Rateless Codes for MIMO Channels

Transcription

1 Rateless Codes for MIMO Channels Marya Modir Shanechi Dept EECS, MIT Cabridge, MA Eail: Uri Erez Tel Aviv University Raat Aviv, Israel Eail: Gregory W Wornell Dept EECS, MIT Cabridge, MA Eail: gww@itedu Abstract Two rateless code constructions are developed for efficient counication over ulti-input ulti-output MIMO Gaussian channels The key ingredients in both architectures are layering, dithering, and repetition Both eploy successive cancellation decoding along with iniu ean-square error MMSE cobining and convert the MIMO channel into a scalar channel to which classical Gaussian base codes can be applied The first construction based on siple layering induces an additive white Gaussian noise AWGN scalar channel and with perfect base codes achieves a substantial efficiency gain over a baseline repetition schee At typical spectral efficiencies, this schee can achieve better than 85% of capacity for a rateless construction with two effective rates The second construction which uses a diagonal layering DL structure is capacity achieving at any SNR and induces a particular tie-varying scalar channel This holds even if the MIMO channel is block constant but tie-varying I INTRODUCTION The design of practical and efficient codes for counicating over ulti-input ulti-output MIMO Gaussian channels, when channel state inforation is not available a priori at the transitter or in broadcast scenarios with ultiple receivers, is of significant interest in a variety of eerging wireless applications and standards These include ulti-antenna systes as well as orthogonal frequency-division ultiplexing OFDM systes For such probles, a rateless approach is rather natural, whereby an encoder aps a essage into an infinite-length codeword for transission The decoder attepts to decode fro successive prefixes of its received sequence When it succeeds, it sends a single-bit acknowledgent ACK to the encoder to terinate the transission Rateless codes are hence instances of hybrid ARQ HARQ schees Of particular interest are rateless codes that achieve rates close to capacity and require low decoding coplexity Exaples of such codes for erasure channels are the wellknown Raptor codes [] developed fro LT codes [2] Such codes have also been applied to other discrete-input channels By contrast, exaple rateless constructions for the scalar Gaussian channel are developed in [3], [4], and involve the transission of sequences of redundancy blocks For MIMO systes, however, the design of HARQ protocols has received coparatively little attention to date in the literature This work was supported in part by NSF under Grant No CCF-5522, Draper Laboratory, by ONR under MURI Grant No N , and by a NSERC scholarship An iportant sub-proble of the rateless design proble for MIMO channels is that of universal coding when capacity is known at the transitter but channel paraeters theselves are not Exaple universal codes for the parallel Gaussian channels, which are of significant interest for exaple in OFDM technologies, have been developed in [5] In this paper, we develop two rateless code architectures for the MIMO Gaussian channel that ap the MIMO channel into an equivalent scalar Gaussian channel to which a standard base code can be applied The ingredients used in these constructions are siilar to those of the rateless codes of [4] In both architectures, layered coding with successive cancellation decoding is used to develop efficient codes and redundancy is introduced using dithered repetition coding with iniu ean-square error MMSE cobining The first construction, which is based on siple layering, was introduced for the case of universal coding for parallel channels in [5] We extend this design forulation to the case of rateless coding for parallel channels and then cobining it with the D-BLAST space-tie codes [6] to MIMO channels This code structure allows standard AWGN channel codes to be used efficiently We show that even with sall nubers of layers, nuerically optiized versions of these rateless codes substantially reduce the gap to capacity The second construction, in contrast to the first construction and the approach in [4], satisfies the additional constraint of tie-invariant encoding, which eans that essage recovery can start with any teporal redundancy block This construction is based on a diagonal layering DL structure and was briefly introduced in [7] In this paper we fully analyze this schee and show that when used with suitable base codes, it is capacity-approaching with design paraeters that are independent of the spectral efficiency of the channel in contrast to the first construction and hence easy to obtain We additionally show that these codes are capacity approaching even if the MIMO channel is block-constant but tie-varying In both architectures, we ephasize the finite signal-to-noise ratio SNR regie As such, our approach differs fro, eg, that of [8] and [9], which focus on high SNR analysis II CHANNEL MODEL AND PROBLEM FORMULATION The channel odel of interest takes the for y = Hx + w, /8/$25 28 IEEE

2 where channel input x and output y at a particular tie are N t - and N r -diensional vectors, respectively, and where the associated noise w is CN, I and independent over tie The channel input is constrained to a total average power of P The transission is arranged in redundancy blocks of N sybols, which is assued to be large but otherwise plays no role in the analysis The channel atrix realization, H, is known to the receiver but not to the transitter Furtherore, our target capacity is the white input capacity C MIMO H = log det I + P HH, 2 with P = P/N t The encoder generates fro the essage a series of increental redundancy blocks and transits the over space the transit antennas and tie The receiver in turn collects as any redundancy blocks as needed to accuulate enough utual inforation to decode the essage The nuber of blocks that need to be collected directly depends on C MIMO H We use C to denote the ceiling rate of the code, ie, a paraeter specifying the highest rate at which the code can operate We require our rateless codes to be capacity-approaching in the following sense For any channel atrix H that satisfies C MIMO H =C / M, 3 for soe integer, we require the essage to be decoded after sending a sequence of sets of N t redundancy blockswe denote this set of channel atrices by HC /, ie, HC / ={H : C MIMO H =C /} 4 Here M is the range of the code, ie, the axiu nuber of teporal redundancy blocks the code is designed for Note that a parallel channel is a special case of the MIMO channel with H = diagα,,α K where K is the nuber of subchannels and where 2 becoes K C H = log +P α k 2 5 k= We denote the set of channel vectors α =α,,α K with C diagα = C / by AC /, ie, AC / ={α : C diagα = C /} 6 III A NEAR-PERFECT RATELESS CONSTRUCTION Here we develop the first class of rateless codes based on a layer-dither-repeat structure first for the case of parallel channels and then extend the to the MIMO channels A Layer-Dither-Repeat Codes for Parallel Channels To construct a rate R code, we first choose the nuber of layers L and the associated codebooks C,,C L that we assue are capacity achieving independent identically distributed Gaussian codebooks We divide the essage into L subessages that are independently coded using the corresponding codebooks We further constrain the codebooks to have equal rate R/L since this has the advantage of allowing the to be derived fro a single base code The redundancy blocks, X,,X M, sent over K subchannels and in M redundancy intervals are constructed fro the codewords c l C l, l =,,L as [ X X M ] T = G [ c c L ] T 7 where G is a KM L atrix of coplex gains and where X for each is a K N atrix and c l for each l is a row vector of length N with N the block length of the code The power constraint enters by liiting the rows of G to have squared nor P and by noralizing the codebooks to have unit power In addition to the layered code structure, we require the code to be successively decodable to keep the coplexity low Specifically, to recover the essage, we decode one layer, treating the reaining layers as colored noise, then subtract its effect fro the received codeword Then we decode another layer fro the residual, treating further reaining layers as noise, and so on We use MMSE cobining of the redundancy blocks during successive interference cancellation SIC which is inforation-lossless Using this construction results in several design degrees of freedo At the encoder, we can choose G, which has KML phase and KML agnitude degrees of freedo available At the decoder, the decoding order can be chosen as a function of the realized channel paraeters α We denote this degree of freedo by a discrete variable n which could take L! possible values corresponding to the L! possible decoding orders We can use these degrees of freedo to axiize the efficiency of the code for a given C which is equivalent to axiizing the rate of the code over these degrees of freedo as follows R = L ax in in ax in I lα, G,,n G G M α AC / n L! l L 8 where G is the set of G satisfying the power constraint, ie, G = {G :[GG ] i,i = P, i =,,KM}, AC / is as given in 6, and I l α, G,,n is the achievable rate in the lth layer with respect to the decoding order specified by n after collecting X,,X The equivalent atrix channel seen after receiving these sets of blocks is given by the block diagonal atrix A = diagα,,α 9 with α repeated ties on the diagonal For l =,,L L I l α, G,,n l =l = log det I + A [G Πn] l:l [G Πn] l:l A, where G is the subatrix consisting of the first K rows of G assigned to the first sets of K redundancy blocks, Πn denotes the atrix that perutes the coluns of G according to n, and [ ] i:j denotes the subatrix consisting of coluns /8/$25 28 IEEE 2

3 i through j of its arguent Finally, using 8, we obtain the resulting efficiency of the optiized code as η L = R/C as a function of the nuber of layers L in the code In the reainder of this section, we consider the case of K =2subchannels for siplicity Moreover, we let P = without loss of generality In this case, we can express the set AC / in the equivalent for AC / ={α t, t } where α t 2 =2 tc /2, a α 2 t 2 =2 +tc /2 b When L =, our construction specializes to a siple repetition code across subchannels and in tie, and serves as a useful baseline Such a code can achieve a rate R = log + α t 2 + α 2 t, 2 after collecting teporal sets of 2 blocks that has its iniu at = M and t =and hence yields an efficiency of η =/C log +2M2 C /2M 2 We will show that our construction achieves an efficiency that is significantly higher than this repetition efficiency Moreover, fixing the decoding order independent of the realized channel significantly degrades the efficiency perforance even in the case of universal coding, ie, M =,aswas deonstrated in [5] Even for M =, such a schee has an efficiency η L that is strictly less than unity for C > even as L In particular, we have the following clai, whose proof we oit due to space constraints Proposition : For the case of K = 2 subchannels, the efficiency of the fixed-decoding-order variant of our rateless code is bounded according to where with R + = 2 log η LC η = R + C /C, + P α 2 + α 2 + log + P α 2 3a 3b α 2 =2 C /2, α 2 =2 C, P = α 2 2 α 2 3c Note that in the case of M =, η L could be ade arbitrarily close to η for large enough L However, for M >, η is only an upperbound on the perforance of fixed-decodingorder variant of the schee since the rateless perforance of the code could at best be equal to its universal perforance It can be verified nuerically that for a code range of M =, the axiu efficiency gain of η over the repetition efficiency η is 38% that occurs at C 376 b/s/hz Hence, without the use of a variable decoding order, our code does not offer large iproveents over a siple repetition code We evaluate the efficiency of our code with variable decoding order by nuerically evaluating 8 In our siulations, both L and C are varied, but we continue to restrict our attention to K =2subchannels We perfor the analysis for the universal case, ie, M =, and also for the rateless case with M =2 The results are shown in Table I As the table shows, for the universal design using only 3 layers, efficiencies in the range of 9% are possible over typical target spectral efficiencies For the rateless design and using only 3 layers, efficiencies are in the range of 82% to 9% in typical spectral efficiencies and in all cases have a substantial gain of around 4% over the siple repetition schee Also the efficiency loss incurred going fro a universal design to a rateless design with M =2is sall and less than 5% In the table we also show the SNR gaps Δ L in db to capacity corresponding to the calculated efficiencies for different nubers of layers Note that the SNR gap for each value of C depends on the realized channel gain pair α,α 2 Inthe table, we indicate the worst-case SNR gap, which corresponds to, eg, the case α 2 = In the rateless case for M =2we show the worst-case SNR gap for channels in both AC and AC /2 denoted by Δ L, Δ2 L Thetablealsoshowsη that corresponds to a lower bound on efficiency loss due to a fixed decoding order Note that this fixed decoding order variant of our schee requires any layers and hence iposes substantially greater coplexity In suary, the efficiency iproveents of our schee relative to a siple repetition code vary fro 33% 48 db,32 db at C =433 b/s/hz, to 42% 53 db,83 db at C =2 b/s/hz in the rateless case with M =2, and fro 23% 33 db at C = 433 b/s/hz, to 29% 5 db at C = 2 b/s/hz in the universal case Moreover, at the typical target spectral efficiency of 433 b/s/hz, our code achieves within approxiately 4 db of capacity in the rateless case and db of capacity in the universal case, neglecting losses due to the base code B MIMO Extension Codes for the MIMO channel can be constructed by concatenating the optiized layered codes for the parallel channels outer code with the D-BLAST architecture inner code and associated MMSE-SIC receiver [6] that aps a MIMO channel into a parallel channel as follows The essage is divided into U subessages that are independently coded using the layer-dither-repeat codes These essages are then transitted over the N t transit antennas according to the following encoding of length NN t + U x x U,,,,, N t 2 N t where x n u denotes the layer-dither-repeat encoding of the uth subessage sent over the nth channel input Since the D-BLAST architecture cobined with MMSE- SIC decoding is capacity achieving, the overall concatenated code has the sae efficiency characteristics as that for the parallel channel assuing U N t such that the efficiency loss due to zero-padding in the D-BLAST is negligible /8/$25 28 IEEE 3

4 TABLE I ACHIEVABLE EFFICIENCIES η L AND THE CORRESPONDING SNR GAP Δ L TO CAPACITY, FOR THE LAYERED-DITHER-REPEAT CODE WITH VARIABLE DECODING ORDER, AS A FUNCTION OF THE RANGE OF THE CODE M, NUMBER OF LAYERS L, AND CEILING RATE C B/S/HZ FOR THE CASE OF K =2SUBCHANNELS TABLE ALSO SHOWS THE EFFICIENCY η FOR THE FIXED-DECODING-ORDER VARIANT OF THE CODE IN THE LIMIT OF LARGE L AND THE GAP TO CAPACITY Δ FOR M = M = C =433 C =8 C =2 η 3 Δ 3 92% db 9% 24 db 87% 47 db η 2 Δ 2 88% 7 db 82% 44 db 77% 83 db η Δ 69% 44 db 62% 93 db 58% 52 db M =2 η 3 9% 86% 82% Δ 3, Δ2 3 db 4, 9 34, 8 65, 33 η 2 78% 7% 65% Δ 2, Δ2 2 db 38, 9 73, 4 26, 65 η 57% 46% 4% Δ, Δ2 db 62, 4 33, 76 28, 6 η Δ 83% 24 db 73% 66 db 66% 23 db IV DIAGONALLY LAYERED RATELESS CODING The layered codes developed in Section III achieve substantial gain over the baseline repetition schee with few layers In this section we develop and fully analyze a class of rateless codes constructed via a diagonal layering approach briefly introduced in [7] that have the following differences with the first construction: First they are tie-invariant Second, they can be directly applied to MIMO channels without the need to be concatenated with the D-BLAST Third, they are capacity achieving when used with suitable base codes as opposed to near-perfect at any SNR, with design paraeters that are independent of the spectral efficiency, at the price of the induced scalar channel being a particular tie-varying one Moreover this even holds for block constant but tie-varying MIMO channels We now develop this DL construction A DL Coding for Scalar Channels With DL coding, the essage is divided into LJ subessages, each of which is independently coded into a corresponding subcodeword These subcodewords are then superiposed in a staggered diagonal anner, where the offset with respect to the previous subcodeword is fixed and deterined by the choice of L The resulting equivalent layered structure of the encoding is as depicted in Fig, where we see that there are J subcodewords per layer, L layers, and L segents per subcodeword that see interference fro different sets of subcodewords Such an encoding can be decoded using successive cancellation In particular, as shown in Fig, we decode starting fro the lower left in a cyclic anner, subtracting the effect θ θ 2 segent θ θ 2 2 θ 4 2 θ θ θ θ 8 2 J 5 θ θ θ θ Fig Encoding structure for diagonally layered counication In this exaple, L =4 Each brick represents a subcodeword, and its associated parenthesized index indicates the order in which decoding proceeds The phases θ l applied within the subcodewords are paraeters exploited to develop a rateless code fro DL encoding of subcodewords as they are decoded, and treating undecoded layers as noise With this approach, the effective channel for all subcodewords are identical Thus, using perfect base codes, the resulting DL encoding is capacity achieving when the sae rate and power is used for each subcodeword provided J is chosen large enough that the efficiency due to zero-padding in the construction is negligible This syetry akes the DL architecture well-suited for the developent of our rateless codes B DL-based Rateless Coding for MIMO Channels A rateless code for the MIMO channel is constructed by creating a collection of redundancy blocks in space and tie that are distinct linear transfored versions of the basic DL-coded block In particular, as Fig reflects, prior to superposition in the DL encoding, the lth segent in each subcodeword is ultiplied by e jθ l, where θ l is a phase paraeter These phases ust be chosen for each redundancy block in the rateless code The redundancy blocks at an arbitrary tie t see Fig can be expressed in the for x P = L Q c 4 x M c L where c l is an arbitrary eleent of a noralized subcodeword in layer l, x is the N t -diensional channel input during increental redundancy interval, and Q is a N t M L atrix whose i, lth coponent e jθi,l describes the phase for the segent at tie t in the lth layer of the redundancy block indexed by i Note that we can write the redundancy blocks at any other tie keeping Q the sae but using the appropriate cyclic shift of c,,c L as shown in Fig The receiver collects as any spatial and/or teporal redundancy blocks as it needs to accuulate enough utual inforation to recover the essage In our rateless setup we require the essage to be decoded for a channel in HC / after collecting sets of N t redundancy blocks The receiver cobines and decodes these redundancy blocks using the inforation-lossless MMSE-SIC front end, creating an equivalent scalar channel Siilar to 9, for a channel realization H, the equivalent channel atrix after receiving θ θ t θ /8/$25 28 IEEE 4

5 teporal sets of blocks is given by the block diagonal atrix, H = diagh,, H 5 The choice of Q deterines the perforance of the rateless code We prove the following proposition Proposition 2: Provided perfect base codes of rate C /L are used, the DL-based rateless coding architecture is perfect, ie, capacity achieving for any channel realization H such that C MIMO H =C / for every {,,M} if and only if L N t M and Q/ L has orthonoral rows Proof: To see that the two conditions are necessary, note that to achieve the white input capacity C MIMO H we need the transitted sybols on the different transit antennas and teporal blocks to be iid Gaussian, which eans that the N t M rows of the G atrix ust be orthogonal Hence Q ust at least have N t M coluns To prove the sufficiency of these conditions note that the achievable rate of the code after collecting redundancy sets is given by the utual inforation as R = Ic,,c L ; y,,y = log det I + P L H Q Q H, 6 with y the channel output corresponding to x and Q the subatrix containing the first N t rows of Q Having L N t M and that the rows of Q/ L are orthonoral, the arguent in 6 reduces to L H Q Q H = H H = diag HH,,HH Substituting back in 6, the achievable rate of the code is given by R = log det I + P H H = log det I + P HH = C where the last equality follows since H HC / The constraints of Proposition 2 are straightforward to satisfy For exaple, it is sufficient to take G to be the N t M N t M discrete Fourier transfor DFT atrix, or, when it exists, the Hadaard atrix Having this, assuing perfect base codes, we can pick the base code rate to be R = C /L C Block Constant Tie-Varying Channels The DL-based rateless code is capacity achieving even for block constant but tie-varying MIMO channels This eans that with this construction, the essage could be decoded after collecting redundancy blocks as long as the sequence of realized channels satisfy log det I + P HiHi = C 7 i= for any =,,M where Hi is the channel realization in the ith teporal redundancy block and M is the code range The proof is exactly the sae as the proof of Proposition 2 only replacing H in 5 with H = diag H,,H D Base Code Design The syetric structure of the DL-based code akes it suitable for rateless coding However, it has the disadvantage of inducing an effective scalar channel in each layer which is tie-varying In particular, this effective channel has a fixed SINR in each of the L segents within a subcodeword that changes fro segent to segent In general, the base code and its decoding structure should be designed having this tievariation in ind However, siulation results in [7], [], [] suggest that the perforance of standard LDPC base codes in these low rate induced channels is not degraded by uch E Coparison to a Baseline Alaouti Schee As a baseline for coparison we consider a rateless code using standard Alaouti code [2] with redundancy blocks generated by repetition We can show that the efficiency of this schee on the 2 2 MIMO channel is given by η ala-rep = /C log +2M2 C /2M For exaple, on the 2 2 channel with spectral efficiency of 24 b/s/hz and assuing perfect base codes, a DL-based code consisting of 2 sets of 2 redundancy blocks and L =4layers has an efficiency gain of 33% over the baseline that has an efficiency of 67% Moreover, the base code rate in the DL code is lower than that in the baseline that requires a good base code of rate 24 67/2 = 8 b/d, the non-binary design of which is ore challenging Thus as a practical MIMO rateless code, the DL-based construction shows considerable proise REFERENCES [] A Shokrollahi, Raptor codes, IEEE Trans Infor Theory, vol 52, no 6, pp , June 26 [2] M Luby, LT-codes, in Proc IEEE Syp Found Cop Sci FOCS, Vancouver, BC, Canada, 22 [3] U Erez, G W Wornell, and M D Trott, Faster-than-Nyquist coding, in Proc Allerton Conf Coun, Contr, Coputing, Monticello, Illinois, 24 [4] U Erez, M Trott, and G Wornell, Rateless coding for Gaussian channels, IEEE Trans Infor Theory, 27, subitted for publication Also available on arxiv; see [5] M M Shanechi, U Erez, and G W Wornell, On universal coding for parallel Gaussian channels, in Proc Int Zurich Seinar Coun, Zurich, Switzerland, Mar 28 [6] G J Foschini, Layered space-tie architectures for wireless counication in a fading environent when using ulti-eleent antennas, Bell Labs Tech J, vol, no 2, pp 4 59, 996 [7] M M Shanechi, U Erez, and G W Wornell, Tie-invariant rateless codes for MIMO channels, in IEEE Intern Syp Infor Theory, Toronto, Canada, July 28, pp [8] L Zhao and S-Y Chung, Perfect rate-copatible codes for the Gaussian channel, in Proc Allerton Conf Coun, Contr, Coputing, Monticello, Illinois, Sep 27 [9] S Tavildar and P Viswanath, Approxiately universal codes over slow fading channels, IEEE Trans Infor Theory, vol 52, no 7, pp , July 26 [] C R Jones, J V T Tian, and R D Wesel, The universal operation of LDPC codes over scalar fading channels, IEEE Trans Coun, vol 55, no, pp 22 32, Jan 27 [] C Jones, T Tian, A Matache, R D Wesel, and J Villasenor, Robustness of LDPC codes on periodic fading channels, in Proc GLOBECOM, Taipei, Taiwan, Nov 22, pp [2] S Alaouti, A siple transitter diversity schee for wireless counications, IEEE J Sel Areas Coun, vol 6, pp , Oct /8/$25 28 IEEE 5