Blind SNR Estimation with Coherent Function for OFDM Systems
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(2) IEICE TRANS. COMMUN., VOL.E91–B, NO.11 NOVEMBER 2008. 3754 L−1 1 ˜ 2 Xl,k − N MMS E L l=0. (7). L−1 2 1 ˜ = Xl,k − H l,k Xl,k L l=0. (8). . S MMS E = . N MMS E . . where S MMS E and N MMS E are MMSE estimates of the signal and AWGN powers, respectively. As can be seen, the ML and MMSE estimators need the estimate of the CFR and the knowledge of the transmitted data; therefore, it is data-aided. This also means that the performances of these SNR estimators are dependent on the channel estimation algorithms. Moreover, when H l,k is derived from the often used least-square (LS) channel estimation, both (5) and (8) will lead to zero, and the SNR estimation is invalidated.. found that, in [6], the SNR estimation of a real-valued single-carrier signal is useful due to its simplicity. It was derived by utilizing the 2nd and 4th moments of the envelope of received frequency-domain data in slow-fading channels. Besides [1], [6] and [9] are also based on high-order moments. We generalize and extend the technique in [6] to the complex-valued signals for the OFDM systems. Since the derivation is similar to [6], it is omitted here. The key parameters and results are written as follows. First, can one define the following 2nd and 4th moments of X˜ l,k at the k-th subcarrier based on time sample averages: 2 Δ 1 X˜ , M= (15) l,k L l 4 Δ 1 X˜ , Q= (16) l,k L l and. 3.2 Error Vector Magnitude (EVM) Algorithm. . The EVM [2] is referred to the magnitude of the error vector and can be expressed as . . ρEV M = . S EV M . N EV M. S EV M . N EV M ⎞2 ⎛ K−1 ⎟⎟⎟ ⎜⎜⎜ 1 ⎜ = ⎜⎜⎝ X l,k,I/Q ⎟⎟⎟⎠ N k=0. ⎞2 ⎛ K−1 K−1 2 ⎜⎜ 1 ⎟⎟⎟ 1 ⎜⎜⎜ = X l,k,I/Q − ⎜⎝ X l,k,I/Q ⎟⎟⎟⎠ N k=0 N k=0. . . N BA L−1 1. S BA = . N BA . (12). . L. ∗ X˜ l,k X˜ l+1,k. (13). l=0. L−1 1 ˜ 2 Xl,k − S BA = L l=0. (14). . where S BA and N BA are Boumard’s estimates of the signal and AWGN powers, respectively. 3.4 Second- and Fourth-Order Moments (M2 M4 ) Algorithm After extensive survey of SNR estimation techniques, we. S MA. (18). . N MA 1/2 2 = M +R−Q . . N MA = M − S MA . . ρBA =. . S MA . In [5], Boumard utilizes the characteristic of the pilot subcarriers; therefore, it is data-aided, and the SNR at the k-th subcarrier can be expressed as follows: S BA. ρ MA =. (11). 3.3 Boumard’s Algorithm. (17). . . (10). where S EV M and N EV M are EVM estimates of the signal and AWGN powers, respectively; and X l,k,I/Q is the estimate of the I/Q component of the transmitted data.. 2 1 ˜ 2 ˜ Xl,k Xl+1,k . L l. Then, the M2 M4 SNR estimate can be written as. (9). . . Δ. R=. (19) (20). . where S MA and N MA are M2 M4 estimates of the signal and AWGN powers, respectively. 3.5 Discussion of Conventional Techniques Summarizing the introduced techniques, first, it is observed that the ML and MMSE estimators need the channel estimates and the knowledge of the transmitted data in addition to the received data. Second, the EVM estimator needs the estimates of the transmitted data. Third, the Boumard’s estimation needs the received data and the pilot subcarriers. Finally, the M2 M4 estimation is NDA and only needs the received data. 4.. Proposed SNR Estimation. The coherence function [7] is often used to determine the degree to which one observed phase-shift keying (PSK) signal is related to another observed signal. If we have uncorrelated noise and the coherent signal, the amount of the powers due to noise and coherent signal can be determined. The concept of coherence function can be applied to the OFDM systems, so that the SNR, at the k-th subcarrier, can be written as 2 γ k ρk = (21) 2 1 − γk .
(3) LETTER. 3755. where 2 Δ γk =. 2 E X˜ l,k X˜ ∗ 2 l+1,k , k ∈ P, 0 ≤ γ ≤ 1 (22) 2 2 E X˜ l,k E X˜ l+1,k . is the magnitude-squared-coherence (MSC) at the k-th subcarrier of the l-th symbol, and P is the pilot set. The MSC is the cross-power spectrum of two consecutive symbols normalized by their auto-power spectra, which can be viewed as the correlation between like frequency components of the received frequency-domain data. Note that ρk can be averaged over subcarriers to improve its accuracy. Originally, the MSC in [7] was realized by the commonly-used technique of time sample average, which is based on the assumption of signal ergodicity. Unfortunately, it is only applicable for data-aided (DA) systems. In OFDM systems, since there is only one data sample at each pilot subcarrier within an OFDM symbol, and the number of pilot subcarriers is a fraction of N, say N/32, the MSC might suffer from large estimation error and long acquisition time. Therefore, to blindly estimate the MSC without known data (or pilots) and derive the SNR estimate accurately in reasonable amounts of symbols, we propose to average the MSC over all N subcarriers within an OFDM symbol as follows ⎛ ⎞ ⎟⎟⎟2 ⎜⎜⎜ 1 ∗ X˜ l,k X˜ l+1,k ⎟⎟⎠ ⎜⎜⎝ N k 2 γ = ⎛ ⎞⎛ ⎞ (23) 2 ⎟⎟⎟ ⎜⎜⎜ 1 2 ⎟⎟⎟ ⎜⎜⎜ 1 X˜ l,k ⎟⎟⎠ ⎜⎜⎝ X˜ l+1,k ⎟⎟⎠ ⎜⎜⎝ N k N k where the bar on top of γ denotes the sample average over N subcarriers. Finally, the SNR can be obtained similar to (21) by 2 γ ρ = 2 . 1 − γ . (24). It should be noted that the proposed method has the same property as the M2 M4 estimation, i.e., NDA and requiring only the received data for the SNR estimation. 5.. Simulations and Discussions. Monte Carlo simulations are conducted to evaluate the performance by means of the estimation’s normalized meansquared error (NMSE) ⎤ ⎡⎛ ⎢⎢⎢⎜⎜ ρ − ρth ⎞⎟⎟2 ⎥⎥⎥ Δ NMSE = E ⎢⎢⎢⎣⎜⎜⎝ th ⎟⎟⎠ ⎥⎥⎥⎦ (25) ρ . where ρ is the SNR estimate. The following parameters of an OFDM system are assumed: there are N = 256 subcarriers; the guard interval is NG = N/8 = 32 samples; the simulated modulation scheme is quadrature phase-shift keying (QPSK); the signal bandwidth is 2.5 MHz; the radio. Fig. 1. SNR estimations in the AWGN channel.. frequency is 2.4 GHz; the subcarrier spacing is 8.68 kHz; the OFDM symbol duration is 115.2 μs; and there are 8 pilot subcarriers. In each simulation run, 1,000 symbols are tested. All the results are obtained by averaging over 2,000 independent channel realizations. To verify the performance of the proposed technique, we compare the proposed method with the following techniques: EVM, Boumard’s, original MSC (21), and M2 M4 in the AWGN and multipath fading channels. 5.1 Performance in the AWGN Channel The performance in the AWGN channel is shown in Fig. 1. As can be seen, the Boumard’s has the best performance when the SNR ranges from very low (< 0 dB) to low (0– 10 dB) values; and the proposed method has the best performance when the SNR ranges from moderate (10–25 dB) to high (> 25 dB) values. In addition, the performance of the proposed method consistently improves when the SNR increases. 5.2 Performance in Multipath Channels To verify the performance of the proposed technique in multipath channels, the channel is assumed to have NG paths. The channel taps are randomly generated by independent zero-mean, complex Gaussian variables with τ σ2hτ = 1, where σ2hτ = E[|h(τ)|2 ] is the power of the τ - th channel tap. The performances of the proposed method in timevariant and — invariant multipath channels are shown in Fig. 2. For clarity, we only show the performances of conventional methods in time-invariant multipath channels. The normalized Doppler frequency in time-variant channels is 0.035 which corresponds to a maximum Doppler frequency of 303.8 Hz, i.e., a mobile speed of 117 km/hr. The theoretical SNR in time-invariant channels is the true SNR. In contrast, in time-variant channels, besides AWGN, the received signal also suffers from inter-carrier interference (ICI). Therefore, with the result in [10], the theoretical SNR can be shown to be.
(4) IEICE TRANS. COMMUN., VOL.E91–B, NO.11 NOVEMBER 2008. 3756. from moderate to high SNRs that we are interested in. Therefore, we can say that the proposed method has the best performance in typical SNR conditions. In addition, the proposed method is less sensitive to multipath fading channels; hence, the proposed method can be used for practical environments. References. Fig. 2. SNR estimations in the multipath fading channel. N−1 . N+2. (N − Δn ) J0 (βΔn ). Δn =1. ρth =. N(N − 1) − 2. N−1 Δn =1. (26) (N − Δn ) J0 (βΔn ) +. 2σ2ν. where J0 (·) is the zeroth-order Bessel function of the first Δ kind; β = 2π fd T/N, fd represents the maximum Doppler shift, T is the symbol duration; and Δn is the time difference due to time selectivity of channels. As can be seen, the performance of the proposed method in multipath channels is similar to that in AWGN channels and is only slightly lower than it. In addition, the performance of the proposed method in time-variant channels is also slightly lower than that in time-invariant channels. 5.3 Discussions In summary, the proposed method has the best performance. [1] D.R. Pauluzzi and N.C. Beaulieu, “A comparison of SNR estimation techniques for the AWGN channel,” IEEE Trans. Commun., vol.48, no.10, pp.1681–1691, Oct. 2000. [2] D. Shin, W. Sung, and I. Kim, “Simple SNR estimation methods for QPSK modulated short bursts,” Proc. IEEE GlobeCom 2001, vol.6, pp.3644–3647, 2001. [3] R.M. Gagliardi and C.M. Thomas, “PCM data reliability monitoring through estimation of signal-to-noise ratio,” IEEE Trans. Commun., vol.COM-16, no.3, pp.479–486, June 1968. [4] S.M. Kay, Fundamentals of statistical signal processing — Estimation, Publishing House of Electronics Industry, Beijing, 2003. [5] S. Boumard, “Novel noise variance and SNR estimation algorithm for wireless MIMO OFDM systems,” Proc. IEEE GlobeCom 2003, pp.1330–1334, 2003. [6] T.R. Benedict and T.T. Soong, “The joint estimation of signal and noise from the sum envelope,” IEEE Trans. Inf. Theory, vol.IT-13, no.3, pp.447–454, July 1967. [7] R.W. Lowdermilk and F.J. Harris, “Synthetic instruments extract masked signal parameters,” IEEE Instrum. Meas. Audio Mag., vol.8, no.3, pp.40–46, Aug. 2005. [8] M. Speth, S. Fechtel, G. Fock, and H. Meyer, “Optimum receiver design for wireless broad-band systems using OFDM-part I,” IEEE Trans. Commun., vol.47, no.11, pp.1668–1677, Nov. 1999. [9] R. Matzner, “An SNR estimation algorithm for complex baseband signals using higher-order statistics,” Facta Universitatis (Nis), no.6, pp.41–52, 1993. [10] J. Li and M. Kavehrad, “Effects of time selective multipath fading on OFDM systems for broadband mobile applications,” IEEE Commun. Lett., vol.3, no.12, pp.332–334, Dec. 1999..
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