- Research
- Open Access
Low complexity frequency domain hybrid-ARQ chase combining for broadband MIMO CDMA systems
- Houda Chafnaji^{1, 2}Email author,
- Tarik Ait-Idir^{1, 2},
- Samir Saoudi^{2} and
- Athanasios V Vasilakos^{3}
https://doi.org/10.1186/1687-1499-2012-134
© Chafnaji et al; licensee Springer. 2012
Received: 15 May 2011
Accepted: 5 April 2012
Published: 5 April 2012
Abstract
In this article, we investigate efficient minimum mean square error (MMSE) frequency domain equalization (FDE)-based iterative (turbo) packet combining for cyclic prefix (CP)-CDMA MIMO with Chase-type ARQ. We introduce two turbo packet combining schemes: (i) In the first scheme, namely "chip-level turbo packet combining", chip-level MMSE-FDE and packet combining are jointly performed at the chip-level. (ii) In the second scheme, namely "symbol-level turbo packet combining", chip-level MMSE-FDE and despreading are separately carried out for each transmission, then packet combining is performed at the level of the soft demapper. The key idea of the proposed schemes is to exploit the diversity among all transmissions with a very low cost by introducing new variables recursively computed. The complexity and performances are evaluated for some representative antenna configurations and load factors (i.e., number of orthogonal codes with respect to the spreading factor) to show the gains offered by the proposed techniques.
Keywords
- code division multiple access (CDMA)
- multi-code transmission
- broadband multiple-input-multiple-output (MIMO)
- automatic repeat request (ARQ)
- packet combining
- frequency domain methods
1. Introduction
Space-time (ST) multiplexing oriented multiple-input-multiple-output (MIMO) and hybrid-automatic repeat request (ARQ) protocols play a key role in the evolution of current wireless systems toward high data rate wireless broadband standards [1]. In ST multiplexing architectures, independent data streams are sent over multiple antennas to increase the transmission rate [2]. In hybrid-ARQ, erroneous data packets are kept in the receiver to help decode the retransmitted packet, using packet combining techniques (e.g., see [3] and references therein). Depending on the retransmitted information, hybrid-ARQ can be classified into Chase-type ARQ and incremental redundancy (IR). Chase-type ARQ is considered as the simplest hybrid-ARQ scheme where the data packet is entirely retransmitted. In the more sophisticated IR hybrid-ARQ scheme, retransmissions only carry portions of the data packet, this presents an efficient technique for increasing the system throughput while keeping the error performance acceptable. In this study, we propose advanced receiver schemes that can be only used for hybrid-ARQ with Chase combining. Combining schemes for IR hybrid-ARQ are out of the scope of the current article.
To support heterogeneous data rates in CDMA systems, multiple spreading codes can simultaneously be allocated to the same user if he requests a high data rate [4]. This method is often referred to as "multi-code transmission," and has been considered in the high speed packet access (HSPA) system [5]. In MIMO CDMA systems, multi-code transmission offers a spectrum efficiency that linearly increases in the order of the number of spreading codes and transmit antennas. This is achieved by assigning the same spreading code group to all transmit antennas. However, in severe frequency selective fading wireless channels, the performance of this scheme can dramatically deteriorate due to co-antenna interference (CAI) and inter-chip interference (ICI). This results in a large delay (due to multiple transmissions) when an ARQ protocol is used in the link layer. Motivated by this limitation, we investigate efficient hybrid- ARQ receiver schemes that allow to reduce the number of ARQ rounds required to correctly decode a data packet in MIMO CDMA ARQ systems with multi-code transmission.
Cyclic-prefix (CP) aided single carrier (SC) CDMA transmission with chip-level minimum mean square error (MMSE)-based frequency domain equalization (FDE) has been introduced in [6]. It is a transceiver scheme that allows to achieve attractive performance with affordable computational complexity cost. Turbo MMSE-FDE for CP-CDMA has then been proposed to cope with severe ICI [7]. In [8], MMSE FDE has been applied to perform packet combining for multi-code CP-CDMA systems with ARQ operating over severe frequency selective fading channels. It has recently been demonstrated that ARQ presents an important source of diversity in MIMO systems [9]. Interestingly, it has been shown in [9] that for both short and long-term static^{a} ARQ channel dynamics, multiple transmissions improve the diversity order of the corresponding MIMO ARQ channel. The case of block-fading MIMO ARQ, i.e., multiple fading blocks are observed within the same ARQ round, has been reported in [10]. Information rates and turbo MMSE packet combining strategies for frequency selective fading MIMO ARQ channel have been investigated in [11]. Turbo MMSE packet combining for broadband MIMO ARQ systems with co-channel interference (CCI) has been reported in [12, 13] using time and frequency domain combining methods, respectively.
In this article, we investigate an efficient turbo receiver schemes for single user multi-code CDMA systems with chase-type ARQ operating over a broadband MIMO channel. We introduce two packet combining where all ARQ rounds are used jointly to decode the data packet. The first packet combining scheme, referred to as chip-level packet combining scheme, is an extension of the combining approach introduced in [11, 13] to the case of multi-antenna multi-code CDMA systems. In this combining scheme, we exploit the fact that both the CP chip-word and data packet are retransmitted at each ARQ round. This allows us to view each transmission as a group of virtual receive antennas and perform combining of multiple transmissions jointly with chip-level soft MMSE FDE. In the second combining scheme, referred to as symbol-level packet combining scheme, frequency domain soft MMSE is performed separately for each transmission then the demapping is jointly performed with packet combining. In this article, our main contribution is to extend the two combining strategies to the case of multi-antenna multi-code CDMA systems and propose a low complexity combining approach based on recursive implementation strategy. Moreover, we present a comparative study of both combining schemes, in term of implementation cost and performance evaluation. Using complexity analysis and performance evaluation, we demonstrate that the choice of the best combining technique depends on the system configuration.
Throughout this article, (.)^{⊤} and (.)^{H} denote the transpose and transpose conjugate of the argument, respectively. diag {x} ∈ ℂ^{ n × n }and $\mathsf{\text{diag}}\left\{{\mathbf{X}}_{1},\dots ,{\mathbf{X}}_{m}\right\}\in {\u2102}^{m{n}_{1}\times m{n}_{2}}$ denote the diagonal matrix and block diagonal matrix constructed from x ∈ ℂ^{ n }and ${\mathbf{X}}_{1},\dots ,{\mathbf{X}}_{m}\in {\u2102}^{{n}_{1}\times {n}_{2}}$, respectively. For x ∈ ℂ^{ TN }, x_{ f }denotes the discrete Fourier transform (DFT) of x, i.e., x_{ f }= U_{ T, N }x, with U_{ T, N }= U_{ T }⊗ I_{ N }, where I_{ N }is the N × N identity matrix, U_{ T }is a unitary T × T matrix whose (m, n)th element is ${\left({\mathbf{U}}_{T}\right)}_{m,n}=\frac{1}{\sqrt{T}}{e}^{-j\left(2\pi mn/T\right)}$, $j=\sqrt{-1}$, and ⊗ denotes the Kronecker product. The rest of this article has the following structure. In Section 2, we present the CP-CDMA MIMO ARQ transmission scheme then provide its corresponding communication model. We also present the architecture of a space-time turbo receiver with no packet combiner. In Section 3, we derive the two iterative soft MMSE FDE-aided packet combining schemes we propose in this article. Section 4, analyzes the complexity and memory size required by both schemes, then focuses on the comparison of their block error rate (BLER) and throughput performances. The article is concluded in Section 5.
2. System description
2.1. CP-CDMA MIMO ARQ transmission scheme
2.2. Communication model
where ${h}_{r,t,l}^{\left(k\right)}$ is the (r, t)th element of ${\mathbf{H}}_{l}^{\left(k\right)}$.
where vectors ${\mathbf{x}}_{f}\triangleq {\left[{\mathbf{x}}_{{f}_{0}}^{\top},\dots ,{\mathbf{x}}_{{f}_{{T}_{c}-1}}^{\top}\right]}^{\top}\in {\u2102}^{{T}_{c}{N}_{T}\times 1}$ and ${\mathbf{n}}_{f}^{\left(k\right)}\triangleq {\left[{\mathbf{n}}_{{f}_{0}}^{{\left(k\right)}^{\top}},\dots ,{\mathbf{n}}_{{f}_{{T}_{c}-1}}^{{\left(k\right)}^{\top}}\right]}^{\top}$ group the DFTs of transmitted chips and thermal noise at round k, respectively. The channel frequency response (CFR) matrix Λ^{(k)}
2.3. Turbo receiver with no packet combining for multi-antenna multi-code CP-CDMA
The conventional receiver for multi-antenna multi-code CP-CDMA, presented in this section, makes use of ARQ principle with no packet combining at the receiver side. At transmission k, the receiver performs soft equalization and computes the extrinsic log-likelihood ratio (LLR) about coded and interleaved bits with the aid of the communication model (12), and the a priori information generated by the soft-input-soft-output (SISO) decoder at the previous iteration. Interference cancelation is performed starting from the first iteration. In fact, this conventional receiver makes use of prior LLRs of coded and interleaved bits generated by the SISO decoder during the last iteration of previous transmission k - 1. This idea was initially introduced in [14] in the context of single antenna coded systems with ARQ.
where ${\mathbf{D}}_{i}^{\left(k\right)}={\mathbf{\Lambda}}_{i}^{{\left(k\right)}^{H}}{\mathbf{\Lambda}}_{i}^{\left(k\right)}$ and $\stackrel{\u0303}{\Xi}$ is the N_{ T } × N_{ T } unconditional covariance of transmitted chips, and is computed as the time average of conditional covariance matrices ${\Xi}_{i}\triangleq \mathsf{\text{diag}}\left\{{\sigma}_{1,i}^{2},\dots ,{\sigma}_{{N}_{T},i}^{2}\right\}$, where ${\sigma}_{t,i}^{2}$ is the conditional variance of x_{ t, i }.
3. Iterative receivers for CP-CDMA MIMO ARQ
In this section, we present two efficient algorithms for performing turbo packet combining for CP-CDMA MIMO ARQ systems: (i) chip-level turbo packet combining, and (ii) symbol-level turbo packet combining. In both schemes, signals received in multiple ARQ rounds are processed using soft MMSE FDE.
3.1. Chip-level turbo packet combining
Summary of the chip-level turbo combining algorithm
0. | Initialization |
---|---|
Initialize ${\underset{\xaf}{\stackrel{\u0303}{\mathbf{y}}}}_{f}^{\left(0\right)}$ and ${\underset{\xaf}{\mathbf{D}}}_{i}^{\left(0\right)}$ with ${\mathbf{0}}_{{T}_{c}{N}_{T}\times 1}$ and ${\mathbf{0}}_{{N}_{T}\times {N}_{T}}$, respectively. | |
1. | Combining at round k |
1.1. Update ${\underset{\xaf}{\stackrel{\u0303}{\mathbf{y}}}}_{f}^{\left(k\right)}$and ${\underset{\xaf}{\mathbf{D}}}_{i}^{\left(k\right)}$according to (25) and (26). | |
1.2. At each iteration, | |
1.2.1 Compute the forward and backward filters using (28) and (29). | |
1.2.2 Compute the MMSE estimate of x_{ f }using (27). | |
1.2.3 Compute extrinsic LLRs ${\varphi}_{t,j,m}^{\left(e\right)}$according to (17). | |
1.3. end 1.2. |
3.2. Symbol-level turbo packet combining
where ${\mathbf{\xi}}_{t,j}^{\left(k\right)}\left(s\right)=\left|{\mathbf{r}}_{t,j}^{\left(k\right)}-{\mathbf{g}}_{t,j}^{\left(k\right)}s\right|{\mathit{\theta}}_{t,j}^{{\left(k\right)}^{-1}}$, with ${\mathbf{g}}_{t,j}^{\left(k\right)}={\left[{g}_{t,j}^{\left(1\right)},\dots ,{g}_{t,j}^{\left(k\right)}\right]}^{\mathsf{\text{T}}}$ is the equivalent channel gain and ${\mathit{\theta}}_{t,j}^{\left(k\right)}=\mathsf{\text{diag}}\left\{{\theta}_{t,j}^{\left(1\right)},\dots ,{\theta}_{t,j}^{\left(k\right)}\right\}$ is the residual interference covariance matrix corresponding to transmissions 1, . . ., k.
Implementation Aspects
Summary of the symbol-level turbo combining algorithm
0. | Initialization: |
---|---|
Initialize ${\stackrel{\u0304}{\mathbf{\xi}}}_{t,j}^{\left(0\right)}\left(s\right)$with 0. | |
1. | Combining at round k |
1.1. At each iteration, | |
1.1.1 Compute the forward and backward filters using (14) and (15). | |
1.1.2 Compute the MMSE estimate on x_{ f }using (13). | |
1.1.3 Update ${\stackrel{\u0304}{\mathbf{\xi}}}_{t,j}^{\left(k\right)}\left(s\right)$according to (31). | |
1.1.4 Compute extrinsic LLRs ${\varphi}_{t,j,m}^{\left(e\right)}$using (30). | |
1.3. end 1.1. |
4. Complexity and performance analysis
4.1. Complexity evaluation
In this section, we briefly analyze both the computational cost and memory requirements of the proposed packet combining schemes. First, note that both combining schemes have identical implementations. The only difference comes from variable updates in steps Table 1(1.1.), and Table 2(1.1.3). Therefore, both techniques approximately have the same implementation cost. In the following, we focus on the number of arithmetic additions and memory required to perform recursions (25), (26), and (31).
summary of the maximum number of arithmetic additions, and memory size
Chip-Level Combining | Symbol-Level Combining | |
---|---|---|
Arithmetic Additions | 2T_{ c } N_{ T } (K - 1) (N_{ T } + 1) | T_{ s } N_{ T } (K - 1) N_{iter} 2^{ M } |
Memory | 2T_{ c }N_{ T } (N_{ T } + 1) | T_{ s }N_{ T }2^{ M } |
4.2. Performance evaluation
In this section, we evaluate the performance of the proposed multi-antenna multi-code CP-CDMA receivers in term of BLER and Throughput η. Following [17], we define the throughput as $\eta \triangleq \frac{\mathbb{E}\left[\mathcal{R}\right]}{\mathbb{E}\left[\mathcal{K}\right]}$, where $\mathcal{R}$ is a random variable (RV) that takes R when the packet is correctly received or zero when the packet is erroneous after K ARQ rounds. $\mathcal{K}$ is a RV that denotes the number of rounds used for transmitting one data packet.
The system used for the evaluation has N_{ T } = 2 transmit antennas, N_{ R } = {1, 2} receive antennas, spreading factor N = 16, Quadrature Phase Shift Keying (QPSK) modulation and 16 states convolutional encoder with polynomial generators (35, 23)_{8}. The length of the coded frame is 1024 bits including tails. We assume short-term static ARQ MIMO channel that has L = 10 chip spaced paths with equally distributed power. The CP length is T_{ C P } = 10. We employ the Max-Log-MAP Version of the MAP decoding algorithm [18] for SISO decoding. The maximum number of transmissions is set to K = 3 and the E_{ c }/N_{0} ratio appearing in all figures is the SNR per chip per receive antenna. We have noticed via simulations that no remarkable performance improvement is obtained when the number of iterations is greater than three. The turbo process is therefore stopped after three iterations for each transmission. The matched filter bound (MFB)^{b} is used to evaluate the diversity achievement of the proposed algorithms. We also use the conventional LLR-level packet combining^{c} as a reference to evaluate the performance gain provided by the proposed combining strategies. In term of complexity, the number of arithmetic additions is relatively insignificant compared with the whole computational cost of the receiver. Therefore, we consider the memory requirements as the major parameter to take into account to evaluate the studied combining schemes in term of implementation cost.
5. Conclusions
In this article, efficient turbo receiver schemes for single user multi-code CP-CDMA transmission with ARQ operating over a broadband MIMO channel were introduced. The key idea of the proposed schemes is to exploit the diversity among all transmissions with a very low cost by introducing new variables recursively computed. Two packet combining algorithms were presented. The first algorithm consists in performing packet combining jointly with frequency domain chip level turbo equalization. The second proposed algorithm performs packet combining jointly with turbo demapping. Complexity evaluation showed that each combining scheme could be the most attractive in term of implementation cost depending on the number of transmit antennas, the factor $\frac{N}{C}$, the constellation length, and the number of turbo iterations. Moreover, simulations demonstrated that both schemes approximately have similar performance for balanced (same number of transmit and receive antennas) MIMO configurations. Hence, for receiver devices that cannot afford large complexity and storage requirements, it may be preferable to use symbol-level combining instead of chip-level combining. In the case of unbalanced configurations (more transmit than receive antennas), we demonstrated that chip-level combining clearly outperforms symbol-level combining. In that case, system configuration should be considered before deciding on the best combining scheme.
Endnotes
^{a}The short-term static ARQ channel dynamic corresponds to the case where two consecutive ARQ rounds observe independent channel realizations. In long-term static channels, all ARQ rounds corresponding to the same data packet observe the same channel realization.
^{b}The MFB curves are obtained for each transmission assuming perfect ICI cancelation and maximum ratio combining (MRC) of all time, space, multipath, and delay diversity branches. ^{c}In LLR-level combining, turbo equalization is separately performed for each transmission, and right before SISO decoding, extrinsic LLRs, at transmission k, are simply added together with those obtained at the last iteration of previous transmission k - 1.
Declarations
Authors’ Affiliations
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