Cramér-Rao Lower Bound and EM Algorithm for ... - COMONSENS

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Cramér-Rao Lower Bound and EM Algorithm for Envelope-Based SNR Estimation of Nonconstant Modulus Constellations Wilfried Gappmair, Member, IEEE, Roberto López-Valcarce, Member, IEEE, and Carlos Mosquera, Member, IEEE

Abstract—Signal-to-noise ratio (SNR) estimation for linearly modulated signals is addressed in this letter, focusing on envelopebased estimators, which are robust to carrier offsets and phase jitter, and on the challenging case of nonconstant modulus constellations. For comparison purposes, the true Cramér-Rao lower bound is numerically evaluated, obtaining an analytical expression in closed form for the asymptotic case of high SNR values, which quantifies the performance loss with respect to coherent estimation. As the maximum-likelihood algorithm is too complex for practical implementation, an expectationmaximization (EM) approach is proposed, achieving a good tradeoff between complexity and performance for mediumto-high SNRs. Finally, a hybrid scheme based on EM and moments-based estimates is suggested, which performs close to the theoretical limit over a wide SNR range. Index Terms—Digital receivers, nonconstant modulus signals, envelope-based SNR estimation, Cramér-Rao lower bound, expectation-maximization algorithm.

I. I NTRODUCTION

M

ANY communication systems require knowledge of the signal-to-noise ratio (SNR), with efficient signal detection [1] and link adaptation [2] as most prominent examples. Algorithms for SNR estimation can be classified according to their operational conditions: i) data-aided (DA) vs. nondata-aided (NDA), ii) coherent vs. envelope-based (EVB), iii) oversampled vs. baud-rate sampled. If pilot symbols are not available, then NDA methods such as moments-based schemes must be applied [3], [4]. These fall within the class of EVB schemes, which are robust to carrier phase uncertainties. Since they need no training symbols and no carrier phase reference, NDA EVB methods are of particular practical interest. Most SNR estimators in the literature assume the symbol timing to be established, although solutions based on continuous-time or oversampled signals can be found in [3], [5]. Paper approved by H. Leib, the Editor for Communication and Information Theory of the IEEE Communications Society. Manuscript received February 26, 2007; revised September 26, 2007, March 19, 2008, and July 30, 2008. This work was supported by SatNEx-II (Satellite Communications Network of Excellence, IST No. 27393), launched by the European Commission for advanced research in satellite communications within the Sixth Framework Programme, and by the Spanish Government under Project SPROACTIVE (Reference TEC2007-68094-C02-01/TCM). Part of the paper was presented at the 8th IEEE Workshop on Signal Processing Advances for Wireless Communications, Helsinki, Finland, June 2007. W. Gappmair is with the Institute of Communication Networks and Satellite Communications, Graz University of Technology, Austria, (e-mail: [email protected]). R. López-Valcarce and C. Mosquera are with the Department of Signal Theory and Communications, University of Vigo, Spain (e-mail: [email protected], [email protected]). Digital Object Identifier 10.1109/TCOMM.2009.06.0700932

With nonconstant modulus (non-CM) constellations, selfnoise effects, caused by the variability of the symbol envelope, are a major challenge for NDA methods. NDA EVB estimation of the SNR for non-CM constellations is addressed in [4], where a family of suboptimal moments-based estimators has been introduced. In this context, it is to be noted that the true Cramér-Rao lower bound (CRLB) for the general non-CM case is still lacking in the open literature: it was obtained in [6] for CM symbols and in [7] for the coherent NDA case, whereas [4] considered an EVB approach with non-CM sources assuming the noise power as known. In this letter, the CRLB for joint NDA EVB estimation of signal and noise powers using general constellations is developed. The resulting expression must be numerically evaluated, although it admits a closed-form solution for asymptotically large SNRs. The EVB maximum-likelihood (ML) SNR estimator requires the numerical solution of a set of nonlinear equations derived from the related likelihood function; the computational complexity motivates the development of the EVB expectation-maximization (EM) algorithm. In view of the results obtained for EM and simpler moments-based techniques, a hybrid approach is suggested, performing very close to the CRLB over a wide SNR range. The rest of the letter is organized as follows. The equivalent baseband model is discussed in Section II. Section III presents the CRLB for NDA EVB estimation of the SNR, whereas Section IV introduces the proposed EM estimator. Simulation results are given in Section V and Section VI concludes the letter. II. S YSTEM M ODEL Assuming perfect recovery of the symbol timing, the equivalent baseband model is given by √ √ k = 1, . . . , K (1) rk = Sck ejθk + N wk , where rk are the observations, S and N the signal and noise power, ck the i.i.d. symbols drawn from an M -ary constellation C, wk are samples of a complex-valued zeromean white Gaussian noise process with unit variance, and θk is an unknown, possibly time-varying phase accounting for carrier offset and phase jitter. . In order to estimate the SNR, defined as ρ = S/N , we . denote the envelope of the observations as yk = |rk |. Furthermore, the alphabet of discrete symbol amplitudes is expressed by R = {R1 , . . . , RI }, where I is the number of different amplitude levels in C. The corresponding a priori probabilities . are given by Pi = Pr{|ck | = Ri }. It is also assumed that the

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 constellation has unit energy: E{|ck |2 } = Ii=1 Pi Ri2 = 1. Finally, the nth-ordersample moment of the observations is . 1 K n defined as Mn = K k=1 yk . Several approaches to SNR estimation are possible, depending on the model adopted for the phase process θk . In some cases, it can be reasonable to assume θk as known. However, situations may arise in which the SNR is not sufficiently high so as to allow reliable phase recovery, or where a large amount of residual phase jitter prevents reliable data demodulation. But even in such situations, SNR estimation still can be of interest, e. g., if adaptive modulation and coding (AMC) is implemented so that the receiver needs to determine the transmission parameters most suitable to the current link conditions. For these reasons, we assume no structure on the phase process θk and focus on EVB SNR estimators throughout this letter. On the other hand, joint SNR and phase estimation methods might be of interest as well, for instance if the frame is affected by a constant phase offset θk = θ; this is considered insofar as some simulation results are presented for comparison purposes. In any case, sensitivity of such joint solutions to practical amounts of phase noise is expected to be an important issue for further research. The EVB approach is robust in this sense, although at the expense of some performance degradation, as shown in the sequel. III. P ERFORMANCE B OUNDS . Conditioned on φ = [S N ] as well as on the nuisance . . parameter vectors c = [c1 · · · cK ] and θ = [θ1 · · · θK ], the . observation r = [r1 · · · rK ] follows a Gaussian distribution given by f (r|φ, c, θ) =

K  1 −|rk −√S ck ejθk |2 /N e . πN

(2)

k=1

. From (2), the conditional pdf of the envelope vector y = [y1 · · · yK ] is obtained as [8, pp. 139–140]   √ K  2yk −(yk2 +S |ck |2 )/N 2 S |ck |yk e f (y|φ, c) = I0 N N =

k=1 K 

f (yk |φ, ck )

(3)

where In (·) denotes the nth-order modified Bessel function of the first kind. It is seen that the yk ’s are mutually independent and identically distributed. They are also independent of both the phase process θk and the argument of the symbols ck . Note in particular that no assumptions on θk have been made. Now (3) can be averaged with respect to |ck |, arriving at K 

f (yk |φ)

(4)

k=1

with I

The log-likelihood function (LLF) of the parameters is then given by K

 . ln f (yk |φ). F (y|φ) = ln f (y|φ) =

(6)

k=1

The CRLB for EVB estimation has been derived in [4], implicitly assuming that the noise power N is known. When both S and N are unknown, the variance of any unbiased SNR estimator is lower-bounded by [9]  ∂Ω 2

Ey ∂N CRLBEVB (ρ) =  2  ∂Ω 2

 ∂Ω

Ey ∂Ω Ey ∂N − E2y ∂Ω ∂ρ ∂ρ ∂N (7) . where Ω(y|ρ, N ) = F (y|[ρN, N ]). It is immediately observed that (7) is larger than 1/Ey {(∂Ω/∂ρ)2 }, the bound from [4]. We note that, in general, (7) must be evaluated by numerical means. Nevertheless, in the asymptotic case of high SNRs, a closed-form solution is available after considering the following facts: √ (i) Since I0 (x) ≈ ex / 2πx for large x [10, Eq. (9.7.1)], the Ricean pdf in (3) √ converges for S  N to a Gaussian pdf with mean S |ck | and variance N/2, i. e.,  √ 2 yk √ e−(yk − S |ck |) /N (8) f (yk |φ, ck ) ≈ πN S |ck | √ 2 1 e−(yk − S |ck |) /N (9) ≈ √ πN where√we have applied the high SNR approximation yk ≈ S |ck | in (8) to obtain (9). Thus, the mixed Ricean distribution f (yk |φ) from (5) becomes a √multimodal Gaussian, with I Gaussian pdfs centered at S Ri . (ii) Albeit looser than the true CRLB, the modified CRLB (MCRLB) is a theoretical limit widely used due to its computational simplicity [11]. As shown in [12], true and modified CRLB coincide asymptotically for high SNRs, provided the parameter to be estimated is not coupled with the nuisance parameters, which is the case in our context. The formal proof, however, is lengthy such that we refer the reader to [12] for details. Using fact (i) above, it is shown in the Appendix that the asymptotic MCRLB is given by 2 (ρ + ρ2 ), ρ  1. (10) K With (10) and fact (ii), the normalized CRLB, defined as . NCRLB(ρ) = CRLB(ρ)/ρ2 , can be written as

2 1 NCRLBEVB (ρ) = 1+ , ρ  1, (11) K ρ MCRLBEVB (ρ) =

k=1

f (y|φ) =

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2 2yk −yk2 /N  e Pi e−Ri S/N I0 f (yk |φ) = N i=1

  √ 2 SRi yk . N (5)

whereas for the coherent DA case [3], it is given by

1 2 NCRLBDA (ρ) = 1+ . K ρ

(12)

Hence, the asymptotic value of (11) is twice the value of (12). This is because EVB schemes effectively discard half of the (real-valued) observations as compared to coherent estimators. Fig. 1 shows the evolution of NCRLBEVB (ρ), computed numerically from (7) for different constellations (due to clarity scaled by K). In all cases, the asymptotic

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K

˜ φ ˆ(n−1) ), with λk (φ;      . ˆ(n−1) , yk . ˜ ck )  φ ˜ φ ˆ(n−1) = Eck ln f (yk | φ, λk φ; (13) To determine (13), we use the high SNR approximation (8). ˜ Hence, up to an additive term independent of φ, k=1

  ˜ φ ˆ(n−1) λk φ; ≈

Fig. 1. Normalized Cramér-Rao lower bounds for SNR estimation (scaled by observation length K).

value is 2/K as predicted by (11). With decreasing SNRs, the bound increases with the number I of constellation levels. Also shown are (12) and the bound obtained in [13] for QPSK in the NDA coherent case. Note that the distance between EVB and coherent NDA diminishes with decreasing SNRs. This is also true for non-CM constellations, as can be seen by comparing the curves in Fig. 1 with the corresponding bounds in [7].

1 1 ˜ − ln S˜ − ln N 4 2  (n−1) (n−1) − 2 S˜ yk ηk yk2 + S˜ ξk − ˜ N (14)

where the a posteriori mean and mean-squared values of the symbol amplitudes have been introduced as    .  ˆ(n−1) (n−1) (15) = Eck |ck |  φ , yk , ηk    .  ˆ(n−1) (n−1) ξk (16) = Eck |ck |2  φ , yk . ˜ φ ˆ(n−1) ) is approximately Averaging now (14) over k, Λ(φ; provided by   1 1 ˜ φ ˆ(n−1) ˜ ≈ − ln S˜ − ln N Λ φ; 4 2  ˜ (n−1) ˜ (n−1) − 2 SB M2 + SA − ˜ N (17) employing the definitions

IV. A N E XPECTATION -M AXIMIZATION A PPROACH From the invariance principle [14], the ML estimate of ρ = S/N is given by the ratio of the ML estimates of S and N , i. e., those values of S and N maximizing the LLF in (6). One must resort to numerical schemes, such as the iterative gradient or Newton’s method, since no closed-form solution is available. Initialization can be achieved by means of simple moments-based estimates [4], [15]. However, gradient methods require step-size tuning, which might be problematic, whereas Newton’s method necessitates the evaluation of firstand second-order partial derivatives of the LLF, which is computationally demanding. But instead of this, the search for a maximum of the LLF might be implemented via an EM algorithm [16]. We note that an EM-based SNR estimator has been reported in [17], which applies only to CM constellations in the coherent case, whereas our EVB method can be used with non-CM constellations. According to the discussion in [18], we consider y as the incomplete and [y |c|] as the complete observation of our problem. Then the EM approach to maximize (6) is an iterative procedure of expectation (E) and maximization (M) steps, at stage n = 1, 2, . . . specified as follows: ˆ(n−1) = [Sˆ(n−1) N ˆ (n−1) ], (i) E-step: Given an estimate φ . ˜ φ ˆ(n−1) ) = ˜ c) | φ ˆ(n−1) , y}, compute Λ(φ; Ec {ln f (y|φ, ˜ = [S˜ N ˜ ] denotes the trial value of φ. where φ ˆ(n) = (ii) M-step: Obtain the estimate in the next step as φ ˜ φ ˆ(n−1) ). arg maxφ˜ Λ(φ; ˜ φ ˆ(n−1) ) as In view of (3), we can in fact express Λ(φ;

A(n−1) B (n−1)

. = . =

K 1  (n−1) ξk , K

(18)

1 K

(19)

k=1 K  k=1

(n−1)

ηk

yk .

Given A(n−1) and B (n−1) , (17) can be straightforwardly ˜ , thus obtaining Sˆ(n) = maximized with respect to S˜ and N (n) 2 (n) (n−1) ˆ(n) ˆ (n) + M2 ), ˆ ˆ S (G ) and N = 2(A − 2B (n−1) G where  (n−1) + 4(B (n−1) )2 − 3A(n−1) M2 2B (n) ˆ G = . (20) 3A(n−1) Finally, for the next iteration the a posteriori statistics must be computed as (n)

ηk

(n)

ξk

= =

I  i=1 I  i=1

(n)

(21)

(n)

(22)

Pi,k Ri , Pi,k Ri2 .

(n) . With Bayes’ rule, the a posteriori probabilities Pi,k = ˆ(n) , yk } are given by Pr{|ck | = Ri | φ

ˆ(n) , |ck | = Ri ) · Pr{|ck | = Ri | φ ˆ(n) } f (yk | φ . ˆ(n) ) f (yk | φ (23) The denominator in (23) does not depend on index i and ˆ(n) } = Pr{|ck | = can be ignored, whereas Pr{|ck | = Ri | φ (n)

Pi,k =

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Fig. 2. Evolution of mean output and success rate for different SNR estimators (16-QAM, K = 500).

Ri } = Pi . Using these intermediate results together with the approximation in (8), we get 2 νk Pi ˆ (n) ˆ (n) (n) (24) Pi,k ≈ √ e−(yk −G Ri ) )/N Ri I (n) with νk a constant to ensure that i=1 Pi,k = 1. In the sequel, we refer to the resulting estimator as EM(1). In the absence of any a priori information (e. g. pilot sym(0) bols), the a posteriori probabilities are initialized as Pi,k = Pi I (0) 2 for all k. Hence, it is clear that ξk = i=1 Pi Ri = I . (0) (0) 1 and ηk = = 1 i=1 Pi Ri = β. This gives A . and B (0) = βM1 , so that the radicand in (20) is α = 2 2 4(B (0) )2 − 3A(0) M2 = 4β √ M1 − 3M2 . For sufficiently high SNRs, one has M1 ≈ β S and M2 ≈ S, which provides α ≈ 4β 4 − 3 S. If this term is negative, then EM(1) will break down in the first iteration. Thus, √ a necessary condition for EM(1) to succeed is that β 2 > 23 ≈ 0.866025, which holds true for most constellations in practice1 . Nevertheless, if approximation (9) is applied instead of (8) to evaluate (13), a variant of EM(1), henceforth denoted as EM(2), is obtained. The only difference is the computation of signal and noise power estimates, which reduces to (n−1) 2 B (n) ˆ S = , (25) A(n−1)   ˆ (n) = 2 M2 − Sˆ(n) A(n−1) . N (26)

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Fig. 3. Evolution of the normalized MSE for different SNR estimators (16QAM, K = 500, ρ0 = 9 dB).

the EM approach needs only 5 KI multiplications and KI LUT accesses. As a further benefit, the EM framework allows to exploit any possible pilot symbols in the frame, whose envelopes (but not necessarily their phases) are known to the receiver: it suffices to suitably change the a posteriori probabilities for those symbols. In this way, semi-blind EVB SNR estimation becomes feasible. Finally, we note that the ML algorithm has to be properly initialized; if a simple technique based on second- and fourth-order moments (M2 M4 ) is used for this purpose [4], [21], the success rate of ML is directly related to that of M2 M4 , which decreases significantly for non-CM constellations operated in the medium-to-high SNR range (see Fig. 2).

ˆ (n) in (26) is nonnegative at every It is easily shown that N iteration. Therefore, EM(2) can be used even with constellations not satisfying the constraint for β. It has been verified that the ML performance is close to the CRLB. In terms of their computational load, it turned out that Newton’s method requires about 40 KI multiplications and 3 KI lookup table (LUT) accesses per iteration, whereas

V. S IMULATION R ESULTS The estimator performance has been tested with 16-QAM signals (I = 3) and a frame length K = 500. For each simulation point, 5000 Monte Carlo runs were executed. Fig. 2 shows the mean behavior of EM and M2 M4 estimates. The latter is unbiased only for SNR ∈ [0, 10] dB, whereas the EM methods exhibit a significant bias solely at very low SNRs, as expected since they hinge on the high SNR approximations in (8) and (9). Also shown are the success rates of M2 M4 and EM(1), i. e., the fraction of trials for which estimations have been successful 2 . As shown in Fig. 3, the mean square error (MSE) of EM(1) and EM(2) is very close to the CRLB for SNR > 10 and 12 dB, respectively. Substantially degrading below these points, EM(2) deteriorates less gracefully than EM(1). On the other hand, M2 M4 provides excellent results in the low SNR range, whereas an increasing departure from the CRLB is observed for SNR > 5 dB. These facts suggest a hybrid approach, choosing different estimators depending on the value of the estimated SNR relative to a predefined threshold ρ0 : if the M2 M4 estimate (evaluated first) is smaller than ρ0 , it is

1 Exceptions include the 16-point star constellation defined in the V.29 modem standard [19], with β 2 = 0.8635, and the 32-APSK constellation recommended in the DVB-S2 standard [20] for ring ratios corresponding to a code rate of 3/4, where β 2 = 0.8588.

2 Since the square root of 2M 2 −M has to be computed [21], the estimate 4 2 does not provide meaningful results if this quantity is negative. On the other hand, EM(1) fails if the radicand in (20) becomes negative at any iteration. As mentioned above, EM(2) always succeeds.

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the exploitation of any available pilot symbols. Together with the moments-based M2 M4 algorithm, the EM estimator can be combined to a hybrid scheme with performance results very close to the CRLB. Although cheaper than numerical maximization of the log-likelihood function, the EM method is computationally more demanding than M2 M4 . A simpler but less powerful alternative, using hard decisions of the symbol envelopes, has been discussed in [23], thus achieving a tradeoff between performance and complexity. The application of more sophisticated EM variants, like parameter-expanded EM (PXEM) [24], is subject of future research. A PPENDIX

Fig. 4. Normalized MSE for joint SNR and phase EM estimation affected by different amounts of phase noise (16-QAM, K = 500).

accepted; otherwise, it is rejected and the EM(1) or EM(2) estimate must be computed. The threshold can be selected as the SNR value for which the individual estimates of M2 M4 and EM(1) or EM(2) achieve the same MSE. As seen in Fig. 3 as well, the hybrid M2 M4 – EM(1) estimator performs close to the CRLB over a wide range of SNR values, with a slight degradation between 6–11 dB; in the 9–15 dB range, slightly worse results are obtained with the hybrid M2 M4 – EM(2) method (not shown due to clarity and space). In order to assess the robustness of EVB estimators to phase uncertainties, Fig. 4 compares the MSE performance of the hybrid M2 M4 – EM(1) scheme with that of the joint SNR and phase EM estimation, assuming an unknown phase offset which remains constant throughout the whole data frame (θk = θ). This joint approach can be derived along the line developed in [17]; details are omitted due to page limitations. In addition, Fig. 4 illustrates the performance of the joint solution, when the carrier phase is affected by noise following the random walk (Wiener) model [22]: θk = θk−1 + Δk , k = 2, . . . , K, where θ1 is uniformly distributed in [−π, +π) 2 and Δk is zero-mean white Gaussian with variance σΔ . Of 2 course for σΔ = 0, a static phase offset is realized; at high SNRs, the normalized MSE goes to 1/K, i. e., the asymptotic value of the CRLB in the DA case (12), which is half that of EVB. However, the joint estimates degrade 2 > 0, especially with increasing SNR substantially if σΔ values. In contrast, the performance of the proposed hybrid M2 M4 – EM(1) method is not affected at all by phase noise. VI. C ONCLUDING R EMARKS We investigated the problem of NDA EVB estimation of the signal-to-noise ratio for non-CM constellations from an ML perspective. The corresponding CRLB was evaluated, with its asymptotic value shown to be twice that of the bound for DA coherent methods. An EM scheme was developed under a high SNR assumption, which iteratively estimates the SNR using the soft values of the symbol envelopes, allowing also

The MCRLB follows from the modified Fisher Information Matrix (FIM), which is the expectation of the conventional FIM (as computed for fixed nuisance parameters) with respect to the nuisance parameters [11]. Using an alternative form based on second-order derivatives [9], one has 2

MCRLB(ρ) =

2

∂ Γ −Ey,c { ∂N 2} 2

2

∂ Γ ∂ Γ 2 Ey,c { ∂∂ρΓ2 }Ey,c { ∂N 2 } − Ey,c { ∂ρ ∂N }

(27)

. where Γ = ln f (y|[ρN, N ], c). Applying (9) and omitting algebraic details due to limited space, we arrive at  K K K 2  2 3 ∂ 2Γ ρ  = − y + yk |ck | k ∂N 2 2N 2 N3 2 N5 k=1 k=1  2  ∂ Γ Kρ K ⇒ Ey,c − (28) =− ∂N 2 2N 2 2N 2 K  ∂2Γ 1  = − yk |ck | ∂ρ2 2 ρ3 N k=1  2  ∂ Γ K (29) ⇒ Ey,c =− 2 ∂ρ 2ρ K  1 ∂2Γ = −  yk |ck | ∂ρ ∂N 2 ρN 3 k=1  2  ∂ Γ K (30) ⇒ Ey,c =− ∂ρ ∂N 2N from which the high SNR approximation in (10) follows immediately. ACKNOWLEDGMENT The authors would like to thank their colleagues for fruitful discussions and the anonymous reviewers for their comments, which helped to substantially improve the quality of this paper. R EFERENCES [1] T. A. Summers and S. G. Wilson, “SNR mismatch and online estimation in turbo decoding,” IEEE Trans. Commun., vol. 46, pp. 421–423, Apr. 1998. [2] T. S. Chung and A. J. Goldsmith, “Degrees of freedom in adaptive modulation: a unified view,” IEEE Trans. Commun., vol. 49, pp. 1561– 1571, Sept. 2001. [3] D. R. Pauluzzi and N. C. Beaulieu, “A comparison of SNR estimation techniques for the AWGN channel,” IEEE Trans. Commun., vol. 48, pp. 1681–1691, Oct. 2000. [4] P. Gao and C. Tepedelenlio˘glu, “SNR estimation for nonconstant modulus constellations,” IEEE Trans. Signal Processing, vol. 53, pp. 865–870, Mar. 2005.

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