Phase Error Criterion Based Adaptive Algorithm for Frequency Estimation

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Introduction
Adaptive method-based frequency estimation can be found in many areas of digital signal processing applications [1,2], such as Doppler efect estimation of radar and sonar systems [3][4][5], clock and carrier synchronization in communication systems, angle of arrival estimation in smart antenna systems, frequency estimation in global navigation satellite systems (GNSS) [6], angle and frequency estimation in cognitive wireless systems [7][8][9], and so on.Depending on the statistical properties of the input signal frequency, say, deterministic or random, the frequency estimation can be classifed into two categories: block-based and sequentialbased estimation techniques.Te multiple signal classifcation (MUSIC) [9], the modifed covariance (MC) [10], the Pisarenko harmonic decomposition (PHD) [11], the reformed PHD [12], and the maximum likelihood estimation (ML) [13] are examples of block estimation that are used for estimating an unknown constant sinusoidal signal frequency.However, since the required computational cost of those techniques is high, many methods have been adopted to overcome this drawback [14,15].For the time-varying sinusoidal signal frequency estimation, sequential-based estimation techniques such as the three recursive leastsquares (RLS) algorithms [16] and the least-mean-square (LMS) family algorithm [17] is required.In [17], So and Ching proposed the real direct frequency estimation (RDFE) adaptive algorithm for a real tone in noise.Te RDFE is based on the linear prediction of real sinusoidal signals [18].Te RDFE is computationally efcient, and it provides unbiased and direct frequency measurements on a sequential basis.In the case of a complex sinusoidal signal frequency estimation, the block-based estimations [19][20][21][22] and the complex adaptive notch flter (CANF)-based adaptive algorithms [23][24][25][26] can be applied.In [23], the modifed complex plain gradient (MCPG) adaptive algorithm was adopted.It was found that the MCPG can improve convergence speed as compared with those of the Regalia method (Regalia) [24] and the complex plain gradient (CPG) algorithm [26] without increasing any computations.However, due to the pole contraction factor of the CANF, the performance of an adaptive algorithm-based adaptive ANF may be poor if the selected value of the pole factor is inappropriate.In [27], the linear prediction-based adaptive algorithm [17] is adopted for the general case of a complex sinusoidal signal, namely, a complex direct frequency estimation (CDFE).Te CDFE is an interesting algorithm because of its simplicity and efciency.However, it provides slow convergence speeds when the signal amplitude is low and is not robust to high impulsive noise.
In this work, we propose a very simple sequential phase error criterion (PEC-)-based adaptive algorithm to estimate the frequency of a complex sinusoid.An instantaneous phase of a complex linear predictor is evaluated at each time instant and used to be the driving function of the algorithm.Te proposed PEC does not require any external or internal signals; only the input and output signals of the system are required.In some conditions, the PEC gives a similar convergence time and MSE to those of the CDFE but is more robust to impulsive noise.In close form, convergence analysis for convergence in the mean of the estimated frequency and steady-state MSE under white Gaussian noise is derived.Extensive simulations under Gaussian and impulsive noise scenarios are evaluated to demonstrate the superiority of the proposed PEC.

Algorithm Derivation
It is assumed that the observation signal of the proposed algorithm takes the form of the following equation: where d(n) � Ae j(ω 0 n+φ) , A > 0, ω 0 ∈ (− π, π), and φ ∈ [− π, π] are, respectively, unknown amplitude, frequency, and phase.A and ω 0 can be constant or time varying whereas φ is uniformly distributed.v(n) � v r (n) + jv i (n) is a zeromean complex white Gaussian noise with variance σ 2 where v r (n) and v i (n) are uncorrelated real white processes with zero mean and identical variance of σ 2 /2.Te input signal to noise ratio (SNR i ) can be computed by SNR i � A 2 /σ 2 .Te objective of this work is to estimate ω 0 from the observation time series x(n), according to the following adaptive rule: where ω0 (n) is the estimate at time n of ω 0 , μ > 0 is the stepsize parameter controlling the speed of convergence; and D(n) is an instantaneous driving function that must satisfy the following criteria: where E[•] is an expectation operator.To explore the aspect of D that satisfes equation (3), we consider the linear prediction of the signal [27] s(n) � − e jω 0 s(n − 1), where the prediction error is defned by and is the estimate of s(n) and ω0 is adjusted using equation (2).Note that the signal s(n) in equation ( 4) and its estimated version ŝ(n) in equation ( 6) are defned as a negative value to obtain a linear phase prediction error signal e(n).Now let us defne a new phase error criterion (PEC) as follows: where the symbol ∠ is the phase operator, and the asterisk ( * ) stands for the complex conjugation.In practice the instantaneous linear phase D(n) can be calculated by where Im(•) is the imaginary part and Re(•) is the real part.
To study the behavior of D(n), we put x(n), e(n), and ŝ(n) into equation (7), yielding (see Appendix A) where δ ω � ω0 − ω 0 is an estimation error.It is obvious that ω0 � ω 0 is a stationary point of equation (10).To assert the validity of equation (10), the estimate of D based on Monte Carlo simulation is studied by using computer simulations.Te parameters used in simulations include ω 0 � 0.5π, φ � 0.2π, and the data length of L � 10 4 .Te driving function D as a function of frequency parameter ω0 for diferent values of SNR i are depicted in Figure 1.It is observed that the analytical and simulated results are well consistent for both low and high values of SNR i and they satisfy the desired criteria in equation (3).As a result, the convergence of equation ( 2) is ensured if the proposed driving function in equation ( 8) is employed.Moreover, the closed form expression for D in equation (10) can be used to study the convergence behaviour of equation ( 2) because it is consistent with the simulations.Finally, by substituting equation ( 8) into (2) the proposed PEC adaptive algorithm is ultimately derived as follows: 2 International Journal of Antennas and Propagation As has been observed, the proposed PEC algorithm is very simple and attractive because no internal and external additional signals are required to update the frequency parameter ω0 of the CLP; only x(n), e(n), and a phase evaluator ∠ are required.Figure 2 shows the block diagram of the proposed estimator.In the next section, the convergence properties of the proposed PEC is addressed.

Mean Analysis
At steady-state ω0 ≈ ω 0 the terms sin a| a⟶0 ≈ a and cos a| a⟶0 ≈ 1, equation (10) can be approximated to be At a stationary point, the term δ ω ≈ 0, therefore, arctan u| u⟶0 ≈ u, equation ( 12) can be simplifed to be In order to study the mean value of equation ( 11), the steady-state expression for D in equation ( 13) is substituted into equation (11).After taking the expectation operator and using δ ω � ω0 − ω 0 , then equation (11) becomes where Equation ( 14) is in the form of a frst-order timeinvariant diference equation in variable � ω 0 (n) whose solution is given by (see Appendix B) where � ω 0 (0) ≡ ω0 (0) is an initial value of the frequency parameter ω0 (n).Since μη 0 must be less than one, the term (1 − μη 0 ) ⟶ 0 as n ⟶ ∞, and equation ( 16) becomes

International Journal of Antennas and Propagation
It is revealed from equation ( 17) that the proposed PEC algorithm converges to the desired solution and is unbiased.In addition, the convergence time of equation ( 11) can be predicted by using the fact that the term (1 − μη 0 ) will exponentially decrease in time, which can indicate that where τ is defned as a time constant.Solving equation (18) for τ yields According to equation ( 19), the approximate convergence time of the proposed PEC will be Moreover, the stability bound of step-size in the mean sense can be easily obtained from equation ( 14) as follows: Te upper bound of step-size in equation ( 21) guarantees monotonic convergence in the mean sense.Te validity of equations ( 16), (17), and ( 20) is tested by using the Monte Carlo simulation technique.A random experiment of 1000 trials for a random phase signal and a random noise sequence with a particular variance is carried out.Te obtained 1000 frequency estimates are ensemble averaged to obtain the mean estimated frequency � ω 0 (n). Figure 3 shows the learning curves of � ω 0 (n) obtainable by equation ( 16) and simulations for SNR i � 0 and 10 dB, ω 0 � 0.5π, data length L � 1000, and μ � 0.05.Note that all samples of the selected parameters used in simulating are defned based on the trialand-error technique to obtain the best results.Te step-size μ is confned within the range of equation (21).It is seen that the analytical result for � ω 0 (n) shown in equation ( 16) can track those of the simulations very well in both low (0 dB) and high (10 dB) values of SNR i and converge to solution as desired.By using equation (20), the convergence time is L i ≈ 298 samples at SNR i � 0 dB and L i ≈ 208 samples at SNR i � 10 dB which are close to those obtained from the simulations.

Steady-State MSE Analysis
In this section, MSE of the frequency estimate ω0 (n) is analyzed.To do this task, the steady-state expression for the prediction error e(n) is required.Te input x(n) to the prediction error e(n) can be modelled by the following transfer function: Substituting z � e jω , ω ∈ [− π, π] in equation (22) yields If we replace ω ⇒ ω0 , equation (23) becomes At steady-state ω0 ≈ ω 0 , the magnitude and phase responses of H(z) can be, respectively, approximated to be and Terefore, the steady-state expression for the prediction error will be where s means steady-state and v 1 (n) is the noise component due to the input noise v(n).Now let us consider the learning increment where is the correlation between v(n) and v * 1 (n) (see Appendix C).Note that the derivation of equation ( 28) is obtained by assuming that the input complex sinusoids and noise components are uncorrelated with each other [28].Now, let us consider the SNR of Q, which is For a high value of SNR Q equation ( 28) can be approximated as [29] Q where Q(n) is the estimate of Q and v 2 (n) is defned as a phase noise of Q with zero mean and variance of [29] Te phase of Q(n) that is equivalent to the driving function D in equation ( 13) can be defned by Using equations (33) in (11) results in Subtracting ω 0 from both sides of equation (34) and using equation (26) yields Squaring on both sides of equation ( 35) and averaging the result gives 4 International Journal of Antennas and Propagation where then equation (36) becomes For slow convergence speed when μ ⟶ 0, equation (38) can be simplifed to be MSE ≈ μσ 2  2 . (39) It is apparent that the MSE approximation in equation ( 39) is valid for a wide range of system parameters, including input frequency, SNR i and step-size as demonstrated in Figures 4-6, respectively.It is observed that the analytical MSE shown in equation (39) can well predict the simulated MSE.It is observed that the MSE is independent of the signal frequency, as shown Figure 4, decreased as SNR i increased, as shown in Figure 5, and increased as step-size increased, as shown in Figure 6.

Numerical Examples
In this section, the performances of the proposed PEC have been revealed and compared with those of the MCPG [23], Regalia [24], CPG [26], and CDFE [27].

Simulation MSE and Estimated Frequency.
To fairly compare the MSE and the estimated frequency by using computer simulation, all examined algorithms are forced to converge at the same time.Tis is done by individually tuning the step-size parameter of each algorithm.In addition, the 1000 complex sinusoids with a specifc frequency and random phase plus noise sequences having the same SNR i are evaluated by using an ensemble average.Te results are shown in Figures 7-10. Figure 7 is an estimation MSE comparison for SNR i = 10 dB (high SNR i ).It is revealed that the proposed PEC and CDFE converge at the same time, and they provide an identical MSE of the estimated frequency, whereas the MCPG converges the fastest and the Regalia shows the slowest.Te CPG speed of convergence is slightly better than that of the Regalia but worse than those of the PEC, CDFE, and MCPG. Figure 8 demonstrates the estimation frequency at the same MSE.As can be seen, the PEC, CDFE, and MCPG provide almost the same convergence speed, whereas the CPG and Regalia show slow convergence speeds.Figure 9 is an estimation MSE comparison for SNR i = − 5 dB (low SNR i ).It is shown that the convergence speed of the proposed PEC is between those of the CDFE and MCPG and the CPG and Regalia.Figure 10 shows the estimation frequency at the same MSE.As has been observed, the PEC provides slower speed than CDFE and MCPG but yields faster speed than the CPG and Regalia.Note that the MCPG, CPG, and Regalia are used with a complex frstorder adaptive IIR notch flter (CANF) [26], which has a zero-pole contraction factor α. In this section, we let α � 0.9.Although the convergence speed of the PEC in the additive complex white Gaussian noise scenario is slow as compared with the MCPG and CDFE, as shown in Figures 9 or 10, it performs well when exposed to a high-impulsive noise environment, as shown in Section 5.2.In addition, the deterioration of convergence speed due to the signal amplitude A does not afect the PEC, as shown in Section 5.3.Moreover, in the comparison of the calculation requirements of all algorithms, the computational complexity is concluded in Table 1.It is found that the proposed PEC requires only 3L multiplications, 2L additions, and L phase evaluations.Te added phase calculation makes PEC tolerant to impulsive noise and insensitive to signal amplitude A, surpassing CDFE, MCPG, CPG, and Regalia.Now let us consider the computational time required for all examined algorithms.Since each algorithm has only one parameter (weight) to be adjusted, for the PEC, it requires six operations per iteration.Tis indicates that the PEC is considered to have a temporal complexity of order O(6L) when its iteration has an input size of L. Because runtime is dependent on input size L, it is said that the time complexity of the PEC has an order of O(L), meaning that it is linear.Similarly for the CDFE, MCPG, CPG, and Regalia, their computation times are also linear and are shown in Table 1.Although the runtime of each comparative algorithms has the same order as that of the PEC, the proposed PEC outperforms all International Journal of Antennas and Propagation comparative techniques in terms of impulsive noise robustness, and its convergence time is not sensitive to the input signal amplitude (see Figure 11).

Impulsive Noise Robustness.
In this section, the impulsive noise robustness of the proposed PEC algorithm is addressed.It is well known that impulsive noise has two main characteristics: random amplitudes and places of occurrence.When impulsive noise is present, the noise power is equal to the strength of the impulse.Tis illustrates the nonstationary character of impulsive noise by looking at the power spectrum of a noise process with a few impulses per second.Impulsive noise is therefore a binary-state, timevarying process, and as a result, its power spectrum and autocorrelation are also binary-state processes.Te expression for an amplitude-modulated binary-state sequence that models an impulsive noise sequence is [30].
where u(n) is a random noise amplitude and b(n) is a binary-state sequence of ones and zeros.Since the binary-state sequence b(n) takes a value of "1" with a probability of p and a value of "0" with a probability of 1 − p, its probability mass function (PMF) can be expressed as follows: which is in the form of a Bernoulli distribution whose mean is p and variance is p(1 − p) whereas the probability density function (PDF ) of u(n) can be modelled as a Gaussian distribution with a zero mean and variance of σ 2 u and is of the form Since u(n) and b(n) are independent random variables, the variance of I(n) can then be defned by (see Appendix D) It is noted that the mean of I(n) is equal to zero because u(n) has a zero mean.Terefore, equation ( 43) is also the impulsive noise power.Under the impulsive noise scenario, the observation signal in equation ( 1) can then be rewritten as follows:  International Journal of Antennas and Propagation Te signal to impulsive noise ratio (SINR) is defned by From equation (45), there are many pairs of p and σ 2 u that yield the same SINR.For a fxed value of σ 2 u , the lower the probability p is, the higher the SINR is achieved.Since the impulsive noise is nonstationary, it is difcult to theoretically study the proposed PEC algorithm under this noise.Terefore, experimentation by means of computer simulation is used to study the PEC performance under impulsive noise instead.Extensive simulations for low/high p and large σ 2 u will be carried out to demonstrate the impulsive noise robustness of the proposed PEC adaptive algorithm.Te     simulations are addressed by considering the following cases: (note that the step-size parameter μ of each algorithm is individually adjusted to obtain the same convergence speed).

C1 Low Probability of Occurrence and Large Noise
Amplitude Variance.For this case, we let p � 0.001 and σ 2 u � 10 5 (SINR = − 20 dB).Te results of the estimation are shown in Figures 12(a) and 12(b).An impulsive noise waveform is shown in Figure 12(a).Te estimated frequency obtained from the PEC, CDFE, MCPG, CPG, and Regalia are shown in Figure 12(b).It is evident that the proposed PEC is robust to very high impulsive noise, whereas the CDFE, MCPG, CPG, and Regalia sufer from impulsive noise; namely, they are not robust.

C2 High Probability and Large Noise Amplitude
Variance.For this case we let p � 0.1 and σ 2 u � 10 5 (SINR = − 40 dB).Te results of the estimation are shown in Figure 13(a) and 13(b).An impulsive noise waveform is shown in Figure 13(a).Te estimated frequency obtained from the PEC, CDFE, MCPG, CPG, and Regalia are shown in Figure 13(b).It is evident that the proposed PEC is robust to very high impulsive noise, whereas the CDFE, MCPG, CPG, and Regalia sufer from impulsive noise; namely, they are unstable.

Efect of Signal Amplitude.
To study the deterioration of convergence speed due to the signal amplitude A, a noise free scenario is assumed.Te parameters used in simulation are A = {0.2,1}, ω 0 � 0.5π, φ � 0.1π, μ � 0.05, α � 0.7 (for MCPG, CPG, Regalia), and single run.Te results are shown in Figure 11.As can be seen, the decrease in input signal amplitude does not afect the convergence speed of the proposed PEC (see Figure 11

Conclusion
We have proposed a phase error criterion adaptive algorithm for estimating the unknown frequency of a complex sinusoidal signal.Te proposed technique provides two main advantages: it is robust to impulsive noise and not very sensitive to the signal amplitude.Te convergence in the mean of the estimated frequency and steady-state expression

Figure 1 :Figure 2 :
Figure 1: Expected value of the PEC driving function D as a function of frequency parameter for (a) SNR i � 0 dB; (b) SNR i � 10 dB.
(a)), whereas the decrease in signal amplitude results in the deterioration of the convergence speed of the CDFE, MCPG, CPG, and Regalia (see Figures11(b)-11(e)).

Table 1 :
Te computational complexity and computational time.