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A nondataaided SNR estimator based on maximum likelihood method for communication between orbiters
EURASIP Journal on Wireless Communications and Networking volume 2020, Article number: 123 (2020)
Abstract
Signaltonoise ratio (SNR) is an important metric for performance assessment in numerous scenerios. In order to ensure the reliability and effectiveness of the system performance, plenty of situations require the information of SNR estimate. At the same time, Mars exploration has been a hot topic in recent years, which leads to the research attention of scholars extending to deep space. In this paper, a new SNR estimator related to deep space scene is proposed. On the one hand, the time of essential data transmission is limited in Mars exploration system. On the other hand, the relative position and condition between orbiters vary quickly all the time, which makes it difficult to obtain the accurate and significant information for Mars exploration. Therefore, it is obvious that the information of SNR can promote the system to adjust the signal transmission rate automatically. Subsequently, the estimation of SNR becomes a fundamental research in automatic digital communications. In this paper, an SNR estimation method based on nondataaided (NDA) with maximum likelihood (ML) estimation is proposed to enhance the accuracy and reliability of Mars exploration process. Additionally, the CramerRao lower bound (CRLB) related to the designed ML algorithm is derived. Finally, the Monte Carlo simulation results demonstrate that the proposed ML estimator algorithm obtains a superior performance when compared to the existing SNR estimators.
Introduction
Since the Earth and Mars are far away from each other, there is an attenuation of signal and exists a delay of time for communication between them, both of which destroy the signal quality in receivers [1, 2]. On the one hand, the time of essential data transmission is limited in Mars exploration system, and on the other hand, the relative position and condition of orbiters vary quickly all the time, which makes it difficult to obtain the accurate and significant information for Mars exploration [3, 4]. Therefore, it is obvious that the information of signaltonoise ratio (SNR) promotes the system to adjust the signal transmission rate automatically [5]. The existing SNR estimation methods can be classified into two groups, namely dataaided (DA) and nondataaided (NDA) [6, 7] estimators, respectively, which apply to the cases that the transmitted signals are known to the receivers for DA estimators and the receivers of the NDA estimators have no information about the transmitted symbols, respectively [8]. Although DA estimation algorithms outperform NDAbased methods in accuracy and effectiveness, it is at the expense of lower transmission efficiency, which is extremely worthwhile for deep space exploration [9]. Therefore, NDAbased estimators are considered for the deep space scene. Recently, many new researches are also investigated [10–16].
Generally, NDAbased SNR estimators include signaltovariation ratio (SVR) [17], second and fourthorder moments (M_{2}M_{4}) [18], squared signaltonoise variance (SNV), and subspacebased method [1]. SVR estimator, which can be applied to not only fading channels but also additive white Gaussian noise (AWGN) channels, is developed to monitor and evaluate the channel quality based on moment operations. It is noticeable that SVR estimator is not applicable to other forms of digital modulated signals except Mary phaseshift keying (PSK) modulated ones. M_{2}M_{4} estimator was first proposed to estimate the strength of carrier and noise in real channels, and we further extended it to complex domain. Due to its independence of the transmitted symbols, namely, it only requires the estimation of the second and fourth order related to the received symbols, M_{2}M_{4} is also one of the NDA estimators. SNV estimator utilizes the first and second moment of the received signals, which is the output of the matched filter (MF). Finally, the subspace method is based on the singular value decomposition (SVD) theory.
This paper aims to develop a NDA SNR estimation method based on maximum likelihood (ML) method to improve the system performance of deep space model; at the same time, it can achieve a higher accuracy than other estimators, such as SVR, SNV, M_{2}M_{4}, and subspacebased method. The CramerRao lower bound (CRLB) of this proposed NDA estimation method is also derived to verify the effectiveness of the proposed MLbased estimator.
The rest of this paper is organized by the following structure. In Section 2, we discuss the system model. The maximum likelihood method based on dataaided is explained in Section 3. Section 4 demonstrates the NDA SNR estimation algorithm through maximum likelihood method. Section 5 shows the experimental figures. Section 6 concludes the contributions made by this paper.
System model
We aim to find the best SNR estimator while consuming the least energy. Table 1 is the notation of variables for this paper. Generally, the statistical properties are averaged to generate the SNR estimate through a large amount of symbols. The discrete and complex binary phaseshift keying (BPSK) [19] signal constellations are adopted to generate the basebandequivalent and bandlimited transmit symbols in real AWGN channels as illustrated in Fig. 1. The length of the transmitted symbols, which is upsampled to L = 16 samples in each symbol, is denoted as N. The root raisedcosine (RRC) filter, whose rolloff R equals to 0.5, and number of tap coefficients Q equals to 65, is assumed [20]. The binary source symbols can be calculated by [21, 22]:
where a represents the signal amplitude whose probability is equal by obtaining values from {−A,A} and α_{n} represents one of the two evenly spaced phases on a unit circle with \( n\in \left \lbrace 1,2, \dots, N\right \rbrace \). The upsampled information sequence can be shown as:
where δ_{ij} denotes the Kronecker delta. The signal after being sampled and pulseshaped can be given as:
where h_{k} is the tap coefficients of the RRC filter with \( k\in \left \lbrace \left (R1\right)/2,\dots,1,0,1,\dots, \left (R1\right)/2 \right \rbrace \) and h_{k} equals to zero for k>(R−1)/2. The received signal can be presented as:
where w_{k} denotes the complex and sampled AWGN with zero mean and variance of σ^{2}. The output of the MF is expressed as:
where ⊗ and ∗ denote the discrete convolution and complex conjugation, respectively, and \( h_{k} = h_{k}^{\ast } \) can be attributed to the assumption that the impulse response of RRC is real and even symmetric. Finally, the downsampled symbols at the receiver can be expressed as:
where f_{0} denotes the maximum impulse response of the full raisedcosine, the samples of which can be written as:
and the downsampled and filtered samples of noise z_{n} can be expressed as:
Subsequently, the SNR is deduced as:
where E{·} represents the expectation and var{·} represents the variance. The SNR can be independent of the channel by normalizing the corresponding square of tap coefficients of the RRC; subsequently, the SNR can be solely related to A and σ^{2}, namely:
Maximum likelihood estimation method based on NDA
In this section, we propose an NDA estimation method [23, 24] based on ML algorithm. The probability density function (PDF) of y_{n} can be expressed as:
where \( P_{+}\left (y_{n}\right) = \frac {1}{\sqrt {2\pi }\sigma }e^{\frac {\left (y_{n}A\right)^{2}}{2\sigma ^{2}}}, P_{}\left (y_{n}\right) = \frac {1}{\sqrt {2\pi }\sigma }e^{\frac {\left (y_{n}+A\right)^{2}}{2\sigma ^{2}}} \), and \( \cosh \left (x\right) = \frac {1}{2}\left (e^{x} + e^{x}\right) \).
The joint PDF of the signal vector at the receiver \( \left [ y_{1},y_{2},\dots,y_{n}\right ] \) is hence given by (due to independence):
The loglikelihood function is described as:
Setting \( \frac {{\partial {L_{N}}\left ({{y_{1}},{y_{2}}, \ldots,{y_{N}}} \right)}}{{\partial A}} = 0 \), we implicitly obtain the ML estimate of A as the solution to the equation:
We consider an iterative algorithm to explore the most suitable value of amplitude that satisfies (13). According to the signal vector at the receiver with N samples of y_{n}, it is obvious to define the function:
By solving the equation above, we can obtain the amplitude estimation of the maximum likelihood method, namely, F(x)=0 at \( x = \hat {A} \), where \( \hat {A} \) represents the estimate of A. We can determine the value of \( \hat {A} \) by the following steps:
Step 1: The received vector is normalized as \(\frac {1}{N}\sum _{n=1}^{N}y_{n}^{2} = 1\). Find the maximum and minimum values of A_{min} and A_{max}. At the same time, determine the total number of iteration I. Let A_{1}=A_{min} and A_{2}=A_{max}. Initialize i = 0.
Step 2: Compute \( A_{m} = \frac {A_{1}+A_{2}}{2} \), and therefore, \( \sigma _{m}^{2} = 1A_{m}^{2} \).
Step 3: Compute \( F\left (A_{m}\right) = A_{m}  \frac {1}{N}\sum _{n=1}^{N}y_{n}\tanh \left (\frac {y_{n}A_{m}}{\sigma ^{2}}\right) \).
Step 4: If F(A_{m})>0, then let A_{2}=A_{m}; otherwise, let A_{1}=A_{m}.
Step 5: Let i = i+1, if i = I, output \( A_{m} = \frac {A_{1}+A_{2}}{2} \) as to the ML estimate of the amplitude and \(\frac {A_{m}^{2}}{1A_{m}^{2}}\) as the estimated SNR. Otherwise, go back to step 2.
Performance comparisons
CramerRao lower bound (CRLB)
We consider CRLB [25, 26] to evaluate whether the estimators work effectively or not. The definition of the SNR has been given in (10). We intend to estimate ρ by N observed samples of y_{n}. The estimation task involves two parameters. The estimated parameter vector is denoted as:
Since the estimation of the SNR is generally expressed in decibel scale, the following form of CRLB is adopted:
where K(θ) is the Fisher information matrix, which can be shown as:
From (17), \( \frac {\partial g\left (\theta \right)}{\partial \theta } \) is determined as:
According to the N observed symbols, the Fisher information matrix is described as:
where \( J\left (\rho \right) = \frac {e^{ \rho }}{\sqrt {2\pi } }\int \limits _{ \infty }^{+ \infty } {\frac {{u^{2}}{e^{ \frac {u^{2}}{2}}}}{\cosh \left ({u\sqrt {2\rho } } \right)}} du \). By substituting (21) to (17), the CRLB of the BPSK signals can be expressed as:
Normalized mean square error (NMSE)
For the above iterative SNR estimator, we provide the required experimental results. To appropriately benchmark our proposed method, the results of SVR, M_{2}M_{4}, SNV, and subspacebased methods are also included. According to the system model given in Section 2, we obtain the NMSE of the aforementioned SNR estimators. The NMSE represents the deviation between estimated and true values and can be calculated as follows:
where \( \hat {\rho } \) describes the estimated SNR while ρ represents the true SNR value. The NMSE of each estimator is calculated by using the Monte Carlo method as follows:
The number of symbols N should be large enough to promise the accuracy and objectivity of the experimental results. In terms of the ML estimator, the NMSE under different simulation numbers is also shown to further compare the performance of the proposed estimator.
Simulation results
In this section, the Monte Carlo simulation results of the NMSE performances of the aforementioned estimators are presented. The CRLB performance, the ML estimator under several different simulation numbers, and the differences among perfect SNR, SNR under subspacebased method, and SNR under MLbased method are also displayed.
Figure 2 shows the NMSE performance among SNV, SVR, M2M4, subspace method, and MLbased estimator with the length of symbols N = 1000. From the simulation result, we can make a conclusion that SNVbased method performs worse than all the other estimators mentioned above. Subspacebased method outperforms SNV, SVR, and M_{2}M_{4} in a reasonable SNR region. However, when compared to the proposed MLbased estimator, it suffers an SNR deficit under the same NMSE, which verifies the efficiency of the proposed MLbased method.
Figure 3 describes the CRLB of the proposed MLbased estimator under two cases, namely, N = 300 and N = 600, respectively. From the simulation result, we can find that the CRLB with N = 300 is higher than that of N = 600. The reason of this phenomenon can be easily understood by the essence that the larger simulation symbols lead to a higher estimation accuracy.
Figure 4 demonstrates the NMSE performance under different lengths of simulation symbols, which are N = 100, N = 200, and N = 500, respectively. It is obvious that with the increase of simulation symbols N, the NMSE performance becomes better, which is consistent with the theoretical analysis. This result is useful for engineering to choose the appropriate length of symbols.
Figure 5 shows the differences among the perfect SNR, SNR under subspacebased method, and SNR under MLbased method under N = 600. We can obtain from the curves that compared to the subspacebased method, the MLbased method achieves a higher estimation accuracy and the gap across the perfect SNR is smaller. Therefore, the proposed MLbased method performs better than all the other estimators mentioned above.
Conclusions
We proposed a novel NDAbased ML estimator, which is based on an iterative algorithm and achieves a higher SNR transmission accuracy compared to several traditional SNR estimators. The simulation results showed that the proposed new ML estimation method achieves a superior NMSE performance compared to the existing ones, which is significant for the researches of Mars exploration. The CRLB of the proposed MLbased method was also derived. Furthermore, the NMSE of the MLbased method under different simulation symbols was compared to further verify the effectiveness of the proposed algorithm.
Abbreviations
 SNR:

Signaltonoise ratio
 NDA:

Nondataaided
 ML:

Maximum likelihood
 CRLB:

CramerRao lower bound
 DA:

Dataaided
 SVR:

Signaltovariation ratio
 M _{2} M _{4} :

Second and fourthorder moments
 SNV:

Squared signaltonoise variance
 AWGN:

Additive white Gaussian noise
 PSK:

Phaseshift keying
 MF:

Matched filter
 RRC:

Root raisedcosine
 PDF:

Probability density function
 NMSE:

Normalized mean square error
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All the three authors have contributed to the proposal of concept, analysis, design, and algorithm of the paper. ZS has contributed to the simulation for this paper. FL has analyzed the theory. XG was a major contributor in writing the manuscript. All the authors have read and approved the final manuscript
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The simulation was performed using MATLAB in Intel Core I5 (64bit).
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Sun, Z., Gong, X. & Lu, F. A nondataaided SNR estimator based on maximum likelihood method for communication between orbiters. J Wireless Com Network 2020, 123 (2020). https://doi.org/10.1186/s13638020017304
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Keywords
 Signaltonoise ratio estimate
 Maximum likelihood
 CramerRao lower bound