Fast-Decodable MIMO HARQ Systems

Transcription

1 1 Fast-Decodable MIMO HARQ Systems Seyyed Saleh Hosseini, Student Member, IEEE, Jamshid Abouei, Senior Member, IEEE, and Murat Uysal, Senior Member, IEEE Abstract This paper presents a comprehensive study on the problem of decoding complexity in a Multi-Input Multi- Output (MIMO) Hybrid Automatic Repeat request (HARQ) system based on Space-Time Block Codes (STBCs) We show that there exist two classes of fast-decodable MIMO HARQ systems: independent and dependent STBC structures For the independent class, two types of protocols namely, fixed and adaptive threshold-based are presented and their effectiveness in both computational complexity reduction and spectral efficiency preservation are discussed For the dependent class, a fast Sphere Decoder (SD) algorithm with a low computational complexity is proposed for decoding process of HARQ rounds Two new concepts are introduced to leverage the fast-decodable notion in MIMO HARQ systems Simulation results show that the proposed fast-decodable MIMO HARQ protocols in both classes provide a significant reduction in the decoding complexity as compared with the original MIMO HARQ method Index Terms Multi-Input Multi-Output (MIMO), Hybrid Automatic Repeat request (HARQ) systems, Space-Time Block Codes (STBCs), Sphere Decoder (SD) algorithm, fast-decodable I INTRODUCTION Space-Time Coding (STC) has been recognized as an effective Multiple-Input Multiple-Output (MIMO) approach in wireless systems that provides a reliable communication without any bandwidth expansion through utilizing multiple transmit and/or receive antennas [1] Space-Time Block Code (STBC) is among STC techniques which achieves full diversity gain with a lower complexity in the implementation [2] The most famous class of STBCs is Orthogonal STBCs (OSTBCs) with linear decoding complexity for the (optimum) Maximum Likelihood (ML) decoder The first OSTBC, which is widely known as the Alamouti code, is designed for two transmit antennas [1] The idea of OSTBCs is extended for any number of transmit antennas in [2], [3] In addition, OSTBCs can be considered as a special case of fast-decodable STBCs class Fast-decodable property is formally defined for STBCs implemented by a lower computational complexity Sphere Decoder (SD) algorithm on the order of O(lM L l+1 ) where M, L, and l are the size of signal constellation, the number of transmitted symbols, and the Complexity Reduction Factor (CRF), respectively [4] Several examples of other fastdecodable STBCs can be found in [4] [9] Manuscript received July 18, 2014; revised November 21, 2014; accepted January 8, 2015 The associate editor coordinating the review of this paper and approving it for publication was Prof Zhaocheng Wang S S Hosseini is with the Department of Computer and Electrical Engineering, Bardsir Branch, Islamic Azad University, Kerman, Iran, ( ssalehhosseini@gmailcom) J Abouei is with the Department of Electrical and Computer Engineering, Yazd University, Yazd, Iran, ( abouei@yazdacir) M Uysal is with the Department of Electrical and Electronics Engineering, Ozyegin University, Istanbul, Turkey, 34794, ( muratuysal@ozyeginedutr) STBC systems are open-loop MIMO settings However, in many wireless applications, there exists a feedback channel between the transmitter and the receiver which allows the implementation of a closed-loop structure to achieve a high reliable communication in a MIMO system [10] Particularly, if an unsuccessful detection is occurred for transmitted data, the receiver requests the same data transmission via a feedback channel This technique is named Automatic Repeat request (ARQ) if the receiver discards previous (re)transmissions in the decoding process A more efficient natural way which combines previous copies of (re)transmitted data is called Hybrid ARQ (HARQ) [11] There exist two types of HARQ techniques namely, Chase Combining (CC) [12] and Incremental Redundancy (IR) [13] schemes The CC scheme coherently combines all received data packets before (or after) performing the interference cancelation process by a linear-based receiver such as Zero-Forcing (ZF) or Minimum Mean Square Error (MMSE) in MIMO systems [11] Since there is no coding technique in all possible retransmission rounds of a data packet, the achievable throughput in the CC scheme is low However, in the IR scheme, the redundant coded packets are transmitted with different lengths to reach a higher throughput efficiency at the cost of a high decoding complexity There are several works in the literature dealing with the combination of STC with HARQ in multiple antenna wireless systems In [14] and [15], a recursive realization of Space- Time Trellis Codes (STTCs) is analyzed for MIMO HARQ systems However, the decoding complexity of a well designed STTC grows exponentially when the transmission rate increases [2] Authors in [16] and [17] present some switchbased MIMO HARQ protocols that work between the spatial multiplexing and the STBC modes In [18], a linear space-time precoder is designed for a MIMO HARQ system to compensate the lack of the time diversity in slow-fading channels The problem of diversity-multiplexing tradeoff [19] in MIMO HARQ systems has been investigated from the informationtheoretic point of view in [20] [22] Recently, the optimal successful average rate of HARQ STBC configurations has been theoretically analyzed for various STBC schemes in [23] Performance of MIMO HARQ systems has been studied and analyzed from various perspectives including packet combining manners [11], [24] [28] and the information-theoretic [20] [23], [29], [30] However, little research has been devoted on the decoding complexity issue in MIMO HARQ systems (eg, [31] [33]) In [31], a suboptimal hard decision decoding algorithm is utilized in MIMO HARQ systems to reduce the decoding complexity at the cost of sacrificing the Bit Error Rate (BER) performance Reference [32] utilizes the SD algorithm along with off-line data at the receiver side to avoid a high computational complexity in a mapping diversity based [34] MIMO HARQ system However, when the data rate or the number of constellation points increases, the stored

2 2 off-line data occupies much more memory space than that of an HARQ setting with a low data transmission rate Thus, the scheme in [32] would not be a desirable protocol for limited memory receivers It is investigated in [33] that a low rate Alamouti-based HARQ protocol has the computational complexity advantage to other conventional HARQ protocols in the decoding process In general, the decoding complexity of a MIMO HARQ round n < N, based on the STBC, with an exhaustive search is of order O(M L ), where L is the number of transmitted signals [4] For the final HARQ round where all N 1 (re)transmissions have a decoding with failure, the decoding complexity depends on the STBC structure, since the full STBC is transmitted If the n th (re)transmission round has a successful decoding, the overall decoding complexity of the MIMO HARQ system will be of order O ( nm L + C STBC (M, L)δ[n N] ), where C STBC (M, L) is the order of decoding complexity for STBC and δ[m] is the discrete delta function Therefore, the decoding complexity of a MIMO HARQ system seems to be a prohibitive task for large signal constellation sizes Even though the spectral efficiency of a MIMO HARQ system is more effective than that of a MIMO system without HARQ, its decoding complexity is much higher than that of a classical MIMO system This drawback makes the use of HARQ very difficult in MIMO systems with high data transmission rates Therefore, it is necessary to find effective solutions to make the decoding procedure faster for an HARQ round n < N Taking the above motivation into account, in this work, we leverage the fast-decodable notion to the MIMO HARQ systems as follows: We first define two classes of fast-decodable MIMO HARQ protocols namely, i) Class I: independent of STBC and ii) Class II: dependent on STBC, in Section III The first class uses a threshold or decision function to avoid performing unnecessary detections in (re)transmission HARQ rounds Using the actual capacity of MIMO channels, two fixed and adaptive methods are presented as two threshold-based protocols of Class I The effectiveness of both protocols in the computational complexity reduction and the spectral efficiency preservation is discussed Class I is independent of STBC structure and suitable for any STBC based MIMO HARQ system For Class II introduced in Section IV, which establishes the fast-decodable notion of MIMO HARQ systems based on the SD algorithm, the new concept fastdecodable MIMO HARQ round is defined and the sufficient conditions for having a fast-decodable HARQ round are derived Since the traditional SD method for fast STBCs [4] cannot directly apply to a fast HARQ round, we propose a new fast SD algorithm which significantly reduces the computational complexity in decoding process of a fast HARQ round The new concept full fast-decodable MIMO HARQ protocol is also presented for HARQ schemes in which all HARQ rounds are fast-decodable We also analytically prove the OSTBCs based MIMO HARQ settings are Lk Input Bits Serial-to-Parallel Convertor M-ary Modulator STBC Encoder Buffer 1 1 t = n STBC Decoder M-ary Demodulator Parallel-to-Serial Convertor t = n-1 Fig 1 A single-user MIMO HARQ wireless system with N t transmit and N r receive antennas full fast-decodable Moreover, we show that the number of full fast-decodable OSTBCs based MIMO HARQ settings increases exponentially when the number of transmit antenna increases In Section V, we conduct some simulations to illustrate the effectiveness of the proposed methods in both classes, for having a significant reduction in the decoding complexity of MIMO HARQ systems Notations: Throughout this paper, we use normal letters for scalars Matrices and column vectors are set in bold capital and lower-case letters, respectively The set of odd Gaussian integers is denoted by Z odd [i] A complex Gaussian random variable with mean µ and variance σ 2 is denoted by CN (µ, σ 2 ) Operation denotes sum modulo-2 Finally, the dot or inner product of two vectors x = [x 1,, x n ] T and y = [y 1,, y n ] T is defined as x, y x H y = n i x i y i II SYSTEM MODEL AND FRAMEWORK DESCRIPTION In this work, we consider a single-user MIMO wireless system consisting of N t transmit and N r receive antennas The model uses an HARQ scheme where it is assumed that a noiseless feedback channel with a negligible delay exists between the transmitter and the receiver This assumption is also used in references [11], [17], [20], [23] A Transmitter The underlying MIMO wireless transmitter is equipped with an STBC encoder that takes L symbols at its input for the data transmission For the system model depicted in Fig 1, the serial-to-parallel convertor divides each packet of size Lk information bits into L parallel binary streams B i (b i1,, b ik ), i,, L, where b ij {0, 1} and k is the cardinality of B i Each bit stream B i is mapped into the symbol s i Z odd [i] using a preassigned M-ary modulation scheme where it is assumed that E[ s i 2 ] = E s The STBC encoder takes L modulated signals as its inputs and generates the codeword matrix X = [x 1 x T ] Nt T for T time slots The stored matrix X in the transmit buffer is divided into N subcodewords {X (n) = [x τ (n 1) +1 x τ (n)] Nt T (n)} N n, each corresponding to the (re)transmitted data at the n th HARQ round and T (n) is the number of assigned time slot(s) at the N n n th HARQ round such that T (n) = T, τ (n) T (i), and n=0 i=0 Lk Output Bits

3 3 T (0) 0 In particular, the first sub-codeword of matrix X, denoted by X (1) = [x τ (0) +1 x τ (1)] Nt T (1), is picked from the transmit buffer and then transmitted through N t transmit antennas at the first HARQ round Once an ACKnowledege (ACK) signal was sent back by the receiver, the data in the buffer is removed and a new packet of information will be transmitted Otherwise, for the case where the receiver feeds back a Negative ACKnowledgment (NACK) signal to the transmitter, the second sub-codeword of matrix X denoted by X (2) = [x τ (1) +1 x τ (2)] Nt T (2) is drawn from the transmit buffer and is sent as the first retransmission packet The procedure of the packet retransmission is still repeated till the ACK signal is fed back to the transmitter or the maximum allowable (re)transmission round, ie, N is reached In the latter case, if the decoding is still with failure, the error is declared and a new packet of information is coded and transmitted In practical situation, the decoding process is only performed in the final retransmission round and error detection is not required B Receiver Let H(n) = [ h (n) i,j ] N r N t denote the channel matrix as- (re)transmission round whose entries sociated with the n th { h (n) i,j }N r,n t i,j CN (0, 1) are independent and identically distributed (iid) random variables It is assumed that the channel matrix is constant over N consecutive (re)transmission rounds, ie, H(n) = H n, 2,, N, and the perfectly estimated channel coefficients are available at the receiver side but not at the transmitter side This assumption is reasonable due to the fact that the variations of the channel are slow and therefore, the channel estimation can be easily performed by transmitting known pilots or training signals to the receiver [35], [36, p 7] In addition, we denote W (n) = [ w (n) i,j ] N r T as the additive (n) white Gaussian noise matrix in the n th (re)transmission round where { w (n) (n) i,j }Nr,T i,j CN (0, 1) are iid random variables Taking the above considerations into account, the N r T (n) received signal matrix Ỹ(n) at the n th (re)transmission round is considered as Ỹ (n) = ρ N t E s HX (n) + W (n), n, 2,, N, (1) where ρ indicates the average received Signal-to-Noise Ratio (SNR) at each receiver antenna, and X (n) is the transmitted sub-matrix at the n th (re)transmission round To decode the transmitted signal vector s = [s 1 s 2 s L ] T with a complex SD algorithm [37], the received signal matrix Ỹ (n) in (1) can be rewritten as the following equivalent form: y (n) = ρ N t E s H (n) s + w (n), n, 2,, N, (2) where the N r T (n) L matrix H (n) is the equivalent channel matrix at the n th (re)transmission round whose entries depend on the structure of the STBC and the channel matrix H (n) The N r T (n) 1 signal vectors y (n) and w (n) denote the equivalent received signal vector and the equivalent noise vector at the n th (re)transmission round, respectively To benefit from the diversity advantage, all the received signal matrices Ỹ(n) are saved in the receive buffer if the NACK signal is sent back in the n th HARQ round Hence, it is necessary to form a transceiver model based on the received signal matrices {Ỹ(i) } n i, where n N is the current (re)transmission round To this end, we collect the equivalent signal vectors {y (i) } n i in an N rτ (n) 1 new vector y (n) as follows: y (1) H (1) w (1) y (n) = ρ = N t E s + s y (n) H (n) w (n) }{{}}{{} H (n) w (n) ρ = H (n) s + w (n), (3) N t E s where the N r τ (n) L matrix H (n) and the N r τ (n) 1 signal vector w (n) are called the equivalent channel matrix and the equivalent noise vector after n (re)transmission rounds, respectively Note that since after N (re)transmission rounds, all sub-matrices of matrix X are transmitted during T time slots, the equivalent channel matrix H (N) is equal to the equivalent channel matrix of a full STBC denoted by H An incomplete decoder [20] is utilized upon the reception of the signal vector y (n), n < N, where it finds the set of signal vectors S = {s y (n) H (n) s 2 F d(n) }, where d (n) = N r τ (n) (1 + log ρ) If S includes only one element, it means that the channel is not in outage and the noise is typical [20] Hence, the decoding is performed successfully and an ACK signal is sent back to the transmitter to request a new packet of data in the next transmission round Otherwise, when S includes more than one signal or is empty, it means that the channel is poor and/or the noise is not typical For this case, the decoding process is failed and a NACK signal is fed back to the transmitter for the retransmission request of the same information packet in the next round Note that in the latter case, the received signal vectors are stored in the receive buffer (in terms of the symbol level) and will be used for the decoding process in the next retransmission round Objective: The main objective of this paper is to reduce the decoding complexity of the MIMO HARQ protocol such that the data transmission rate of the proposed system is preserved in all SNR ranges To address this issue, we consider the average Number of Visited Points (NVPs) [5], [6] by a decoder as a complexity measure throughout of the paper In addition, the spectral efficiency of a transmitted packet is defined as the ratio of the transmitted information bits, ie, L log 2 (M), per the number of reached channel use intervals in a (re)transmission round with successful decoding or the final retransmission round, ie, τ (ns) where n s is the number of possibly (re)transmission rounds for a successful decoding or N Therefore, the spectral efficiency for a transmitted packet, denoted by S, could be written as S = L log 2 (M) τ (ns) III STBC-INDEPENDENT FAST-DECODABLE MIMO HARQ PROTOCOLS The decoding complexity of a MIMO HARQ system could be categorized into two cases: i) the decoding complexity

4 4 before the final retransmission round, and ii) the decoding complexity at the final retransmission round For the first case, the computational complexity is of order O(nM L ) for the n th < N (re)transmission rounds when an exhaustive search is applied The computational complexity of the second case is the same as that of the STBC since the whole STBC is transmitted in the final retransmission round Therefore, it is reasonable to seek various solutions which reduce the amount of decoding complexity for the first case Since the proposed MIMO HARQ protocols are applicable to STBCs either with general structure or especial structure, we classify our methods into two classes: i) Class I: Independent of STBC and ii) Class II: Dependent on STBC Before presenting the details of Class I, we first elaborate the key idea behind the proposed protocol Let C MIMO, C STBC, and R (n) denote the actual channel capacity of an N t N r MIMO system, the maximum mutual information achieved by an N t N r STBC, and the data transmission rate at the n th HARQ round, respectively, for a specific value of the SNR It is a well known fact that C STBC C MIMO as shown in [38] Let us consider the case where the instantaneous outage is occurred, ie, R (n) > C MIMO It is reasonable to say that the error is occurred with a high probability since R (n) > C MIMO > C STBC On the other hand, the error is not certainly occurred when R (n) < C MIMO, since i) the data transmission rate at the n th (re)transmission round may be greater than C STBC, and ii) the error may be caused by the atypical noise Hence, a decision needs to be made whether the ACK signal should be fed back or NACK in the non-outage case Based on the above observation, we present the following threshold or decision function for the Class I: { Send NACK without decoding, F ( H) < γ (n), Determine ACK or NACK with decoding, F ( H) > γ (n), (4) where F ( H) is a scalar function of H and γ (n) represents the amount of threshold in the n th < N (re)transmission round When the channel is poor at the current (re)transmission round n, ie, F ( H) < γ (n), the MIMO HARQ system feedbacks a NACK signal without using the incomplete decoder On the other hand, the incomplete decoder performs the common error detection process when the channel is in a good condition The parameter γ (n) has a crucial effect on both decoding complexity and the average spectral efficiency preservation Depending on the choice of γ (n), two threshold-based MIMO HARQ protocols are proposed: Fast-Decodable MIMO HARQ Protocol with a Fixed Threshold: In this type of threshold-based MIMO HARQ system, one can choose the Frobenius norm F ( H) = H 2 F or the condition number Con( H H H ) if N r N t or Con( H H H) if Nr > N t as the threshold functions and the parameter γ (n) as a constant parameter We will show numerically that a considerable reduction in the computational complexity, without affecting on the BER performance, would be provided if the parameter is properly selected Although a large value of γ (n) provides a lower decoding complexity, the spectral efficiency has more losses due to the increase in the γ (n) probability of the event H 2 F < γ(n) In this paper, we prefer to use the Frobenius norm based on the following reasons: Both methods could achieve the same results if the parameter γ (n) is properly selected Parameter γ (n) has a crucial effect on the performance metrics of the fixed threshold based MIMO HARQ protocols The condition number is more restrictive than the Frobenius norm since it cannot be utilized for the MISO channels The computation of Frobenius norm is simpler than that of the condition number since the Frobenius norm needs O(N r N t ) flops, while the condition number needs O(max(N r, N t ) min(n r, N t ) 2 ) flops due to the singular value decomposition [39, Lecture 31] Fast-Decodable MIMO HARQ Protocol with an Adaptive Threshold: For this case, we choose the parameter γ (n) as a function of transmission rate in the n th HARQ round A threshold function based on the outage capacity is the best choice for this protocol since the average spectral efficiency is preserved for all SNR ranges Numerical results show that the amount of the complexity reduction is negligible for the high SNR regimes Remark 1: In practical situations where the MIMO HARQ system works at a fixed SNR value, it is desirable to use an adaptive threshold method if the SNR is low or moderate due to the spectral efficiency preservation with a considerable reduction in the decoding complexity For high SNR values, one can use the fixed threshold method with a proper choice of the parameter γ (n) such that the computational complexity reduces, while the spectral efficiency can be still preserved by the system IV STBC-DEPENDENT FAST-DECODABLE MIMO HARQ PROTOCOLS So far, we have presented two protocols for the computational complexity reduction of MIMO HARQ systems Our proposed methods are independent of the STBC structure and are useful for any STBC In this section, we present the second class of fast-decodable transmission protocols for MIMO HARQ systems based on the SD algorithm In particular, we call a MIMO HARQ round fast-decodable if the worst case decoding complexity with the SD algorithm is less than the order of O(M L ) In the sequel, a fast SD algorithm and the new notion of full fast-decodable for this class are introduced A SD Algorithm for MIMO HARQ Systems Before introducing the fast-decodable notion for MIMO HARQ systems, we need to describe how to apply the SD algorithm for HARQ rounds For the n th HARQ round, we define matrix D (n) H (n)h H (n) where it could be fullcolumn-rank (positive definite) or semi-positive definite We first consider the positive definite case and then describe

5 5 the methodology for semi-positive definite case Due to fullcolumn-rank, we have N r τ (n) L and the Cholesky factorization of matrix D (n) can be written as D (n) = R (n)h R (n), where n < N and R (n) = [ i,j ] L L is an upper-triangular matrix Regarding the fixed radius of the incomplete decoder, the standard SD algorithm [40] cannot be directly applied To overcome this problem, the modified standard SD algorithm for HARQ rounds is presented in five steps as follows: Preliminaries: R (n), with n < N, denotes an L L upper-triangular matrix with i,j being its (i, j) th entry ỹ (n) = F(y (n) ) is an L 1 vector for the n th HARQ round, where F() is a matrix transformation which can be determined by the Cholesky method used for the SD decomposition For this definition, ỹ (n) i is the i th entry of vector ỹ (n) d (n) = N r τ (n) (1 + log ρ) is the radius of incomplete decoder at the n th HARQ round Step 1: Set i = L and ŝ L+1 = [] 0 0 Find the set Ŝ i = {ŝ i = [ŝ i ŝ i+1 ŝ L 1 ŝ L ] T i,i (ỹ(n) i ŝ i ) + i,i+1 (ỹ(n) i+1 ŝ i+1) + + i,l (ỹ(n) L ŝ L) 2 d (n) L j,j (ỹ(n) j ŝ j ) + j,j+1 (ỹ(n) j+1 ŝ j+1) + j=i+1 + j,l (ỹ(n) L ŝ L) 2 } for each vector ŝ i+1 = [ŝ i+1 ŝ i+2 ŝ L 1 ŝ L ] T Set Nŝi = Ŝi and go to the next step Step 2: If Nŝi > 0, decrease i = i 1 and go to the next step Otherwise, go to step 4 Step 3: If i 0, go back to step 1 Otherwise, if Nŝi+1 = 1, go to step 5, else go to step 4 Step 4: The algorithm is terminated and a NACK signal should be fed back to the transmitter Step 5: The algorithm is terminated Ŝi is the solution and an ACK signal should be fed back to the transmitter Due to the fixed radius of the incomplete decoder, two important differences exist between the standard SD algorithm and the above scheme i) If there is no admissible (L+1 k)- dimensional, k = L, L 1,, 1, point in the sphere, ie, Ŝ k to be an empty set, the algorithm will be terminated by the NACK signal instead of increasing the sphere radius in the standard SD algorithm (Steps 1 and 2) ii) If there exist more than one L-dimensional point in the sphere, ie, Ŝ1 has at least two points, the algorithm again will be terminated by the NACK signal instead of decreasing the sphere radius (Step 3) Due to the rank-deficiency of matrix D (n) in some HARQ rounds, the modified SD algorithm cannot be directly utilized In particular, the matrix D (n) is rank-deficient in the n th HARQ round if N r τ (n) < L and therefore, the modified standard SD algorithm is not applicable due to the zero diagonal entries N r τ (n) +1,N r = = τ (n) r(n) +1 L,L = 0 To address this issue, we use the Generalized SD (GSD) algorithm in [41], [42] which first transforms the rank-deficient matrix D (n) into a full-column-rank matrix G (n) = F(D (n), λ), where F is the transform operator and λ is a constant parameter Then, the modified standard SD algorithm is applied to determine whether the feedback signal should be ACK or NACK B A Fast SD Algorithm for MIMO HARQ Systems Inspired by the fast-decodable definition for STBCs in [4], we first provide a definition for HARQ rounds with the fast decoding property as follows: Definition 1 (Fast-Decodable HARQ Rounds): The HARQ round n is called fast-decodable if the upper-triangular matrix R (n) in the Cholesky factorization of matrix D (n) or G (n) has the following condition: i,i+1 = r(n) i,i+2 = = r(n) i,l (n) 1 = r(n) = 0, i, 2,, l (n) 1, i,l (n) (5) where i,j is the (i, j) th entry of matrix R (n) and l (n) L The parameter l (n) is called the CRF in the n th HARQ round Now, we are ready to present the fast SD algorithm for MIMO HARQ systems in the following eight steps: Preliminaries: s = [s 1 s 2 s L 1 s L ] T is an L 1 vector where s i s belong to a signal constellation with size M ŝ is the estimated vector of s is an L l s (n) 1 sub-vector that includes (l (n) + 1) th to L th elements of vector s R (n), with n < N, denotes an L L uppertriangular matrix which satisfies the condition in (5) with 1 l (n) L R (n) is an L l (n) L sub-matrix that includes (l (n) + 1) th to L th rows of matrix R (n) ỹ (n) = F(y (n) ) is an L 1 vector where F() is a matrix transformation which can be determined by the Cholesky method used for the SD decomposition It is (n) l easy to write R (n) (y (n) s) 2 F = G(s ) + F i (s i, s ) i where F i (s i, s ) 0 for i, 2, l (n) and G(s ) 0 2 min F refers to the second minimum of a function F S over the discrete set S Step 1: Use the modified standard SD algorithm with L l (n) levels 1 to find set = {s S R (n) (ỹ (n) s) 2 F d (n) } Set N = S and go to the next step Step 2: If N s 1, set j, p = 0 and go to the next step Otherwise, s go to step 7 l (n) Step 3: If G(s j ) + min i (s i, s i Z [i]f s j ) d (n), where i odd s j is the jth element of set S, set p = p 1 and go to the next step Otherwise, go to step 6 Step 4: If p, save each F i (s i, s j ), each min i (s i, s i Z [i]f s j ), G(s j ), ŝ, and go to the next step odd Otherwise, go to step 7 1 Since all elements of vector s should not be estimated by the modified standard SD algorithm, the conditions i 0 and Nŝi+1 in Step 3 of the modified standard SD algorithm change to i l (n) and Nŝi+1 1, respectively Steps 4 and 5 are removed and the output of algorithm is all satisfied vectors in set Ŝi

6 6 TABLE I DIFFERENT FULL FAST-DECODABLE MIMO HARQ PROTOCOLS BASED ON OSTBC FOR THREE TRANSMIT ANTENNAS Protocol No First Round Second Round Third Round x 1 x 2 x X (1) = x 2 x 1 X (2) = 0 X (3) = x 3 x 3 0 x 1 x 2 x 1 x 2 x X (1) = x 2 x 1 X (2) = 0 x 3 x 3 0 x 1 x 2 x 1 x 2 x X (1) = x 2 x 1 0 X (2) = x 3 x 3 0 x 1 x 2 Step 5: If l (n) i i k min k,2,l {G(s j ) + (n) 2 min k (s k, s k Z [i]f s j ) + odd min i (s i, s i Z [i]f s j )} > d (n), go to the next step odd Otherwise, go to step 7 Step 6: Increase j = j + 1 If j N s go back to step 3 Otherwise, if p go to step 8, else go to step 7 Step 7: The algorithm is terminated and a NACK signal should be fed back to the transmitter Step 8: The algorithm is terminated ŝ is the solution and an ACK signal should be fed back to the transmitter Remark 2: The key idea behind the proposed fast SD algorithm is to avoid exhaustive search through all candidate vectors for the error detection procedure in MIMO HARQ systems In particular, after the set is found by the modified standard SD algorithm with L l S (n) levels, l (n) ML metrics are separately computed to find the minimum of functions F i (s i, s j ) for each s j S Clearly, if the inequality in step 3 is not true, other possible combinations of functions F i (s i, s j ) would not be in the hyper-sphere On the other hand, when the inequality in step 3 is satisfied, the uniqueness of the solution will be checked in step 5 To this end, the second minimum of cost function R (n) (y (n) s) 2 F is found and compared with the radius d (n) If it is greater than d (n), the obtained vector would be unique; otherwise, an error would be detected Remark 3: Based on discussions in Remark 2, we can now determine the worst-case computational complexity of the HARQ round n < N by the following considerations: i) In Step 1, the worst-case computational complexity of the SD algorithm, which is used to find the candidate vectors s j s, is of order O(M L l(n) ) [4] ii) For a selected candidate vector s j, l (n) linear ML metrics are utilized to examine whether s j lies in the hyper-sphere or not Clearly, the computational complexity is of order O(l (n) M) for each examined s j iii) To obtain the complexity order in Step 5, we first note that two terms of the expression are computed and saved in Steps 2 3 and 4 Since min k (s k, s k Z [i]f s j ) is the only term calculated odd l (n) times in Step 4, the computational complexity of this step is of order O(l (n) M) Note that the terms in Step 4 and the calculations in Step 5 are saved and performed at most one time, respectively According to above considerations, the overall computational complexity of the algorithm, in the worst-case, is of order O(l (n) M(M L l(n) + 1)) For a large signal constellation, this amount of complexity approaches to O(l (n) M L l(n) +1 ) which is the same result as found in [4] for a classical STBC with the fast decoding property Remark 4: As previously mentioned, the decoding procedure for the final retransmission round is the same as that of the STBC case Hence, if the utilized STBC for a MIMO HARQ system satisfies the conditions in Definition 1, the same fast-decodable algorithms for the STBCs (eg, [4]) can be directly applied when n = N In order to examine the fast decodable property of a MIMO HARQ round, we need sufficient conditions in terms of the equivalent channel matrix H (n) Theorem 1 The MIMO HARQ round n with the equivalent channel matrix H (n) is fast-decodable if only elements of matrix H (n) yield the following equations: h (n) i, h (n) i+1 = h(n) i, h (n) i+2 = = h(n) i, h (n) l (n) 1 = h (n) i, h (n) = 0, i, 2,, l (n) 1, (6) l (n) where h (n) i denotes the i th column of matrix H (n) Proof: See Appendix A C Examples of Fast-Decodable MIMO HARQ Protocols We begin the subsection by giving a definition for fastdecodable MIMO HARQ protocols as follows: Definition 2 (Fast-Decodable MIMO HARQ Protocols): A MIMO HARQ protocol is fast-decodable of degree K with 1 K N, if it has K fast-decodable HARQ rounds For the case where K = N, the protocol is called full fast-decodable To elaborate more deeply on the fast-decodable notion in MIMO HARQ systems, we provide some examples based on the OSTBCs To generalize the results, the OSTBCs in [43] are considered due to providing the maximum rate with a systematic approach Two constraints are used in our examples: i) all information symbols must be transmitted at the first transmission round This constraint is not necessary

7 7 for other retransmission rounds, and ii) we only study the transmission strategies which are full fast-decodable Table I displays different full fast-decodable MIMO HARQ protocols based on the OSTBC for three transmit antennas To show the full fast-decodable property for these protocols, one can readily use the results of Theorem 1 It is seen that the worst case decoding complexity is of order O((M +1)M L 1 ) for all (re)transmission rounds However, the spectral efficiency performance of these protocols are different due to the difference in the number of HARQ rounds and the number of assigned time slots in each HARQ round In particular, the provided average spectral efficiency by protocol No 1 is more efficient than that of protocols No 2 and 3 since one time slot is assigned at retransmission rounds As a result extracted from Table I, the problem of constructing the full fast-decodable protocols from OSTBCs could be considered as choosing different sub-matrices from the original matrix X such that the condition in (5) for each HARQ round n with a CRF 1 < l (n) L is satisfied It is worth mentioning that the proper selection of the first sub-matrix X (1) for n guarantees the fast-decodability of other retransmission rounds apart from the selection manner of other sub-matrices The following theorem proves this assertion: Theorem 2 An N t N r MIMO HARQ protocol with N HARQ rounds which uses an N t T OSTBC with N t 3 is full fastdecodable if only the first two columns of matrix OSTBC is chosen for the first transmission round Proof: See Appendix B Although an exact description of all different full fastdecodable MIMO HARQ protocols for three transmit antennas has been provided in Table I, it is not an easy task for more than three transmit antennas due to the exponential increase in the number of full fast-decodable protocols This claim is proved by the following theorem: Theorem 3 Consider an N t N r full fast-decodable MIMO HARQ protocol which uses an N t T OSTBC matrix X with N t 3 It is also assumed that two first columns of matrix X are assigned in the first transmission round For such a protocol, the number of different full fast-decodable MIMO HARQ protocols, in the sense of the number of assigned time (1) T T (1) slots, is equal to 2 min 1 where T min is the minimum required number of time slots for having a fast-decodable round when n Proof: See Appendix C Theorem 3 expresses that the number of different full fastdecodable MIMO HARQ protocols increases exponentially with the difference of the parameters T and T (1) min 2, where the lower bound on the T (1) min is obtained as a consequence of Theorem 2 A numerical description of the number of full fastdecodable MIMO HARQ protocols including the parameters N t, T, and T (1) min are listed in Table II As seen from this table, an increase in the number of transmit antennas N t leads to a large difference between parameters T and T (1) min, therefore, a huge number of different full fast-decodable protocols are possible even for a moderate value of N t TABLE II NUMBER OF DIFFERENT FULL FAST-DECODABLE MIMO HARQ PROTOCOLS IN TERMS OF N t, T, AND T (1) min N t T T (1) (1) T T min 2 min V SIMULATION RESULTS In this section, we present some simulation results to evaluate the average spectral efficiency, in terms of the Bit Per Channel Use (BPCU), the average NVPs and the BER of the proposed HARQ schemes for 3 1 Multi-Input Single-Output (MISO) and 3 2 MIMO systems In all simulations, we use the Monte-Carlo approach to derive the average spectral efficiency, BER, and NVPs versus the SNR To preserve the SNR value ρ for each receive antenna, we assume that the average power of each symbol s i, the variances of the channel coefficients h i,j s and the noise samples n i,j s are normalized to unit The channel matrix H (n) is also assumed to be constant during N (re)transmission rounds of a packet and changes independently from one packet to the next TABLE III SIMULATION PARAMETERS γ (1) Protocol Name N r N r Fixed Threshold HARQ I 1 5 Fixed Threshold HARQ II 2 6 We compare the the decoding complexity, the spectral efficiency and the BER performances of the original HARQ with those of the MIMO HARQ schemes in Table I under different threshold-based protocols For simplicity, the case without any transmission protocol is called the original HARQ It should be noted that the threshold-based MIMO HARQ protocols are not restricted to the OSTBCs schemes and can be applied to all other STBCs As proved and compared in [23], using the capacity lossless STBCs in the STBC based HARQ settings achieves the best spectral efficiency For example, Alamouti based HARQ system provides the best spectral efficiency among other 2 1 MIMO system Obviously, for more than two transmit antennas, the MIMO HARQ systems based on the full-rate STBCs, such as QOSTBCs [44], provide a better spectral efficiency than the OSTBCs For the adaptive case, we use threshold function F ( H) = det(i Nr + ρ N t H HH ) with γ (n) L log 2 M τ (n) and we test the threshold function F ( H) = H 2 F with γ(1) s of Table III and γ (2) = 01γ (1) for the fixed threshold scheme 2 In the (re)transmission round n < N, the exhaustive search is utilized for Class I such that upon 2 The thresholds of the fixed protocol are derived using the try and error method

8 8 visiting the second point, the decoding procedure is stopped and a NACK signal is fed back to the receiver Moreover, a linear ML decoder is used at the final retransmission round due to the orthogonality property of the OSTBCs Average Spectral Efficiency (BPCU) age No Visited Points by the Exhaustive Search BER Original HARQ; N r Adaptive HARQ; N r Threshold HARQ I; N r Threshold HARQ II; N r Original HARQ; N r Adaptive HARQ; N r Threshold HARQ I; N r Threshold HARQ II; N r SNR (db) (a) Average spectral efficiency (BPCU) versus SNR Original HARQ; N r Adaptive HARQ; N r Threshold HARQ I; N r Threshold HARQ II; N r Original HARQ; N r Adaptive HARQ; N r Threshold HARQ I; N r Threshold HARQ II; N r SNR (db) (b) Average NVPs by the exhaustive search versus SNR Original HARQ; N r Adaptive HARQ; N r Threshold HARQ I; N r Threshold HARQ II; N r Original HARQ; N r Adaptive HARQ; N r Threshold ARQ I; N r Threshold ARQ II; N r SNR(dB) (c) BER versus SNR Fig MISO and 3 2 MIMO HARQ systems based on the OSTBC under different threshold-based protocols with 16-QAM constellation Figs 2(a), (b) and (c) compare the average spectral efficiency, the average NVPs and the BER of 3 1 MISO and 3 2 MIMO HARQ systems with different thresholdbased protocols under the transmission scheme 1 from Table I, respectively In comparison with the original HARQ method, the average spectral efficiency is preserved by the adaptive case in all SNR ranges, while it is lost when a fixed threshold is utilized On the other hand, in the low and moderate SNR regimes, the adaptive protocol provides the lowest decoding complexity among other threshold-based methods Therefore, the adaptive protocol yields the best performance for a MIMO HARQ system which works in the low and moderate SNR ranges For high SNR regimes, the average NVPs of adaptive threshold-based tends to that of the original HARQ To prove this fact, we consider the lower bound [45, p 55] for the threshold function of adaptive protocol as follows: det(i Nr + ρ N t H HH ) (1 + ( ρ N t ) P P i µ i ), (7) H where µ i s> 0 are the non-zero eigenvalues of matrix H H and P min(n t, N r ) Obviously, the lower bound in (7) tends to infinity when SNR goes to infinity Hence, F ( H) is always greater than γ (n) and the average NVPs of both the original HARQ and the adaptive HARQ coincide with each other in this asymptotic regime Since the lower bound increases as the P th power of the SNR, the average NVPs of the adaptive HARQ tends to that of the original HARQ with a faster rate when N r < N t increases till N r < N t (see Fig 2(b)) Overall, a fixed threshold protocol which brings a lower decoding complexity with an acceptable loss of spectral efficiency yields the best choice for high SNR ranges Fig 2(c) shows that all the proposed schemes have a similar BER performance as compared to that of the original scheme To clarify this, we note that the spectral efficiency of the adaptive method is the same as that of the original case in all SNR ranges (see Fig 2(a)) Therefore, the BER performances of both methods are not different since outage and non-outage events can be correctly identified by the original (with a more decoding complexity) and the adaptive (with a less decoding complexity) methods For the fixed threshold method, the same scenario as that of the adaptive case is repeated in the low SNR ranges In the moderate and high SNR regimes, the spectral efficiency has a loss due to identifying some nonoutage events as outage However, since the undetected error of the incomplete decoder is very small in moderate and high SNR ranges, the BER performances of both the original and the fixed threshold HARQ methods are same To give a comparison between the transmission schemes in Table I, Figs 3(a) and (b) compare the average spectral efficiency and the average NVPs of these schemes under the adaptive protocol, respectively As seen from the figures, the scheme 1 provides the best spectral efficiency in comparison to other schemes in all SNR ranges However, this advantage is gained at the cost of more decoding complexity as shown in Fig 3(b) In addition, the scheme 2 achieves a better spectral efficiency than that of the scheme 3 for high SNR values, while it keeps a lower decoding complexity than that of the scheme 3 In order to resolve this contradiction, an analysis is performed for both average spectral efficiency and average complexity order in the MISO case For the i th scheme, i, 3, let denote the probabilities P{ H 2 F > γ i}

9 9 Average No Visited Points by the Exhaustive Search Average Spectral Efficiency (BPCU) BER Scheme No 1; N r Scheme No 2; N r Scheme No 3; N r Scheme No 1; N r Scheme No 2; N r Scheme No 3; N r SNR (db) (a) Average spectral efficiency (BPCU) versus SNR Scheme No 1; N r Scheme No 2; N r Scheme No 3; N r Scheme No 1; N r Scheme No 2; N r Scheme No 3; N r SNR (db) (b) Average NVPs by the exhaustive search versus SNR Scheme No 1; N r Scheme No 2; N r Scheme No 3; N r Scheme No 1; N r Scheme No 2; N r Scheme No 3; N r SNR(dB) (c) BER versus SNR Fig 3 OSTBC MISO/MIMO HARQ systems under different transmission schemes of Table I with 16-QAM constellation and P{successful decoding H 2 F > γ i} by p i and q i, respectively 3 It is not hard to see that the average spectral efficiency, denoted by S i, and the average complexity order, 3 For the adaptive protocol in MISO case, one can write F ( H) = H 2 F and γ (n) = N L log2 M t ρ (2 τ (n) 1) TABLE IV THE NUMERICAL VALUES OF p i S AND q i S IN DIFFERENT SNRS ALL NUMBERS ARE ROUNDED BY THE ACCURACY 10 4 OR 10 5 SNR (db) p 2 q 2 p 3 q 3 p 2 q p 3q denoted by C i, of the i th scheme could be written as follows: S i = 3 log 2 M 4 (1 p i q i ) + 3 log 2 M p i q i, (8) i C i = 3M(1 p i q i ) + M 3 p i (9) Using equation (8) and after some manipulations, we have S 2 > S 3 p 2 q 2 > 1 3 p 3q 3 (10) Expression (10) provides a comparison measure for finding the scheme with a higher spectral efficiency A numerical evaluation of the above condition is given in Table IV where it is seen that p 2 q 2 < 1 3 p 3q 3 when the SNR is equal or smaller than 22 and p 2 q 2 > 1 3 p 3q 3 if the SNR is greater than 22 The result is completely in agreement with our simulation in Fig 3(a) It is also interesting to investigate the average complexity order for schemes 2 and 3 With a similar calculation for S i s, we have C 2 < C 3 p 2 (M 2 3q 2 ) < p 3 (M 2 3q 3 ) (11) For a relatively large constellation size, it is reasonable to approximate M 2 3q i by M 2 and get to the following result: C 2 < C 3 p 2 < p 3 (12) Due to the greater value of threshold for scheme 2, p 2 < p 3 and therefore, C 2 < C 3 However, as the SNR tends to infinity, the threshold value of both schemes goes to zero Hence, p i s tend to one and C i s asymptotically coincide each other in high SNR regimes Fig 3(c) compares the BER performances of different HARQ schemes in Table I under the adaptive threshold method As seen from this figure, the BER performances of different schemes are same again in all SNR ranges To verify the result, we express the error probability of i th scheme, denoted by P i (e), i, 2, 3, as follows: P i (e) = P i (e F ( H) < γ 1 )P(F ( H) < γ 1 ) + P i (e γ 1 < F ( H) < γ 2 ) P(γ 1 < F ( H) < γ 2 ) + P i (e F ( H) > γ 2 )P(F ( H) > γ 2 ), where γ 1 4 γ 2 6 The error probability P i (e F ( H) < γ 1 ) is same for all three schemes since the outage event occurs and the HARQ schemes retransmit data, similar to a full OSTBC, with the same data rate This observation shows that why

10 10 the spectral efficiencies and the BER performances of three schemes are same for low SNR ranges where most of data are retransmitted in the outage event (see Fig 3(a) where the spectral efficiency of all three schemes are 3 bits/s/hz in low SNR ranges) The error probability P i (e γ 1 < F ( H) < γ 2 ) is same for i, 3, since schemes 1 and 3 face with the non-outage event and utilize a similar incomplete 3 3 OSTBC matrix (see Tables I and III) for the retransmission in the second and transmission in the first HARQ rounds, respectively Note that due to the non-typical noise or codeword error, a failure decoding maybe happened for both schemes and hence, the data is retransmitted again in the final retransmission round Under this condition, the outage event is still observed by the second scheme and its data rate is the same as that of the final retransmission round However, the only event, which can be made a difference between the error probabilities of schemes 1, 3 and scheme 2, is the undetected error probability of the incomplete decoder in schemes 1 and 2 Since the undetected error probability is very low in moderate SNR values, where the event γ 1 < F ( H) < γ 2 is more probable, its influence on the error probability is not significant By a similar argument, it is easy to show that the error probabilities P i (e F ( H) > γ 2 ) is same for i, 2 and are very close to that of the scheme 3 due to the negligible undetected error of the incomplete decoder in high SNR values As seen from the above figures, the average NVPs are not monotonic with SNR This behavior could be verified using C i in (9) and the values p i, q i, and p i q i of Table IV In low SNR ranges, since the outage event is occurred in most of transmissions, the values of p i and p i q i are very small, hence, the linear term in (9) is dominant C i monotonically increases when SNR grows due to increase in the coefficient of the cubic term in (9) However, the coefficient of the linear term decreases such that a maximum value is imposed on the average NVPs When the maximum value of C i reaches, C i decreases such that it can be only expressed by the cubic term in (9) due to the vanished linear coefficient Finally, Fig 4 compares the spectral efficiency and the complexity metrics between the original HARQ, fast SD HARQ protocol, and the combination protocol of adaptive thresholdbased and fast SD protocols, respectively As seen from the figure, for all cases, the fast-decodable approach provides a significant reduction in the average NVPs in comparison to the original HARQ In addition, the provided average NVPs of the combination method is lower than those of both original HARQ and fast SD HARQ protocol in low and moderate SNR ranges For high SNR ranges, the average NVPs of fast SD HARQ and combined HARQ protocols get closer together with respect to the performed analysis for Fig 2 VI CONCLUSION In this paper, we presented two classes of fast-decodable MIMO HARQ systems For the first class, namely independent of the STBC, two threshold-based protocols were introduced and their effectiveness in the computational complexity reduction and the spectral efficiency preservation were discussed Average Spectral Efficiency (BPCU) Average No of Visited Points Orig ARQ; NO 1; N r SD ARQ; NO 1; N r Orig ARQ; NO 2; N r SD ARQ; NO 2; N r Orig ARQ; NO 1; N r SD ARQ; NO 1; N r Orig ARQ; NO 2; N r SD ARQ; NO 2; N r Combined ARQ; NO 1; N r Combined ARQ; NO 2; N r Combined ARQ; NO 1; N r Combined ARQ; NO 2; N r SNR (db) (a) Average spectral efficiency (BPCU) versus SNR Orig ARQ; NO 1; N r SD ARQ; NO 1; N r Orig ARQ; NO 2; N r SD ARQ; NO 2; N r Orig ARQ; NO 1; N r SD ARQ; NO 1; N r Orig ARQ; NO 2; N r SD ARQ; NO 2; N r Combined ARQ; NO 1; N r Combined ARQ; NO 2; N r Combined ARQ; NO 1; N r Combined ARQ; NO 2; N r SNR (db) (b) Average NVPs versus SNR Fig 4 A comparison between the fast implementation and the exhaustive search of a MISO/MIMO HARQ system; 4-QAM constellation For the dependent class, a fast SD algorithm was proposed and it was shown that a significant reduction in the computational complexity of a MIMO HARQ system can be achieved by this fast algorithm Two new concepts namely, fast-decodable HARQ round and full fast-decodable HARQ protocol were introduced and the conditions, in terms of the equivalent channel matrix, under which a MIMO HARQ protocol is full fastdecodable were obtained We also presented an example of full fast-decodable MIMO HARQ system based on the OSTBCs For a MIMO HARQ system with the OSTBC scheme, it was analytically proved that a proper selection of the first transmission codeword leads to the fast-decodability of other retransmission rounds We also showed that the number of different full fast-decodable MIMO HARQ settings increases exponentially with the difference of parameters T and T (1) min such that it is not feasible to simply provide all full fastdecodable protocols for a MIMO HARQ system except for

11 11 the case of three transmit antennas Finally, simulation results showed that the proposed protocols, including both dependent and independent classes, provide a significant reduction in the decoding complexity of a MIMO HARQ system Due to the added zero elements in (19), this transformation cannot also change the condition in (6) or equivalently in (15) Hence, the theorem is still valid for the semi-definite case and the proof is completed A Proof of Theorem 1 APPENDIX We first prove the theorem for the case where D (n) = H (n)h H (n) is a positive definite matrix and then show that the theorem is still valid for the semi-positive definite case Consider the Cholesky factorization of the L L matrix D (n) = H (n)h H (n) = R (n)h R (n) where n < N, R (n) = [ i,j ] L L is an upper-triangular matrix and D (n) = [d (n) i,j ] L L is a positive definite matrix Using the definition of Cholesky decomposition [46], we have i,j i 1 d (n) i,j i,i k k,i r (n) k,j i < j (14) It is not hard to see that the condition in (6) is equivalent to the following equation: d (n) i,i+1 = d(n) i,i+2 = = d(n) i,l (n) 1 = d(n) = 0, (15) i,l (n) where i, 2,, l (n) 1 By using (14) and (15) for i, one can readily conclude that 1,2 = r(n) 1,3 = = r(n) 1,l (n) 1 = r(n) = 0 (16) 1,l (n) We assume that for i p, 1 p l (n) 2 i,j = 0 p + 1 j l(n) 1 (17) We now show that (17) is also true for i = p + 1 To this end, we rewrite (14) for i = p + 1 as follows: p+1,j p d (n) p+1,j (n) k,p+1 r k,j p + 1 < j p+1,p+1 k (18) Obviously, two terms d (n) p+1,j, p + 2 j l(n), and k,p+1, 1 k p, are zero due to the assumptions in (15) and (17), respectively Therefore, the condition in (5) satisfies and the proof is completed for the positive definite case Now, we assume that D (n) is semi-positive definite As previously discussed in Section IV-A, we apply a semi-positive to positive definite transformation method [41], [42] to matrix D (n) and use the positive definite matrix G (n) = F(H (n), λ) for the SD algorithm For the method used in [41], since G (n) = D (n) + λi L, it is clear that non-diagonal entries of matrix D (n) do not change and the condition in (15) is still satisfied by G (n) For the transformation used in [42], matrix H (n) = [H (n) 1 H (n) 2 ], where H(n) 1 and H (n) 2 are N r τ (n) N r τ (n) and N r τ (n) L N r τ (n) matrices, respectively, is directly transformed to a full-column-rank matrix, denoted by H (n) t, as follows: ( H (n) H (n) t = 1 H (n) ) 2 0 L Nrτ (n) N λi (19) rτ (n) L N rτ (n) B Proof of Theorem 2 For simplicity, the theorem is proved for the MISO case where N r and N t is arbitrary We then show that the result can be straightforwardly generalized to the MIMO case To have a full fast-decodable MISO HARQ protocol which is independent of the sub-matrix selection for retransmission rounds, it is sufficient that i) the sub-matrix H (1) = H (1) of H satisfies (6) with the CRF l (1), and ii) for any arbitrary value of N, the resulted matrix H (n) from adding any arbitrary sub-matrices H (n) of H to H (n 1) satisfies (6) for n = 2, 3,, N Without loss of generality, we assume T (1) If two first columns of a maximal rate N t T OSTBC [43] is selected for the first transmission round, the equivalent channel matrix H (1) would have the following general form: ( ) H (1) h1 h H, h 2 h 1 (20) where H is a 2 (L 2) matrix It is easy to show that H (1) satisfies (5) with the CRF l (1) and, therefore, the first transmission round is fast-decodable Now, we consider the special case where T (2) = T (3) = = T (N), since other HARQ protocols with different values of T (i), i, 3,, N, and N can be constructed from this special case By adding the T (2) L sub-vector H (2), the resulted τ (2) L matrix H (2) satisfies (6) if and only if h (2) 1 h(2) 2 = 0 With a similar argument as for n, one can deduce the following equations: h (n) 1 h(n) 2 = 0, n, 3,, N (21) By comparing the relationshhip between the equivalent channel matrix H and the codeword matrix X and considering (21), we conclude that the T 1 vector X (n) cannot contain both of symbols s 1 and s 2 for n, 3,, N Hence, the problem is to prove that the two first columns of a maximal rate OSTBC matrix are only the columns in which both of symbols s 1 and s 2 appear To this end, we turn our focus on the structural algorithm of the OSTBC with the maximal rate [43] The algorithm is recursive in the sense that the N t T Nt OSTBC is built from the OSTBC matrix with N t 1 transmit antennas and T Nt 1 time slots Let us denote the N t 1 T Nt 1 OSTBC matrix with L NT 1 information symbols by X Nt 1 To build an N t T Nt OSTBC, the 1 T Nt 1 row vector x new, which contains the L NT L NT 1 new information symbols and T Nt 1 L NT + L NT 1 zeros, adds to X Nt 1 and results the N t T Nt 1 matrix: ( ) XNt 1 X Nt = (22) x new To preserve the orthogonality between the new row x new and other N t 1 rows, T Nt T Nt 1 column vectors of length N t

12 12 are added to matrix X Nt and results the following matrix: ( ) XNt 1 X Nt = x 1 x 2 x TNt T Nt 1 (23) x new Since each of L Nt information symbols must appear only one time in a row of matrix X Nt, one can readily conclude that the last element of L Nt 1 column vectors x i s, i = 1, 2,, L Nt 1, is the information symbols of matrix X Nt 1 and that of other T Nt T Nt 1 L Nt 1 are zero In addition, the top N t 1 elements of vectors x i s are elements of vector x new or zero Hence, there is not a new column x i that has a combination of information symbols of matrix X Nt 1 Since, the information symbols s 1 and s 2 are in the OSTBC matrix with two transmit antennas, they do not appear both in the new columns of the OSTBC matrix with N t 3 In order to generalize the theorem to the MIMO case, three issues should be noted: i) the N r N t L equivalent channel matrix H MIMO can be constructed from N r H MISO of dimension N t L such that 1 i th N r matrix H MISO involves N t path gains associated with the i th receive antenna and N t transmit antennas, ii) all H MISO s have the same structures, and iii) choosing the j th column of matrix X is equivalent to j th row selection of all N r matrices H MISO By considering the above points and the proof steps in MISO case, one can readily generalize the theorem for the MIMO systems C Proof of Theorem 3 Since the T (1) min of total time slots T are assigned to the first transmission round, our problem is equivalent to the number of ways of dividing the T T (1) min time slots between two to T T (1) min + 1 (re)transmission rounds Let us denote by t i the number of assigned time slots in the i th (re)transmission round The problem can be formulated as follows: t 1 + t 2 = T t 1 + t 2 + t 3 = T m t 1 + t t (1) T T min 1 + t = T T T (1) min t 1 + t t (1) T T + t (1) min T T = T, min +1 min m (24) where t 1 T (1) min, t i 1, and 2 i T T (1) min + 1 It is not hard to show ( that the number ) of solutions for m th (1) T T equation in (24) is min where 1 m T m T (1) min, and hence, the total number of solutions is equal to T T (1) ( ) min (1) T T (1) T T min 1 REFERENCES [1] S M Alamouti, A simple transmit diversity technique for wireless communications, IEEE Journal on Selected Areas in Communications, vol 16, no 8, pp , Oct 1998 [2] V Tarokh, H Jafarkhani, and R Calderbank, Space-time block codes from orthogonal designs, IEEE Trans on Information Theory, vol 45, no 5, pp , July 1999 [3] V Tarokh, H Jafarkhani, and R Calderbank, Space-time block codes for high data rate wireless communication: Performance results, IEEE Journal on Selected Areas in Communications, vol 17, no 3, pp , March 1999 [4] E Biglieri, Y Hong, and E Viterbo, On fast-decodable space-time block codes, IEEE Trans on Information Theory, vol 55, no 2, pp , Feb 2009 [5] J Paredes, A B Gershman, and M G-Alkhansari, A new fullrate full-diversity space-time block code with nonvanishing determinants and simplified maximum-likelihood decoding, IEEE Trans on Signal Processing, vol 56, no 6, pp , June 2008 [6] S Sezginer, S Sari, and E Biglieri, On high-rate full-diversity 2 2 space-time codes with low-complexity optimum detection, IEEE Trans on Communications, vol 57, no 5, pp , May 2009 [7] M Samuel and M Fitz, Multi-strata codes: Space-time block codes with low detection complexity, IEEE Trans on Communications, vol 58, no 4, pp , April 2010 [8] S S Hosseini, S Talebi, and M Shahabinejad, A full-rate full-diversity 2 2 space-time block code with linear complexity for the maximum likelihood receiver, IEEE Communications Letters, vol 15, no 8, pp , Aug 2011 [9] S S Hosseini, S Talebi, and J Abouei, Comprehensive study on a 2 2 full-rate and linear decoding complexity space-time block code, IET Communications, vol 9, no 1, pp , Jan 2015 [10] J Abouei, H Bagheri, and A K Khandani, An efficient adaptive distributed space-time coding scheme for cooperative relaying, IEEE Trans on Wireless Communications, vol 8, no 10, pp , Oct 2009 [11] E N Onggosanusi, A G Dabak, Y Hui, and G Jeong, Hybrid ARQ transmission and combining for MIMO systems, in Proc IEEE International Conference Communications (ICC 03), Dallas, TX, USA, May 2003, pp [12] D Chase, Code combining-a maximum-likelihood decoding approach for combining an arbitrary number of noisy packets, IEEE Trans on Communications, vol 33, no 5, pp , May 1985 [13] S Sesia, G Caire, and G Vivier, Incremental redundancy hybrid ARQ schemes based on low-density parity- check codes, IEEE Trans on Communications, vol 52, no 8, pp , Aug 2004 [14] A V Neguyen and M A Ingram, Hybrid ARQ protocols using spacetime codes, in Proc IEEE Vehicular Technology Conference (VTC 01), Atlantic City, NJ, USA, Oct 2001, pp [15] V Gulati and K R Narayanan, Concatenated codes for fading channels based on recursive space-time trellis codes, IEEE Trans on Wireless Communications, vol 2, no 1, pp , Jan 2003 [16] E de Carvalho and P Popovski, Strategies for ARQ in 2 2 MIMO systems, IEEE Communications Letters, vol 12, no 6, pp , June 2008 [17] S Ko, H Seo, and B G Lee, Estimation-based retransmission mode selection for reinforced HARQ operation in MIMO systems, IEEE Trans on Wireless Communications, vol 8, no 9, pp , Sept 2009 [18] K Zheng, H Long, L Wang, W Wang, and Y-Il Kim, Design and performance of space-time precoder with hybrid ARQ transmission, IEEE Trans on Vehicular Technology, vol 58, no 4, pp , May 2009 [19] L Zheng and D Tse, Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels, IEEE Trans on Information Theory, vol 49, no 5, pp , May 2003 [20] H El Gamal, G Caire, and M O Damen, The MIMO ARQ channel: Diversity-multiplexing-delay tradeoff, IEEE Trans on Information Theory, vol 52, no 8, pp , Aug 2006 [21] T Kim and M Skoglund, Diversity-multiplexing tradeoff in MIMO channels with partial CSIT, IEEE Trans on Information Theory, vol 53, no 8, pp , Aug 2007 [22] A Chuang, A Guillen i Fabregas, L K Rasmussen, and I B Collings, Optimal throughput-diversity-delay tradeoff in MIMO ARQ blockfading channels, IEEE Trans on Information Theory, vol 54, no 9, pp , Sept 2008 [23] C Shen and P Fitz, Hybrid ARQ in multiple-antenna slow fading channels: Performance limits and optimal linear dispersion code design, IEEE Trans on Information Theory, vol 57, no 9, pp , Sept 2011 [24] X Liang, C-M Zhao, and Z Ding, Piggyback retransmissions over wireless mimo channels: Shared Hybrid-ARQ (SHARQ) for bandwidth efficiency, IEEE Trans on Wireless Communications, vol 12, no 8, pp , Aug 2013

13 13 [25] Z Zhang, J Xu, and L Qiu, Robust ARQ precoder optimization for AF MIMO relay systems with channel estimation errors, IEEE Trans on Wireless Communications, vol 12, no 10, pp , Oct 2013 [26] J Wu, G Wang, and Y R Zheng, Energy efficiency and spectral efficiency tradeoff in type-i ARQ systems, IEEE Journal on Selected Areas in Communications, vol 32, no 2, pp , Feb 2014 [27] B Makki and T Eriksson, On the performance of MIMO-ARQ systems with channel state information at the receiver, IEEE Trans on Communications, vol 62, no 5, pp , May 2014 [28] M Zia and Z Ding, Bandwidth efficient variable rate HARQ under orthogonal space-time block codes, IEEE Trans on Signal Processing, vol 62, no 13, pp , July 2014 [29] K D Nguyen, L K Rasmussen, A Guillen i Fabregas, and N Letzepis, MIMO ARQ with multibit feedback: Outage analysis, IEEE Trans on Information Theory, vol 58, no 2, pp , Jan 2012 [30] M S Nafea, D Hamza, KG Seddik, M Nafie, and H El Gamal, On the ARQ protocols over the Z-interference channels: Diversitymultiplexing-delay tradeoff, in Proc IEEE International Symposium on Information Theory (ISIT 12), Cambridge, MA, USA, July 2012, pp [31] M-K Oh, Y-H Kwon, and D-J Park, Efficient hybrid ARQ with space-time coding and low-complexity decoding, in Proc IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP 04), Montreal, Quebec, Canada, May 2004, pp [32] H Samra and Z Ding, New MIMO ARQ protocols and joint detection via sphere decoding, IEEE Trans on Signal Processing, vol 54, no 2, pp , Feb 2006 [33] C-M Chen, J-Y Hsu, P-H Kuo, and P-A Ting, MIMO hybrid- ARQ utilizing lower rate retransmission over mobile WiMAX system, in Proc IEEE Mobile WiMAX Symposium (MWiMAXS 09), Napa Valley, CA, USA, Dec 2009, pp [34] H Samra, Z Ding, and P M Hahn, Symbol mapping diversity design for packet retransmission through fading channels, in Proc IEEE Global Telecommunications Conference (GLOBECOM 03), San Fransisco, CA, USA, Dec 2003, pp [35] V Tarokh, A Naguib, N Seshadri, and A R Calderbank, Space-time codes for high data rate wireless communication: Performance criteria in the presence of channel estimation errors, mobility, and multiple paths, IEEE Trans on Communications, vol 47, no 2, pp , Feb 1999 [36] F Oggier, J-C Belfiore, and E Viterbo, Cyclic division algebras: A tool for space-time coding, Foundations and Trends in Communications and Information Theory, vol 4, no 1, pp 1 95, 2007 [37] B M Hochwald and S T Brink, Achieving near-capacity on a multiple-antenna channel, IEEE Trans on Communications, vol 51, no 3, pp , March 2003 [38] B Hassibi and B M Hochwald, High-rate codes that are linear in space and time, IEEE Trans on Information Theory, vol 48, no 7, pp , July 2002 [39] L N Trefethen and D Bau III, Numerical linear algebra, vol 50, Siam, 1997 [40] B Hassibi and H Vikalo, On the sphere-decoding algorithm I Expected complexity, IEEE Trans on Signal Processing, vol 53, no 8, pp , Aug 2005 [41] T Cui and C Tellambura, An efficient generalized sphere decoder for rank-deficient MIMO systems, IEEE Communications Letters, vol 9, no 5, pp , May 2005 [42] P Wang and T Le-Ngoc, A low-complexity generalized sphere decoding approach for underdetermined linear communication systems: Performance and complexity evaluation, IEEE Trans on Communications, vol 57, no 11, pp , Nov 2009 [43] W Su, X-G Xia, and K J R Liu, A systematic design of high-rate complex orthogonal space-time block codes, IEEE Communications Letters, vol 8, no 6, pp , June 2004 [44] H Jafarkhani, A quasi-orthogonal space time block code, IEEE Trans on Communications, vol 49, no 1, pp 1 4, Jan 2001 [45] H Lütkepohl, Handbook of Matrices, John Wiley & Sons, New York, first edition, 1996 [46] J E Gentle, Numerical linear algebra for applications in statistics, Springer, New York, first edition, 1998 Seyyed Saleh Hosseini (S 12) received the BSc and the MSc degrees in communication systems engineering both from the Shahid Bahonar University of Kerman, Kerman, Iran, in 2009 and 2012, respectively From 2013 to 2014, he was a faculty member (lecturer) in the Department of Electrical Engineering, Javid University, Iran Currently, S Hosseini is an lecturer in the Department of Electrical and Computer Engineering, at the Azad University, Bardsir Branch, Kerman, Iran He also has served as a reviewer for a number of IEEE conferences and international journals including IEEE Transactions on Communications and IET communications His research interests are in general areas of communication theory and wireless communications with particular reference to space-time coding, MIMO HARQ, and error probability analysis Jamshid Abouei (S 05-M 11-SM 13) received the BSc degree in electronics engineering and the MSc degree in communication systems engineering (with the highest honor) both from the Isfahan University of Technology (IUT), Iran, in 1993 and 1996, respectively, and the PhD degree in electrical engineering from the University of Waterloo in Waterloo, ON, Canada, in 2009 From 2009 to 2010, he was a Postdoctoral Fellow in the Multimedia Lab, in the Department of Electrical & Computer Engineering, at the University of Toronto, ON, Canada Currently, Dr Abouei is an Assistant Professor in the Department of Electrical & Computer Engineering, at the Yazd University, Iran His research interests are in general areas of wireless ad hoc and sensor networks, with particular reference to energy efficiency and optimal resource allocation, multi-user information theory, cooperative communication in wireless relay networks, applications of game theory, and orthogonal codes in CDMA systems Dr Abouei has received numerous awards and scholarships, including FOE and IGSA awards for excellence in research in University of Waterloo, Canada, MSRT PhD Scholarship from the Ministry of Science, Research and Technology, Iran in 2004, and distinguished researcher award in Province of Yazd, Iran, December 2011 Murat Uysal (SM 07) received the BSc and the MSc degree in electronics and communication engineering from Istanbul Technical University, Istanbul, Turkey, in 1995 and 1998, respectively, and the PhD degree in electrical engineering from Texas A&M University, College Station, Texas, in 2001 Dr Uysal is currently a Professor and Chair of the Department of Electrical and Electronics Engineering at Ozyegin University, Istanbul, Turkey Prior to joining Ozyegin University, he was a tenured Associate Professor at the University of Waterloo, Canada, where he still holds an adjunct faculty position Dr Uysal s research interests are in the broad areas of communication theory and signal processing with a particular emphasis on the physical layer aspects of wireless communication systems in radio, acoustic and optical frequency bands He has authored some 200 journal and conference papers on these topics and received more than 3500 citations Dr Uysal is a Senior IEEE member and an active contributor to his professional society He currently leads the EU COST Action OPTICWISE which is a consolidated European scientific platform for interdisciplinary research activities in the emerging area of optical wireless communications Dr Uysal serves on the editorial boards of IEEE Transactions on Communications, IEEE Transactions on Vehicular Technology, Wiley Wireless Communications and Mobile Computing (WCMC) Journal, and Wiley Transactions on Emerging Telecommunications Technologies (ETT) In the past, he served as an Editor for IEEE Transactions on Wireless Communications ( ), IEEE Communications Letters ( ), Guest Co-Editor for WCMC Special Issue on MIMO Communications (October 2004) and IEEE Journal on Selected Areas in Communications Special Issue on Optical Wireless Communications (December 2009) Over the years, he has served on the technical program committee of more than 80 international conferences and workshops in the communications area Dr Uysal is the recipient of several awards including the University of Waterloo Engineering Research Excellence Award, Natural Science and Engineering Research Council of Canada (NSERC) Discovery Accelerator Supplement (DAS) Award, Turkish Academy of Sciences Distinguished Young Scientist Award, and Ozyegin University Best Researcher Award among others