Training design for precoded BICMMIMO systems in blockfading channels
 Zohreh Andalibi^{1, 2}Email author,
 Ha H Nguyen^{1, 2} and
 Joseph E Salt^{1, 2}
DOI: 10.1186/16871499201280
© Andalibi et al; licensee Springer. 2012
Received: 7 August 2011
Accepted: 4 March 2012
Published: 4 March 2012
Abstract
In order to improve bandwidth efficiency and error performance, a new training scheme is proposed for bitinterleavedcoded modulation in multipleinput multipleoutput (BICMMIMO) systems. Typically, in a blockfading channel, the training overhead used for obtaining channel knowledge is proportional to a power of 2 of the number of transmit antennas. However, this overhead can be reduced by embedding pilot symbols within data symbols before precoding. The values, positions, and the number of pilot symbols are found by minimizing the CramerRao bound on the channel estimation error. Computer simulations are presented to demonstrate the advantage of the proposed scheme over other training methods, in terms of both the meansquareerror of the channel estimation and the system's frameerrorrate.
Keywords
BICMMIMO block fading channel estimation training design pilot symbols CramerRao bound iterative receiver1 Introduction
The pioneering work on multipleinput multipleoutput (MIMO) systems [1] shows that a MIMO system can provide a multiplexing gain and accordingly high spectral efficiency over slow fading channels. On the other hand, to achieve a high diversity order, spacetime transmission techniques can be implemented at the transmitter [2, 3]. To achieve both high diversity order and coding gain in coded modulation systems, the concept of spacetime transmission has also been applied [4, 5]. In such systems, spacetime transmission is typically implemented using a linear spacetime matrix, or equivalently a linear precoder, so that a single modulation symbol is efficiently transmitted across multiple transmit antennas. Among many research works on precoder design for coded modulation systems with multiple antennas, the design that considers all the relevant components of the transmitter, namely precoding, modulation, and interleaver, can be found in [5–7]. Specifically, a fullrate precoder with any size and for any number of transmit antennas is designed in [6] to maximize the achievable diversity order and coding gain in MIMO blockfading channels.
It is shown in [6] that the maximum achievable diversity order can be realized by an iterative receiver that employs a softinput softoutput detector [5] and under the assumption of having the perfect channel state information (CSI) at the receiver. In practice, however, CSI has to be estimated using a channel estimator and it is never perfect. Two types of channel estimators have been used for MIMO blockfading channels in coded modulation systems, i.e., trainingbased and semiblind channel estimators [8, 9]. In both types of channel estimators, known signals are used to estimate the CSI at the first iteration of the iterative receiver.
Conventionally, for blockfading channels, known signals or the training sequence is included at the beginning of each data block, which is called timemultiplexed training or pilot symbolassisted modulation (PSAM) scheme [10]. This scheme however reduces bandwidth efficiency of MIMO systems, since the amount of training overhead needed is at least a power of 2 of the number of transmit antennas [11] to ensure the identifiability of the MIMO channel. A straightforward application of the PSAM scheme to a BICMMIMO system would be timemultiplex data information with the training information after the precoder.
As an alternative to the above conventional PSAM scheme, a potential benefit can be sought by timemultiplexing data information with the training information before the precoder in the transmitter. This new approach shall reduce the required training overhead compared to the conventional PSAM, since the transmitted training symbols are spread over more time periods; thanks to the precoder. This approach shall be referred to as precoded PSAM (PPSAM). Investigating power and time allocations of the training symbols in PPSAM scheme is the main objective of this article.
Moreover, by multiplexing the training sequence before precoder, training symbols can be exploited in both the initialization and iteration phases of the iterative channel estimation process. This is different from a conventional iterative channel estimator using PSAM scheme, in which training sequence is only used at the initialization phase. A natural question is whether the optimal training design for the initialization phase using PPSAM scheme is still optimal for subsequent iterations of an iterative channel estimator. On the one hand, the channel estimation error at the initialization phase translates to an SNR shift in the BER performance [8]. On the other hand, the channel estimation error from the last iteration of the iterative estimator has a strong impact on the error floor of the BER performance [12]. Therefore, optimal training sequence should be designed carefully that considers both initialization and iteration phases.
One of different criteria that have been used to design training sequences is the minimization of the CramerRao bound (CRB) of the channel estimation error [10]. This criterion shall be used in this article due to two main reasons. First, it is directly related to the channel estimation error. Second, since the CRB is a lower bound on the meansquarederror (MSE) of any unbiased estimator, designing training sequences using this criterion would be applicable to many estimation algorithms. Other design criteria, such as maximizing the channel capacity [8] and minimizing the outage probability [13], are based on some specific channel estimation algorithms.
The article is organized as follows. The system model of BICMMIMO is presented in Section 2. In Section 3 a lower bound on the MSE of the channel estimator is obtained and the training sequence is designed by minimizing this bound. Section 4 provides numerical results and comparisons. Section 5 concludes the article.
2 System model
Every group of N supersymbols is then spread over N time periods using a linear precoder G. The Nn_{t} × Nn_{t} matrix G multiplies a vector of Nn_{t} QAM symbols at the precoder input, and generates Nn_{t} symbols to be transmitted over n_{t} antennas, over N time periods.
where ${\mathbf{y}}_{k}=\left[{y}_{\left(k1\right)N{n}_{\mathsf{\text{r}}}+1},\phantom{\rule{2.77695pt}{0ex}}{y}_{\left(k1\right)N{n}_{\mathsf{\text{r}}}+2},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{y}_{\left(k1\right)N{n}_{\mathsf{\text{r}}}+N{n}_{\mathsf{\text{r}}}}\right]$ is the received vector at the k th precoding time period and w_{ k } is the noise vector with size 1 × Nn_{r} whose components are i.i.d zeromean circularly symmetric Gaussian random variables with variance N_{0}. It is noted from (2) that although both data and pilot symbols are precoded, the part of the precoder that multiplies the pilot symbols depends on the positions of the pilot symbols in x _{ k }. Equivalently, the design of the pilot symbols is governed by the properties of the precoder used. Since this study adopts the transmission framework and precoder design in [6], it is useful to review the properties of the precoder proposed in [6].
In general, the properties of the precoder in [6] are established by the maximumlikelihood decoding analysis and an assumption of ideal channel interleaving. Specifically, this linear precoder which achieves full diversity order and maximum coding gain satisfies the following two conditions:

A genie condition, which guarantees orthogonal and equal norm subrows in the linear precoding matrix. Each subrow has size n_{t} in a precoding matrix with size Nn_{t}× Nn_{t}.

Dispersive nucleo algebraic (DNA) condition, which is based on Proposition 2 in [6], forces null and orthogonal nucleotides with size s' = N/n_{ s }. Nucleotides refer to subparts of subrows with size s'.
where Φ^{[i ] [j]}is the i th subrow of the j th row of Φ with size 1 × s', I_{ n } is an identity matrix with size n × n and ⊗ denotes the Kronecker product.
The properties that shall be useful for the problem considered in this article, which are implied directly from the genie and DNA conditions, are ΦΦ^{ H } = I_{ Ns ' } and ${\mathbf{\Phi}}^{\left[i\right]\phantom{\rule{0.3em}{0ex}}\left[t\right]}{\left({\mathbf{\Phi}}^{\left[j\right]\phantom{\rule{0.3em}{0ex}}\left[t\right]}\right)}^{H}=\frac{1}{N}\delta \left(ij\right)$. It is also useful to point out that each component of Φ has an exponential form with a scaling factor of $\frac{1}{\sqrt{N\phantom{\rule{0.3em}{0ex}}{s}^{\prime}}}$.
The iterative receiver is also shown in Figure 1. The channel estimator produces an estimate of the channel using the minimum MSE (MMSE) criterion based on the training sequence. Details about channel estimation with the proposed method of inserting training sequence shall be given in Section 3. After channel estimation is performed using the training signal, the softinput softoutput demodulator uses the MMSE criterion to demodulate the data. The softoutput MMSE demodulator computes the extrinsic information for the interleaved bits, $\left\{{\mathrm{\Lambda}}_{\mathsf{\text{ext}}}^{\left({\stackrel{\u0303}{c}}_{l}\right)}\right\}$, from the received symbols. To obtain Λvalues, the demodulator exploits the a priori information of the coded bits coming from the decoder, $\left\{{\mathrm{\Lambda}}_{\mathsf{\text{ap}}}^{\left({\stackrel{\u0303}{c}}_{l}\right)}\right\}$, and the channel estimate ${\widehat{\mathbf{H}}}_{k}$. In the first iteration, the demodulator assumes that the a priori Λvalues are zero, except for the pilot symbols. For the corresponding bits of the pilot symbols, the demodulator uses a large number, say ± 100 as their a priori Λvalues. The deinterleaved outputs, i.e., $\left\{{\mathrm{\Lambda}}_{\mathsf{\text{ap}}}^{\left({c}_{l}\right)}\right\}$, become the a priori Λvalues used in the channel decoder shown in Figure 1 after removing the information of pilot symbols. The channel decoder uses the maximum a posteriori probability (MAP) algorithm to compute the extrinsic Λvalues $\left\{{\mathrm{\Lambda}}_{\mathsf{\text{ext}}}^{\left({c}_{l}\right)}\right\}$. for all coded bits, which are used again in the next iteration in the demodulator. In subsequent iterations, soft information from the decoder is used to improve the performance of the channel estimator. The detailed operation of the iterative channel estimator is discussed in the following sections.
3 Training design and channel estimator
where y^{[i,t]}= y^{[(t 1)s'+i]}represents the ((t  1)s' + i)th received symbol during N time periods, with size n_{r} × 1. Moreover, h^{[t]}is the column vector formed by vertically stacking the columns of an n_{t} × n_{r} channel realization matrix H^{[t]}and x^{[τ]}'s are constructed by splitting x in Ns' subvectors with size 1 × n_{t}/s'. In the following, we call these subvectors x^{[τ]}'s nucleo symbols.
It is quite obvious from (4) that, to have all the received supersymbols, y^{[i,t]}, contain training information, there should be at least one pilot nucleo (i.e., n_{t}/s' pilot symbols) in each group of Ns' nucleos to be precoded.
where ${\mathcal{I}}_{\mathsf{\text{d}}}$ and ${\mathcal{I}}_{\mathsf{\text{p}}}$ are sets of indexes from {1, . . . , Ns'}, that are assigned to data and pilot nucleos, respectively, and $\left{\mathcal{I}}_{\mathsf{\text{d}}}\right+\left{\mathcal{I}}_{\mathsf{\text{p}}}\right=\left(N\phantom{\rule{0.3em}{0ex}}{s}^{\prime}{n}_{\mathsf{\text{p}}}\right)+{n}_{\mathsf{\text{p}}}=N\phantom{\rule{0.3em}{0ex}}{s}^{\prime}$. Note that the subscripts "d" and "p" are used to differentiate between data and pilot nucleos. For convenience, the notations ${\mathbf{\Phi}}_{\mathsf{\text{p}}}^{\left[i,t\right]\phantom{\rule{0.3em}{0ex}}\left[\tau \right]}$ and ${\mathbf{\Phi}}_{\mathsf{\text{d}}}^{\left[i,t\right]\phantom{\rule{0.3em}{0ex}}\left[\tau \right]}$ are used to refer to subrows of Φ that are multiplied by pilot and data nucloes, i.e., ${\mathbf{x}}_{\mathsf{\text{p}}}^{\left[\tau \right]}$ and ${\mathbf{x}}_{\mathsf{\text{d}}}^{\left[\tau \right]}$, respectively. Furthermore, in the following the notation T^{[i,t]}is used for ${\mathbf{I}}_{{n}_{\mathsf{\text{r}}}}\otimes \left({\sum}_{\tau \in {\mathcal{I}}_{\mathsf{\text{p}}}}{\mathbf{x}}_{\mathsf{\text{p}}}^{\left[\tau \right]}\otimes {\mathbf{\Phi}}_{\mathsf{\text{p}}}^{\left[i,t\right]\phantom{\rule{0.3em}{0ex}}\left[\tau \right]}\right)$.
The derivation of FIM is given in the next section. Pilot symbols are exploited at the initialization phase and in subsequent iterations considering the special structure of the training sequence. In general, training design can be investigated for these two phases separately. However, for the precoder adopted in this article, the optimal training design obtained for the initialization phase turns out to also be optimal for the iteration phase. Nevertheless, the optimal numbers of pilot nucleos in these two phases of channel estimation are not the same.
3.1 Fisher information matrix
where $\mathcal{H}={\left({\mathbf{H}}^{\left[1\right]}\right)}^{T}$ and ${\mathbf{\Phi}}_{d}^{\left[i\right]}$ is the i th submatrix of Φ with size (N s'  n_{p}) × s' that is assigned to data symbols.
where e_{ l } is an n_{ t }n_{r}× 1 null vector with a single element 1 at position l.
where ${\mathbf{A}}^{\left[i\right]}\equiv \left({\sigma}_{x}^{2}{I}_{{n}_{t}/{s}^{\prime}}\otimes \left({\left({\mathbf{\Phi}}_{d}^{\left[i\right]}\right)}^{T}{\left({\mathbf{\Phi}}_{d}^{\left[i\right]}\right)}^{*}\right)\right)$.
where ${\mathbf{X}}_{\mathsf{\text{p}}}^{\left[i\right]}={\sum}_{\tau \in {\mathcal{I}}_{\mathsf{\text{p}}}}{\mathbf{x}}_{\mathsf{\text{p}}}^{\left[\tau \right]}\otimes {\mathbf{\Phi}}_{\mathsf{\text{p}}}^{\left[i\right]\left[\tau \right]}$, and ${\mathbf{Q}}_{i}={\left({\mathbf{A}}^{\left[i\right]}\right)}^{T}{\mathcal{H}}^{T}{\mathbf{R}}_{i}^{1}{\mathcal{H}}^{*}{\left({\mathbf{A}}^{\left[i\right]}\right)}^{*}$.
In general, the second term in (11) depends on ${\mathcal{I}}_{\mathsf{\text{p}}}$, but not on the training symbols, whereas the first term depends on both x_{p} and ${\mathcal{I}}_{\mathsf{\text{p}}}$. Although both terms depend on n_{p}, how FIM^{init} depends on n_{p} is determined by ${\mathcal{I}}_{\mathsf{\text{p}}}$. Therefore, in the following ${\mathcal{I}}_{\mathsf{\text{p}}}$ and x_{p} are first optimized. Then n_{p} is determined for the optimized ${\mathcal{I}}_{\mathsf{\text{p}}}$.
3.2 Optimization of training symbols and their positions
where ${\left({\mathbf{x}}_{\mathsf{\text{p}}}^{\left[\tau \right]}\right)}_{j}$ is the j th pilot symbol in the τ th pilot nucleo and the FIM is given in (11).
Because of the shiftinvariant property of (15) with respect to τ, τ can be any value in the set {1, 2, . . . , Ns'}. For simplicity, set τ = 1 and the superscript τ is omitted. Using the fact that if X > 0 then tr (X^{1}) ≥ ∑ _{ i } 1/(X)_{ i },_{ i }, the original optimization problem is simplified by minimizing the lower bound of the objective function.
On the other hand, ${\sum}_{i=1}^{{s}^{\prime}}\left({\left({\mathbf{\Phi}}_{\mathsf{\text{p}}}^{\left[i\right]\phantom{\rule{0.3em}{0ex}}\left[\tau \right]}\right)}^{H}{\mathbf{\Phi}}_{\mathsf{\text{p}}}^{\left[i\right]\left[\tau \right]}\right)=\frac{1}{N}{\mathsf{\text{I}}}_{{s}^{\prime}}$, ${\sum}_{i=1}^{{s}^{\prime}}{\left({\mathbf{A}}^{\left[i\right]}\right)}^{T}{\left({\mathbf{A}}^{\left[i\right]}\right)}^{*}=\frac{{\sigma}_{x}^{4}}{{s}^{\prime}}\left({\left(\frac{N{s}^{\prime}1}{N}\right)}^{2}+{\left(\frac{1}{N}\right)}^{2}\right){\mathbf{I}}_{{n}_{\mathsf{\text{t}}}}$ and the constraint is $\frac{{s}^{\prime}}{N}{\mathbf{x}}_{\mathsf{\text{p}}}{\mathbf{x}}_{\mathsf{\text{p}}}^{H}=\frac{{s}^{\prime}}{N}{\sum}_{j=1}^{{n}_{t}/{s}^{\prime}}{\left({\mathbf{x}}_{\mathsf{\text{p}}}\right)}_{j}{}^{2}$. Therefore, it is not hard to see that the solution of the simplified optimization problem is ${\left({\mathbf{x}}_{\mathsf{\text{p}}}\right)}_{1}{}^{2}=\phantom{\rule{2.77695pt}{0ex}}{\left({\mathbf{x}}_{\mathsf{\text{p}}}\right)}_{2}{}^{2}=\cdots =\phantom{\rule{2.77695pt}{0ex}}{\left({\mathbf{x}}_{\mathsf{\text{p}}}\right)}_{{n}_{t}/{s}^{\prime}}{}^{2}=\frac{N{P}_{t}}{{n}_{t}}$. It means that all pilot symbols should have the same power. For example, one can select corner points of the QAM constellations for the training symbols.
Case 2 (n_{p} ≥ 2): In this case there are two options for the placements of pilot nucleos. The first option is to group all pilot nucleos in one single cluster and the second option is to spread pilot nucleos. It can be shown that the CRB is invariant with respect to a shift of the placements of pilot nucleos in both options. Therefore, it suffices to select one cluster or one spread placement. However, the precoder has been designed such that the softoutput demodulator works with uncorrelated inputs and putting pilot nucleos between data nucleos may violate this condition. That condition is satisfied when A^{[i]}has a diagonal form. The implication of this property is to place pilot nucloes equispaced in x_{ k } and ${\mathcal{I}}_{\mathsf{\text{p}}}=\left\{{i}_{0}+kn;\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}k=0,\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{n}_{\mathsf{\text{p}}}1\right\}$, where n = Ns'/n_{p} and i_{0} ∈ {1, . . . , n}, which leads to ${\mathbf{A}}^{\left[i\right]}={\sigma}_{x}^{2}\frac{N{s}^{\prime}{n}_{\mathsf{\text{p}}}}{N{s}^{\prime}}{\mathbf{I}}_{{s}^{\prime}}$. In this selection it is supposed that n_{p} is divisible by Ns'.
the solution is given by ${\sum}_{\tau}{\left({\mathbf{x}}_{\mathsf{\text{p}}}^{\left[\tau \right]}\right)}_{j}{}^{2}=\frac{N{P}_{t}}{{n}_{t}};\phantom{\rule{2.77695pt}{0ex}}j=1,\dots ,{n}_{\mathsf{\text{t}}}/{s}^{\prime}$.
Now consider the training design for the iteration phase. Observe that all the terms in (12) have diagonal forms with equal diagonal elements, except ${\sum}_{\tau \in {\mathcal{I}}_{\mathsf{\text{p}}}}{\sum}_{{\tau}^{\prime}\in {\mathcal{I}}_{\mathsf{\text{p}}}}{\left({\mathbf{x}}_{\mathsf{\text{p}}}^{\left[\tau \right]}\right)}^{H}{\mathbf{x}}_{\mathsf{\text{p}}}^{\left[{\tau}^{\prime}\right]}\otimes {\sum}_{i=1}^{{s}^{\prime}}{\left({\mathbf{\Phi}}_{\mathsf{\text{p}}}^{\left[i\right]\phantom{\rule{0.3em}{0ex}}\left[\tau \right]}\right)}^{H}{\mathbf{\Phi}}_{\mathsf{\text{p}}}^{\left[i\right]\phantom{\rule{0.3em}{0ex}}\left[{\tau}^{\prime}\right]}$. This means that the solution of problem (14), but with $\mathsf{\text{FI}}{\mathsf{\text{M}}}^{\mathsf{\text{init}}}\left({n}_{\mathsf{\text{p}}},\phantom{\rule{2.77695pt}{0ex}}{\mathbf{x}}_{\mathsf{\text{p}}},\phantom{\rule{2.77695pt}{0ex}}{\mathcal{I}}_{\mathsf{\text{p}}}\right)$ replaced by $\mathsf{\text{FI}}{\mathsf{\text{M}}}^{\mathsf{\text{iter}}}\left({n}_{\mathsf{\text{p}}},\phantom{\rule{2.77695pt}{0ex}}{\mathbf{x}}_{\mathsf{\text{p}}},\phantom{\rule{2.77695pt}{0ex}}{\mathcal{I}}_{\mathsf{\text{p}}}\right)$, is to choose equal diagonal elements for this term. Therefore, the training sequence designed for the initialization is also optimal for the iteration phase.
In summary, by selecting pilot nucleos such that the sum of the powers of their corresponding pilot symbols with the same indexes are equal, the bound on CRB is minimized. The above condition can give different selections for pilot symbols from a twodimensional constellation. It should be pointed out, however, that not all selections guarantee that pilot symbols belong to standard QAM constellations.
3.3 Determination of the number of the training symbols
For blockfading channels, the number of pilot nucleos, i.e., n_{p}, should be as small as possible that meets the power constraint. Using a larger value for n_{p} wastes bandwidth and does not change the system performance.
Optimum n_{p} for several sets of parameters {n_{t}, n_{r}, N}
n _{t}  n _{r}  N  n _{p} 

2  2  2  1 
4  2  2  2 
4  2  4  4 
4  4  2  1 
4  4  4  1 
3.4 Channel estimation
For the channel estimation task, one can view the received vector during one block length as ${\mathit{\phi}}^{\left[t\right]}={\left[{\left({\mathbf{y}}^{\left[1,t\right]}\right)}^{T},\phantom{\rule{2.77695pt}{0ex}}{\left({\mathbf{y}}^{\left[2,t\right]}\right)}^{T},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{\left({\mathbf{y}}^{\left[{s}^{\prime},t\right]}\right)}^{T}\right]}^{T}$.
where T = [(T^{1}) ^{ T } , . . . , (T^{[s']}) ^{ T } ] ^{ T } .
In the subsequent iterations, soft information from the decoder is used to improve the performance of the channel estimator. The channel estimator uses such information to compute new estimates of the channel coefficients using expected values of the data symbols. Therefore, the interleaved $\left\{{\mathrm{\Lambda}}_{\mathsf{\text{ext}}}^{\left({c}_{l}\right)}\right\}$ from the decoder are fed back to the estimator to calculate the expected values of the data symbols, i.e., E{x_{d}}. The entries of E{x_{d}} are calculated using $\left\{{\mathrm{\Lambda}}_{\mathsf{\text{ap}}}^{\left({\stackrel{\u0303}{c}}_{l}\right)}\right\}$ at each iteration by E{(x_{d})_{ i }} = ∑_{ x }_{∈Ω}x · p((x_{d}) _{ i } = x). The detailed derivations of the probability p((x_{d}) _{ i } = x) from Λvalues are given in [15] (note that the calculation depends on the mapping rule in Ω).
To verify the results obtained in this section, Section 4 compares numerically the MSE performance of the above channel estimator obtained with the optimal and suboptimal training sequences.
4 Illustrative results
In this section, the frameerrorrate (FER) and MSE performances of BICMMIMO systems using a MMSE iterative channel estimator are presented. The spacetime precoder is the DNAcyclo precoder that satisfies the properties outlined in Section 2. We consider quadrature phaseshift keying (QPSK) modulation with Gray mapping.
and when the setting for N, n_{ s }, n_{ p } and P_{ t } in Figure 3 are used. The channel is generated randomly and is assumed to be Rayleigh distributed. For the purpose of comparison, the results for MSE performances of the optimal PPSAM, denoted by OPPSAM and the suboptimal PPSAM, denoted by SOPPSAM as well as the CRB are shown in Figure 4. For SOPPSAM, two pilot nucleos are inserted as one cluster in front of data nucleos in a symbol to be precoded. In contrast, in the case of OPPSAM, the optimized training sequence embeds the pilot nucleos at the first and third positions of Ns' = 4 positions for nucleos. The MSE curves show that the performance of the optimal scheme is better than the suboptimum scheme for the first iteration (i.e., initialization). In fact the MSE performance of the proposed scheme closely approaches the CRB at high E_{ b } /N_{0} after 5 iterations.
5 Conclusion
In this article, a new training design for a BICMMIMO system over a blockfading channel has been proposed. The design inserts pilot symbols into the data symbols before precoding. The new training sequence improves bandwidth efficiency as compared to the conventional PSAM scheme and can also be used by the demodulator in the receiver. In order to design the optimal training symbols and their positions, the CRB on the channel estimations at the initialization and at the iteration phases are minimized. Compared to PSAM, performance improvement achieved with the proposed training is about 1.5 dB at a FER level of 10^{2}.
Endnotes
^{a}In practice, since n_{ s } is typically an approximated value over some range and since N can be selected, such an assumption can be fulfilled. ^{b}Using the matrix inversion lemma, one has ${\mathbf{R}}_{i}^{1}={\left(\mathcal{H}{\mathbf{A}}^{\left[i\right]}{\mathcal{H}}^{H}+{N}_{0}{\mathbf{I}}_{{n}_{\mathsf{\text{r}}}}\right)}^{1}={N}_{0}^{1}{\mathbf{I}}_{{n}_{r}}+{N}_{0}^{2}\mathcal{H}{\mathbf{A}}^{\left[i\right]}{\mathcal{H}}^{H}{\left({I}_{{n}_{r}}+{N}_{0}^{1}\mathcal{H}{\mathbf{A}}^{\left[i\right]}{\mathcal{H}}^{H}\right)}^{1}$. Therefore, for high SNR, $E\left\{{\mathbf{R}}_{i}^{1}\right\}$ can be approximated by ${N}_{0}^{1}{\mathbf{I}}_{{n}_{r}}$.
Declarations
Authors’ Affiliations
References
 Caire G, Shamai S: On the achievable throughput of a multiantenna Gaussian broadcast channel. IEEE Trans Inf Theory 2003, 49(7):16911706. 10.1109/TIT.2003.813523MathSciNetView ArticleMATHGoogle Scholar
 Alamouti SM: A simple transmit diversity technique for wireless communications. IEEE J Sel Areas Commun 1998, 16(8):14511458. 10.1109/49.730453View ArticleGoogle Scholar
 Tarokh V, Seshadri N, Calderbank AR: Spacetime codes for high data rate wireless communication: performance criterion and code construction. IEEE Trans Inf Theory 1998, 44(2):744765. 10.1109/18.661517MathSciNetView ArticleMATHGoogle Scholar
 Boutros J, Viterbo E: Signal space diversity: a power and bandwidth eficient diversity technique for the Rayleigh fading channel. IEEE Trans Inf Theory 1998, 44(4):14531467. 10.1109/18.681321MathSciNetView ArticleMATHGoogle Scholar
 Boutros J, Gresset N, Brunel L: Turbo coding and decoding for multiple antenna channels. In International Symposium on Turbo Codes and Related Topics. Brest, France; 2003:18.Google Scholar
 Gresset N, Brunel L, Boutros J: Spacetime coding techniques with bitinterleaved coded modulations for MIMO blockfading channels. IEEE Trans Inf Theory 2008, 54(5):21562178.MathSciNetView ArticleMATHGoogle Scholar
 Gresset N, Boutros JJ, Brunel L: Optimal linear precoding for BICM over MIMO channels. In ISIT, 66. Chicago, IL; 2004.Google Scholar
 Coldrey M, Bohlin P: Trainingbased MIMO systems, Part I: performance comparison. IEEE Trans Signal Process 2007, 55(11):54645476.MathSciNetView ArticleGoogle Scholar
 Nicoli M, Ferrara S, Spagnolini U: Softiterative channel estimation: methods and performance analysis. IEEE Trans Signal Process 2007, 55(6):29933006.MathSciNetView ArticleGoogle Scholar
 Dong M, Tong L, Sadler BM: Optimal insertion of pilot symbols for transmissions over timevarying flat fading channels. IEEE Trans Signal Process 2004, 52(5):14031418. 10.1109/TSP.2004.826182MathSciNetView ArticleGoogle Scholar
 Taricco G, Biglieri E: Spacetime decoding with imperfect channel estimation. IEEE Trans Wirel Commun 2005, 4(4):18741888.View ArticleGoogle Scholar
 Huang Y, Ritcey JA: Joint iterative channel estimation and decoding for bitinterleaved coded modulation over correlated fading channels. IEEE Trans Wirel Commun 2005, 4(5):25492558.MathSciNetView ArticleGoogle Scholar
 Piantanida P, Sadough SM: On the outage capacity of a practical decoder accounting for channel estimation inaccuracies. IEEE Trans Commun 2009, 57(5):13411350.View ArticleGoogle Scholar
 Kay SM: Fundamentals of Statistical Signal Processing: Estimation Theory. PrenticeHall PTR, New Jersey; 1993.MATHGoogle Scholar
 Khalighi MA, Boutros JJ: Semiblind channel estimation using the EM algorithm in iterative MIMO APP detectors. IEEE Trans Wirel Commun 2006, 5(11):31653173.View ArticleGoogle Scholar
 Kraidy GM, Rossi P: Fulldiversity iterative MMSE receivers with spacetime precoders over blockfading MIMO channels. In Proc IEEE Int Conf Wireless Commun and Signal Processing. Suzhou; 2010:15.Google Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.