 Research
 Open Access
Two accurate phasedifference estimators for dualchannel sinewave model
 Hing Cheung So^{1}Email author and
 Zhenhua Zhou^{1}
https://doi.org/10.1186/168761802013122
© So and Zhou; licensee Springer. 2013
 Received: 30 August 2012
 Accepted: 16 June 2013
 Published: 27 June 2013
Abstract
Two algorithms for estimating the phasedifference between two real sinusoids with common frequency in the presence of white noise are proposed. The first estimator utilizes the maximum likelihood criterion to find the phase of each channel output in a separable manner, and the phaseshift estimate is then given by their difference. The algorithm extension to unknown frequency and/or noise powers is also studied. On the other hand, the development of the second method is based on the linear prediction approach with a properly chosen sampling frequency. Furthermore, variance expressions for the two estimators are derived. Computer simulations are included to corroborate the theoretical calculations and to contrast the performance of the proposed schemes with several existing phasedifference estimators as well as CramérRao lower bound.
Keywords
 Phasedifference; Maximum likelihood estimation; Linear prediction; Time delay estimation; Sinewave model
1 Introduction
The A _{ i }>0 and ϕ _{ i }∈ [0,2π) denote the sinusoidal amplitude and initial phase at the ith channel, respectively, while ω=Ω T∈ [0,π) is the common frequency with Ω and T being the frequency of the continuoustime counterpart and sampling period. The additive noises v _{1}(n) and v _{2}(n) are uncorrelated white Gaussian processes with variances ${\sigma}_{1}^{2}$ and ${\sigma}_{2}^{2}$. The task is to estimate the phaseshift, denoted by θ=ϕ _{1}−ϕ _{2}, from x _{1}(n) and x _{2}(n).
Handel [4] has proposed the nonlinear least squares approach for (1) and (2), where there is an additional DC offset term at each channel output, which corresponds to the estimation of seven unknown parameters. Alternatively, the seven unknowns can also be solved by the ellipsefitting [5] technique. When Ω in (2) is known, we have proposed the unbiased quadratic delay estimator (UQDE) and discretetime Fourier transformbased method in [6] for accurately estimating θ. The derivation of both algorithms is based on utilizing the inphase and quadraturephase components of {s _{ i }(n)}, but they can give optimum estimation performance only when π/(2Ω T)=π/(2ω) is a positive integer. Using the idea of [6], the modified simple algorithm (MSAL) [3] has recently been developed for phasedifference estimation even if π/(2ω) is not an integer.
The main contribution of this work is to develop and analyze two accurate phaseshift estimation approaches for the signal model of (1) and (2). In Section 2, the maximum likelihood (ML) estimator for θ is presented. It is proved that its variance is equal to the CramérRao lower bound (CRLB) in the presence of white Gaussian noises. We also extend the ML algorithm to the scenarios of unknown frequency and/or noise powers. By properly choosing T such that ω=π/2, a linear prediction (LP)based phasedifference estimator is derived and its performance is investigated in Section 3. Section 4 contains numerical examples for corroborating our theoretical development and comparing the performance of the ML and LP algorithms with the UQDE and MSAL, as well as CRLB. Finally, conclusions are drawn in Section 5.
2 Maximum likelihood estimator
where ${\widehat{\alpha}}_{i}$ and ${\widehat{\beta}}_{i}$ are the ML estimates of α _{ i } and β _{ i }, respectively. Solving (6) and (7), we obtain:
Note that this estimation procedure corresponds to a simplified form for the ML formulation of the sevenparameter model [4] with unknown frequency and additional DC offsets.
where $C={\sum}_{n=1}^{N}\stackrel{2}{sin}(\mathrm{n\omega})\xb7{\sum}_{n=1}^{N}\stackrel{2}{cos}(\mathrm{n\omega})({\sum}_{n=1}^{N}$ sin(n ω)cos(n ω))^{2}, ${C}_{i}=\stackrel{2}{cos}({\varphi}_{i}){\sum}_{n=1}^{N}\stackrel{2}{sin}(\mathrm{n\omega})+\stackrel{2}{sin}({\varphi}_{i})$ ${\sum}_{n=1}^{N}\stackrel{2}{cos}(\mathrm{n\omega})+2sin({\varphi}_{i})cos({\varphi}_{i}){\sum}_{n=1}^{N}sin(\mathrm{n\omega})cos(\mathrm{n\omega})$, i=1,2, which is identical to the CRLB (see Appendix 2).
where ${\text{SNR}}_{i}={A}_{i}^{2}/(2{\sigma}_{i}^{2})$, i=1,2, is the signaltonoise ratio at the ith channel.
Once $\widehat{\omega}$ is obtained by solving (19), the phasedifference is estimated as in (8) to (11).

Step 1. Set r=1.

Step 2. Compute $\widehat{\omega}$ using (19).

Step 3. Use $\omega =\widehat{\omega}$ to estimate the noise powers as ${\sigma}_{i}^{2}={\mathbf{x}}_{i}^{T}\left({\mathbf{I}}_{N}\mathbf{\Xi}{({\mathbf{\Xi}}^{T}\mathbf{\Xi})}^{1}{\mathbf{\Xi}}^{T}\right){\mathbf{x}}_{i}/N$, i=1,2, where I _{ N } is the N×N identity matrix.

Step 4. Repeat steps 2 and 3 until a stopping criterion is reached.

Step 5. Compute the phasedifference estimate $\widehat{\theta}$ based on (8) to (11).
3 Linear prediction estimator
where E denotes the expectation operator, A=[Toeplitz([−c _{1} 0 _{1×(N−2)}]^{ T },[−c _{1} −c _{2} 0 _{1×(N−3)}])] with 0 _{ M×N } being the M×N zero matrix. Moreover, Toeplitz (u,v ^{ T })stands for the Toeplitz matrix with u and v ^{ T } being the first column and first row, respectively.

Step 1. Set W=I _{ N−1}.

Step 2. Compute $\widehat{\mathbf{c}}$ using (25).

Step 3. Use $\mathbf{c}=\widehat{\mathbf{c}}$ to construct W.

Step 4. Repeat steps 2 and 3 until a stopping criterion is reached.

Step 5. Compute the phasedifference estimate as $\widehat{\theta}={tan}^{1}({\u0109}_{1}/{\u0109}_{2})$.
When N→∞, (29) is equivalent to (13). That is, the performance of the LP estimator is optimum in the asymptotic sense. Note that when r is not known a priori, we can use the procedure in Section 2 to perform the noise level estimation.
4 Simulation results
Extensive computer simulations are carried out to evaluate the mean square error (MSE) performance of the two proposed phasedifference estimators by comparing with the UQDE and MSAL as well as CRLB. The validity of the theoretical calculations of (12), (13), and (29) is also investigated. For the LP method, we use the number of iterations, which is selected to be 15, as the stopping criterion. The amplitude and phase parameters are assigned as α _{1}=3, α _{2}=1, ϕ _{1}=2, and ϕ _{2}=1. The noises {v _{1}(n)} and {v _{2}(n)} are uncorrelated zeromean white Gaussian processes with variances ${\sigma}_{1}^{2}$ and ${\sigma}_{2}^{2}$, respectively. Unless stated otherwise, N=10, ${\sigma}_{1}^{2}=0.5{\sigma}_{2}^{2}$, and the frequency as well as power ratio are assumed known. All the results provided are averages of 500 independent runs.
5 Conclusion
Two algorithms for accurate phasedifference estimation between two discretetime realvalued sinusoids with common frequency have been developed and analyzed. The first estimator first computes the ML solution for phase at each channel output, and the phaseshift is given by the difference between the two ML estimates. We have also extended the method to work for scenarios when the frequency and/or noise powers are unknown. When the discretetime frequency is properly chosen as π/2, one channel output can be represented as a linear combination of another channel output, where the corresponding LP coefficients have simple relationship with the phasedifference parameter. The second estimator utilizes this LP relationship and applies the weighted least squares for phasedifference estimation. The variance expressions for the two methods are derived and confirmed by computer simulations. It is shown that the ML and LP estimators perform comparably with conventional methods when the frequency is equal to π/2. For other frequencies, even if they are unknown, the ML algorithm can still achieve optimum performance when the noise is sufficiently small. Nevertheless, the proposed algorithms are more computationally demanding, particularly for a larger data length.
Appendices
Appendix 1
where cov(l _{1},l _{2})denotes the covariance of l _{1} and l _{2}.
Substituting the corresponding entries of (33) into (32) yields (12).
Appendix 2
Substituting (40) into (39) yields (12).
Declarations
Authors’ Affiliations
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