The advantages and applications of iterative channel estimation and decoding receivers have been extensively investigated in literature [1]–[3]. The advancement in wireless communication applications encourage the use of iterative channel estimation and decoding receivers because of the varying nature of the Rayleigh fading channel. Traditionally, in iterative receiver structures, the initial Channel State Information (CSI) are computed using the properties of the known pilot symbols. The quality of the CSI is improved after each iterative decoding using the feedback information from the decoder. Different iterative channel estimation and decoding receivers have been proposed over the years, where the focus of these receiver designs have been to deliver efficient and reliable data transmission over fading channels at low complexity. Most techniques on iterative channel estimation and decoding receivers use Turbo and Low Density Parity Check (LDPC) codes as the Forward Error Correction (FEC) scheme in the receiver structure [1], [2], [4]–[6]. In this paper, attention is centred on Reed-Solomon ($RS$
) codes as the underlying FEC scheme in the receiver structure. In addition, this paper focuses on symbol-wise $RS$
decoders.
The $RS$
codes discovered in [7], have been shown to meet the singleton bound, which makes the code maximum distance separable [8], [9]. Efficient soft decision $RS$
decoding algorithms include [14]–[17]; however, [16], [17] are bit-wise based $RS$
decoding algorithms. These $RS$
soft decision decoding algorithms have been shown to perform beyond the conventional $RS$
hard decision decoding algorithms such as [10]–[13]. Notwithstanding, the performance of $RS$
codes and other FEC schemes degrade over Rayleigh fading channels due to the varying nature of the channel. This therefore motivated the use of FEC decoding schemes in the form of joint iterative channel estimation and decoding receivers.
We provided a survey in [3] and proposed the first joint iterative channel estimation and $RS$
decoding receiver using the Koetter and Vardy ($RS$
-KV) soft decision decoder [15] as the FEC scheme in the receiver structure. The importance of deploying $RS$
codes in an iterative receiver structure was emphasized in [3]. Additionally, we divulged in [3] that $RS$
codes are suitable in iterative receiver structures as in the case with Turbo and LDPC codes. The main drawback of the joint iterative receiver structure in [3] is the computational delay and time complexity of the receiver, majorly due to the $RS$
-KV algorithm used in the receiver structure. The $RS$
-KV decoder applied to the Guruswami and Sudan algorithm (GS) [18] standing alone can provide a practical $\mathcal {O}(M^{2}n^{2}I^{4})$
complexity to the receiver structure, where $n$
is the codeword length and $I$
is the adjustable interpolation multiplicity specified in the $GS$
algorithm [19]. In this regard, we propose a low complexity iterative channel estimation and decoding receiver structure based on the $RS$
Parity-check Transformation Algorithm ($RS$
-PTA). This proposed $RS$
-PTA iterative receiver structure is a low complexity and high performance efficient modification of the $RS$
-KV iterative receiver structure that was proposed in [3].
The paper adopts $M$
-ary Quadrature Amplitude Modulation ($M$
-QAM) systems as the underlying modulation scheme, where $M=16$
. The CSI is derived using the Pilot Symbol Assisted Modulation (PSAM) Linear Minimum Mean Square Error (LMMSE) estimator [1], [20]. More so, we assume a flat Rayleigh fading channel with normalized Doppler frequency $F_{d}T$
(where, $F_{d}$
is the Doppler frequency and $T$
is the symbol period). The Jakes’s isotropic scattering model [21] is deployed for the complex Rayleigh fading. Firstly, we show in Fig. 11, that the $RS$
-PTA iterative receiver structure offers better Symbol Error Rate (SER) performance as compared to $RS$
-KV iterative receiver structure in [3]. Secondly, as expounded in [14], the $RS$
-PTA is a much less computationally intensive decoding algorithm in comparison with the $RS$
-KV algorithm. Consequently, it exhibits low computational delay and time complexity when deployed in a joint iterative receiver structure as shown in Table 1. Besides, the proposed $RS$
-PTA iterative receiver structure showcases a simple symbol remapping technique in comparison with most iterative receiver structures found in the literature. Thus, the $RS$
-PTA iterative receiver structure can be deployed in real time communication systems.
A. Problem Statement
The PTA [14] was proposed as a soft decision decoding FEC scheme for $RS$
codes. The PTA is a symbol-wise $RS$
soft decoder that iteratively transforms the parity-check matrix in order to perform a syndrome check. With each iteration, the $RS$
-PTA fine-tune the derived soft reliability information from the channel output based on some rules. The $RS$
-PTA has an adjustable decimal parameter $\delta $
which specifies the decoding SER versus Signal-to-Noise Ratio (SNR) performance, and the decoding time complexity of the algorithm. In all cases, the smaller the degree of $\delta $
the better the SER performance of the $RS$
-PTA, which in turns increase the decoding time complexity of the algorithm. For reference, see Fig. 2.
The results in Fig. 2 are compiled from a simple $RS$
-PTA coded 16-QAM system set-up as shown in Fig. 1. From Fig. 2, it can be seen that the SER for perfect CSI ($\delta =0.001$
) is $10^{-2}$
at a SNR of approximately $22dB$
. Perfect CSI means that the CSI is known and do not need to be estimated. On the other hand, the SER degrades to $10^{-1}$
when the CSI is estimated using LMMSE estimator with known pilot symbols. This discrepancy is due to the varying nature of the Rayleigh fading channel which cannot be easily tracked, most especially at high $F_{d}T$
. This in summary plummets the performance of the $RS$
-PTA decoder and all FEC decoders. One way of solving this problem; that is, improving the SER performance of this $RS$
-PTA over Rayleigh fading channels is by iteratively combining the $RS$
-PTA decoder with an optimal channel estimation technique as proposed in this paper.
B. Contribution and Related Work
As earlier mentioned, [3] proposed the first symbol-wise $RS$
iterative channel estimation receiver using the $RS$
-KV decoder. The iterative receiver proposed in [3] exhibits high computational delay and time complexity largely due to the $RS$
-KV decoder. Thus, the $RS$
-PTA iterative channel estimation receiver is proposed in this paper as an alternative to the work in [3]. The proposed iterative channel estimation and $RS$
-PTA decoder exhibits low computational delay and time complexity, and a better SER performance in comparison to the work in [3] as presented in Section IV-B. The $RS$
codes are deployed in wireless communication applications such as LTE (Long-Term Evolution), DVB (Digital Video Broadcasting), DAB (Digital Audio Broadcasting), and in long distance satellite communications. Also, the use of $RS$
code in DAB and DVB applications is recommended by the European Telecommunications Standard Institute (ETSI) [22], [23]. Furthermore, because $RS$
codes are capable of correcting both burst and random errors, the Consultative Committee for Space Data Systems (CCSDS) suggest the adoption of $RS$
codes in data link [24], [25]. Therefore, the proposed $RS$
-PTA iterative receiver can be used in these applications because they require efficient and low complexity channel estimation and $RS$
decoding scheme over Rayleigh fading channels. Note that the primary purpose of this paper is not to compare between codes ($RS$
codes, Turbo codes or LDPC codes), rather, the proposal of a less complex and performance efficient iterative receiver structure based on Reed-Solomon codes and a fair comparison with existing $RS$
iterative receiver structure.
The remainder of this paper is organised as follows. The discrete-time transmitter and channel model is described in Section II. Section III explains the proposed $RS$
-PTA iterative receiver structure in a block-wise fashion. Results with discernible comments are presented in Section IV. In particular, Section IV presents a comparative SER simulation result and computational time complexity analysis between the $RS$
-KV iterative receiver structure and this proposed $RS$
-PTA iterative receiver structure. This paper is summarised with notable remarks in Section V.
SECTION III.
$RS$
-PTA Iterative Receiver Structure
The proposed $RS$
-PTA iterative receiver structure is as illustrated in Fig. 5. The input to the iterative receiver after down-converting and matched filtering is defined as:\begin{equation} y_{v}=cx_{v}z_{v}+ \phi _{v}, \quad v=1,2, \cdots , n, \end{equation}
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\begin{equation} y_{v}=cx_{v}z_{v}+ \phi _{v}, \quad v=1,2, \cdots , n, \end{equation}
where $y_{v}$
is the noisy received symbol, $\phi _{v}$
is the complex AWGN with variance $\sigma ^{2}=N_{o}/2$
($N_{o}$
is the noise power), $c$
is a positive real number, and $z_{v}$
is the complex discrete-time zero mean Gaussian variable representing the fading distortion at both the data and pilot symbol positions. In practice, $z_{v}$
is only known at the pilot symbol positions by dividing the received pilot symbols with the known transmitted pilot symbol. In order to derive the CSI ($z_{u}$
) at the data symbol positions, this known properties of the pilot symbols are used.
In Fig. 5, the known properties of the pilot symbols (that is, the CSI at the pilot symbol positions) are firstly used to derive the initial CSI at the data symbol positions. Afterwards, the pilot symbols are separated from the data symbols. Therefore, a soft reliability information matrix is initially derived from the data symbols. The initial soft reliability information is fed into the $RS$
-PTA decoder which iteratively operate on this matrix to produce a final reliability information matrix. The output of the $RS$
-PTA decoder (the final reliability information matrix) is simply remapped into symbols in order to fine-tune the initial CSI. The $RS$
-PTA iterative receiver process is further explained as follows.
A. Initial Lmmse Estimate
Given that the properties at the pilot symbol position only is defined as:\begin{equation} y_{p}=cx_{p}z_{p}+ \phi _{p}, \quad p=1, \cdots , 5, \end{equation}
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\begin{equation} y_{p}=cx_{p}z_{p}+ \phi _{p}, \quad p=1, \cdots , 5, \end{equation}
where $y_{p}$
, $x_{p}$
, and $\phi _{p}$
are defined as above but in terms of the pilot symbols only. The initial CSI is derived using the optimal Linear Minimum Mean Square Error (LMMSE) estimator proposed in [1] and [20]. Hence, the LMMSE estimate of the fading distortion at the data symbol positions is expressed as [1], [20]:\begin{equation} z_{u}=\psi _{u}^{\dagger }y_{p}. \end{equation}
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\begin{equation} z_{u}=\psi _{u}^{\dagger }y_{p}. \end{equation}
The notation $\psi _{i_{u}}$
is derived using the Wiener-Hopf equation defined as [1], [20]:\begin{equation} {\mathcal {C}}_{u}={\mathcal {A}}\psi _{u}, \end{equation}
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\begin{equation} {\mathcal {C}}_{u}={\mathcal {A}}\psi _{u}, \end{equation}
where ${\mathcal {A}}$
and ${\mathcal {C}}_{u}$
are the autocorrelation matrix and covariance vectors respectively defined as [1], [20]:\begin{equation} {\mathcal {A}}=E\Big [y_{p}y_{p}^{\dagger }\Big ]~\mbox {and}~{\mathcal {C}}_{u}=E\Big [z_{p}^{*}y_{p}\Big ]. \end{equation}
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\begin{equation} {\mathcal {A}}=E\Big [y_{p}y_{p}^{\dagger }\Big ]~\mbox {and}~{\mathcal {C}}_{u}=E\Big [z_{p}^{*}y_{p}\Big ]. \end{equation}
The sign ${\dagger }$
and $*$
denotes the conjugate transpose and the complex conjugate respectively. As a key note, $\psi _{u}$
is interpolated to derive the CSI at the each data symbol position. See [1], [20] for straightforward derivations of the LMMSE estimator.
B. Initial ${\mathcal {R}}$
-Matrix
The noisy received data symbols ($y_{v}$
) are firstly separated from the pilot symbols. Subsequently, we scaled the noisy received data symbols by the derived CSI ($z_{u}$
) in order to obtained the scaled received symbols ($y_{u}$
) as defined by:\begin{equation} y_{u}=\dfrac {y_{v}}{z_{u}}, \quad u=1,2, \cdots , n. \end{equation}
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\begin{equation} y_{u}=\dfrac {y_{v}}{z_{u}}, \quad u=1,2, \cdots , n. \end{equation}
Hence, the initial reliability matrix (${\mathcal {R}}$
-matrix) is computed from the scaled received symbols ($y_{u}$
) and the symbols in the output 16-QAM constellation ($q_{u}$
) as [8], [26], [28]:\begin{equation} {\mathcal {D}}=c.\Bigg (e^{-{\sqrt {(y_{u_{in}}-{{q_{u_{in}}}})+(y_{u_{im}}-{q}_{u_{im}})}}}\Bigg ), \end{equation}
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\begin{equation} {\mathcal {D}}=c.\Bigg (e^{-{\sqrt {(y_{u_{in}}-{{q_{u_{in}}}})+(y_{u_{im}}-{q}_{u_{im}})}}}\Bigg ), \end{equation}
where $q_{u}=(q_{u_{in}},q_{u_{im}})$
is the in-phase and quadrature parts of the symbols in the output 16-QAM constellation, and $y_{u}=(y_{u_{in}},y_{u_{im}})$
is the in-phase and quadrature parts of the scaled received symbols. Literally, (7) measures the distance from $y_{u}$
to all the output 16-QAM constellation points $q_{u}$
to form the ${\mathcal {R}}$
-matrix (${\mathcal {R}}_{i}$
) defined as:\begin{equation} {\mathcal {R}}_{i}= \begin{pmatrix} \alpha _{0,0} & \alpha _{0,1} & \alpha _{0,2} & \cdots & \alpha _{0,n-1} \\ \alpha _{1,0} & \alpha _{1,1} & \alpha _{1,2} & \cdots & \alpha _{1,n-1} \\ \alpha _{2,0} & \alpha _{2,1} & \alpha _{2,2} & \cdots & \alpha _{2,n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \alpha _{n,0}& \alpha _{n,1} & \alpha _{n,2} & \cdots & \alpha _{n,n-1}\end{pmatrix}\!. \end{equation}
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\begin{equation} {\mathcal {R}}_{i}= \begin{pmatrix} \alpha _{0,0} & \alpha _{0,1} & \alpha _{0,2} & \cdots & \alpha _{0,n-1} \\ \alpha _{1,0} & \alpha _{1,1} & \alpha _{1,2} & \cdots & \alpha _{1,n-1} \\ \alpha _{2,0} & \alpha _{2,1} & \alpha _{2,2} & \cdots & \alpha _{2,n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \alpha _{n,0}& \alpha _{n,1} & \alpha _{n,2} & \cdots & \alpha _{n,n-1}\end{pmatrix}\!. \end{equation}
The notation $\alpha _{n,n-1}$
is the reliability value denoting the likelihood of receiving a QAM symbol. Also, the entries of ${\mathcal {R}}_{i}$
are normalized and quantized to $\log _{2} (n+1)$
bits of precision. For a performance efficient way of obtaining (8), refer to [26]. This initial soft reliability information matrix (${\mathcal {R}}_{i}$
) is fed into the $RS$
-PTA decoder. The $RS$
-PTA decoder [14] iteratively fine-tunes ${\mathcal {R}}_{i}$
based on some rules (as briefly explained in Section III-C) to output a final reliability information matrix (${\mathcal {R}}_{f}$
).
C. $RS$
-PTA Decoder
As explained in Section I, the $RS$
-PTA decoder iteratively operate on the derived ${\mathcal {R}}$
-matrix (${\mathcal {R}}_{i}$
) to produce a final ${\mathcal {R}}$
-matrix ${\mathcal {R}}_{f}$
defined as:\begin{align} {\mathcal {R}}_{f}= \begin{pmatrix} \alpha ^{\prime }_{0,0} & \alpha ^{\prime }_{0,1} & \alpha ^{\prime }_{0,2} & \cdots & \alpha ^{\prime }_{0,n-1} \\[0.2pc] \alpha ^{\prime }_{1,0} & \alpha ^{\prime }_{1,1} & \alpha ^{\prime }_{1,2} & \cdots & \alpha ^{\prime }_{1,n-1} \\[0.2pc] \alpha ^{\prime }_{2,0} & \alpha ^{\prime }_{2,1} & \alpha ^{\prime }_{2,2} & \cdots & \alpha ^{\prime }_{2,n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \alpha ^{\prime }_{n,0}& \alpha ^{\prime }_{n,1} & \alpha ^{\prime }_{n,2} & \cdots & \alpha ^{\prime }_{n,n-1}\end{pmatrix}\!, \end{align}
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\begin{align} {\mathcal {R}}_{f}= \begin{pmatrix} \alpha ^{\prime }_{0,0} & \alpha ^{\prime }_{0,1} & \alpha ^{\prime }_{0,2} & \cdots & \alpha ^{\prime }_{0,n-1} \\[0.2pc] \alpha ^{\prime }_{1,0} & \alpha ^{\prime }_{1,1} & \alpha ^{\prime }_{1,2} & \cdots & \alpha ^{\prime }_{1,n-1} \\[0.2pc] \alpha ^{\prime }_{2,0} & \alpha ^{\prime }_{2,1} & \alpha ^{\prime }_{2,2} & \cdots & \alpha ^{\prime }_{2,n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \alpha ^{\prime }_{n,0}& \alpha ^{\prime }_{n,1} & \alpha ^{\prime }_{n,2} & \cdots & \alpha ^{\prime }_{n,n-1}\end{pmatrix}\!, \end{align}
where ${\mathcal {R}}_{f}$
and ${\mathcal {R}}_{i}$
are of the same dimension.
To obtain (9), the PTA decoder [14] firstly finds the $k$
most reliable symbols from each column of ${\mathcal {R}}_{i}$
(in this case, we select the symbol with the highest reliability value). The indices of these $k$
reliable symbols are denoted as $\beta $
while the indices of the $n-k$
remaining symbols are denoted as $\gamma $
. Subsequently, the systematic parity check equation $H$
is transform to $\bar {H}$
in such a way that $\bar {H}$
contains a partitioned identity matrix where the indices of the parity column vectors are equal to $\beta $
, and indices of the unit column vectors $a_{t}$
are equal to $\gamma $
. Finally, the PTA decoder calculates the $n-k$
parity check equations using $\bar {H}$
. This is simply achieved by performing a hard decision detection on ${\mathcal {R}}_{i}$
to obtain a receive vector $r$
. Afterwards, its finds the dot product $r \cdot \bar {H}_{w}$
(syndrome check), where $\bar {H}_{w}$
denotes the $w$
-th row of $\bar {H}$
. If the result of the dot product is equal to zero, all the elements of ${\mathcal {R}}_{i_{b,e}}$
is incremented by $\delta $
. Otherwise, all the elements of ${\mathcal {R}}_{i_{b,e}}$
is decremented by $\delta $
, where $e \in a_{t} \cup \beta $
and $b$
such that ${\mathcal {R}}_{i_{b,e}}$
is the highest element in column ${\mathcal {R}}_{i_{e}}$
. Thus, if all the parity check equations are satisfied, the iterative process is stopped, else, the new decision matrix ${\mathcal {R}}_{f}$
is used as the input. The process is repeated until ${\mathcal {R}}_{f}$
seats an element $-{\mathcal {R}}_{f_{\varphi ,\theta }}$
. The sign $\varphi $
and $\theta $
represents the row and column of ${\mathcal {R}}_{f}$
respectively. The decoded symbol $\bar {g}$
is therefore obtained from the hard decoding decision of ${\mathcal {R}}_{f}$
by selecting the most reliable symbols in each column of ${\mathcal {R}}_{f}$
. However, for more reliable symbol decoding over fading channels, the entries of ${\mathcal {R}}_{f}$
are remapped into symbols in order to improve the quality of the previous CSI. Refer to [14] for mathematical expression of the PTA decoder.
D. Symbol Remapping
The output of the $RS$
-PTA decoder (${\mathcal {R}}_{f}$
) prior to the hard decoding decisions, is simply remapped into symbols. This is performed by taking the mean of the elements in ${\mathcal {R}}_{f}$
over all the output 16-QAM constellation points $q_{u}$
as defined by (10):\begin{equation} \bar {y}_{u_{j}}=\sum \limits _{l=1}^{n+1} q_{u}{\mathcal {R}}_{f_{l,j}}, \end{equation}
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\begin{equation} \bar {y}_{u_{j}}=\sum \limits _{l=1}^{n+1} q_{u}{\mathcal {R}}_{f_{l,j}}, \end{equation}
where $l$
and $j$
denotes the row and column of ${\mathcal {R}}_{f}$
respectively. The remapped symbols $\bar {y}_{u}$
are subsequently fused with the pilot symbols in preparation for the next iterative estimate. Key remark, the $RS$
-PTA iterative receiver structure do not require the Log-Likelihood Ratio (LLR) and the code bit probability calculation steps to derive the a priori information which is used in the symbol remapping process. Accordingly, the symbol remapping process of the $RS$
-PTA iterative receiver structure is far much less time complex in comparison with most iterative receiver structures in the literature. For more details on the LLR and the code bit probability calculation steps of iterative receiver structures, see [2], [5], [6].
E. Iterative LMMSE Estimate
The iterative CSI estimate is derived using a similar approach as the initial CSI estimate. Thus, it is derived as:\begin{equation} z_{\mu _{u}}=\psi _{\mu _{u}}^{\dagger }y_{u}, \end{equation}
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\begin{equation} z_{\mu _{u}}=\psi _{\mu _{u}}^{\dagger }y_{u}, \end{equation}
where $\mu $
represents the number of iterative estimates, and $y_{u}$
is defined as:\begin{equation} y_{u}=cx^{p}_{u}z_{u}+ \phi _{u}. \end{equation}
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\begin{equation} y_{u}=cx^{p}_{u}z_{u}+ \phi _{u}. \end{equation}
The combination of the remapped symbol $\bar {y}_{u}$
and the pilot symbols is denoted as $x^{p}_{u}$
. Moreover, the autocorrelation matrix ${\mathcal {A}}$
and the covariance vector are defined in terms of the received data symbols as:\begin{equation} {\mathcal {A}}=E\Big [y_{u}y_{u}^{\dagger }\Big ]~\mbox {and}~ {\mathcal {C}}_{u}=E\Big [z_{u}^{*}y_{u}\Big ]. \end{equation}
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\begin{equation} {\mathcal {A}}=E\Big [y_{u}y_{u}^{\dagger }\Big ]~\mbox {and}~ {\mathcal {C}}_{u}=E\Big [z_{u}^{*}y_{u}\Big ]. \end{equation}
The iterative process continues until the set number of iterative estimates is reached. Thus, we summarise the $RS$
-PTA iterative receiver structure with Algorithm 1.
Opinions expressed and conclusions arrived at, are those of the authors and are not necessarily to be attributed to the National Research Foundation.