extension to L parae channe 9], the DMT in Theorem hods for any coherence boc-ength M N r +N t. In a simiar fashion our resut in Theorem hods for any M N r + N t. In particuar the converse in Section IV hods for any M. The intereaved coding scheme in Section V-A reduces the setup to parae channes and appies to any M N r + N t. However this scheme requires the nowedge of T at both the encoder and decoder. Our second coding scheme, which is based on a tree code and ony requires the nowedge of T at the decoder, does require M to be sufficienty arge. In particuar our anaysis for this scheme requires that M must increase as og SNR to achieve the optima DMT. IV. CONVERSE In this section we estabish a ower bound on the error probabiity for any streaming code in Definition. We thereby 3

4 upper bound the achievabe DMT. In particuar we show that Prerror]. T dr ρ where d r is the DMT tradeoff associated with a singein MIMO channe. For the purpose of estabishing a contradiction, we wi assume that a DMT better than d T r is achievabe, say T d r. We show that for any > 0 a contradiction wi buid up if we operate the system over a sufficienty arge number of bocs N. The smaer is the onger it taes the contradiction to buid up. The steps in our proof are the foowing, iustrated in Fig.. FANO: Appy Fano s inequaity to each w individuay. A decision on w must be made at time T = +T. GENIE: Condition the decoding of w on a previous messages w0. This can ony hep the decoder thereby increasing the DMT because the decoder nows exacty the vaue of a earier messages. This step can be thought of as a genie-heper. 3 SUFFIX OUTAGE: Next we condition on the event that the suffix of the codeword is in outage. By suffix we mean the symbos transmitted in bocs, +,..., T = T +. We bound this event using the standard DMT anaysis. 4 COMBINE EVENTS: Finay, using standard information manipuations, we combine events up to message w N T + for some arge N to be determined. 5 CONTRADICTION: Finay, using the statistica description of the channe aw, we find that for any > 0 we can identify a finite N sufficienty arge such that a contradiction arises and this demonstrates that a DMT of T d r is not achievabe. Foowing the approach outined above, in our first step we appy Fano s inequaity, Chapter ] to ower bound the error probabiity associated with message w. To do this, we define E to be the event that ŵ w and note that Prerror] = sup 0 PrE ]. We start by indexing the error events pointwise in the possibe channe reaizations. Stricty speaing, the summation over the channe gains must be an integra, since the channe gains are continuous vaued. We however use summations, so that the expressions are easier to foow. A the steps in this section easiy foow when the summations are repaced by corresponding muti-integras. PrE ] = H T 0 PrH T 0 = H T 0 ] PrE H T 0 = H T 0 ]. 9 Message w needs to be decoded at time T = +T. The observations accumuated to that time are the channe outputs Y T 0. Recognizing that the og-cardinaity of the message set from which w is chosen is Mr og ρ, we appy Fano s inequaity to each channe reaization to get 0 PrE H T 0 = H T 0 ] + Hw Y T 0, HT 0 = H T Mr og ρ = + Hw Hw + Hw Y T 0, HT 0 = H T 0 Mr og ρ Mr og ρ Hw 0, H T 0 = H T 0 + Mr og ρ Hw Y T 0, HT 0 = H T Mr og ρ 0 where the atter inequaity foows since Hw = Hw 0, H T 0 = H T 0. The second step in our proof is the genie-aided step. We condition the ast term in the above on a previous messages yieding the further ower bound: PrE H T 0 = H T 0 ] 0 Mr og ρ Hw 0, H T 0 = H T 0 Mr og ρ + Hw 0, Y T 0, HT 0 = H T 0 Mr og ρ = Mr og ρ Iw ; Y T 0 0, H T 0 = H T 0 Mr og ρ = Mr og ρ Iw ; Y T 0, H T = HT Mr og ρ, To get the ast equaity we note the foowing conditions. First, since message and channe reaizations are independent Hw 0, H T 0 = H T 0 = Hw 0, H T = H T. Second, we note the foowing Marov reationship: w w0, Y T, HT Y0, H 0. This reation hods due to the causa nature of the encoder and the i.i.d. nature of the channe. In particuar, note that causa encoding means that the channe inputs X T are a function of w and w0 whie past channe inputs are a function ony of w0. Thus, since the channe is memoryess the past channe output and state information Y0, H 0 provides no information about w that w0, Y T, HT does not provide. In the third step we condition on the suffix being in outage. In particuar, define the singe-boc outage set H = { H : Ix; y H = H r og ρ } 3 By the cassic outage anaysis 3], 4], which underies the DMT of Theorem, we now that P = PrH H ]. = ρ dr 4 where d is the DMT specified in Theorem and the exponentia equaity is at high SNR. By suffix outage we mean that H j is in outage for every boc j =,..., T, in other words T j= H j H. Using H T to denote the T fod Cartesian product of the set H, and recaing that the channe gains are samped in an i.i.d. fashion across bocs, we have ] ] Pr T j= H j H = Pr H T HT = P T. = ρ T d r. 5 We next incorporate the effect of outage into our ower bound. In Appendix A we show that PrE ] PrH T HT ] Mr og ρ Iw ; Y T 0, H T Mr og ρ HT., HT 6 where the expression H T HT in the conditioning indicates that the sequence H T beongs to the outage set H T. 4

5 suffix outage T N T... N observations when decode msg combine errors to here Fig.. In the converse, we consider a tota of N coherence bocs and N T messages. For each message, we consider the event that the coherence bocs,..., + T are in outage and ower bound the error probabiity in 6. We then combine the error probabiities associated with a the messages to obtain a ower bound on the maximum error. Since a the terms in the mutua information expression in 6 are independent of the channe gains: {H0, H N T + } we can express Iw ; Y T = Iw ; Y T 0, H T, HT HT 0, H N 0, H N 0 H N 7 Iw ; Y0 N 0, H N 0, H N 0 H N. 8 where the ast step foows from the fact that the mutua information is non-negative. We wi see that the fina oosening in 8 doesn t weaen our bound for two reasons. The first is because we study the max error probabiity. The second is because the information about the messages embedded in ater channe uses, i.e., Y N T +, must be used to decrease the entropy of ater messages. It cannot be focused excusivey on reducing the uncertainty of w without detrimenta effects on the abiity to estimate ater messages. This couping of errors across time is what we ca the outage ampification effect. in 0 into a simpe sum max PrE ] 0 N T Mr og ρ P T P T. = N j=0 IX j; Y j H j, H j H N T Mr og ρ NMr og ρ Mr og ρ N T Mr og ρ ] Nr Mr og ρ N T r ] ] 3 4 ρ T dr 5 where 4 foows from the definition 3 of H and we reca that there are M channe uses in each coherence interva. To see the contradiction we assume high SNR, so that the second term vanishes. Then, for any > 0, by seecting N > T r the term Nr ] N T r is stricty positive. Since > 0 is arbitrary, it foows that a diversity order greater than T d r cannot be achieved. In the fourth step we combine events. Substituting 5 and We observe that the N required to reaize a contradiction 8 into 6 we find is inversey proportiona to. This means that if you operate your system to exceed the DMT by a very sma amount it PrE ] P T wi tae some time for a contradiction to buid up. A coding ] scheme can be designed so that eary message can borrow Mr og ρ Iw ; Y0 N 0, H N 0, H N 0 H N channe resources from ater message to ensure their reiabiity.. Mr og ρ But, eventuay, the borrowing buids up and ater generations 9 cannot meet their obigations. The parameter N indexes the generation that runs into difficuty. And, since the max error is at east as arge as the average error, we can arrive at max PrE ] P T 0 N T ] Mr og ρ IXN 0 ; Y0 N H N 0, H N 0 H N N T Mr og ρ 0 as shown in the steps between. where 0 foows from the data processing inequaity since w0 N T X N 0 Y0 N hods regardess of the channe reaization. In the fina step we appy the channe statistics to get a contradiction. In particuar, since fading across different bocs is independent we can brea the mutua information term V. CODING THEOREM We present two approaches for achieving the DMT stated in Theorem. As mentioned in the introduction, the first approach is based on intereaving the ast T messages across coherence boc whie the second approach is based on a deay-universa tree code construction. A. Intereaving Scheme We show that a simpe intereaving based scheme suffices to achieve the DMT stated in Theorem. Our codeboo C maps each message w I M {,,..., Mr og ρ } to T codewords {X 0 w, X w,..., X T w } where each X j C Nt M T. Thus the overa code is a Cartesian product: C = C 0 C... C T, where X C. We wi assume that 5

6 max PrE ] 0 N T N T Mr og ρ N T =0 PrE ] ] N T P T =0 Iw ; Y0 N 0, H N 0, H N 0 H N N T Mr og ρ ] = P T Mr og ρ Iw N T 0 ; Y0 N H N 0, H N 0 H N N T Mr og ρ ] P T Mr og ρ IXN 0 ; Y0 N H N 0, H N 0 H N N T Mr og ρ each codeboo C j is samped i.i.d. according to a compex ρ norma CN 0, N t distribution 3 For transmission of each message, we assume that each coherence boc of ength M is further divided into T subbocs of ength M T, as indicated in Fig. 3. Suppose that I,0,..., I,T denote these intervas. The codeword X 0 w is transmitted in the first sub-boc I,0 of coherence boc. The codeword X w is transmitted in the sub-boc I +, of coherence boc + and iewise X j w is transmitted in the j-th sub-boc, I,j, of coherence boc + j. The corresponding output sequences associated with message w are denoted by: Y,j = H +j X j w + Z,j, j = 0,..., T. 6 The decoder finds the message ŵ such that for each j {0,..., T } the sequence pair X j ŵ, Y,j is jointy weay typica ]. The shaded boxes in Fig. 3 denote the intervas used for the decoding of w. The outage event at the decoder, associated with ŵ, is given by: { +T } C j ρ r og ρ 7 T j= where C j ρ = og det I + ρ N t H j H j. Since 7 precisey corresponds to the outage event of a quasi-static parae MIMO fading channe, with T channes and a mutipexing gain of T r, the achievabiity of the DMT in Theorem foows from Coroary. B. Sequentia Tree Codes Our second scheme is based on a sequentia tree code construction. This approach has the advantage that the encoder does not require the nowedge of T. The deay constraint ony needs to be reveaed to the decoder, and yet the optima DMT is attained. Our proposed construction consists of a sequence of codeboos {C 0, C,..., C,...}, where C is the codeboo to be used in coherence boc when messages w 0,..., w are reveaed to the encoder. Codeboo C consists of a tota of 3 We note however that any space-time code that achieves the DMT for independent parae MIMO fading channes can be used for the subcodeboos C 0,..., C T. In particuar the non-vanishing determinant NVD code in 8] can be used for these sub-codeboos instead of the random Gaussian codeboo. MR+ codeword sequences, with one codeword for each eement in the set: I + M = { w 0,..., w : w 0 I M,..., w I M }. 8 where I M {,,..., MR }. A codewords are samped ρ i.i.d. from CN 0, and are reveaed to both the encoder N t and the decoder in advance 4. In coherence boc, the encoder maps w 0,..., w to the codeword X w 0 C Nt M in C, and transmits it over M channe uses. The entire transmitted sequence up to and incuding boc is denoted by X 0w 0 { X 0 w 0, X w 0,..., X w 0 }, X 0w 0 C Nt +M 9 The decoder uses a sequentia, decision-directed decoding rue. We focus on the decoding of message w at the end of coherence boc T = + T, which corresponds to the deadine of message w. The decoder considers the entire received sequence Y T 0 = Y 0,..., Y T and computes a fresh estimate of a the messages up to time in + steps as foows. In the first step, the decoder searches over a message sequences ŵ T 0 such that the pair X T 0 ŵ T 0, YT 0 is jointy typica. If each such message sequence has a unique prefix, say w 0, then w 0 is seected as the message in boc 0. Otherwise an error is decared. Once the message w 0 is fixed in the first step, the decoder then proceeds to the second step. It searches for the message sequences ŵ T such that the pair X T w 0, ŵ T, YT is jointy typica. If each such message sequence has a unique prefix, say w then it is seected as the message in boc. Otherwise an error is decared. Once the message w is fixed the decoder proceeds sequentiay, producing,..., w. In determining w, with, the decoder fixes w 0 and searches for a sequence of messages ŵ T such that the corresponding transmit sequence X T 0 w 0, ŵ T has the property that the sub-sequence between to T the suffix satisfies where the set T, X T w 0, ŵ T, Y T T,T, 30 is the set of a jointy typica se- 4 We wi mae the assumption that the communication terminates after a sufficienty arge but fixed number of coherence bocs. 6

7 Coherence Boc: K Coherence Boc: K+ Coherence Boc: K+ Coherence Boc: K+3 Coherence Boc: K+4 Coherence Boc: K+5 Coherence Period X 0 w X w - X w - X 3 w -3 X 4 w -4 X 5 w -5 X 0 w + X w X w - X 3 w - X 4 w -3 X 5 w -4 X 0 w + X w + X w X 3 w - X 4 w - X 5 w -3 X 0 w +3 X w + X w + X 3 w X 4 w - X 5 w - X 0 w +4 X w +3 X w + X 3 w + X 4 w X 5 w - X 0 w +5 X w +4 X w +3 X 3 w + X 4 w + X 5 w Fig. 3. Intereaving based coding scheme for T = 6. Each coherence boc is divided into T sub-intervas and each sub-interva is dedicated to transmission of one message. The transmission of message w spans coherence bocs, +,..., + T using codewords of X 0 w,..., X T w as shown by the shaded bocs. quences ], { T, = X, Y : X T p X, Y T p Y, = og p X,Y X, Y hp X,Y ] M + }. ε 3 In 3 T p X and T p Y denotes the set of typica {X } and {Y } sequences respectivey and hp X,Y denotes the differentia entropy of jointy Gaussian random variabes. If the ist of a message sequences ŵ T that satisfy 30 have a unique prefix w then we concatenate w with w 0 to get w 0, otherwise an error is decared. When the process continues to step + without decaring an error, w is decared to be the output message estimate, i.e., ŵ = w. Fig. 4 iustrates the codeboo construction and the proposed sequentia decoding rue. The figure on the eft hand side iustrates the sequentia tree code. The right figures iustrate the sequentia decoding of w 0, w and. When decoding w 0, we consider a possibe paths in the tree typica with the received sequence. If a such paths ead to a unique prefix w 0, we decare this to be the message. Otherwise an error is decared. Once w 0 is fixed, we move aong the path of w 0 in the tree. Thereafter we search for a paths in the tree from eve to + T that are typica with the received sequence. This process continues unti eve is reached and w is determined. Remar : Our decoder is a decision directed decoder. In estimating w 0, it first estimates w 0 based on Y T 0. It next maes a conditiona estimate of w based on Y T with w 0 fixed, and continues aong in + steps. One may be tempted to try a simper decoding scheme that avoids the + steps and directy search for a unique prefix ŵ0 such that the resuting transmit sequence X T 0 is jointy typica with the received sequence Y T 0 i.e., { T =0 og p X,Y X, Y hp X,Y ] M + T }. ε 3 Such an approach wi not guarantee the recovery of the true w with high probabiity. This is because for the contribution of the terms before ŵ wi dominate. Even when ŵ w but ŵ 0 = w 0, the pair ˆX T, Y T wi in genera satisfy 3 as for the contribution of the suffix associated with ŵ wi be negigibe. In other words, our proposed decision directed decoder in 30 fixes the oder messages and guarantees that when decoding w we do not incude the bias introduced by w0 in 3. Anaysis of error probabiity: We show that for any > 0 and 0 < r < minn r, N t, the error probabiity averaged over the ensembe of codeboos satisfies Prŵ w ρ T dr. In our anaysis, we expoit symmetry in the code construction, as we as the encoding and decoding functions. To ay out the anaysis assume, without oss of generaity, that a particuar message sequence w0 = w0 has been transmitted. Define the events 5 E = and note that { w 0 : w 0,..., w = w 0,..., w, w w }, 0 33 Pr { w w } PrE, 34 where E corresponds to the event that our proposed decoder fais in step of the decoding process. We deveop an upper 5 A the error events E are defined for the decoder at time T +. However we suppress this dependence to eep the notation compact. =0 7

8 w w w w w 0 w 0 w Decoding of w 0 w w w 0 w Codeboo Construction w 0 Decoding of w Decoding of Fig. 4. The eft figure iustrates the tree-codeboo. The message w 0 is mapped to one of MR codewords in the first eve, the message pair w 0, w is mapped to one of MR codewords in the second eve etc., Whie decoding w the decoder starts at the root of the tree. It first finds a possibe transmit paths of depth + T in the tree, typica with the received sequence. If a unique prefix codeword w 0 is determined then the decoder moves aong the path of w 0 and finds a possibe codewords from eve to + T that are typica with the received codeword. A unique message w is determined if there is a unique prefix codeword in this eve. This process continues ti eve is reached and w is determined. bound on E for each 0 and substitute these bounds in 34. We further express E = A B, where { A = X T w T 0, YT } : X T w T 0, YT / T,T 35 denotes the event that a decoding faiure happens because the transmitted sub-sequence starting from position fais to be typica with the received sequence, whereas B = { w T 0 : w 0 = w 0, w w, X T w T 0, YT 36 denotes the event that the decoding faiure happens because a transmit sequence corresponding to a message sequence with w w appears typica with the received sequence. As shown in the Appendix B, using an appropriate Chernoff bound we can express, PrA MT +fε = MT + fε 37 where fε is a function that satisfies fε > 0 for each ε > 0. To bound PrB we begin by noting that by our code construction, we are guaranteed that whenever w w, the associated transmitted subsequence X T w T 0 is samped independenty of Y T. Hence from the joint typicaity anaysis ], we have that for any sequence w T 0 with w w Pr X T w T 0, YT T,T H T M T j= Ixj;yj Hj=Hj 3ε = H T where = M T j= Cjρ;Hj 3ε w T I T + M C j ρ; H j og det I + ρ H j H j N t 38 is the associated mutua information between the input and output in the j-th coherence boc when the channe matrix H j = H j. Appying the union bound we have that } PrB H T = H T 39 T,T Pr X T w T 0, YT T,T H T = H T w T I T + M M T j= Cjρ;Hj 3ε MT +R M T j= Cjρ;Hj 3ε M T j= Cjρ;Hj T +R 3ε To bound PrB we define { T i= O = H,..., H T : C i ρ; H i +T r og ρ+ r og ρ+4ɛ og ρ } 44 8

9 where r = d r d r 45 where we reca that d r denotes the quasi-static DMT 7 of the MIMO fading channe and we use d r to denote its right derivative. Note that d r < 0 for a r 0, minn t, N r ] as the DMT is a decreasing function of r. Thus it foows that r > 0. Note that PrB PrB H T From 44 and 43 we have O c + PrH T O. 46 PrB H T O c M r og ρ Mε og ρ 47 = ρ Mε M r. 48 We next upper bound the second term in 46. Note that O precisey corresponds to the parae MIMO channe in Coroary with L T + = + T channes, and mutipexing gain s = Lr + r + 4ε. The associated DMT satisfies: s L d = L d r + L L r + 4ε 49 L = L d r + L r + o ε 50 L d r + d r L r + o ε 5 = Ld r d r + o ε 5 = T d r + d r d r + o ε 53 = T d r + d r + o ε 54 where we use the continuity of d r in 50 and et o ε be a function of ε that vanishes as ε 0. We use the convexity of d r in 5. We substitute 45 for r in 5 and substitute L = T + in 53. Thus we have PrO T dr+ ρ d r+o ε. 55 From 46 and substituting 48 and 55 and using E = A B we have PrE PrA + PrB 56 MT + fε + ρ Mε M r T dr + ρ 57 From the union bound, PrE PrE 58 =0 =0 MT + fε + =0 ρ Mε M r + =0 T dr ρ d r+o ε. 59 d r+o ε. We upper bound the first term in 59 as M+T fε = M+T fε 60 =0 =0 M+T fε MT fε+, 6 =0 which vanishes as M. By a simiar argument we can upper bound the second term as ρ Mε M r = ρ Mε ρ M r 6 =0 ρ Mε =0 =0 ρ M r ρ Mε 63 for sufficienty arge ρ and M such that ρ M r. In a simiar fashion we can upper bound the third term in 59 as =0 T dr ρ From 59 we have that d r+o ε ρ T dr+oε =0 ρ dr 64 ρ T dr+oε. 65 PrE MT fε + ρ T dr+oε + ρ Mε 66 By seecting M dr og ρ fε, we have that MT fε ρ T dr+oε. Finay since ε > 0 can be seected as sma as required and o ε 0 as ε 0 we have that PrE ρ T dr. This competes the error anaysis for the sequentia tree code. VI. MULTIPLE MESSAGES PER COHERENCE BLOCK Our focus so far has been on the case where ony one message arrives in each coherence boc. In this section, we consider some specia cases when two messages, say w, and w, arrive in each coherence boc. Each message must be decoded after M T channe uses, where T denotes the deay in coherence bocs. In such a setup, the number of coherence bocs seen by w, before its deadine, wi be different from w,. Thus a simpe intereaving techniques such as is presented in Section V-A is no onger optima; more sophisticated coding techniques that expoit the asymmetry between w, and w, wi be required. Assume that w, arrives at time t, = M + M, whie w, arrives at time t, = M + M + where 0, /] denotes the offset reative to the start of the coherence boc when w, arrives. Fig. 5 denotes the streaming setup with two messages per boc corresponding to = 0 and = / respectivey. We obtain the optima DMT for the SISO channe with T = and = 0, which corresponds to the first case in Fig. 5, in Section VI-A. By the symmetry of the probem, the same resut aso appies when = /, iustrated in the second case in Fig 5. In subsection VI-B we show that if either = 0 or = /, but its actua vaue is not nown to the encoder, the DMT is stricty smaer. 9

10 Δ = 0 Δ = / w w 3 w 3 w w 3 h h h 3 h h h 3 ŵ ŵ ŵ ŵ ŵ 3 ŵ ŵ ŵ ŵ Fig. 5. Streaming setup with two messages arriving in each coherence boc. In coherence boc two messages, w, and w,, arrive as shown in the figure. We assume that w, arrives M symbos after the start of the coherence boc, whie w, arrives + M symbos from the start of coherence boc. We assume a decoding deay of one coherence boc for each message as shown. The eft figure iustrates the case when = 0, whie the right figure shows the case when =. A. DMT when = 0. We assume that each message w,i is uniformy distributed in the set I M = {,,..., MR/ }, so that a tota of MR information bits arrive in each coherence boc of ength M. We consider a SISO channe mode. Let h denote the channe gain in coherence boc, and denote the corresponding input sequence as x M. We spit x M = x M/,, x M/, x,, x, into two subsequences, each of ength M/. The input sequence x, can depend on w,, w,, w,,..., whie x, can depend on w,, w,, w,, w,,.... The received sequence y M is aso partitioned into y M/,, y M/, y,, y,. Reca that for the SISO channe, we have y,i = h x,i +z,i where the additive noise sequence z,i is samped i.i.d. from CN 0,. We wi assume that each message w,i must be decoded with a deay of T = coherence period. Thus, w, must be decoded at the end of coherence boc whereas w, must be decoded in the midde of coherence boc +, as is iustrated in Fig. 5. The foowing resut shows how to expoit the asymmetry in channe conditions experienced by w, and w, to attain a higher DMT than that which can be obtained through simpe intereaving. 6 Proposition : The optima DMT of the SISO streaming setup with two messages per coherence boc, = 0, and T =, is: dr = min r, r, r 0, ]. 67 Converse: The upper bound is based on two genie aided arguments. The bound dr = r/ foows by reveaing every message w, to the destination. Thus message w, needs to be decoded at the end of coherence boc. Since w, is uniformy distributed in {,,..., MR/ } and has a rate of R/, it foows that the DMT for this genie aided channe equas dr = r/. To estabish the other upper bound of dr = r we consider another genie aided channe. We revea message w, at the start of coherence boc and reax the deadine of w, and w, such that both ony need to be decoded at the end of the coherence boc +. Such an assumption can ceary ony improve the DMT. However the setup now is identica to that considered in Theorem where the message 6 We wi drop the subscript d T in this section for the DMT since we fix T =. w = w,, w, arrive at the start of coherence boc and must be decoded with a deay of T = coherence bocs. The associated DMT, dr = r for this channe, is thus an upper bound for the origina setup. This competes the justification of the converse. Achievabiity: We next present a coding scheme that attains the DMT in 67. We first spit each message w, into two equa sized messages w, = w,, w,, where each submessage is of rate R 0 = R/4. Thus we can assume that both w, and w, are independent and samped uniformy from J M = {,,..., MR/4 }. We do not spit the messages w, and assume that it is samped uniformy from I M = {,,..., MR/ }. We sampe three Gaussian codeboos as foows: The codeboo C A consisting of 3MR0 codewords x M/ A samped i.i.d. from CN 0, ρ. Each pair w,, w, is mapped to a unique codeword x A w,, w, i.e., C A = { x A w,, w, } w, J M,w, I M 68 The codeboo C B consisting of MR0 codewords x M/ B samped i.i.d. from CN 0, ρ. Each message w, is mapped to a unique codeword x B w, i.e., C B = { x B, }, J M 69 The codeboo C C consisting of MR0 codewords x M/ C samped i.i.d. from CN 0, ρ β. Each message w, is mapped to a unique codeword x C w,. C C = {x C w, } w, I M 70 We wi seect β = r/. Note that the tota power in the second boc is ρ + ρ r/. = ρ, since ρ r/ ρ. In coherence boc, the transmitter transmits x, = x A w,, w, in the first haf of the coherence boc and x, = x B, + x Cw, in the second haf of the coherence boc. The receiver observes y,i = h x,i + z,i for i {, }. The decoding of the messages is as foows. In the second haf of coherence boc, the receiver decodes w, using y,, by treating x C w, as additiona noise. It searches for a unique message ŵ, J M such that x B ŵ,, y, T M/ ε,. The error event E, 0

11 w w w,,,,, w,,,, w, w w w Channe Input X A w, w,, X w B, X w X A w,, w, C, X w B, X C w, Boc Boc + Channe Output Y Y,, Y, Y, wˆ ˆ,, w, wˆ, wˆ ˆ,, w ˆ, w, Fig. 6. Coding scheme for two messages per coherence boc with = 0. The first message w, is spit into two sub-messages w,, w, of equa size, whie the second message w, is not spit. In the first haf of the coherence boc, we transmit the codeword x A w,, w, whie in the second haf of the coherence boc we transmit the sum x B w, + x Cw,. Pr denotes the event that ŵ, w, and et O, denote the outage event: { og h ρ + + h ρ β r4 } og ρ. 7 After decoding ŵ, the decoder subtracts x Bŵ, from y, i.e., ỹ, = y, h x B ŵ,. The decoder uses the second haf of coherence boc and the first haf of coherence boc + to decode the message pair w,, w+,. In particuar, it searches for a pair ŵ,, ŵ+, such that x Cŵ,, ỹ, T M/ ε,3 and x A ŵ+,, ŵ,, y +, T M/ ε,4 are jointy typica. The error anaysis invoves two events, E, and E+,, associated with the error in decoding w, and w+, respectivey. In particuar, et E, = {ŵ, w, } and the outage event O, be given by: { og + ρ β h + og+ρ h + 3 } 4 r og ρ. 7 Simiary et E+, = {{ŵ, = w, } {ŵ+, w+, }} be the event that the message w, is decoded correcty, but an error occurs in the decoding of w+, and et O, be the corresponding outage event: { og + ρ h r } 4 og ρ. 73 It suffices to show that the error probabiity satisfies E, E, E+, ρ dr + o M where dr is defined in 67 and o M approaches zero as M. By seecting M sufficienty arge for each fixed ρ, the proposed DMT is then achievabe. We first consider the event E, = {ŵ, w, } and use the foowing upper bound: Pr E, Pr E, O,c + PrO, 74, where O, is defined in 7, with β = r/ as: { og h ρ + r4 } + h ρ og ρ. 75 r/ From standard arguments, the first term in 74 decreases to zero as M, and thus we ony need to upper bound the second term. Letting h = ρ α we have that 75 is equivaent to og + ρ α r og ρ 76 + ρ α r/ which in turn impies that α < r as ρ. Thus we have that and in turn PrO, ρ r/ 77 PrE, ρ r/ + o M 78 hods. Next we upper bound the probabiity of the event E, = {w, ŵ, }. We can express PrE, = PrE, O c, + PrO, 79 where reca that the event O, in 7 is defined, with β = r/ as: { og + ρ r/ h + og + ρ h r og ρ }. 80 Note that whenever {w, ŵ, }, we have that x C ŵ,, ỹ, are mutuay independent and furthermore x A ŵ+,, ŵ,, y +, are mutuay independent. It can be shown through a standard union bound argument that PrE, O, c vanishes to zero as M. To upper bound

12 O, we et h = ρ α and h + = ρ α and note that 80 reduces to: og + ρ α r/ + og + ρ α 3r og ρ. 8 The associated DMT is given by d r = min α,α A α + + α + 8 where A = {α, α 0 : α r/ + + α 3r/} and v + equas 0 is v < 0. It can be deduced that d r = r and thus PrO, ρ r 83 hods. Thus we have that PrE, o M + ρ r 84 hods. Finay we consider the event E+, = {{ŵ, = w, } {ŵ+, w +, }} which corresponds to an error in message estimate ŵ+, in the first haf of the coherence boc. Under this event the codeword x C ŵ, is decoded correcty however the pair x A ŵ+,, ŵ,, y +, is mutuay independent. Using O, be defined in 73, we can express Pr E, PrO, + Pr E, O,c,. 85 It foows from standard arguments that PrO, =. ρ r/ and furthermore Pr E, O,c, vanishes to zero as M. Thus we have that Pr E, This competes our achievabiity. B. Unnown Offset ρ r/ + o M. 86 We consider the case when either = 0 or = /, but when the actua vaue of is not nown to the transmitter. Such a setup appies when simutaneousy transmitting to two users whose coherence bocs are offset by M/ symbos. The foowing resut shows that we cannot have a universa coding scheme obivious of that achieves the same DMT. Proposition : Consider the SISO channe mode with two messages in each coherence boc as in Prop.. Assume that either = 0 or = /, but where the actua vaue of is nown ony to the receiver. The DMT for this setup equas dr = r. Proof : The achievabiity is straightforward. Each message w,j is mapped to a codeword of ength M/ of a Gaussian codeboo and transmitted immediatey. Since each message is of rate r og ρ a DMT of dr = r is achievabe. For the converse, we consider a muticast setup with two receivers. In coherence boc the transmitter transmits x, in the first haf of the coherence boc and transmits x, in the second haf, i.e., x = x, x, ], where both x,, x, C M/. Receiver observes y = y, y, ] in coherence boc as foows: y, = h x, + n,, 87 y, = h x, + n,. 88 Receiver observes v = v, v, ] in coherence boc as foows: v, = h x, + z,, 89 v, = h + x, + z,. 90 The noise variabes n,j and z,j have i.i.d. CN 0, entries. For both receivers we have T = and message deadines are shown in Fig. 5. Note that the duration of w, spans ony one coherence boc for receiver, but w, spans two coherence bocs. Liewise w, spans ony one coherence boc for receiver, but w, spans two bocs. We show that under this constraint the maximum possibe DMT is dr = r We begin by considering Fano s inequaity for receiver for message w 0, and rate Mr og ρ: PrE 0, h 0 = h 0 Mr og ρ Iw 0,; y h 0 = h 0. Mr og ρ 9 Ignoring the second term, which goes to zero as M, and using the same sequence of steps eading to 6 we have. with P = ρ r+ PrE 0, P Iw 0,; y 0 h 0, h 0 H Mr og ρ = P Iw 0,; y 0 h0 N+, h0 N+ H N+ Mr og ρ 9 93 P Iw 0,; y0 N, v0 N h0 N+, h0 N+ H N+ Mr og ρ 94 where 93 foows from the fact that h N+ is independent of w 0, y 0, h 0. Simiary, appying Fano s inequaity to receiver for message w 0, we have PrE 0, P Iw 0,; v 0,, v, 0,, h, h H Mr og ρ 95 = P Iw 0,; v 0,, v, 0,, h0 N+, h0 N+ H N+ Mr og ρ 96 P Iw 0,; y0 N, v0 N 0,, h0 N+, h0 N+ H N+. Mr og ρ 97

13 Liewise we can show that for each N The second scheme pairs a sequentia decoder with a tree code. This scheme aso attains that DMT, and in a deayuniversa fashion, but is more computationay compex and PrE, P Iw,; y0 N, v0 N h0 N+, h0 N+ H N+, w0 appears to require sufficienty ong coherence bocs. Finay, Mr og ρ we discuss some extensions when mutipe messages arrive in 98 each coherence boc. The fundamenta imits of deay-constrained streaming over PrE, fading channes are not we understood in genera. We hope P Iw,; y0 N, v0 N h0 N+, h0 N+ H N+, w0, w, that the techniques deveoped in this wor can serve as a usefu. Mr og ρ starting point for other investigations. 99 Thus we have that max max {PrE,, PrE, } 0 N N {PrE, + PrE, } 00 N =0 N Iw0 ; y0 N, v0 N h0 N+ H N+ P 0 NMr og ρ P IxN 0 ; y0 N, v0 N h0 N+ H N+ NMr og ρ 0 P N =0 Ix,; y,, v, h + Ix, ; y,, v, h, h + NMr og ρ 03 ρ r+ N + + N + r og ρ. 04 Nr og ρ The steps eading to 04 are simiar to 5 and hence are not eaborated. For N sufficienty arge the expression in the bracets in 04 is positive. This estabishes that dr r + must hod. Since > 0 is arbitrary this concudes the converse in Prop.. We concude this section with the foowing remar. When there are mutipe messages that arrive at equa intervas in each coherence boc, different messages observe different channe conditions. Prop. shows that coding schemes that expoit this asymmetry between messages can improve the DMT. On the other hand such schemes depend cruciay on where the messages arrive in each boc. If such information is not avaiabe the DMT is, in genera, smaer, as is estabished in Prop.. VII. CONCLUSIONS In this paper we study the probem of deay-constrained streaming over a boc fading channe. We estabish the diversity mutipexing tradeoff when there is one message arriving in each coherence boc. The converse is based on a nove outage-ampification argument that buids up a contradiction, over a sufficienty arge duration, if we assume a arger DMT. We propose two coding schemes for achieving the optima DMT. The first uses an intereaving scheme that reduces the system to a set of parae independent channes. The advantage of this scheme is its simpicity. The disadvantage is that the paybac deadine T must be nown in advance. 3 APPENDIX A PROOF OF THE LOWER BOUND IN 6 In this Appendix we incorporate the effect of outage into our ower bound. We continue from 9, dividing the channe reaizations into sets in which the suffix is in outage, and when it is not, and dropping the atter. Thus, PrE ] H T 0 :HT HT PrH T 0 = H T 0 ] PrE H T 0 = H T 0 ]. Appying and marginaizing out over the prefixes {H 0 } we get PrE ] H T :HT HT PrH T = HT ] Mr og ρ Iw ; Y T 0, H T Mr og ρ = HT. 05 We can further express the right hand side of 05 as foows. Note that PrH T = HT ] 06 Mr og ρ H T H T :HT HT = = Mr og ρ Mr og ρ :HT HT H T :HT HT For the second term in 05, note that PrH T = HT ]Iw ; Y T = H T :HT HT Iw ; Y T = PrH T HT ] = PrH T Iw ; Y T PrH T H T PrH T ] 07 PrH T HT ] 08 = HT ] 0, H T :HT HT 0, H T HT ]Iw ; Y T 0, H T = HT 09 = HT, HT HT 0 PrH T = HT, HT 0, H T = HT HT HT ] HT, HT HT, where in 0, we use the fact that the indicator random variabe H T HT is a deterministic function of HT, and

14 hence can be added as in 0. In, we note that for each H T HT we have: PrH T = PrHT = HT HT HT ] 3 = HT ] PrHT HT HT = HT ] PrH T HT ] 4 = PrHT = HT ] PrH T HT ]. 5 Thus substituting 5 in 0 we have that foows. Substituting 08 and into 05, we have PrE ] PrH T HT ] Mr og ρ Iw ; Y T This competes the justification of 6. APPENDIX B PROOF OF 37. 0, H T Mr og ρ HT., HT 6 Our proof is based on the Chernoff-Cramer theorem of arge deviations stated beow. Theorem 3: Suppose that x,..., x N are i.i.d. random variabes with a rate function f x defined as f x t = sup θ { θ t og E x expθ x] }, 7 and et M n = n n i= x i. Then there exists a constant N > 0 such that for a n N PrM n t e nfx t. 8 Reca that A, is the event that the true codeword is not jointy typica with the received sequence. To upper bound the probabiity we can ignore the margina typicaity constraints and use PrA, = Pr og p X,Y X, Y hp X,Y ] M + > ε. 9 Note that as Y = H X + Z, the H are nown to the decoder, and the noise sequence {Z } is independent, p X,Y X, Y = p X X p Y X Y X 0 = p X X p Z Y H X = p X X p Z Z, where the ast equaity hods since the codewords are samped i.i.d. and the noise is aso i.i.d. Thus hp X,Y = hp X + hp Z. And so og p X,Y X, Y hp X,Y ] 3 = = = og p X X og p Z Z hp X hp Z ] 4 = og p X X hp X + og p Z Z hp Z ] = 4 where the ast step foows from the triange inequaity. Substituting 4 into 9 and using using the union bound we have where we define A X, = { X : A Z, = { PrA, PrA X, + PrAZ, 5 Z = og p XX hp X ] M + } ε, 6 : = og p } ZZ hp Z M + ε. 7 Note that X is a sequence of M i.i.d. random vectors each ρ samped from CN 0, N t I and E og p X X ] = hp X. Simiary, E og p Z Z ] = hp Z. Then using Theorem 3, there exist functions f X ε and f Z ε such that for sufficienty arge N = M + PrA X, exp{ M + f X ε}, PrA Z, exp{ M + f Z ε}. Furthermore by directy using 7 we can show that f X ε > 0 and f Y ε > 0. Setting fε = maxf X ε, f Z ε estabishes 37. REFERENCES ] A. Khisti and S. C. Draper, Streaming data over fading wireess channes: The diversity-mutipexing tradeoff, in ISIT, 0, pp. 5. ] S. V. Hany and D. N. C. Tse, Mutiaccess Fading Channes-Part II: Deay-Limited Capacities, IEEE Transactions on Information Theory, vo. 44, no. 7, pp , ] G. Caire, G. Taricco, and E. Bigieri, Optimum power contro over fading channes, IEEE Transactions on Information Theory, vo. 45, no. 5, pp , ] E. Bigieri, J. G. Proais, and S. Shamai, Fading channes: Informationtheoretic and communication aspects, IEEE Transactions on Information Theory, vo. 44, no. 6, pp , ] D. Tse and P. Viswanath, Fundamentas of Wireess Communication. Cambridge University Press, ] A. Lozano, A. Tuino, and S. Verdu, Mutiantenna capacity: Myths and reaities, in Space-Time Wireess Systems: From Array Processing to MIMO Communications, C. P. H. Bocsei, D. Gesbert and A. J. van der Veen, Eds. Cambridge University Press, 006, pp ] L. Zheng and D. Tse, Diversity and mutipexing: A fundamenta tradeoff in mutipe antenna channes, IEEE Trans. Inform. Theory, vo. 49, pp , May ] S. Yang and J. Befiore, Optima space-time codes for the mimo ampify-and-forward cooperative channe, IEEE Trans. Inform. Theory, vo. 53, no., pp , ] H. Lu, Constructions of mutiboc space-time coding schemes that achieve the diversity-mutipexing tradeoff, IEEE Trans. Inform. Theory, vo. 54, no. 8, pp , ] L. Mroueh, Space time coding in mutipe input mutipe output systems: chaenges and appications, in Communication systems: New research. Nova Pubisher, 0. ] P. Corone and H. Bocsei, Diversity-Mutipexing tradeoff in seective-fading MIMO channes, in Proc. Int. Symp. Inform. Theory, 007, pp