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Fig. 1. Theoretical performance of SLN estimate é for different values of 7, with (a) p = 0.2 and (b) p = 0.9.

Figure 1 Theoretical performance of SLN estimate é for different values of 7, with (a) p = 0.2 and (b) p = 0.9.

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correlations of the nonstationary process x(n) are given for k = 0,...,P —1 by the expressions [3], [14] where H,(F') denotes the continuous-time Fourier transform of h(t), and the notation 6(-) stands for the Kronecker’s delta. The frequencies k/P (or simply k), for & = 0,...,P — 1, are referred to as cyclic frequencies or cycles [3]. Due to the symmetry property of the raised-cosine function h,(t), it is easy to check that G(k;7) is real-valued, and G(k;7) (thus, Ro,(k;7)) is nonzero only for cycles k = 0,+1. Since Rox(k;T) = exp{2iakt /P}R3,.(—k; —7), it follows that only the subset {Ro,(1;7)}, V7, represents all the second-order statistics that may be used for estimating the unknown timing epoch e. In practice, the cyclic correlations R2.,(k; 7) have to be estimated from a finite number of samples N, and the standard sample estimate of R2,(k;7) is given by (see, e.g., [1] and [3])
correlations of the nonstationary process x(n) are given for k = 0,...,P —1 by the expressions [3], [14] where H,(F') denotes the continuous-time Fourier transform of h(t), and the notation 6(-) stands for the Kronecker’s delta. The frequencies k/P (or simply k), for & = 0,...,P — 1, are referred to as cyclic frequencies or cycles [3]. Due to the symmetry property of the raised-cosine function h,(t), it is easy to check that G(k;7) is real-valued, and G(k;7) (thus, Ro,(k;7)) is nonzero only for cycles k = 0,+1. Since Rox(k;T) = exp{2iakt /P}R3,.(—k; —7), it follows that only the subset {Ro,(1;7)}, V7, represents all the second-order statistics that may be used for estimating the unknown timing epoch e. In practice, the cyclic correlations R2.,(k; 7) have to be estimated from a finite number of samples N, and the standard sample estimate of R2,(k;7) is given by (see, e.g., [1] and [3])
where « stands for the kurtosis of w(l) and So,(k; f) := >>, Rox(k; 7) exp{—2iar f} denotes the cyclic spectrum of x(n) at cyclic frequency k. respectively, whose closed-form expressions of the (u,v) en- tries, u,U = —Tm,---;Tm, are established in [14], and with the notations adopted so far, can be expressed as Next, we transform the complex Gaussian probability density function (pdf) CN'(0, T° T) into its equivalent algebraic form of the real Gaussian pdf f.(U2,.) by defining the (47), + 2) x 1-di- mensional vector: Us, := [re(Rex)! im(Roz)” 1’. Simple calculations show that the covariance matrix of Uo, is given by
where « stands for the kurtosis of w(l) and So,(k; f) := >>, Rox(k; 7) exp{—2iar f} denotes the cyclic spectrum of x(n) at cyclic frequency k. respectively, whose closed-form expressions of the (u,v) en- tries, u,U = —Tm,---;Tm, are established in [14], and with the notations adopted so far, can be expressed as Next, we transform the complex Gaussian probability density function (pdf) CN'(0, T° T) into its equivalent algebraic form of the real Gaussian pdf f.(U2,.) by defining the (47), + 2) x 1-di- mensional vector: Us, := [re(Rex)! im(Roz)” 1’. Simple calculations show that the covariance matrix of Uo, is given by
Since ¢; and ¢ do not depend on P, MSE(é) is independent of P whenever P > 3. One can observe that in the noiseless case (No — 0), G1 = a, therefore, the asymptotic variance of the O&M estimate is equal to zero, which means that asymptotically in SNR and N, the variance of the O&M estimate converges to zero faster than O(1/N). MSE of é€. Under the assumption of P > 3, some straightfor- ward calculations yield MSE(é) = 8(¢, — C2)/m?p?L, where
Since ¢; and ¢ do not depend on P, MSE(é) is independent of P whenever P > 3. One can observe that in the noiseless case (No — 0), G1 = a, therefore, the asymptotic variance of the O&M estimate is equal to zero, which means that asymptotically in SNR and N, the variance of the O&M estimate converges to zero faster than O(1/N). MSE of é€. Under the assumption of P > 3, some straightfor- ward calculations yield MSE(é) = 8(¢, — C2)/m?p?L, where
TABLE I Fig. 2. MSE of the O&M estimator versus P (9 = 0.5, SNR = 10 dB). totic variance of € in (13) is minimized. By adopting the deriva- tion presented in Section III, one can obtain the closed-form ex- pression for the asymptotic variance of € in (13), and hence, a, can be found accordingly. Due to the space limitations, their de- tailed derivations will not be presented in this letter, and the in- terested reader is suggested to consult [15]. The most commonly used one is the FLN nonlinearity, and the corresponding estimator takes the form
TABLE I Fig. 2. MSE of the O&M estimator versus P (9 = 0.5, SNR = 10 dB). totic variance of € in (13) is minimized. By adopting the deriva- tion presented in Section III, one can obtain the closed-form ex- pression for the asymptotic variance of € in (13), and hence, a, can be found accordingly. Due to the space limitations, their de- tailed derivations will not be presented in this letter, and the in- terested reader is suggested to consult [15]. The most commonly used one is the FLN nonlinearity, and the corresponding estimator takes the form
In this section, we conduct computer simulations to confirm the analysis presented in the previous sections and to illustrate the performance of the proposed OPT estimator. All the ex- perimental results are obtained by performing a number of 10° Monte Carlo trials assuming a QPSK constellation, and the nor- malized timing epoch € = 0.3. Unless otherwise noted, the over- sampling rate P = 4 is adopted.
In this section, we conduct computer simulations to confirm the analysis presented in the previous sections and to illustrate the performance of the proposed OPT estimator. All the ex- perimental results are obtained by performing a number of 10° Monte Carlo trials assuming a QPSK constellation, and the nor- malized timing epoch € = 0.3. Unless otherwise noted, the over- sampling rate P = 4 is adopted.
Fig. 4. Improvement of OPT over SLN versus p (SNR = 20 dB). mator. Fig. 4 shows the improvement exhibited by the OPT es- timator with respect to the SLN estimator versus rolloff factor p assuming SNR = 20 dB. It appears that the improvement is negligible whenever p > 0.6.
Fig. 4. Improvement of OPT over SLN versus p (SNR = 20 dB). mator. Fig. 4 shows the improvement exhibited by the OPT es- timator with respect to the SLN estimator versus rolloff factor p assuming SNR = 20 dB. It appears that the improvement is negligible whenever p > 0.6.
Fig. 5. Comparison of MSEs with (a) p = 0.1 and (b) p = 0.9.
Fig. 5. Comparison of MSEs with (a) p = 0.1 and (b) p = 0.9.
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