Signal-to-noise ratio (SNR) estimation has been a recurrent research topic, due to the relevance of SNR for a variety of mobile communication systems. The a priori knowledge of the communication channel conditions is an important issue as long as those systems become more complex and widely required. In  for instance, the a priori knowledge of the channel, by means of the SNR, is proposed for evaluating the effective transmission rate (throughput) in a communication system with adaptive modulation and coding, while in  its use is considered in adaptive transmission systems.
In , the knowledge of the SNR is necessary for assessing the time-varying channel condition of an adaptive system with frequency hopping, and in  it is necessary for planning relay communication systems. In , SNR is a useful parameter in the scenario of turbo decoding systems, while in  it is useful in the context of low-density parity check (LDPC) codes.
Usually, the choice of the estimation technique depends on the complexity of the mathematical model of the signal in the receiver [4, 7]. Depending on that complexity, one can use a method belonging to the class of estimators that use a training sequence (that is, a data-aided method (DA)) [8, 9], or a method belonging to the class of estimators that do not have a priori knowledge of the transmitted sequence of symbols (that is, a non-data-aided method (NDA)) .
In , for example, a new SNR estimator is derived, DA and NDA, for a slow time-varying channel with impulse response characterized by a polynomial function of order Lc and phase-shift keying (PSK) signals. In the NDA scenario, the expectation maximization (EM) algorithm is proposed for the calculation of the estimates and improvement of performance is observed in relation to the estimator with channel considered constant throughout the time of observation, at the cost of a moderate increase in computational complexity. The difference of that approach in relation to the method of moments (MOM) NDA technique considered in our article is that the MOM is not efficient in reaching the Cramér-Rao bound.
In , the least-squares (LS) technique is proposed for estimation of SNR in a received signal model composed by M-QAM symbols (M=16) multiplied by samples of the impulse response of a time-varying channel added to samples of white Gaussian noise. The impulse response of the channel is approximated by a polynomial as a function of time with constant coefficients calculated by means of the LS technique, with the aid of a pilot sequence sent periodically and an array of antennas of Na elements. The performance of the technique is evaluated by means of the normalized root mean square error (NRMSE), without comparison with the Cramér-Rao bound, and requires a much smaller number of samples than the required by the MOM without training sequence, in order to reach NRMSE values below 0.1 for an SNR above 10 dB.
In , one considers the maximum likelihood estimation (MLE) of the SNR in the demodulator of an orthogonal and non-coherent frequency-shift keying (FSK) signal, in which the transmitted signal is affected by Rayleigh fading and that this fading is constant in a block of k symbols, while in , the authors consider a binary frequency-shift keying (BFSK) affected by constant Rayleigh fading in a block of symbols whose BFSK carrier has frequency deviation. The MLE with and without pilot sequences are presented for the SNR, assisted by estimators by the MOM for the frequency deviation. Although the MLE is optimum in the sense of the minimum variance criterion, it depends on the expression of the probability density function (PDF) of the samples of the observed signal, and in the case of generalized fading models, such as those considered in this work, their PDFs make the PDFs of the received signal complicated for treatment by MLE.
In , the authors consider a model of received signal for a signal with frequency modulated differential chaos shift keying (FMDCSK) transmitted by a channel with multipath propagation and Rayleigh fading. The authors analyze the SNR estimators and their performance considering that the channel coefficients that characterize the fading are constant in a sequence of K symbols of the FMDCSK signal. The estimators are calculated for the data-aided (DA), non-data-aided (NDA), and joint DA-NDA cases, presenting good results in relation to the proximity of the Cramér-Rao Lower Bound (CRLB) for values of SNR above 20 dB.
In , an estimation method of SNR is presented for linearly modulated signals captured by an array of antennas in an environment with complex white additive Gaussian noise with spatial uncorrelation between the elements. In this NDA and single input multiple output (SIMO) estimator, based on the MOM, the performance is assessed in terms of the Normalized Mean Square Error (NMSE) for QAM signals and improves with the increase of the number of elements. The estimation of SNR in SIMO systems is also addressed in , in which the authors consider signal samples captured by an array of antennas in a channel model of constant gain by the time of observation of a sequence of symbols. A MLE based on the I/Q components of the received signal is evaluated, for the cases DA and NDA, that reach the CRLB for a wide range of SNR. In , the problem of estimation of the SNR is extended to a multiple input multiple output (MIMO) system in a channel model with block flat fading, in which the channel gain matrix is considered constant by a block of N symbols. From this consideration and reduction of the channel model to a Gaussian channel to each block of N symbols, the authors then present ML estimators for the DA and NDA cases, as well as the CRLB.
In , the main contribution of the authors is the analysis of the CRLB of SNR estimates of signals with minimum shift keying (MSK) modulation and QAM with turbo encoding. Even in a constant gain channel model over a K symbols window, the task of calculating the CRLB for the M-QAM symbols is the most laborious of the article and is solved from the analysis of the structure of Gray’s mapping. The authors then show that the proposed DA ML estimator has a lower CRLB than the NDA CRLB case. The CRLB is also evaluated in  for a channel model of constant gain over the whole set of symbols observed under QAM modulation, for the cases DA and NDA. In the channel models of the aforementioned references, the channel gain is considered constant by a sequence of symbol intervals, which contributes to the mathematical treatment of estimation by maximizing the likelihood function. In our proposed model, in which fading can vary at each time interval in which a sample is obtained at the output of the receiver’s matched filter, the PDF of the received signal modulus makes the likelihood function calculation more difficult.
In , the proposed solution consists of using the goodness-of-fit test of Kolmogorov-Smirnov (K-S) for the evaluation of the distance between empirical cumulative distribution function (CDF) generated from observed samples of the received signal and the theoretical curves of CDFs generated and stored in a file for different configurations of channel parameters. The proposed estimator was presented for a Gaussian channel, of constant gain by a sequence of symbol intervals, and was evaluated by means of the normalized root mean square error (NRMSE). For fading channels, such as the models considered in this manuscript, the method would depend on the calculation of the CDF of the envelope of the signal received, which would be a challenging task.
It is worth mentioning that in the paper by Bellili et al. , the authors do not make any other consideration about the channel coefficients than they are deterministic and unknown. In our article, the only consideration is that the fading is constant during the interval of the impulse response of the matched filter in the receiver. The main merit for our work is that the random behavior of the channel is taken into account by the fading probability distributions and the estimators obtained have simple expressions, encompass the estimators for other fading models, and are not aided by data sequence. In the flat fading channel model considered in , the NDA estimator needs the EM algorithm that converges for an optimum solution with a reduced number of iterations at cost of a greater computational complexity.
In propagation environments in which the fading is characterized by probability distributions such as η−μ and κ−μ , the SNR estimation is difficult by using estimators such as MLE, because the likelihood function for the problem becomes complicated. In these cases, the MOM is a good alternative.
Despite raising the mathematical complexity of the SNR estimation problem, general probability distributions, such as the η−μ and κ−μ distributions, model a wider variety of fading signals, such as signals received on Nakagami-m, Hoyt (Nakagami-q), Rice (Nakagami-n), unilateral Gauss, and Rayleigh channels. According to , the κ−μ distribution is better suited for line-of-sight applications and the η−μ distribution is more appropriate for non-line-of-sight applications. The SNR estimators obtained for the signal received under these two fading models encompass all the estimators obtained for the other fading models mentioned. The estimation of SNR by the MOM has been the considered technique, as shown, for example, in references [24, 25] and more recently in reference  in which the transmitted signal is modeled by a complex Gaussian random variable with zero mean by component. In , the MOM is proposed in a process of joint estimation of both the K parameter of the Rice fading distribution and SNR in a SIMO communication system.
In the present paper, the method of moments for SNR estimation is applied to channel models in which the fading is characterized by η−μ and κ−μ distributions and modulation scheme M-QAM  is considered.
1.1 Main contributions
The main contributions of this paper are as follows:
New expressions for the SNR estimates by NDA MOM for a received signal model for M-QAM signal under η−μ and κ−μ fading.
New expressions for the evaluation of the mean, variance, and NMSE of the estimates, obtained from a statistical linearization procedure.
Comparative analysis between the estimates obtained for the models η−μ, κ−μ, and Nakagami-m.
Exact expressions for the moments of order k of the envelope of the observed signal.
The remaining of the paper is organized as follows. In Section 2, methods used in the work and the problem definition are presented. In Section 3, the derivation of the kth moment of the absolute value of the samples of the observed signal is presented. In Section 4, the derivation of the SNR estimates is presented. In Section 5, it is shown how to obtain the SNR estimates for the signal under Nakagami fading from the estimates for the models η−μ and κ−μ. In Section 6, the derivation of the mean, variance, and NMSE of the estimates is presented. In Section 7, the moments of order 2, 4, 6, and 8 of M-QAM are presented. In Section 8, a proposal is presented for evaluating the CRLB. In Section 9, theoretical curves corroborated by curves obtained by simulations are presented, and in Section 10, the conclusions of the work are provided.