Performance analysis on joint channel decoding and state estimation in cyberphysical systems
 Liang Li^{1}Email authorView ORCID ID profile,
 Shuping Gong^{3},
 Ju Bin Song^{2} and
 Husheng Li^{1, 2}
https://doi.org/10.1186/s136380170943y
© The Author(s) 2017
Received: 1 February 2017
Accepted: 8 September 2017
Published: 22 September 2017
Abstract
We propose to use an mean square error (MSE) transfer chart to evaluate the performance of the proposed belief propagation (BP)based channel decoding and state estimation scheme. We focus on two models to evaluate the performance of BPbased channel decoding and state estimation: the sequential model and the iterative model. The numerical results show that the MSE transfer chart can provide much insight about the performance of the proposed channel decoding and state estimation scheme.
Keywords
1 Introduction

Does the proposed algorithm converge and help to improve channel decoding and system state estimation?

How much gain can be obtained by using redundancy of observations in time domain to assist channel decoding?
 1.
Does the iterative channel decoding and estimation converge, and how many iterations are sufficient?
 2.
How much gain can be obtained by utilizing priori information from the previous time slots to assist state estimation at the current time slot?
In the area of wireless communication, the purpose of performance analysis for decoding scheme is to find out if, for any given encoder, decoder, and channel noise power, a messagepassing iterative decoder can correct the errors or not.
To analyze the performance, in [3–5], Ten Brink proposed using an extrinsic information transfer (EXIT) chart to track the iterative decoding performance. Based on the assumption that the distribution of the extrinsic loglikelihood ratios (LLRs) is a Gaussian distribution, the EXIT chart tracks mutual information from the extrinsic LLRs through an iterative decoding process. Compared with the previously used method of density evolution, the EXIT chart is computationally simplified, and it also allows to visualize the evolution of mutual information through iterative decoding process in a graph. The details of the EXIT chart can be found in [6].
The EXIT chart has two useful properties as shown in [7]. One is the necessary condition for the convergence of iterative decoding that the flipped EXIT chart curve of the outer decoder for iterations lies below the EXIT chart curve of the inner coder. The other is that the area under the EXIT curve of outer code relates to the rate of inner coder. In [8], the authors demonstrated that if the priori channel is an erasure channel, for any outer code of rate R, the area under the EXIT curve is 1−R. To our best knowledge, the area property of the EXIT chart has been proved only for cases where the priori channels are erasure channels.
The mean square error (MSE) transfer chart improves the EXIT chart as shown in [7], because the area property of the MSE transfer chart corresponding to the area property of the EXIT charts has been proven in both erasure channels and AWGN channels. Instead of tracking mutual information, the MSE transfer chart, as an alternative to evaluate decoding performance, has been proposed in [9] to track the iterative decoding performance based on the relationship between mutual information and the minimum mean square error (MMSE) for the additive white Gaussian noise (AWGN) channel.
In this paper, we use the MSE transfer chart to analyze the message passing procedure of channel decoding and system state estimation by assuming that the priori information is an AWGN channel. Compared with [9], our hybrid model prioritizes practicality because the system states and observations considered are continuous values while the information transmitted in wireless system are quantized information bits. Unlike previous research, our algorithm addresses the message passing between continuous values from the state estimator and quantized information bits from the channel decoder with the condition that the system state is correlated over different time slots. In addition, in order to view the evolution of the estimation error, we analyze the performance of state estimation in two cases: within two time slots and more than two time slots.
Our work is also informed by other areas in wireless communication, which have faced similar issues in source coding (quantization) [10–13] and joint source and channel decoding [14–19]. The idea of source coding (quantization) in the context of this work is combining the side information available at the controller to assist system state estimation, and the route of joint source and channel decoding in [20–25] is to utilize redundancy in the source to assist channel decoding. Our work can also be considered as one special case of joint source and channel decoding. However, there are two major differences. One difference is that most works focus on the source with binary values and use the EXIT chart [26, 27] or the protograph EXIT (PEXIT) [16] for performance analysis, while [28, 29] considered the case with the source of nonbinary values, but the performance analysis of decoding was not provided. The other difference is that most works, such as [29], considered the joint source and channel decoding within two time slots. For instance, only the estimation from the previous time slot is used to calculate the estimation of current time slot. In our work, the dynamic state changes over more than two time slots, and the performance of channel decoding and system estimation at the current time slots also impacts its performance in all future time slots. Therefore, we study the performance of iterative estimation and decoding across multiple time slots.
This paper is organized as following. Following the review of the literature on iterative decoding performance analysis method in Section 1, Section 2 briefly introduces on the EXIT chart and the MSE transfer chart for the performance evaluation of iterative channel decoding. Section 3 describes the system models for performance analysis. Section 4 presents the message passing framework between system observation and channel decoding. Section 5 describes the MSE transfer chart, and Section 6 presents how to use the MSE transfer chart to evaluate BPbased sequential and iterative channel decoding and state estimation. Finally, a brief conclusion is given in Section 7.
2 Preliminaries on the EXIT Chart
In this section, we review the concept of the EXIT chart and the MSE transfer chart by iterative decoding the output of a serially concatenated encoder. In Section 2.1, we use an example to illustrate the serially concatenated coding scheme and its iterative decoding process. Then, in Section 2.2, we review how to use the EXIT chart and the MSE transfer chart to analyze the performance of iterative decoding.
2.1 A serially concatenated encoding scheme and corresponding iterative decoding algorithm
where \(\text {SNR}=\frac {E_{b}}{N_{0}}\) is the signal power to noise power ratio and v _{ i } is a zero mean and unit variance Gaussian noise.
where \(\mathbf {L}_{A,i}^{\text {out}, k}\) means the priori information of \(\mathbf {L}_{A,i}^{\text {out}, k}\) for all S except S _{ i }.
2.2 The EXIT chart and the MSE transfer chart
where L _{ i } is the extrinsic information, i.e. \(\mathbf {L}_{E,i}^{\text {out}, k}, \mathbf {L}_{E,i}^{\text {in}, k}, \mathbf {L}_{A,i}^{\text {out}, k}\), and \(\mathbf {L}_{A,i}^{\text {in}, k}\).
The measure used by the EXIT chart is mutual information, i.e., F(S , L)=I(S , L), which is based on the observation that the PDF of extrinsic LLRs can be approximated by a Gaussian distribution [3–5].
3 System model
In this section, we describe the system model for the analysis.
3.1 Linear dynamic system and communication system
where x(t) is the Ndimensional vector of system state at time slot t, u(t) is the Mdimensional control vector, y(t) is the Kdimensional observation vector, and n(t) and w(t) are noise vectors, which are assumed to be Gaussian distributions with zero mean and covariance matrix Σ _{ n } and Σ _{ w }, respectively. For simplicity, we do not consider u(t).
where the additive white Gaussian noise e(t) has a zero expectation and variance Σ _{ c }. Note that we consider the AWGN channel, ignore the fading and normalize the transmit power to be 1. The algorithm and conclusion in this work can be easily extended to the cases with different channels and different types of fading.
3.2 Models for belief propagation based channel decoding and state estimation
In this section, we firstly introduce the Bayesian network structure and then use the following two models to evaluate BPbased channel decoding and state estimation: BPbased sequential processing and BPbased iterative processing.
3.2.1 Bayesian network structure and the message passing
Message passing in BPbased channel decoding and state estimation system
Step  Distribution  Gaussian distribution  Details 

x(t−1)→x(t)  π _{ x(t−1),x(t)}(x(t−1))  \(\mathcal {N}\left (\mathbf {x}_{t1}, \mathbf {x}_{\pi _{x},t1}, \mathbf {P}_{\pi _{x},t1}\right) \)  * 
x(t)→y(t)  π _{ x(t)}(x(t))  \(\mathcal {N} \left (\mathbf {x}_{t}, \mathbf {x}_{l, t}, \mathbf {P}_{l, t}\right)\)  \(\begin {array}{l}\mathbf {x}_{l, t}= \mathbf {Ax}_{\pi _{x}, t1}+\mathbf {Bu}_{t1}; \\ \mathbf {P}_{l, t} = \mathbf {A} \mathbf {P}_{\pi _{x},t1} \mathbf {A}^{T} +\mathbf {\Sigma }_{n}\end {array}\) 
–  π _{ x(t),y(t)}(x(t))  \(\mathcal {N}\left (\mathbf {x}_{t}, \mathbf {x}_{\pi _{y},t}, \mathbf {P}_{\pi _{y}, t}\right) \)  \(\mathbf {x}_{\pi _{y},t}= \mathbf {x}_{l, t}; \mathbf {P}_{\pi _{y}, t}= \mathbf {P}_{l, t} \) 
y(t)→b(t)  π _{ y(t)}(y(t))  \(\mathcal {N}\left (\mathbf {y}_{t}, \mathbf {y}_{l,t}, \mathbf {S}_{l, t}\right)\)  \( \mathbf {y}_{l,t}= \mathbf {C} \mathbf {x}_{\pi _{y},t} ;\mathbf {S}_{l, t} =\mathbf {C} \mathbf {P}_{\pi _{y}, t} \mathbf {C}^{T} +\mathbf {\Sigma }_{w}\) 
–  π _{ y(t),b(t)}(y(t))  \(\mathcal {N}(\mathbf {y}_{t}, \mathbf {y}_{\pi, t}, \mathbf {S}_{\pi, t})\)  y _{ π,t }=y _{ l,t };S _{ π,t }=S _{ l,t } 
b(t)→y(t)  λ _{ b(t),y(t)}(y(t))  \(\mathcal {N}(\mathbf {y}_{t}, \mathbf {y}_{\lambda, t}, \mathbf {S}_{\lambda, t})\)  The y _{ λ,t } and S _{ λ,t } is provided in Section 4. 
y(t)→x(t)  λ _{ y(t),x(t)}(x(t))  \(\mathcal {N}(\mathbf {x}_{t}, \mathbf {x}_{\lambda _{y},t}, \mathbf {P}_{\lambda _{y},t})\)  \(\mathbf {x}_{\lambda _{y},t}= \mathbf {C}^{1}\times \mathbf {y}_{\lambda, t};{\newline } \mathbf {P}_{\lambda _{y},t} = \mathbf {C}^{1} (\mathbf {S}_{\lambda, t}+\mathbf {\Sigma }_{w})(\mathbf {C}^{1})^{T}\) 
x(t)→x(t+1)  π _{ x(t),x(t+1)}(x(t))  \(\mathcal {N}(\mathbf {x}_{t}, \mathbf {x}_{\pi _{x},t}, \mathbf {P}_{\pi _{x},t}) \)  \( \mathbf {P}_{\pi _{x},t}=(\mathbf {P}_{l, t}^{1}+ \mathbf {P}_{\lambda _{y},t}^{1})^{1}; \mathbf {x}_{\pi _{x},t} = \mathbf {P}_{\pi _{x},t}(\mathbf {P}_{l, t}^{1}\mathbf {x}_{l,t} + \mathbf {P}_{\lambda _{y},t}^{1}\mathbf {x}_{\lambda _{y},t})\) 
3.2.2 Model for BPbased sequential processing
As noted in Section 1, two objectives here are to evaluate how much gain can be obtained by utilizing π _{ x(t−1),x(t)}(x(t−1)) to assist channel decoding and state estimation at time slot t and to evaluate the performance of state estimation over multiple time slots, i.e., the evolution of \(\mathbf {P}_{\pi _{x},t1}\) as shown in Fig. 7(b).
3.2.3 Model for BPbased iterative processing between two time slots
The goal is to evaluate the performance of iterative channel decoding and state estimation for different realizations of the two distributions for π _{ x(t−1),x(t)}(x(t−1)). For instance, when π _{ x(t−1),x(t)}(x(t−1)) (say, \(\mathbf {P}_{\pi _{x},t1}\)) equals to 0×I, x(t−1) is a determined state estimation. Therefore, this reference model can be converted to the model shown in Fig. 8 (b) by setting π _{ x(t−1),x(t)}(x(t−1) (say, \(\mathbf {P}_{\pi _{x},t1}\)) as 0×I. Or if π _{ x(t−1),x(t)}(x(t−1)) (say, \(\mathbf {P}_{\pi _{x},t1}\)) is set as ∞×I, x(t−1) is unknown. Then, the reference model is transformed to the model shown in Fig. 8 (c).
4 Message passing between state estimator and channel decoder
Quantizing the message between channel decoder and state estimator
where [Q _{min},Q _{max}] is the range for quantization and Q _{ I } is the quantization interval, which is given as \(Q_{I}=\frac {Q_{\text {max}}Q_{\text {min}}}{2^{B}1}\). Note that when i≠j, \(E\{[ \tilde {\mathbf {b}}_{(k1)B+i}(t)  {\mathbf {b}}_{(k1)B+i}(t)][ \tilde {\mathbf {b}}_{(k1)B+j}(t)  {\mathbf {b}}_{(k1)B+j}(t)]\}=0\), which is obtained based on the independence of channel noise and required for the derivation of (14).
Thus, we finalize the computation of \(\mathcal {N}(\mathbf {y}_{t}, \mathbf {y}_{\lambda, t}, \mathbf {S}_{\lambda, t})\) based on one realization, y ^{′}(t), which is obtained from the PDF of π _{ y(t),b(t)}(y(t)).
Approximation for the message passing between state estimator and channel decoder
Note that the computation of S _{ λ,t }(S _{ π,t }) in (17) would require the averaging over a sufficient number of realizations (i.e., large enough to show the probability distribution based on a fixed y _{ π,t }) of y ^{′}(t) and the averaging over all possible y _{ π,t } based on its PDF \(\phantom {\dot {i}\!}f_{\mathbf {y}_{\pi, t}}\) to be computed sequentially, i.e., Forward Process and Backward Process should be calculated sequentially. The Forward Process generally refers to the message passing from a node to its children, and the Backward Process generally refers to the message passing from a node to its parents. This arises because y _{ π,t } not only impacts priori information L _{ A }(y _{ π,t },S _{ π,t }) but also defines the set of codewords which are generated by quantizing y ^{′}(t).
 1.Compute the PDF of L _{ A }(y _{ π,t },S _{ π,t }) and \(\phantom {\dot {i}\!}f_{\mathbf {y}_{\pi,t}}\).
 2.
Compute the extrinsic information from the channel decoder, i.e., the PDF of L _{ E }(L _{ A }(y _{ π,t },S _{ π,t }),y ^{′}(t),e ^{′}(t)).
 3.
Compute S _{ λ,t }(S _{ π,t }) from the extrinsic information L _{ E }(L _{ A }(y _{ π,t },S _{ π,t }),y ^{′}(t),e ^{′}(t)).
 1.
Compute the mutual information I _{ A } based on the PDF of L _{ A }(y _{ π,t },S _{ π,t }) corresponding to S _{ π,t } and \(\phantom {\dot {i}\!}f_{\mathbf {y}_{\pi,t}}\).
 2.
Compute the extrinsic information from the channel decoder, i.e., mutual information I _{ E } or \(\text {MMSE}_{\text {ext}}^{B}\) based on the PDF of L _{ E }(L _{ A }(y _{ π,t },S _{ π,t }),y ^{′}(t)),e ^{′}(t)).
 3.
Compute S _{ λ,t }(S _{ π,t }) from the extrinsic information of the channel decoder I _{ E } or \(\text {MMSE}_{\text {ext}}^{B}\).
5 The MSE transfer chart for channel decoding and state estimation
In this section, we show how to obtain the MSE transfer chart for evaluating BPbased sequential channel decoding and state estimation as shown in Fig. 7(b) and BPbased iterative channel decoding and state estimation as shown in Fig. 8(a).
5.1 The MSE transfer chart for sequential channel decoding and state estimation
and the PDF y _{ π,t }, i.e., \(\phantom {\dot {i}\!}f_{\mathbf {y}_{\pi, t}}\), is assumed to keep the same in all time slots and all iterations. Similarly, \(\text {MMSE}_{\text {ext}}^{S}\) represents the averaging S _{ π,t+1} corresponding to S _{ π,t } and y _{ π,t } with the PDF of \(\phantom {\dot {i}\!}f_{\mathbf {y}_{\pi, t}}\).
Compared with the model shown in Fig. 7 (b), the MSE transfer chart has two differences. First, the starting node for the message passing is changed from x(t−1) to y(t). The reason for this changing is to keep alignment with the structure of the MSE transfer chart for BPbased iterative channel decoding and state estimation. Note that since there is no extra information added from node x(t−1) to y(t), x(t−1) and y(t) provide the same amount of information for channel decoding in the context of information theory. From this point, these two models are equivalent. The second modification is that the scalar measures, to draw the MSE transfer chart \(\text {MMSE}_{\text {ap}}^{S}\) and \(\text {MMSE}_{\text {ext}}^{S}\), rather than the matrix measures are used to evaluate the performance of the sequential message passing.
 1.
y(t)→b(t), we have \(\mathbf {S}_{\pi, t}=\text {MMSE}_{\text {ap}}^{S}\mathbf {I}_{K}\) and the PDF of y _{ π,t } is \(\phantom {\dot {i}\!}f_{\mathbf {y}_{\pi, t}}\).
 2.
b(t)→y(t), S _{ λ,t }, the detailed derivation is provided in Section 4.
 3.
y(t)→x(t), we have \( \mathbf {P}_{\lambda _{y},t} = \mathbf {C}^{1} (\mathbf {S}_{\lambda, t}+\mathbf {\Sigma }_{w})(\mathbf {C}^{1})^{T} \).
 4.
x(t)→x(t+1), we have \( \mathbf {P}_{\pi _{x},t}=(\mathbf {P}_{l,t}^{1}+\mathbf {P}_{\lambda _{y},t}^{1})^{1} \), where \( \mathbf {P}_{l, t} = \mathbf {A} \mathbf {P}_{\pi _{x},t1} \mathbf {A}^{T} +\mathbf {\Sigma }_{n}\).
 5.
x(t+1)→y(t+1), we have \( \mathbf {P}_{\pi _{y},t+1}=\mathbf {A}\times \mathbf {P}_{\pi _{x},t+1}\times \mathbf {A}^{T} +\mathbf {\Sigma }_{n} \).
 6.y(t+1)→b(t+1), we have \( \mathbf {S}_{\pi,t+1}=\mathbf {C}\times \mathbf {P}_{\pi _{y},t+1}\times \mathbf {C}^{T} +\mathbf {\Sigma }_{w} \). Note that S _{ π,t+1} is a matrix, \(\text {MMSE}_{\text {ext}}^{S}\) is calculated by solving the following equation:$$ I_{A}(\text{MMSE}_{\text{ext}}^{S}\mathbf{I}_{K})=I_{A}(\mathbf{S}_{\pi,t+1}) $$(20)
where we first obtain the priori information I _{ A } for each ith diagonal variance of S _{ π,t+1}, where i∈{1,..,K}. Next, to achieve (20), we compute the average value of all the variances, as \(\text {MMSE}_{\text {ext}}^{S}\). An example of the calculation will be illustrated Section 6.1 in Fig. 15. The physical meaning of (20) is that \(\text {MMSE}_{\text {ext}}^{S}\) is the value such that \(\text {MMSE}_{\text {ext}}^{S}\mathbf {I}_{K}\) can provide the same amount of priori information for the channel decoder as S _{ π,t+1}.
5.2 The MSE transfer chart for BPbased iterative channel decoding and state estimation
 1.The flow from y(t) to y(t+1): the starting node is y(t), and \(\text {MMSE}_{\text {ap}}^{t}\) represents the approximated S _{ π,t }, i.e.,$$ \mathbf{S}_{\pi, t} = \text{MMSE}_{\text{ap}}^{t} \mathbf{I}_{K} $$(21)
and the PDF of y _{ π,t }, i.e., \(\phantom {\dot {i}\!}f_{\mathbf {y}_{\pi, t}}\), is assumed to keep the same in all time slots and all iterations and is denoted by \(\phantom {\dot {i}\!}f_{\mathbf {y}_{\pi }}\). \(\text {MMSE}_{\text {ext}}^{t}\) represents the average S _{ π,t+1} corresponding to \(\mathbf {S}_{\pi, t}=\text {MMSE}_{\text {ap}}^{t}\mathbf {I}_{k}\) and y _{ π,t } with the PDF, \(\phantom {\dot {i}\!}f_{\mathbf {y}_{\pi }}\).
 2.The flow from y(t+1) to y(t): the starting node is y(t+1), and \(\text {MMSE}_{\text {ap}}^{t+1}\) represents the approximated S _{ π,t+1}, i.e.,$$ \mathbf{S}_{\pi, t+1} = \text{MMSE}_{\text{ap}}^{t+1} \mathbf{I}_{K} $$(22)
and the PDF of y _{ π,t+1} is \(\phantom {\dot {i}\!}f_{\mathbf {y}_{\pi }}\).
\(\text {MMSE}_{\text {ext}}^{t+1}\) represents the averaging S _{ π,t } corresponding to \(\mathbf {S}_{\pi, t}=\text {MMSE}_{\text {ap}}^{t+1} \mathbf {I}_{K}\) and y _{ π,t+1} with the PDF, \(\phantom {\dot {i}\!}f_{\mathbf {y}_{\pi }}\).
Then, we obtain two curves: one curve with \(\text {MMSE}_{\text {ap}}^{t}\) versus \(\text {MMSE}_{\text {ext}}^{t}\) for the flow from y(t) to y(t+1); the other flipped curve with \(\text {MMSE}_{\text {ext}}^{t+1}\) versus \(\text {MMSE}_{\text {ap}}^{t+1}\) for the flow from y(t+1) to y(t). Finally, by following the steps in Table 1: (1) y(t)→b(t), (2) b(t)→y(t), (3) y(t)→x(t), (4) x(t)→x(t+1), (5) x(t+1)→y(t+1), (6) y(t+1)→b(t+1), (7) y(t+1)→b(t+1), (8) b(t+1)→y(t+1), (9) y(t+1)→x(t+1), (10) x(t+1)→x(t), (11) x(t)→y(t), and (12) y(t)→b(t), we can calculate the values of \(\text {MMSE}_{\text {ext}}^{t}\) and \(\text {MMSE}_{\text {ext}}^{t+1}\) by \(I_{A}(\text {MMSE}_{\text {ext}}^{t}\mathbf {I}_{K})=I_{A}(\mathbf {S}_{\pi,t+1})\) in step 5 and \(I_{A}(\text {MMSE}_{\text {ext}}^{t+1}\mathbf {I}_{K})=I_{A}(\mathbf {S}_{\pi,t})\) in step 11, respectively.
6 Numerical results
We consider an electric generator dynamic system for verification. Each dimension of the observation y(t) is quantized with 14 bits, and the dynamic range for quantization is [−432,432]. A \(\frac {1}{2}\)rate recursive systemic convolution (RSC) code is used as the channel encoding scheme, and the code generator is set as g=[1,1,1;1,0,1].
6.1 Message passing between state estimator and channel decoder
In this section, we show the performance results of the proposed channel decoding and state estimation algorithm, especially the message passing between the state estimator and channel decoder. In addition, we illustrate how much gain can be obtained by using the redundancy of system dynamics to assist channel decoding. To achieve above, the approximate framework 2 shown in Fig. 12 (b) is considered, but different from that we set S _{ π,t } and S _{ λ,t } as \(\text {MMSE}_{\text {ap}}^{Y}\mathbf {I}_{K}\) and \(\text {MMSE}_{\text {ext}}^{Y}\mathbf {I}_{K}\).
6.2 Performance analysis for sequential channel decoding and state estimation
 1.When \(\text {MMSE}_{\text {ap}}^{S}\) is less than 1, the corresponding \(\text {MMSE}_{\text {ext}}^{S}\) for all \(\frac {E_{b}}{N_{0}}\) are equal. Note that \(\text {MMSE}_{\text {ap}}^{S}\) is used to model the amount of priori information from x(t−1). Therefore, the smaller \(\text {MMSE}_{\text {ap}}^{S}\) is, the higher amount of priori information attained from x(t−1) is. Although the extrinsic information from the channel decoder at time slot t can also contribute to the prediction of y(t+1), with small \(\text {MMSE}_{\text {ap}}^{S}\), the priori information from x(t−1) is dominant in the prediction of y(t+1). Therefore, the difference of gains from channel decoder with different \(\frac {E_{b}}{N_{0}}\) is not seen in \(\text {MMSE}_{\text {ext}}^{S}\).
 2.
With a large \(\text {MMSE}_{\text {ap}}^{S}\), the higher \(\frac {E_{b}}{N_{0}}\) is, the higher \(\text {MMSE}_{\text {ext}}^{S}\) is. With the increasing of \(\text {MMSE}_{\text {ap}}^{S}\), x(t−1) provides less amount of priori information for the prediction y(t), and the extrinsic information from channel decoder becomes dominant in the prediction of y(t). Then, the channel gains with different \(\frac {E_{b}}{N_{0}}\) are seen.
 3.
When \(\text {MMSE}_{\text {ap}}^{S}\) is around 1.0e −2, the values \(\text {MMSE}_{\text {ext}}^{S}\) are flat. This arises because, based on the information from the time slots priori to t+1, the prediction of y(t) is not reliable since state dynamics n(t) and the noise of system observation w(t) are not predictable. This leads that the minimum value of \(\text {MMSE}_{\text {ext}}^{S}\) is limited by the covariance matrix of n(t) and w(t), i.e., Σ _{ n } and Σ _{ w }, respectively.
Following the same idea of the EXIT chart and the MSE transfer chart for iterative channel decoding, we use the MSE transfer chart to evaluate the performance of BPbased sequential channel decoding and state estimation over mutiple time slots. In the MSE transfer chart, we have two curves: one is \(\text {MMSE}_{\text {ap}}^{S,t}\) versus \(\text {MMSE}_{\text {ext}}^{S,t}\), which is equivalent to the curve for \(\text {MMSE}_{\text {ap}}^{S}\) versus \(\text {MMSE}_{\text {ext}}^{S}\)(\(\text {MMSE}_{\text {ap}}^{S}\) and \(\text {MMSE}_{\text {ext}}^{S}\) at time slot t); the other is \(\text {MMSE}_{\text {ext}}^{S,t+1}\) versus \(\text {MMSE}_{\text {ap}}^{S,t+1}\), which is equivalent to the flipped curve for \(\text {MMSE}_{\text {ap}}^{S}\) versus \(\text {MMSE}_{\text {ext}}^{S}\)(\(\text {MMSE}_{\text {ap}}^{S}\) and \(\text {MMSE}_{\text {ext}}^{S}\) at time slot t+1).
6.3 Performance analysis for iterative channel decoding and state estimation
In this section, we show how to use the proposed MSE transfer chart to evaluate the performance of BPbased iterative channel decoding and state estimation within two time slots. As we stated, the node x(t−1) and the node y(t) provide the same amount of information in the context of information theory. From this point, the priori information at the node y(t), as same as the node x(t−1), will be \( \pi _{\mathbf {x}(t1), \mathbf {x}(t)}(\mathbf {x}(t1))= \mathcal {N}(\mathbf {x}_{t1}, \mathbf {x}_{\pi _{x}, t1}, P_{\pi _{x}, t1}) \) with \(\mathbf {x}_{\pi _{x}, t1}=\mathbf {0}\) and \(P_{\pi _{x}, t1}=\text {MMSE}_{\text {ap}}^{t1,x}\mathbf {I}_{k}\).
 1.With the same \(\text {MMSE}_{\text {ap}}^{t1,x}\), the higher \(\text {MMSE}_{\text {ap}}^{t}\) is, the lower gain of \(\text {MMSE}_{\text {ext}}^{t}\) can be obtained from \(\text {MMSE}_{\text {ap}}^{t}\).
 2.
The higher the \(\text {MMSE}_{\text {ap}}^{t1,x}\) is, the higher gain of \(\text {MMSE}_{\text {ext}}^{t}\) can be obtained from \(\text {MMSE}_{\text {ap}}^{t}\). This exists because the prediction of y(t+1) is contributed by both priori information from x(t−1) and the extrinsic information from the channel decoder at time slot t. When \(\text {MMSE}_{\text {ap}}^{t1,x}\) is small, the information from x(t−1) is dominant in predicting y(t+1). Although the prediction of y(t) can increase the amount of extrinsic information from channel decoder at time slot t, it cannot contribute much to the gain of \(\text {MMSE}_{\text {ext}}^{t}\) as the priori information from x(t−1) is dominant.
 1.With the same \(\text {MMSE}_{\text {ap}}^{t1,x}\), the higher \(\text {MMSE}_{\text {ap}}^{t+1}\) is, the lower gain of \(\text {MMSE}_{\text {ext}}^{t+1}\) can be obtained from \(\text {MMSE}_{\text {ap}}^{t}\).
 2.
The higher the \(\text {MMSE}_{\text {ap}}^{t1,x}\) is, the higher gain of \(\text {MMSE}_{\text {ext}}^{t+1}\) can be obtained from \(\text {MMSE}_{\text {ap}}^{t+1}\).
 1.BPbased iterative channel decoding can decrease \(\text {MMSE}_{\text {ext}}^{t}\) and help improve the estimation of x(t+1). The values of the crossing point between the curve (\(\text {MMSE}_{\text {ap}}^{t}\) versus \(\text {MMSE}_{\text {ext}}^{t}\)) and the flipped curve (\(\text {MMSE}_{\text {ap}}^{t+1}\) versus \(\text {MMSE}_{\text {ext}}^{t+1}\)) for \(\text {MMSE}_{\text {ap}}^{t}\) and \(\text {MMSE}_{\text {ext}}^{t}\) are 1.0e + 1.33 and 1.0e + 1.25, respectively, while the values of \(\text {MMSE}_{\text {ap}}^{t}\) and \(\text {MMSE}_{\text {ext}}^{t}\) with no priori information are 1.0e + 5 and 1.0e + 1.43, respectively. Therefore, the total gain of \(\text {MMSE}_{\text {ext}}^{t}\) for BPbased iterative channel decoding and state estimation is 10∗(1.43−1.25)=1.8 dB.
 2.The gain of \(\text {MMSE}_{\text {ext}}^{t}\) with only three steps is close to the gain of \(\text {MMSE}_{\text {ext}}^{t}\) at the convergence point. The trace of BPbased iterative channel decoding and state estimation with the mentioned three steps is shown by the arrows with blue color in the figure, and the details are listed as below:
 1)
From y(t) to y(t+1): As there is no priori information, the starting point is \(\text {MMSE}_{\text {ap}}^{t}\) with the value of 1.0e + 5, and the corresponding value of \(\text {MMSE}_{\text {ext}}^{t}\) is 1.0e + 1.43.
 2)
From y(t+1) to y(t): We have \(\text {MMSE}_{\text {ap}}^{t+1}\) with the value of 1.0e + 1.43, and the corresponding value of \(\text {MMSE}_{\text {ext}}^{t+1}\) is 1.0e + 1.34.
 3)
From y(t) to y(t+1): We have \(\text {MMSE}_{\text {ap}}^{t}\) with the value of 1.0e + 1.34, and the corresponding value of \(\text {MMSE}_{\text {ext}}^{t}\) is 1.0e + 1.255. Compared with step (1), 10∗(1.43−1.255)=1.75 dB is obtained for \(\text {MMSE}_{\text {ext}}^{t}\) while it is 1.8 dB for \(\text {MMSE}_{\text {ext}}^{t}\) from convergence point. Thus, the gain loss for BPbased iterative channel decoding and state estimation with these three steps is just 1.8−1.75=0.05 dB.
In summary, we can implement BPbased iterative channel decoding and state estimation with above three steps to obtain the gain of \(\text {MMSE}_{\text {ext}}^{t}\), which is close to the gain at convergence point.
 1)
Note that when \(\text {MMSE}_{\text {ap}}^{t1,x}\) equals to 1 or 10, the BPbased iterative channel decoding and state estimation scheme cannot improve the performance of state estimation. This is because with a small \(\text {MMSE}_{\text {ap}}^{t1,x}\) the priori information from x(t) is dominant in estimating y(t+1), which leads that the prediction of y(t) in channel decoder at time slot t is negligible in predicting y(t+1).
The MSE transfer charts for \(\frac {E_{b}}{N_{0}}=3\) and 1 dB have similar observations as that for \(\frac {E_{b}}{N_{0}}=\)5 dB, so we will not list the results here.
6.4 Performance analysis for Kalman filteringbased heuristic approach
Similarly, by utilizing the redundancy of system dynamics, a Kalman filteringbased heuristic approach is evaluated in this section. In the Kalman filteringbased heuristic approach, the prediction of y(t) based on Kalman filtering is used as the priori information for b(t), and instead of using only the extrinsic information of the channel decoder to obtain a soft estimation of y(t), the total information including both the priori information and the extrinsic information generated by the channel decoder is used to obtain a hard estimation of y(t).
The corresponding framework used for the Kalman filteringbased heuristic approach is similar to the BPbased channel decoding and state estimation as shown in Fig. 12 (b). The priori information from y(t) is modeled with \(\mathbf {S}_{\pi, t}=\text {MMSE}_{\text {ap}}^{Y}\mathbf {I}_{K}\), and the priori information for b(t) is represented by mutual information I _{ A }. Finally, the total information from the channel decoder for estimating of b(t) is modeled by \(\text {MMSE}^{B}_{\text {tot}}\), which means the MMSE of estimating b(t) based on the total information including both the priori information and the extrinsic information from the channel decoder.
7 Conclusions
We propose to use the MSE transfer chart to evaluate the performance of BPbased channel decoding and state estimation. We focus on two models, the BPbased sequential processing model and the BPbased iterative processing model, for channel decoding and state estimation. The former can be used to evaluate the performance of sequential processing over multiple time slots, and the latter can be used to evaluate the performance of iterative processing within two time slots. The numerical results show by utilizing the MSE transfer chart the proposed channel decoding and state estimation algorithm can decrease the MSE and improve performance of channel decoding and state estimation. Specifically, a total 1.75 dB gain can be earned through threestep BPbased iterative channel decoding and state estimation process when no prior information is given.
Declarations
Acknowledgements
The authors would like to thank the support of National Science Foundation under grants ECCS1407679, CNS1525226, CNS1525418, and CNS1543830.
Authors’ contributions
HL conceived the Brainbow strategies for this work. JBS and HL supervised the project. SG built the initial constructs and LL validated them, analyzed the data, and wrote the paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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