Credibility Test for Frequency Estimation of Sinusoid Using Chebyshev ’ s Inequality

Estimation of sinusoid frequency is a key research problem related to radar, sonar, and communication systems. The results of numerous investigations on frequency estimation have been reported in the literature. Nevertheless, to the best of our knowledge, none of themhave dealt with credibility evaluation, which is used to decidewhether an individual frequency estimate of the sinusoid is accurate or not. In this study, the credibility problem is modeled as a hypothesis test based on Chebyshev’s inequality (CI). The correlation calculated from the received signal and the reference signal generated according to the frequency estimate is used as a test statistic. A threshold is determined based on CI, and the analytical expression for the frequency estimation credibility detection performance is derived. Simulations show that the proposed method performs well even at low signal-to-noise ratios.


Introduction
Frequency estimation of a sinusoid signal is an important problem concerning applications related to commercial and military signal processing systems.For some methods, estimation of frequency is the precondition for estimating the other parameters of sinusoid signals [1] as well as frequency estimation of modulated signals [2,3].Many algorithms have been proposed for estimation of frequency from received signals [4][5][6][7][8][9][10][11][12].Performance evaluation of frequency estimation algorithms is also a key operation in practical signal processing systems and can be considered from two points of view.Algorithm designers focus on the overall statistical performance that can be evaluated by comparing the mean square error with the Cramér-Rao lower bound (CRLB).On the other hand, users consider the credibility (or confidence) of individual frequency estimation important.In a noncooperative context, especially at low signal-to-noise (SNR) ratios, it is important to decide whether an individual frequency estimate is accurate or not when the true value of the frequency is unknown at the receiver side.For example, in the pulse sorting system, pulse radio frequency is one of the five key parameters that are pulse duration (PD), pulse amplitude (PA), pulse radio frequency (RF), angle of arrival (AOA), and time of arrival (TOA).Therefore, the sorting performance may be affected by the accuracy and credibility of pulse radio frequency.As another example in electronic intelligence (Elint), it is helpful for the users to determine the parameter limit and to remove the outlier estimates by using the credibility checking of each estimate [13].
Recently, investigations conducted have been devoted to the confidence evaluation of blind modulation recognition results to enhance the reliability of the overall signal processing units and conserve both software and hardware resources.The confidence measurement of modulation recognition becomes the key output information of the signal processing system in military applications and is used to identify unknown radar signals [14].As IEEE 1990.6 for cognitive radio (CR) [15] expressed, the modulation recognition confidence rating is regarded as additional output information in some civilian signal processing devices; an example is Agilent's option MR1 for E3238S signal detection and monitoring systems.Nevertheless, the detailed method of evaluating the reliability of modulation recognition is not described in both [10,11].Fehske et al. [16] defined the half value of the maximum and the second maximum output of the back propagation-(BP-) based classifier as the confidence metric of modulation classifying results for 2 Mathematical Problems in Engineering CR.Lin and Liu [17] proposed a confidence measurement method based on the information entropy to measure the confidence of modulation recognition results in single-input and single-output (SISO) as well as multiple-input multipleoutput (MIMO) channels of CR.Still, credibility evaluation of frequency estimation of a sinusoid remains an inadequately addressed research problem.
In this paper, we propose a method to automatically decide the credibility of an individual frequency estimate of a sinusoid without any a priori knowledge about the parameters of the received signal.Section 2 of this paper presents the sinusoid signal model and a hypothesis test for credibility evaluation.The null hypothesis is defined as that when the absolute frequency estimation error of the sinusoid is less than a quarter of a discrete frequency interval and the alternative one is the contrary.In Section 3, the statistic is defined by calculating the magnitude of the correlation between the received signal and the reference signal generated according to a frequency estimator.If the absolute estimation error is less than a quarter of the discrete frequency interval, it shows that the mean of the correlation under the null hypothesis is different from that under the alternative hypothesis.By using this property, Section 4 presents a decision rule for the credibility test for frequency estimation of sinusoid based on Chebyshev's inequality (CI).Finally, Section 5 summarizes the proposed algorithm, and Section 6 reports the simulation results.

Signal Model and Basic Assumptions
2.1.Signal Model.A complex sinusoid contaminated by noises can be described by the following signal model: where ,  0 , and  denote, respectively, the amplitude, carrier frequency, and initial phase of the sinusoid signal ().Δ is the discrete sampling interval, and  corresponds to the length of the samples.The additive noise () is supposed to be a white complex Gaussian process with a zero mean and variance  2 whose real and imaginary parts are independent of each other.

Hypothesis Model for Credibility Evaluation.
In the noncooperative environment, both the modulation format and signal parameters are unknown at the receiver side.Generally, processing of the sinusoid is performed in two main steps: modulation recognition and carrier frequency estimation.For a certain processing cycle, credibility evaluation of the frequency estimator is aimed at detecting whether the individual frequency estimate is accurate or not.In practice, for the widely used fast Fourier transform-(FFT-) based estimators, if the signal-to-noise ratio (SNR) is greater than the moderate threshold, the maximum absolute bias of the frequency estimation (|Δ|) is less than a quarter of the discrete sampling frequency interval (Δ) [18].Therefore, we describe the credibility assessment as the following hypothesis test: Δ     ≤ 0.25Δ,  1 :     Δ     > 0.25Δ. (2)

Statistic Selection and Analysis
3.1.Feature Analysis.Assuming that the observed signal is a single-tone sinusoid, the reference signal can be constructed by the sinusoid model as follows: where f0 is estimated by the maximum likelihood (ML) method or the other suboptimal estimators.The correlation between the observed signal () and the reference signal () can be expressed as where  and  are the signal part and noise part of , respectively.The signal part  can be further derived as where  = ( − 1)ΔΔ +  is the phase of .Letting  = Δ/Δ, the magnitude of  is given by  =           sin (ΔΔ) sin (ΔΔ)          ≈  sinc () , where  is the factor of the frequency estimation error and sinc() = sin / is the sinc function.
Rewrite  to the algebraic form as where   ,   are the real and imaginary parts of , respectively.They are independent Gaussian distributions.Now, the mean and variance of   and   can be expressed as follows: For the appropriate frequency estimator, if the SNRs are above a certain threshold [5,14], the absolute estimation error is less than the discrete sampling frequency interval Δ; that is, 0 ≤  ≤ 1.Consider the magnitude of the mean of  as Under hypothesis  0 , assuming that the absolute error is relatively small, that is, |Δ| ≤ 0.25Δ and  ≤ 0.25, it follows that Under hypothesis  1 , because of the low SNR or other possible reasons, the maximum magnitude of the observed signals' spectrums might not be located in the correct point or the discrete frequency interval and the estimation error Δ is larger than Δ/4; that is,  ≥ 0.25.Thus,      ( 1 )     =  sinc () ≤ 0.9003.
In order to explain the relationship between the frequency estimation error and the magnitude of the mean of , we define the ratio as where |()|  is the magnitude of the mean of  at a given .According to the sinc function, the ratio decreases monotonically with the frequency estimating error factor  supposing 0 ≤  ≤ 1.As Δ increases,  increases and the ratio  decreases.This relationship is shown in Figure 1.
For  ≤ 0.25, the ratio  > 0.9003 and it decreases slowly departing from 1 with the increase of .For  > 0.25, the ratio decreases dramatically.Hence, we can infer the absolute error of certain frequency estimation depending on the ratio  and obtain the credibility metric of the estimation.
In order to facilitate convenient handling of the formula derivation, the statistic used to decide  0 and  1 can be defined as follows: Clearly, the random variable  under   ,  = 0, 1, follows a standard noncentral law and its contribution can be represented as [19] where  0 () is the zero-order modified Bessel function and   = ( 2   +  2   )/ 2  denotes the noncentrality.Under the assumption relating to  0 , that is,  ≤ 0.25, we obtain where   is the noncentrality parameter when  = 0.25.
Similarly, under the  1 assumption, we obtain From the above discussion, we find that, under each assumption,  0 or  1 , the statistic  follows a standard noncentral law with different parameters.Therefore, the hypothesis test defined by ( 2) can be transformed to a parameter test involving  that is given by

Decision Rule and the Threshold.
We first consider the degradation form of (17) given as Consequently, letting  = 2 + ,  = /, the statistic  can be approximated to a Gaussian distribution by  = (/) 1/3 [20] whose mean and variance are given by respectively.Thus, the standard variable and the hypothesis test expressed by ( 17) can be simplified by where   denotes the mean of .
As the means of  between the two assumptions are different, CI can be used as the decision rule to perform the hypothesis test.Hence, we write CI as where  > 0 is a certain real number.As  approximately follows a standard Gaussian law, the probability of the samples located in the range of threefold the standard error is greater than 0.9973 and we can adjust  to obtain the desired probability.Hence, we decide   1 if Consequently, the hypothesis test of (2) can be rewritten as Therefore, we decide  1 if where  is the threshold.The probability of false alarm can be defined as where is the distribution function of a standard Gaussian variable and  is a given false alarm.Taking the inverse function of (26) leads to the threshold where Φ −1 () denotes the inverse of the distribution function of a standard Gaussian probability density function (pdf).
Figure 2 shows the histogram and the fitted pdfs of the statistic  for different frequency estimator factors.It can be seen that, under a certain SNR, the proposed statistic based on CI approximated to the standard Gaussian and the selected threshold can be used to distinguish  0 and  1 assumptions effectively.

Algorithm Summary
The proposed credibility test algorithm is composed of the following steps.
(1) Estimation of the frequency of the signal by using a certain estimation method.(2) Construction of the reference signal () with (3) by using the estimated frequency f0 .(3) Computation of the correlation between the received signal and the reference signal and contraction of the statistic  with (20).( 4) Computation of the test threshold  from the probability of false alarm with (27).( 5) Evaluation of the credibility of frequency estimation by comparing the statistic  to the threshold  with (25).The hypothesis  0 is chosen if  < .Otherwise  1 is accepted.

Performance Analysis
Under   0 assumption with  =   , the statistic defined in ( 20) is expressed as where   = 2 +   and  =   /  , respectively.For  ≥   , the statistic is We note that, for  >   , it can be rewritten as where  =   / and  ≥ 1.According to the property of the Gaussian distribution, the statistic is given by where and the variance is Hence, we obtain the probability of false alarm as Similarly, under  1 assumption with  <   , the statistic where  =   / and 0 ≤  ≤ 1.From (35), with the mean and variance are, respectively, given by Thus, the detection probability can be expressed by ) . (37) From ( 34) and (37), we observe that the detection performance depends largely on the parameter , which is the function of the number of samples received, the SNR, and the frequency estimation error factor.

Simulation Results
Monte Carlo simulations were carried out to study the behavior of the proposed algorithm in different environments.The simulations were aimed at detecting whether the absolute error of a certain estimated frequency is greater than a quarter of the discrete sampling frequency.Ten thousand Monte Carlo trials were performed for each condition.Monte Carlo trials were run in each of the following conditions: (1) the sinusoid signal contaminated by additive white Gaussian noises is received; (2) SNR = 10 log 10  2 / 2 ; (3) the sampling frequency is set to 100 MHz except in 6.6.

Influence of False Alarm.
Figure 3 shows the detection behavior for the credibility test by using the CI method with respect to the false alarm and the comparisons with the values theoretically calculated using (37).The carrier frequency of the sinusoid is 19.801MHz, while the sample size is 1024.The initial phase is /6, and the frequency estimation error factor is 0.5.The SNR is varied from −15 dB to 6 dB in steps of 3 dB.The values of probability of false alarm are fixed as 0.1, 0.01, and 0.001, respectively.In the three cases, the probability of detection (  ) is close to 1 at an SNR greater than 0 dB.The detection probability is increased by a higher value of probability of false alarm.With  = 0.1, the probability of detection is close to 1 for an SNR of −6 dB.Moreover, the derived theoretical   is close to the simulated value.

Influence of the Frequency Estimation
Error. Figure 4 shows the probability of detection with respect to the carrier frequency estimation error and the comparisons with the theoretically calculated values.The carrier frequency estimation error is, respectively, equal to 0.3, 0.4, and 0.5 for a sinusoid with 1024 samples and the probability of false alarm equals 0.01.The rest of the parameter settings of the signal in the simulation remain the same as that stated in Section 6.1.The detection probability is increased by increasing the carrier frequency estimation error factor.For  = 0.5 and an SNR close to −9 dB, the probability of detection is approximately equal to 1.
In fact, a larger frequency estimation error factor improves the mean of the proposed statistic , and the discrimination between nonnull and null is easier.In addition, it can be observed from Figure 3 that the derived theoretical   is very close to the simulated values in the three cases.settings are the same as that stated in Section 6.2.In the three cases with different sample numbers, the detection probability is close to 1 for an SNR of 0 dB.The credibility test performance is enhanced by the greater number of received samples at the same SNR.When the number of samples is 2048, the detection probability is close to 1 for an SNR of −6 dB.We can observe from Figure 5 that the derived theoretical   is very close to the simulated values in the three cases.

Influence of Frequency Discretization.
We evaluated the performance of our method in the case of different carrier frequencies.Figure 6 shows the performance of detection versus the carrier frequency selected in the simulations.We defined the discretization factor as  = ( 0 −  0 Δ) where  0 = 199 and five frequencies were generated by setting  = 0, 0.1, 0.3, 0.5, 0.7.Clearly, if  = 0, the frequency equals 19.53125 MHz, which is just located on the discrete frequency bin.When  ̸ = 0, it indicates that the selected frequencies are not located on the discrete frequency bin.The parameters are  = 1024,  = 0.4,  = /6, and  = 0.01. Figure 6 shows that the detection probability does not depend on the frequencies we selected by using different discretization factors, and the same levels of performance are achieved in the five cases and are approximately identical to the theoretically calculated performance in all the cases.6.5.Influence of the Initial Phase.We investigated the impact of the initial phase on the performance of our method.Figure 7 shows the performance of detection versus the initial frequency used in the simulations.We set three initial phases as /12, /6, and /3 and the sample size as 1024.The rest of the parameters set in each simulation are the same as that mentioned in Section 6.3.Figure 7 shows that the initial phase of the sinusoid does not affect the detection probability  and the same levels of performance are achieved in the two cases and are identical to the theoretical performances as the statistic defined in (13) is not relevant to the phase because of the use of the modulus operation.
6.6.Influence of the Sampling Frequency.We now clarify the impact of the sampling frequency on the performance of the proposed method.Figure 8 shows the performance of detection versus the sampling frequency selected in the simulations.The other parameters are  0 = 19.081MHz,  = 1024,  = 0.4,  = /6, and  = 0.01.From the figure, we observe that the performance of detection is not affected by the sampling frequency and is identical to the theoretical performance.

Comparison to Other Methods.
Comparative experiments were also carried out for the proposed statistic, Aboutanios-Mulgrew (AM) method-based [21] detector, and a combination of them by OR and AND rules, respectively.We have described the AM-based detector that can also be used to test the credibility of the frequency estimate, in the appendix.The parameters are  0 = 19.081MHz,  = 1024,  = 0.4,  = /6, and  = 0.01.The curves of probabilities of detection for different SNRs are shown in Figure 9.We can observe that, under an SNR less than −3 dB, the probabilities of detection of the AMbased method are greater than those of the CI-based method.
For the fusion method, the probabilities of detection using the "AND" rule are slightly greater than those of the AMbased method, and vice versa for the "OR" rule; that is, the probability of detection is the smallest when the SNR is less than −3 dB.When the SNRs are greater than −3 dB, the probabilities of detection of the four detectors all approach 1.Therefore, under lower SNRs, the more efficient fusion methods combining the CI and AM methods need to be further developed.
It should be noted that calculating the threshold is difficult because the exact probability distribution function of the AM-based statistic  AM , defined in the appendix, is hard to be expressed analytically.Therefore, here we introduce the bootstrap-based method [22,23] to obtain the threshold without depending on the analytical probability of the statistic based on AM, but the computational load increases many times because of repeated resampling operations in the bootstrap-based hypothesis test procedures.

Case Study.
In this section, we use the CI-based credibility test statistic to evaluate three commonly used frequency estimators: the Rife algorithm [4], the maximum likelihood estimator [6] by Newton iteration, and the Aboutanious-Mulgrew estimator [21].The carrier frequency of the sinusoid is 19.081MHz, the initial phase is /6, and the sample size is 1024.The range of SNR is from −18 dB to −6 dB.The value of probability of false alarm is set to 0.05.If we let   ( = 0, 1) be the event associated with the decision of choosing   ( = 0, 1),   stands for the number of times that deciding   when hypothesis   is correct.Therefore, Type I error probability and Type II error probability can be calculated using  01 /( 00 +  01 ) and  10 /( 10 +  11 ), respectively.The error probability of the test can be expressed by   = ( 01 +  10 )/10000, while   =  11 /( 11 +  10 ) indicates the ability of the proposed method to detect the unreliable frequency estimate, which means that the maximum absolute bias of a certain frequency estimate (|Δ|) is less than a quarter of the discrete sampling frequency interval (Δ).
Table 1 illustrates the performance of the credibility test for the ML frequency estimator of the sinusoid by using the proposed CI-based statistic.It should be remarked that the credibility test performance is enhanced by increasing the SNR.If the SNR is −6 dB, all the 10000 frequency estimates based on ML are reliable, and the error probability of the proposed test approximates to 0. When the SNR decreases to −12 dB, 61 estimates are unreliable among 10000 simulations.By using the proposed statistic, all the 61 unreliable estimates can be detected, indicating a detection rate of about 100%.Among the 9939 reliable estimates, there are 21 times when estimates are mistakenly decided as unreliable; in other words, the error probability is about 0.21%.When the SNR is decreased to −15 dB, the error probability is about 0.81% and the detection rate is greater than 95%.
Table 2 depicts the performance of the credibility test for the Rife-based sinusoid frequency estimator.The credibility test performance is enhanced by a larger SNR.For an SNR of −6 dB, all the 10000 times, the frequency estimates obtained Table 3 provides the performance of the credibility test for the AM-based sinusoid frequency estimator.The credibility test performance is enhanced by a larger SNR.Similar to that data in Tables 1 and 2, the test performance is improved by a larger SNR.With the SNR decreased to −15 dB, the error probability is about 0.69% and the detection rate is greater than 80%.
Overall, from Tables 2 and 3, we can observe that the proposed credibility test statistic can be used for effectively identifying unreliable frequency estimates by using ML, AM, and Rife algorithms with a high detection probability and a low error probability when the SNR is greater than −15 dB.Compared to the estimator performances summarized in Tables 1 and 2, the estimator performance summarized in Table 3 is the lowest because AM-based estimation has a better performance than that of both ML and Rife algorithms and the frequency estimation factors are small.
In practice, for the detected unreliable estimate, we can abandon the results or estimate the frequency of the signal again, thereby enhancing the reliability of the entire signal processing unit.For example, if the FFT-based coarse frequency estimator is used in practice and we set the parameters as   = 100 MHz,  0 = 19.55MHz,  = 1000, and the number of DFT  = 1000, there are two cases.
(1) Case 1.When the SNRs (e.g., SNR = −23 dB) are below the SNR threshold, the maximum spectrum line cannot be detected correctly.In this case, the estimate error is larger than 0.25Δ and the estimate will be judged as incredible by the proposed CI-based statistic.If we perform resampling over the frequency grid by setting  = 2000, the maximum spectrum line still cannot be detected.As the absolute estimation errors are the same under the two scenarios, the resampling action cannot enhance the credibility of the estimate and this kind of incredible estimate should be abandoned.
(2) Case 2. When an SNR is greater than the SNR threshold, an incredible frequency estimate is detected because of the displacement of the maximum spectrum line by the discretization.In this case, if we enlarge  to 2000 by resampling in the frequency domain,  0 can be just moved to the discrete frequency bin.Hence, the estimation error can be decreased and the credibility can be enhanced.However, in practice, the true frequency cannot be known by the user; hence it is difficult to adjust the resampling parameter to match the estimation exactly.In this case, we need to select the more accurate estimating method or abandon the result obtained by original estimation.

Conclusion
The paper presented a CI-based credibility test algorithm for frequency estimation of a sinusoid.A credibility assessment testing model is defined to analyze the mathematical characteristics of the correlations between the observed signal and the reference signal under different hypotheses.The test is performed with the proposed threshold that is based on CI.Experimental results revealed that a good performance was achieved even at low SNRs.Credibility tests were performed with the ML frequency estimator using the proposed threshold based on CI as well with the Rife frequency estimator and the Aboutanious-Mulgrew estimator.Experimental results revealed that a good performance was achieved by the proposed method even at low SNRs.

4 𝛾Figure 2 :
Figure 2: Distribution of the statistic  for  = 0.1, 0.25, 0.4 at SNR = 0 dB.The carrier frequency is set to 19.081 MHz, the length of received samples is 1024, the initial phase is /6, and the number of simulations is 1000.

Figure 3 :
Figure 3: Effect of the false alarm probability on the detection probability obtained by the credibility test of a frequency estimate.

Figure 4 :
Figure 4: Effect of the frequency estimation error factor on the detection probability obtained by the credibility test of a frequency estimate.

Figure 5 :
Figure 5: Effect of the number of received samples on the detection probability obtained by the credibility test of a frequency estimate.

Figure 6 :
Figure 6: Effect of the frequency discretization of the signal on the detection probability obtained by the credibility test of a frequency estimate.

Figure 7 :
Figure 7: Effect of the initial phase of the signal on the detection probability obtained by the credibility test of a frequency estimate.

Figure 8 :DFigure 9 :
Figure 8: Effect of the sampling frequency of the signal on the detection probability obtained by the credibility test of a frequency estimate.

Table 1 :
Credibility test performance of ML frequency estimator.

Table 2 :
Credibility test performance of Rife frequency estimator.