Shrinkage-based biased signal-to-noise ratio estimator ... - IEEE Xplore
IET Communications Research Article
Shrinkage-based biased signal-to-noise ratio estimator using pilot and data symbols for linearly modulated signals
ISSN 1751-8628 Received on 1st October 2014 Revised on 6th January 2015 Accepted on 9th February 2015 doi: 10.1049/iet-com.2014.0943 www.ietdl.org
Chee-Hyun Park, Soojeong Lee, Joon-Hyuk Chang ✉ School of Engineering Hanyang University, Seoul 133-791, Korea ✉ E-mail: [email protected]
Abstract: In diverse engineering problems including wireless communications, the estimate of the signal-to-noise ratio (SNR) is required. In this study, the authors develop a shrinkage-based SNR estimator in the data-aided and non-dataaided schemes for higher M-ary phase-shift-keying (M ≥ 8) and quadrature amplitude modulations. The observed Cramér-Rao lower bound is used as the variance of the expectation maximisation estimator to determine the optimal shrinkage factor. Simulation results show that the normalised mean-squared error of the proposed method is lower than that of the expectation maximisation method for low and moderate SNR conditions.
1
Introduction
Estimation of the signal-to-noise ratio (SNR) is important in modern communication systems since various systems use techniques that depend on an SNR estimate for proper operation. For example, both the determined signal power and the noise power are required in turbo decoders. Also, rate-adaptive transmission systems to adapt the modulation scheme or the coding rate use the SNR estimation. The types of the SNR estimation can be classified into data-aided (DA) and non-data-aided (NDA) methods. DA estimators rely on the insertion of pilot symbols to the data frame, whereas NDA estimators ignore the statistical information about the transmitted data, which typically leads to poor performance when the SNR is low. Several studies related to the SNR estimation exist. Pauluzzi and Beaulieu [1] investigated the SNR estimators of baseband M-ary phase-shift-keying (MPSK) TxDA and RxDA symbols. Alagha [2] also derived a Cramér-Rao lower bound (CRLB) for SNR estimation with binary phase-shift-keying (BPSK) and quadrature phase-shift-keying (QPSK) modulation schemes. A significant difference between the bounds for the NDA and DA methods was observed for low SNR conditions; however, both methods showed identical performances in high SNR regimes. Gappmair [3] extended Alagha’s work to be usable for any two-dimensional M-ary phase-shift-keying (PSK) systems and linear modulation schemes. This allows for SNR estimation based on an NDA estimator by exploiting the axis/half plane symmetry of conventional modulation schemes although this method still adopts a numerical method for the computation of integrals. Bellili et al. [4] developed an exact closed-form CRLB for the NDA SNR estimator in square quadrature-amplitude modulation (QAM) formulation. Chen and Beaulieu [5] derived a maximum likelihood (ML)-based SNR estimator using pilot and data symbols simultaneously; however, this algorithm is restricted to the BPSK modulation scheme. Gappmair et al. [6] developed the expectation maximisation (EM)-based SNR estimation algorithm for linearly-modulated signals. Das and Rao [7] proposed the SNR estimation algorithm for multiple-input–multiple-output communications systems. Bellili et al. [8, 9] dealt with the ML-based SNR estimation method over the fast time-varying channels. Fu et al. [10] utilised the Kolmogorov–Smirnov test to estimate SNR for multilevel constellations. Also, Chitte et al. [11] adopted the biased estimation to improve the ML distance estimator and Park et al. [12] derived the shrinkage-based closed-form DA-NDA SNR estimation algorithm for BPSK and
QPSK modulation schemes. In statistics and signal processing, the constraint of unbiasedness is often a practical issue, because the mean-squared error (MSE) can be minimised with the MSE criterion, which does not depend on the unknown true parameter. However, in some problems, restricting attention to unbiased estimation leads to unreasonable results [13, 14]. The variance can be decreased at the cost of increasing the bias, while ensuring the reduced overall estimation error. Hence, unbiasedness does not necessarily lead to a low estimation error [15] and the estimator is typically determined using the biased estimation in which a trade-off between variance and bias is utilised. Biased estimation has been used in a various signal processing problems, for example, image restoration, smoothing algorithms in time series analysis, spectrum estimation, wavelet denoising and so on. In this paper, we propose a shrinkage-based biased estimator in the context of SNR estimation for higher MPSK modulation (M ≥ 8) and QAM schemes. While the variance of the EM estimate is needed to find the optimal shrinkage factor used in the biased estimation, the variance of the EM estimator in higher MPSK modulation (M ≥ 8) and QAM schemes cannot be obtained. As the EM estimator is an iteration-based ML estimator, the variance of the EM estimator would be asymptotically equal to the CRLB. However, the CRLB of the SNR estimator cannot be found in practice since the true parameters are unknown. To resolve this problem, the observed CRLB (OCRLB) is utilised, where the EM estimate is inserted into the unknown true parameter as in [16, 17]. We show that the OCRLB is similar to the variances of the higher MPSK modulation and QAM schemes via simulation results. As a result, the OCRLB can be used instead of the variance of the EM estimator and thus the optimal shrinkage factor based on the variance of the EM estimator can be determined. It is discovered that the proposed estimator outperforms the conventional EM-based estimator in low and moderate SNR regimes in terms of the MSE under various experimental conditions.
2 EM-based SNR estimation using the data symbols Throughout this paper, it is assumed that the symbol timing and phase have been perfectly recovered. The data symbols ck’s (k = 1, …, L) are modelled as the zero mean, independent and identically distributed symbols ck ∈ C with unit variance E[|ck|2] = 1, where C denotes the M-ary symbol alphabet. The received samples zk are IET Commun., 2015, Vol. 9, Iss. 11, pp. 1388–1395
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then given by
estimators, and thus the SNR estimate is obtained as follows zk = h · ck + nk ,
k = 1, . . . , L
(1)
gˆ EM =
where h is the positive channel gain and nk is the additive white Gaussian noise (AWGN). Real and imaginary parts of the AWGN samples are assumed to be independent, each with the same variance s2n /2. The EM algorithm is performed to find the SNR estimate [6], which can be divided into the following two procedures: (2) E-step: Calculate the conditional expected value of the log-likelihood function with respect to ci given zk using the current estimate of Φ such that ˆ (q−1) ) = Q(F|F
L
ˆ (q−1) , z ] Ec [ log fz (zk |F, ci )|F k
(2)
k=1
(3) M-step: Find the parameter that maximises (2), where F = (h, s2n ), Ec[·] denotes the expectation with respect to ci ∈ H, H denotes the subset of symbols in C located in the right or the left half of the complex plane, and q = 0, 1, 2, …, qmax is the iteration number. fz(·) is the probability density function (pdf) of zk given by fz (zk |F, ci ) ≃
1 −(|zk |2 −2h|Re[zk c∗i ]|+h2 |ci |2 )/s2n e 2ps2n
(3)
The conditional expectation in E-step can then be represented by using the high SNR assumption and Bayes theorem as follows ˆ (q−1) ) = L{−log 2p − log s2 − Q(F|F n
1 s2n
× (M2 − 2hA(q−1) + h2 B(q−1) )}
The rigorous proof of this ML estimator for SNR was given in [18] by showing that the second-order derivative for γ is negative to verify whether gˆ EM truly maximises the likelihood function in the value of (6).
3 Shrinkage method-based SNR estimation using pilot and data symbols The data symbols ck’s (k = 1, …, L) are given in the previous section, and the pilot symbols dk’s (k = L + 1, …, L + P) are assumed to be cos(π/4) + jsin(π/4). The received samples zk are then given by zk = h · ck + nk , zk = h · dk + nk ,
ˆ (q−1) ) = Q(F|F
L+P
ˆ (q−1) , z ] Ec [ log fz (zk |F, ci )|F k
k=1
=
(4)
L
ˆ (q−1) , z ] Ec [ log fz (zk |F, ci )|F k
k=1 L+P
log fz (zk |F, dk )
(8)
k=L+1
1 1 |zk |2 , A(q−1) = h(q−1) L k=1 L k=1 k M /2
where fz(·) is the pdf of zk given by
|Re[zk c∗i ]|Pik(q−1)
fz (zk |F, ci ) ≃
i=1
B(q−1) =
L 2 (q−1) 1 ˆ (q−1) , z ] = z(q−1) , z(q−1) = Ec [|c2i ||F |ci |Pik k k k L k=1 i=1
Pik(q−1) =
ˆ (q−1) , c )Pr(c ) fz (zk |F i i ˆ (q−1) ) f (z |F
M /2
z
(7)
(1) E-step: Calculate the conditional expected value of the log-likelihood function with respect to ci given zk under the current estimate of Φ
L
ˆ (q−1) , z ] = h(q−1) = Ec [|Re[zk c∗i ]||F k k
k = 1, . . . , L k = L + 1, . . . , L + P
The EM algorithm used in the proposed algorithm consists of two procedures as follows:
+
M2 =
(6)
n
where L
( h(qmax ) )2 s2 (qmax )
1 −(|zk |2 −2h|Re[zk c∗i ]|+h2 |ci |2 )/s2n e , k = 1, . . . , L 2ps2n
fz (zk |F, dk ) =
1 −(|zk |2 −2h|Re[zk dk∗ ]|+h2 |dk |2 )/s2n e , 2ps2n
k = L + 1, . . . , L + P
(9)
k
Differentiating (4) with respect to h and s2n yields the following results (q−1) ˆh(q) = A and sˆ2n (q) = M2 − (hˆ (q) )2 · B(q−1) B(q−1)
(5)
where hˆ (q) denotes the estimate of h in the qth iteration. Note that the invariance property of the ML estimator, where the ML estimator of a ratio of two parameters equals the ratio of their respective ML
(2) M-step: Find the parameter that maximises (8), where Φ and Ec[·] are the same as those defined in the previous section. The conditional expectation in E-step can then be represented in the same manner as in the previous section (see (10)) where M2,d = 1/L Lk=1 |zk |2 and 2 M2,p = 1/P L+P k=L+1 |zk | . Differentiating (10) with respect to h and 2 sn leads to the following results ∗ L · A(q−1) + L+P k=L+1 |Re[zk dk ]| hˆ (q) = 2 L · B(q−1) + L+P k=L+1 |dk |
1 × (M2,d − 2hA(q−1) + h2 B(q−1) )} s2n L+P L+P 1 2h h2 2 ∗ 2 |Re[zk dk ]| + |d | + P − log 2p − log sn − 2 × M2,p − P k=L+1 sn P k=L+1 k
(11)
ˆ (q−1) ) = L{ −log 2p − log s2 − Q(F|F n
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(10)
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and
sˆ2n (q) = M2,d,p − (hˆ (q) )2 · B(q−1) where M2,d,p = 1/L + P found as
L+P k=1
gˆ
EM
(12)
|zk |2 . As a result, the SNR estimate is (hˆ (qmax ) )2 = sˆ2 (qmax )
F DA,MPSK(M ≥8) (g, s2n ) = F DA,QAM (g, s2n ) =
(13)
We adopt the shrinkage estimation to improve the MSE performance of the EM method. The conventional optimal estimators, such as ML and least squares estimator, have a tendency to overestimate parameters to be estimated because of the measurement noise. The shrinkage estimator has been used to counteract the overestimation and the radius of the parameter set is required to determine the shrinkage factor in the minimax estimation [19]. This procedure is not an easy work and requires some complex optimisation method. For this reason, we consider the blind minimax estimator, which does not require the radius of the parameter set to be determined in the conventional minimax estimation. In the blind minimax estimation, the shrinkage factor is obtained as follows
(gˆ EM )2
−1 CRLBDA-NDA,MPSK (g, s2n ) = F DA,MPSK (g, s2n ) + FNDA,MPSK (g, s2n ) (19) −1 CRLBDA-NDA,QAM (g, s2n ) = FDA,QAM (g, s2n ) + FNDA,QAM (g, s2n )
(20) The CRLBs for γ of the DA-NDA SNR estimation are represented as follows
(14)
CRLBDA-NDA,QAM (g) = CRLBDA-NDA,QAM (g, s2n )
(22)
(15)
EM
EM , sˆ 2n MPSK ) = OCRLBDA-NDA,MPSK (gˆ MPSK −1 EM EM EM F DA,MPSK (gˆ EM ˆ 2n MPSK ) + F NDA,MPSK (gˆ MPSK , sˆ 2n MPSK ) MPSK , s
(23) EM EM OCRLBDA-NDA,QAM (gˆ QAM , sˆ 2n QAM )
(16)
and the EM estimator is asymptotically efficient, so that (16) can be rewritten as MSE(gˆ s ) = (ms )2 var(gˆ EM ) + (ms − 1)2 g2
(21)
where [·]i, j denotes the (i, j)th component of [·]. Since the true CRLB cannot be obtained in practice, the OCRLB is adopted as given below
where gˆ s turns out to be the multiplication of the shrinkage factor and the EM estimator. Accordingly, it is discovered that the MSE can be reduced compared to the conventional unbiased estimator by appropriately selecting the shrinkage factor. Indeed, the MSE of the shrinkage estimator can be obtained as follows MSE(gˆ s ) = var(gˆ s ) + bias2 (gˆ s )
CRLBDA-NDA,MPSK (g) = CRLBDA-NDA,MPSK (g, s2n ) 1,1 1,1
By using m s, the shrinkage estimator for the SNR estimation is found as follows
gˆ s = ms · gˆ EM
Then, the CRLBs of the DA-NDA SNR estimation are obtained, respectively, as follows
EM 2
ms =
L/2s2n L/2g, L/2s2n , L(2 + g)/2s4n
(18)
n
(gˆ ) + var[gˆ EM ]
NDA FIM is found by inserting the EM estimates (11), (12) into the true parameter of the FIM. Also, the DA FIM for higher MPSK modulation and QAM schemes is obtained as follows
=
−1 EM EM EM EM F DA,QAM (gˆ QAM , sˆ 2n QAM ) + F NDA,QAM (gˆ QAM , sˆ 2n QAM )
(24)
The OCRLBs for gˆ EM of the DA-NDA SNR estimator are obtained as given below
(17)
This implies that the MSE is quantified in a fashion of the equation containing the shrinkage factor and thus can be smaller than that when using the unbiased optimal estimator by controlling the shrinkage factor. Specifically, the MSE of the shrinkage estimator can be adjusted to be smaller than that of the EM estimator if (ms )2 var(gˆ EM ) + (1 − ms )2 g2 , var(gˆ EM ). When rearranging (17) for the shrinkage factor m s, the MSE of the shrinkage estimator is smaller than that of the unbiased optimal estimator if the shrinkage factor lies within g2 − var(gˆ EM )/g2 + var(gˆ EM ) , ms , 1. The EM algorithm has been shown to iteratively obtain the ML estimate for the regular exponential family distribution [20]. Hence, the variance of the EM estimator is equal to that of the ML estimator. The variance of the NDA ML estimate is infeasible in the higher MPSK modulation and QAM schemes since the likelihood equation to find the NDA ML estimator cannot be solved in an analytical form. It is known that the ML estimate is asymptotically efficient, and thus its variance is equal to the CRLB in the high SNR or the large number of samples condition [21]. However, the exact CRLB cannot be obtained because the true parameters are unknown. In this case, the OCRLB can be used instead where the ML estimate is substituted into the unknown true parameter [16, 17]. The NDA Fisher information matrix (FIM) for higher MPSK modulation and QAM schemes can be computed using the numerical method [3]. The observed
EM ) OCRLBDA-NDA,MPSK (gˆ MPSK EM EM , sˆ 2n MPSK ) = OCRLBDA-NDA,MPSK (gˆ MPSK
(25)
EM ) OCRLBDA-NDA ,QAM(gˆ QAM EM EM = OCRLBDA-NDA,QAM (gˆ QAM , sˆ 2n QAM )
(26)
1,1
1,1
The OCRLB is the optimal estimate of the true CRLB because the CRLB based on the ML estimate is the ML estimate of the CRLB by the invariance property of the ML estimator.
4
Simulation results
In this section, the performance of the proposed shrinkage SNR estimator was compared with that of the EM estimator. In the simulation, the sample sizes of the pilot and data symbols were fixed to 10 and 50, respectively and the SNR was varied from 0 to 30 dB. The number of iterations was 3000 for all simulations. Fig. 1 shows the comparison of the square of the SNR, the true variances of the EM estimator (obtained from the sample variance), true CRLB, and OCRLB for 8-PSK modulation scheme as a function of SNRs. It was shown that IET Commun., 2015, Vol. 9, Iss. 11, pp. 1388–1395
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Fig. 1 Comparison of the square of the SNR, the variance of the EM estimator, true CRLB and observed CRLB for 8-PSK scheme as a function of the SNR
the OCRLB using (25) is similar to the variances of the EM estimator for the 8-PSK modulation scheme in most SNR regimes excluding the high SNR interval. Also, Figs. 2 and 3 show the comparisons of the true variances of the EM
estimator, OCRLBs and true CRLBs in 16-PSK modulation and 16-QAM scheme as a function of the SNR. Although there were differences between the OCRLB using (25), (26) and the true variances of the EM estimator for 8-PSK, 16-PSK and
Fig. 2 Comparison of the square of the SNR, the variance of the EM estimator, true CRLB and observed CRLB for 16-PSK scheme as a function of the SNR
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Fig. 3 Comparison of the square of the SNR, the variance of the EM estimator, true CRLB and observed CRLB for 16-QAM scheme as a function of the SNR
16-QAM schemes, the differences were much small when compared to the square of the SNR; the squared SNR was 60 dB while the difference between the OCRLB of the EM estimator and the true variance of the EM estimator for 8-PSK scheme was about 5 dB at 30 dB of the SNR as shown in
Fig. 1. Thus, the differences between the approximated variance (OCRLB) and the true variance did not cause the large error in the determination of the optimal shrinkage factor, as can be seen from (14). The shrinkage factors of the proposed method for 8-PSK, 16-PSK and 16-QAM schemes are
Fig. 4 Shrinkage factor of the proposed method for 8-PSK, 16-PSK modulation and 16-QAM schemes
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Fig. 5 NMSE performance comparison of the proposed method and the EM method for 8-PSK modulation scheme
shown in Fig. 4. As the SNR increases, the shrinkage factors are closer to one because the square of the SNR grows faster than the OCRLB. Also, the slopes of the shrinkage factor curves were comparatively larger than those over the moderate SNR intervals because the increasing rate of the OCRLBs was
smaller than that of squared SNR as can be seen from Figs. 1 and 2. Next, the normalised MSE (NMSE), normalised with respect to the square of the true value of the SNR, was compared for 8-PSK, 16-PSK modulation and 16-QAM schemes to investigate the superiority of the proposed method.
Fig. 6 NMSE performance comparison of the proposed method and the EM method for 16-PSK modulation scheme
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Fig. 7 NMSE performance comparison of the proposed method and the EM method for 16-QAM scheme
The following NMSE criterion was used in the simulation N NMSE =
(g − gˆ (i))2 N × g2
i=1
(27)
where N is the total number of Monte Carlo simulations, γ is the SNR and i is the iteration number. It can be seen from Fig. 5 that the NMSE performance of the shrinkage SNR estimation is superior to that of the EM method in low and moderate SNR regimes with an 8-PSK modulation scheme. The averaged NMSE gain of the shrinkage EM estimator over the existing EM method was about 0.5 dB in the low SNR regime and the difference between the averaged NMSEs was smaller as the SNR increased. Also, Figs. 6 and 7 show that the NMSE performance of the shrinkage EM-based SNR estimation method outperforms that of the EM method for low and moderate SNR regimes with a 16-PSK modulation and 16-QAM schemes. Again, the averaged NMSE of the proposed method was smaller than that of the existing method by about 0.5 dB in the low and moderate SNR regimes. The averaged NMSEs of the investigated methods are closer to the NCRLB as the SNR increases because the NDA-SNR estimation algorithms are effective in the high SNR conditions. It can be concluded that the adoption of the shrinkage factor using the OCRLB can be utilised even for the higher MPSK modulation and QAM schemes from the above simulation results.
5
Conclusions
A shrinkage-based SNR estimation method, in which pilot and data symbols were used, was compared with the EM-based SNR estimator for the higher MPSK modulation (M ≥ 8) and QAM schemes. Since the variance of the EM SNR estimator for higher MPSK modulation and QAM systems in the DA-NDA scheme cannot be derived in a closed form, the OCRLB was adopted as the variance of the EM SNR estimator for the MPSK modulation and QAM schemes when determining the shrinkage factor. The
simulation results showed that the variance of the EM SNR estimator can be approximated by the OCRLB. As a result, the NMSE average obtained by employing the shrinkage EM estimator was smaller than that of the EM estimator for low and moderate SNR regimes with various MPSK modulation and QAM schemes. The gain of the averaged NMSE of the proposed method over the existing method was about 0.5 dB in the low and moderate SNR regimes. As a future work, the non-data-aided SNR estimation method under the non-Gaussian noise situations will be developed.
6
Acknowledgment
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2014R1A2A1A10049735) and this research was supported by the MSIP, Korea, under the ITRC support program (NIPA-2014H0301-14-1019) supervised by the NIPA.
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References
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Shrinkage-based biased signal-to-noise ratio estimator using pilot and data symbols for linearly modulated signals
ISSN 1751-8628 Received on 1st October 2014 Revised on 6th January 2015 Accepted on 9th February 2015 doi: 10.1049/iet-com.2014.0943 www.ietdl.org
Chee-Hyun Park, Soojeong Lee, Joon-Hyuk Chang ✉ School of Engineering Hanyang University, Seoul 133-791, Korea ✉ E-mail: [email protected]
Abstract: In diverse engineering problems including wireless communications, the estimate of the signal-to-noise ratio (SNR) is required. In this study, the authors develop a shrinkage-based SNR estimator in the data-aided and non-dataaided schemes for higher M-ary phase-shift-keying (M ≥ 8) and quadrature amplitude modulations. The observed Cramér-Rao lower bound is used as the variance of the expectation maximisation estimator to determine the optimal shrinkage factor. Simulation results show that the normalised mean-squared error of the proposed method is lower than that of the expectation maximisation method for low and moderate SNR conditions.
1
Introduction
Estimation of the signal-to-noise ratio (SNR) is important in modern communication systems since various systems use techniques that depend on an SNR estimate for proper operation. For example, both the determined signal power and the noise power are required in turbo decoders. Also, rate-adaptive transmission systems to adapt the modulation scheme or the coding rate use the SNR estimation. The types of the SNR estimation can be classified into data-aided (DA) and non-data-aided (NDA) methods. DA estimators rely on the insertion of pilot symbols to the data frame, whereas NDA estimators ignore the statistical information about the transmitted data, which typically leads to poor performance when the SNR is low. Several studies related to the SNR estimation exist. Pauluzzi and Beaulieu [1] investigated the SNR estimators of baseband M-ary phase-shift-keying (MPSK) TxDA and RxDA symbols. Alagha [2] also derived a Cramér-Rao lower bound (CRLB) for SNR estimation with binary phase-shift-keying (BPSK) and quadrature phase-shift-keying (QPSK) modulation schemes. A significant difference between the bounds for the NDA and DA methods was observed for low SNR conditions; however, both methods showed identical performances in high SNR regimes. Gappmair [3] extended Alagha’s work to be usable for any two-dimensional M-ary phase-shift-keying (PSK) systems and linear modulation schemes. This allows for SNR estimation based on an NDA estimator by exploiting the axis/half plane symmetry of conventional modulation schemes although this method still adopts a numerical method for the computation of integrals. Bellili et al. [4] developed an exact closed-form CRLB for the NDA SNR estimator in square quadrature-amplitude modulation (QAM) formulation. Chen and Beaulieu [5] derived a maximum likelihood (ML)-based SNR estimator using pilot and data symbols simultaneously; however, this algorithm is restricted to the BPSK modulation scheme. Gappmair et al. [6] developed the expectation maximisation (EM)-based SNR estimation algorithm for linearly-modulated signals. Das and Rao [7] proposed the SNR estimation algorithm for multiple-input–multiple-output communications systems. Bellili et al. [8, 9] dealt with the ML-based SNR estimation method over the fast time-varying channels. Fu et al. [10] utilised the Kolmogorov–Smirnov test to estimate SNR for multilevel constellations. Also, Chitte et al. [11] adopted the biased estimation to improve the ML distance estimator and Park et al. [12] derived the shrinkage-based closed-form DA-NDA SNR estimation algorithm for BPSK and
QPSK modulation schemes. In statistics and signal processing, the constraint of unbiasedness is often a practical issue, because the mean-squared error (MSE) can be minimised with the MSE criterion, which does not depend on the unknown true parameter. However, in some problems, restricting attention to unbiased estimation leads to unreasonable results [13, 14]. The variance can be decreased at the cost of increasing the bias, while ensuring the reduced overall estimation error. Hence, unbiasedness does not necessarily lead to a low estimation error [15] and the estimator is typically determined using the biased estimation in which a trade-off between variance and bias is utilised. Biased estimation has been used in a various signal processing problems, for example, image restoration, smoothing algorithms in time series analysis, spectrum estimation, wavelet denoising and so on. In this paper, we propose a shrinkage-based biased estimator in the context of SNR estimation for higher MPSK modulation (M ≥ 8) and QAM schemes. While the variance of the EM estimate is needed to find the optimal shrinkage factor used in the biased estimation, the variance of the EM estimator in higher MPSK modulation (M ≥ 8) and QAM schemes cannot be obtained. As the EM estimator is an iteration-based ML estimator, the variance of the EM estimator would be asymptotically equal to the CRLB. However, the CRLB of the SNR estimator cannot be found in practice since the true parameters are unknown. To resolve this problem, the observed CRLB (OCRLB) is utilised, where the EM estimate is inserted into the unknown true parameter as in [16, 17]. We show that the OCRLB is similar to the variances of the higher MPSK modulation and QAM schemes via simulation results. As a result, the OCRLB can be used instead of the variance of the EM estimator and thus the optimal shrinkage factor based on the variance of the EM estimator can be determined. It is discovered that the proposed estimator outperforms the conventional EM-based estimator in low and moderate SNR regimes in terms of the MSE under various experimental conditions.
2 EM-based SNR estimation using the data symbols Throughout this paper, it is assumed that the symbol timing and phase have been perfectly recovered. The data symbols ck’s (k = 1, …, L) are modelled as the zero mean, independent and identically distributed symbols ck ∈ C with unit variance E[|ck|2] = 1, where C denotes the M-ary symbol alphabet. The received samples zk are IET Commun., 2015, Vol. 9, Iss. 11, pp. 1388–1395
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then given by
estimators, and thus the SNR estimate is obtained as follows zk = h · ck + nk ,
k = 1, . . . , L
(1)
gˆ EM =
where h is the positive channel gain and nk is the additive white Gaussian noise (AWGN). Real and imaginary parts of the AWGN samples are assumed to be independent, each with the same variance s2n /2. The EM algorithm is performed to find the SNR estimate [6], which can be divided into the following two procedures: (2) E-step: Calculate the conditional expected value of the log-likelihood function with respect to ci given zk using the current estimate of Φ such that ˆ (q−1) ) = Q(F|F
L
ˆ (q−1) , z ] Ec [ log fz (zk |F, ci )|F k
(2)
k=1
(3) M-step: Find the parameter that maximises (2), where F = (h, s2n ), Ec[·] denotes the expectation with respect to ci ∈ H, H denotes the subset of symbols in C located in the right or the left half of the complex plane, and q = 0, 1, 2, …, qmax is the iteration number. fz(·) is the probability density function (pdf) of zk given by fz (zk |F, ci ) ≃
1 −(|zk |2 −2h|Re[zk c∗i ]|+h2 |ci |2 )/s2n e 2ps2n
(3)
The conditional expectation in E-step can then be represented by using the high SNR assumption and Bayes theorem as follows ˆ (q−1) ) = L{−log 2p − log s2 − Q(F|F n
1 s2n
× (M2 − 2hA(q−1) + h2 B(q−1) )}
The rigorous proof of this ML estimator for SNR was given in [18] by showing that the second-order derivative for γ is negative to verify whether gˆ EM truly maximises the likelihood function in the value of (6).
3 Shrinkage method-based SNR estimation using pilot and data symbols The data symbols ck’s (k = 1, …, L) are given in the previous section, and the pilot symbols dk’s (k = L + 1, …, L + P) are assumed to be cos(π/4) + jsin(π/4). The received samples zk are then given by zk = h · ck + nk , zk = h · dk + nk ,
ˆ (q−1) ) = Q(F|F
L+P
ˆ (q−1) , z ] Ec [ log fz (zk |F, ci )|F k
k=1
=
(4)
L
ˆ (q−1) , z ] Ec [ log fz (zk |F, ci )|F k
k=1 L+P
log fz (zk |F, dk )
(8)
k=L+1
1 1 |zk |2 , A(q−1) = h(q−1) L k=1 L k=1 k M /2
where fz(·) is the pdf of zk given by
|Re[zk c∗i ]|Pik(q−1)
fz (zk |F, ci ) ≃
i=1
B(q−1) =
L 2 (q−1) 1 ˆ (q−1) , z ] = z(q−1) , z(q−1) = Ec [|c2i ||F |ci |Pik k k k L k=1 i=1
Pik(q−1) =
ˆ (q−1) , c )Pr(c ) fz (zk |F i i ˆ (q−1) ) f (z |F
M /2
z
(7)
(1) E-step: Calculate the conditional expected value of the log-likelihood function with respect to ci given zk under the current estimate of Φ
L
ˆ (q−1) , z ] = h(q−1) = Ec [|Re[zk c∗i ]||F k k
k = 1, . . . , L k = L + 1, . . . , L + P
The EM algorithm used in the proposed algorithm consists of two procedures as follows:
+
M2 =
(6)
n
where L
( h(qmax ) )2 s2 (qmax )
1 −(|zk |2 −2h|Re[zk c∗i ]|+h2 |ci |2 )/s2n e , k = 1, . . . , L 2ps2n
fz (zk |F, dk ) =
1 −(|zk |2 −2h|Re[zk dk∗ ]|+h2 |dk |2 )/s2n e , 2ps2n
k = L + 1, . . . , L + P
(9)
k
Differentiating (4) with respect to h and s2n yields the following results (q−1) ˆh(q) = A and sˆ2n (q) = M2 − (hˆ (q) )2 · B(q−1) B(q−1)
(5)
where hˆ (q) denotes the estimate of h in the qth iteration. Note that the invariance property of the ML estimator, where the ML estimator of a ratio of two parameters equals the ratio of their respective ML
(2) M-step: Find the parameter that maximises (8), where Φ and Ec[·] are the same as those defined in the previous section. The conditional expectation in E-step can then be represented in the same manner as in the previous section (see (10)) where M2,d = 1/L Lk=1 |zk |2 and 2 M2,p = 1/P L+P k=L+1 |zk | . Differentiating (10) with respect to h and 2 sn leads to the following results ∗ L · A(q−1) + L+P k=L+1 |Re[zk dk ]| hˆ (q) = 2 L · B(q−1) + L+P k=L+1 |dk |
1 × (M2,d − 2hA(q−1) + h2 B(q−1) )} s2n L+P L+P 1 2h h2 2 ∗ 2 |Re[zk dk ]| + |d | + P − log 2p − log sn − 2 × M2,p − P k=L+1 sn P k=L+1 k
(11)
ˆ (q−1) ) = L{ −log 2p − log s2 − Q(F|F n
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(10)
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and
sˆ2n (q) = M2,d,p − (hˆ (q) )2 · B(q−1) where M2,d,p = 1/L + P found as
L+P k=1
gˆ
EM
(12)
|zk |2 . As a result, the SNR estimate is (hˆ (qmax ) )2 = sˆ2 (qmax )
F DA,MPSK(M ≥8) (g, s2n ) = F DA,QAM (g, s2n ) =
(13)
We adopt the shrinkage estimation to improve the MSE performance of the EM method. The conventional optimal estimators, such as ML and least squares estimator, have a tendency to overestimate parameters to be estimated because of the measurement noise. The shrinkage estimator has been used to counteract the overestimation and the radius of the parameter set is required to determine the shrinkage factor in the minimax estimation [19]. This procedure is not an easy work and requires some complex optimisation method. For this reason, we consider the blind minimax estimator, which does not require the radius of the parameter set to be determined in the conventional minimax estimation. In the blind minimax estimation, the shrinkage factor is obtained as follows
(gˆ EM )2
−1 CRLBDA-NDA,MPSK (g, s2n ) = F DA,MPSK (g, s2n ) + FNDA,MPSK (g, s2n ) (19) −1 CRLBDA-NDA,QAM (g, s2n ) = FDA,QAM (g, s2n ) + FNDA,QAM (g, s2n )
(20) The CRLBs for γ of the DA-NDA SNR estimation are represented as follows
(14)
CRLBDA-NDA,QAM (g) = CRLBDA-NDA,QAM (g, s2n )
(22)
(15)
EM
EM , sˆ 2n MPSK ) = OCRLBDA-NDA,MPSK (gˆ MPSK −1 EM EM EM F DA,MPSK (gˆ EM ˆ 2n MPSK ) + F NDA,MPSK (gˆ MPSK , sˆ 2n MPSK ) MPSK , s
(23) EM EM OCRLBDA-NDA,QAM (gˆ QAM , sˆ 2n QAM )
(16)
and the EM estimator is asymptotically efficient, so that (16) can be rewritten as MSE(gˆ s ) = (ms )2 var(gˆ EM ) + (ms − 1)2 g2
(21)
where [·]i, j denotes the (i, j)th component of [·]. Since the true CRLB cannot be obtained in practice, the OCRLB is adopted as given below
where gˆ s turns out to be the multiplication of the shrinkage factor and the EM estimator. Accordingly, it is discovered that the MSE can be reduced compared to the conventional unbiased estimator by appropriately selecting the shrinkage factor. Indeed, the MSE of the shrinkage estimator can be obtained as follows MSE(gˆ s ) = var(gˆ s ) + bias2 (gˆ s )
CRLBDA-NDA,MPSK (g) = CRLBDA-NDA,MPSK (g, s2n ) 1,1 1,1
By using m s, the shrinkage estimator for the SNR estimation is found as follows
gˆ s = ms · gˆ EM
Then, the CRLBs of the DA-NDA SNR estimation are obtained, respectively, as follows
EM 2
ms =
L/2s2n L/2g, L/2s2n , L(2 + g)/2s4n
(18)
n
(gˆ ) + var[gˆ EM ]
NDA FIM is found by inserting the EM estimates (11), (12) into the true parameter of the FIM. Also, the DA FIM for higher MPSK modulation and QAM schemes is obtained as follows
=
−1 EM EM EM EM F DA,QAM (gˆ QAM , sˆ 2n QAM ) + F NDA,QAM (gˆ QAM , sˆ 2n QAM )
(24)
The OCRLBs for gˆ EM of the DA-NDA SNR estimator are obtained as given below
(17)
This implies that the MSE is quantified in a fashion of the equation containing the shrinkage factor and thus can be smaller than that when using the unbiased optimal estimator by controlling the shrinkage factor. Specifically, the MSE of the shrinkage estimator can be adjusted to be smaller than that of the EM estimator if (ms )2 var(gˆ EM ) + (1 − ms )2 g2 , var(gˆ EM ). When rearranging (17) for the shrinkage factor m s, the MSE of the shrinkage estimator is smaller than that of the unbiased optimal estimator if the shrinkage factor lies within g2 − var(gˆ EM )/g2 + var(gˆ EM ) , ms , 1. The EM algorithm has been shown to iteratively obtain the ML estimate for the regular exponential family distribution [20]. Hence, the variance of the EM estimator is equal to that of the ML estimator. The variance of the NDA ML estimate is infeasible in the higher MPSK modulation and QAM schemes since the likelihood equation to find the NDA ML estimator cannot be solved in an analytical form. It is known that the ML estimate is asymptotically efficient, and thus its variance is equal to the CRLB in the high SNR or the large number of samples condition [21]. However, the exact CRLB cannot be obtained because the true parameters are unknown. In this case, the OCRLB can be used instead where the ML estimate is substituted into the unknown true parameter [16, 17]. The NDA Fisher information matrix (FIM) for higher MPSK modulation and QAM schemes can be computed using the numerical method [3]. The observed
EM ) OCRLBDA-NDA,MPSK (gˆ MPSK EM EM , sˆ 2n MPSK ) = OCRLBDA-NDA,MPSK (gˆ MPSK
(25)
EM ) OCRLBDA-NDA ,QAM(gˆ QAM EM EM = OCRLBDA-NDA,QAM (gˆ QAM , sˆ 2n QAM )
(26)
1,1
1,1
The OCRLB is the optimal estimate of the true CRLB because the CRLB based on the ML estimate is the ML estimate of the CRLB by the invariance property of the ML estimator.
4
Simulation results
In this section, the performance of the proposed shrinkage SNR estimator was compared with that of the EM estimator. In the simulation, the sample sizes of the pilot and data symbols were fixed to 10 and 50, respectively and the SNR was varied from 0 to 30 dB. The number of iterations was 3000 for all simulations. Fig. 1 shows the comparison of the square of the SNR, the true variances of the EM estimator (obtained from the sample variance), true CRLB, and OCRLB for 8-PSK modulation scheme as a function of SNRs. It was shown that IET Commun., 2015, Vol. 9, Iss. 11, pp. 1388–1395
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Fig. 1 Comparison of the square of the SNR, the variance of the EM estimator, true CRLB and observed CRLB for 8-PSK scheme as a function of the SNR
the OCRLB using (25) is similar to the variances of the EM estimator for the 8-PSK modulation scheme in most SNR regimes excluding the high SNR interval. Also, Figs. 2 and 3 show the comparisons of the true variances of the EM
estimator, OCRLBs and true CRLBs in 16-PSK modulation and 16-QAM scheme as a function of the SNR. Although there were differences between the OCRLB using (25), (26) and the true variances of the EM estimator for 8-PSK, 16-PSK and
Fig. 2 Comparison of the square of the SNR, the variance of the EM estimator, true CRLB and observed CRLB for 16-PSK scheme as a function of the SNR
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Fig. 3 Comparison of the square of the SNR, the variance of the EM estimator, true CRLB and observed CRLB for 16-QAM scheme as a function of the SNR
16-QAM schemes, the differences were much small when compared to the square of the SNR; the squared SNR was 60 dB while the difference between the OCRLB of the EM estimator and the true variance of the EM estimator for 8-PSK scheme was about 5 dB at 30 dB of the SNR as shown in
Fig. 1. Thus, the differences between the approximated variance (OCRLB) and the true variance did not cause the large error in the determination of the optimal shrinkage factor, as can be seen from (14). The shrinkage factors of the proposed method for 8-PSK, 16-PSK and 16-QAM schemes are
Fig. 4 Shrinkage factor of the proposed method for 8-PSK, 16-PSK modulation and 16-QAM schemes
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Fig. 5 NMSE performance comparison of the proposed method and the EM method for 8-PSK modulation scheme
shown in Fig. 4. As the SNR increases, the shrinkage factors are closer to one because the square of the SNR grows faster than the OCRLB. Also, the slopes of the shrinkage factor curves were comparatively larger than those over the moderate SNR intervals because the increasing rate of the OCRLBs was
smaller than that of squared SNR as can be seen from Figs. 1 and 2. Next, the normalised MSE (NMSE), normalised with respect to the square of the true value of the SNR, was compared for 8-PSK, 16-PSK modulation and 16-QAM schemes to investigate the superiority of the proposed method.
Fig. 6 NMSE performance comparison of the proposed method and the EM method for 16-PSK modulation scheme
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Fig. 7 NMSE performance comparison of the proposed method and the EM method for 16-QAM scheme
The following NMSE criterion was used in the simulation N NMSE =
(g − gˆ (i))2 N × g2
i=1
(27)
where N is the total number of Monte Carlo simulations, γ is the SNR and i is the iteration number. It can be seen from Fig. 5 that the NMSE performance of the shrinkage SNR estimation is superior to that of the EM method in low and moderate SNR regimes with an 8-PSK modulation scheme. The averaged NMSE gain of the shrinkage EM estimator over the existing EM method was about 0.5 dB in the low SNR regime and the difference between the averaged NMSEs was smaller as the SNR increased. Also, Figs. 6 and 7 show that the NMSE performance of the shrinkage EM-based SNR estimation method outperforms that of the EM method for low and moderate SNR regimes with a 16-PSK modulation and 16-QAM schemes. Again, the averaged NMSE of the proposed method was smaller than that of the existing method by about 0.5 dB in the low and moderate SNR regimes. The averaged NMSEs of the investigated methods are closer to the NCRLB as the SNR increases because the NDA-SNR estimation algorithms are effective in the high SNR conditions. It can be concluded that the adoption of the shrinkage factor using the OCRLB can be utilised even for the higher MPSK modulation and QAM schemes from the above simulation results.
5
Conclusions
A shrinkage-based SNR estimation method, in which pilot and data symbols were used, was compared with the EM-based SNR estimator for the higher MPSK modulation (M ≥ 8) and QAM schemes. Since the variance of the EM SNR estimator for higher MPSK modulation and QAM systems in the DA-NDA scheme cannot be derived in a closed form, the OCRLB was adopted as the variance of the EM SNR estimator for the MPSK modulation and QAM schemes when determining the shrinkage factor. The
simulation results showed that the variance of the EM SNR estimator can be approximated by the OCRLB. As a result, the NMSE average obtained by employing the shrinkage EM estimator was smaller than that of the EM estimator for low and moderate SNR regimes with various MPSK modulation and QAM schemes. The gain of the averaged NMSE of the proposed method over the existing method was about 0.5 dB in the low and moderate SNR regimes. As a future work, the non-data-aided SNR estimation method under the non-Gaussian noise situations will be developed.
6
Acknowledgment
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2014R1A2A1A10049735) and this research was supported by the MSIP, Korea, under the ITRC support program (NIPA-2014H0301-14-1019) supervised by the NIPA.
7
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