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Iterative SNR estimation using a priori information
Digital Signal Processing 19 (2009) 278–286
Contents lists available at ScienceDirect
Digital Signal Processing www.elsevier.com/locate/dsp
Iterative SNR estimation using a priori information Nie Yuanfei ∗ , Ge Jianhua, Wang Yong State Key Laboratory of Integrated Service Networks, Xidian University, No. 2 Taibainan Street, Xi’an, China
a r t i c l e
i n f o
Article history: Available online 17 July 2008 Keywords: Signal-to-noise ratio (SNR) Expectation maximization (EM) Cramer–Rao bound (CRB) Turbo-like system
a b s t r a c t A class iterative signal-to-noise ratio (SNR) estimation algorithm is proposed in this paper. The data samples are governed by a given distribution with a priori. The expectation maximization (EM) algorithm is applied to iteratively maximize the likelihood function so as to realize the SNR estimation. Cramer–Rao bounds (CRB) with different a priori are compared for binary phase shift keying and orthogonal phase shift keying systems, which show the potential of the SNR estimator in turbo-like systems. In high-order modulations, simulation results show that the reduced-complexity iterative method with equal a priori has better performance in middle or high SNR region than the foregone ones. Moreover, the new method with feedback information is the best when its iteration number is 4 and extrinsic information larger than 0.4. These methods are applied in the bit-interleaved coded modulation with iterative decode (BICM-ID) system to validate the effect of the proposed methods. © 2008 Elsevier Inc. All rights reserved.
1. Introduction Estimation of signal-to-noise ratio (SNR) is essential to the successful of any communications receiver, especially in adaptive coded modulation and turbo-like systems including turbo-like decoder, bit-interleaved coded modulation with iterative decode (BICM-ID) and turbo-equalization. Mismatched SNR parameter degrades the performance of the receiver [1,2]. For the turbo-like detector or decoder, overestimation of SNR is less detrimental than underestimation, and the required SNR accuracy to prevent large performance degradation is in a parochial region. A number of SNR estimators based on non-data-aided (NDA) or data-aided (DA) have been discussed in the literature [2–6]. Numerical evaluation of Crame–Rao bounds (CRBs) was given for BPSK and QPSK [7]. These methods are in two style approaches that recur to maximum-likelihood (ML) or moment method (MM). In NDA algorithms, MM has better performance in the low SNR region, while ML has better performance in the high region. As for the DA, ML seems to be the only choice. In the literature [5], a NDA SNR estimator for BPSK based on expectation maximization (EM) algorithm was proposed to iteratively maximize the NDA likelihood function-offering and improve the performance over a wide range of SNR and data lengths. Another iterative method using numerical computation was proposed in [6]. In this paper, the algorithm in literature [5] is extended to the high-order modulations and a reduced-complexity algorithm presented. Then an iterative SNR estimator is proposed in turbo-like systems. In such systems, extrinsic information fed back from the soft decoder can be used to improve the estimation accuracy, which has been few studied before according to our survey. Such a new algorithm abridges the ML algorithm in the DA mode and that in the NDA. This paper will be organized as follows: in Section 2 we give the system model briefly; in Section 3, the iterative algorithm is presented for high-order modulations with reduced-complexity. Section 4, the CRBs are compared with different a priori information and the iterative algorithm with extrinsic information is put forward. In Section 5 the simulation results are shown for different
*
Corresponding author. E-mail addresses: [email protected] (N. Yuanfei), [email protected] (G. Jianhua), [email protected] (W. Yong).
1051-2004/$ – see front matter doi:10.1016/j.dsp.2008.07.003
© 2008 Elsevier Inc.
All rights reserved.
N. Yuanfei et al. / Digital Signal Processing 19 (2009) 278–286
279
SNR estimators. Finally, we apply these methods in the BICM-ID systems to validate such results. Section 6 concludes this paper. 2. System model In the absence of errors in timing, phase, and frequency offset, symbol spaced samples at the matched filter output (assumed inter-symbol interference free) can be expressed as rk = hk xk + nk ,
k = 1, 2, . . . N ,
(1)
where xk ∈ χ = {s1 , . . . , s M } is a modulated symbol, and hk is the channel fading coefficient. nk is additional white Gaussian noise with zero mean and variance σ 2 . log2 M bits map onto a constellation symbol. The vth bit in a modulated symbol xk is labeled by bk, v , where v = 1, . . . , log2 M. Similarly, siv represents the vth bit in si , where i = 1, . . . , M. The transmitted and received samples in a frame can be conveniently represented in N × 1 column vector form, such as X = [x1 · · · x N ] T and R = [r1 · · · r N ] T respectively, where T is the transpose operator. Since the received samples are assumed with no interference to each other, for rk under the condition of its intrinsic probability pk,i and θ k = [hk σ 2 ] T , the probability density function (PDF) is expressed as
p k ,i
f A (rk | θ k ) =
s i ∈χ
2πσ 2
exp −
|rk − hk si |2
σ2
(2)
,
where pk,i represents the original probability of which xk equals to si . And more, the conditional PDF of the received signal vector is
f A (R | θ ) =
f A (rk | θ).
(3)
k=1,..., N
In flat fading channels, hk can approximate to a constant h in a frame, and θ = [h σ 2 ] T . Without loss of generality, SNR in the received signal can be expressed as α = |h|2 /σ 2 when the constellation is unitary. Also h can be assumed as a real number. 3. CRB for BPSK/QPSK with equiprobable a priori Before proposing the new iterative algorithm, the CRBs of BPSK and QPSK with equiprobable a priori are presented to illustrate the potential in turbo-like scenarios. The CRB is a lower bound for the mean square error (MSE) of any unbiased estimator satisfying a number of regularity conditions. To present the estimated SNR in decibels scale, we use g (α ) = 10 log10 (α ) to replace the decimal one. For the SNR estimation problem, it can be expressed as CRB =
∂ g (α )T −1 ∂ g (α ) , J T θθ ∂θ ∂θ
JθθT = E
∂ 2 f A (R; θ ) ∂θ∂θ T
(4)
,
where J θθ T is the Fisher information matrix. The model of a priori is random process for given signal sequence and channel condition. To make this problem simple, we assume it equal-probability. For BPSK, the original symbol probability is p 1 . Then substitute (3) and the decibel expression into (4), after some operations, the CRB for BPSK is obtained as CRBBPSK (α ; p 1 ) = where
200( α1 − ϕ (α , p 1 ) + 1) N (ln 10)2 (1 − ϕ (α , p 1 ) − 4αϕ (α , p 1 ))
ϕ (α , p 1 ) is a integral function of α and p 1 as +∞
ϕ (α , p 1 ) = A −∞
and
(5)
,
A=2
2
π
ξ 2 exp( −ξ2 ) dξ 2
p 1 exp(ξ
√
2α ) + (1 − p 1 ) exp(−ξ
√
2α )
p 1 (1 − p 1 ) exp(−α ).
Note that in the above equations, α is in linear scale but CRBBPSK is in (dB)2 . In formulation (5), p 1 = 0 and p 1 = 0.5 correspond to the DA and NDA cases, respectively. Similarly, for the QPSK constellation with Gray labeling, if the signal is sampled in the two-dimensional form, not the one-dimensional, the CRB is CRBQPSK (α ; p 1 ) =
100( α2 − ϕ ( α2 , p 1 ) + 1)
N (ln(10))2 (1 − ϕ ( α2 , p 1 ) − 2αϕ ( α2 , p 1 ))
.
(6)
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N. Yuanfei et al. / Digital Signal Processing 19 (2009) 278–286
Fig. 1. CRBs of BPSK and QPSK with different a priori.
For the QPSK with anti-Gray labeling or those higher-order constellations, their CRBs are not easy to calculate. Therefore, only the two CRB in (5) and (6) are drawn in Fig. 1, which illustrate the potential of the estimation methods with some a priori information. When the bit error ratio (BER) arrives 0.1, the bound is much lower than that of 0.5. And when BER is 0.01, it is close to DA. Since low BER can be obtained in turbo-like systems, it is necessary to make use of extrinsic information fed back from the soft decoder in a new SNR estimator. 4. Iterative estimation without soft decoder This section deals with the iterative SNR estimation with no extrinsic information fed back from the soft decoder. Under such condition, the random symbol PDF with no extrinsic information is f N (rk | θ) = f A (rk | θ )| pk (si )=1/ M .
(7)
Similarly, we have PDF of the received vector, f N ( R | θ ). According to the ML principle, a sub-optimal solution of the parameter vector θ is
θ˜ = arg max f N (R | θ).
(8)
θ
But there is no closed-formed analytical solution when f N (rk | θ) is a mixed distribution [7]. As an alternative to potentially costly exhaustive two-dimension search, EM algorithm can be used to obtain suboptimal result through iteratively processing [8]. Define θ (m) = [h(m) σ(2m) ] T the estimated parameter vector after m − 1 iterations. Let Q (θ (m) | θ (m−1) ) denote the expectation of the samples’ logarithmic likelihood PDF, f N (R | θ (m) ), at parameter vector θ (m) with respect to X conditioned on R and θ (m−1) , which is
Q (θ (m) | θ (m−1) ) = E X|R;θ (m−1) ln f N (R | θ (m) ) ,
(9)
where ln(·) is the natural logarithm function and E [·] denotes the expectation operator. Based on R and θ (m−1) , the cost function can be Q (θ (m) | θ (m−1) ) =
M N
−1 pm ln f N (rk | θ (m) ) k ,i
k=1 i =1
= C − N ln σ(2m) −
N |rk |2 − rk∗ h(m) x¯ k,m−1 − rk h∗(m) x¯ k∗,m−1 − |h(m) |2 xˆ k,m−1
σ(2m)
k=1
,
(10)
where ∗ is the complex conjugate operator and C is a constant. x¯ k,m−1 and xˆ k,m−1 stand for the conditional mean and power, respectively, which are x¯ k,m−1 =
M i =1
−1 si pm , k ,i
xˆ k,m−1 =
M i =1
−1 | s i |2 p m . k ,i
(11)
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The original probability of which xk equals to si under known θ (m−1) can be expressed as −1 pm = p (si | rk ; θ (m−1) ) = p (rk | si ; θ (m−1) ) p (si ; θ (m−1) )/ p (rk ; θ (m−1) ). k ,i
(12)
Since no extrinsic information provided by the soft decoder, (12) can be rewritten as M
−1 pm = p (rk | si ; θ (m−1) )/ p (rk | θ m−1 ) = p (rk |si ; θ (m−1) ) k ,i
p (rk | si ; θ (m−1) ).
(13)
i =1
And the conditional probability is
p (rk | si ; θ (m−1) ) ∼ exp −
|rk − h(m−1) si |2
σ(2m−1)
(14)
.
Similarly to [5], straight-forward differentiation yields θ (m) which minimizes Q (θ (m) ; θ (m−1) ), ∂ Q (θ (m) ; θ (m−1) )/∂θ (m) = 0, then
N
∗
¯ k,m−1 rk k=1 x
hm = N
ˆ k,m−1 k−1 x
1
2 m
σ =
N
N
2
|rk | −
k=1
, |
N
∗
¯ k,m−1 rk | k=1 x
N ˆ k,m−1 k=1 x
2
(15)
.
The initial parameter vector can be set as θ (0) = [1 ∞] T , which responds to the ML method based on hard-decision. In this algorithm, the symbol reliability is updated through (13) and (14) and the vector is iteratively obtained by (11) and (15). For high-order modulations, complexity of such processing is direct ratio to the size of the constellation. To simplify the method, a main way is to decrease the searching space. Like the listed sphere algorithm, we only consider part of the constellation. Define βk,i = |rk − hm−1 si |, and the reduced-complexity method is described as follows. (i) Initialization: obtain θ (1) according to the original ML with hard-decision; −1 (ii) Selection: calculate βk,i , order them in ascent, select the first M r value and set the other M − M r infinite, to get pm k ,i through substituting them into (13) and (14); −1 into (11); (iii) Update: achieve the new parameter vector θ (m) by importing pm k ,i (iv) Repetition: repeat (ii) and (iii) until it satisfies such conditions as given iteration number or other stop criterions, and then estimated SNR is found. Different from the previous method, another iterative method was proposed in [6]. Since ML method can converge to a local optimal result, which means lim h(m−1) = h(m) ,
m→∞
lim
m→∞
σ(2m−1) = σ(2m) .
(16)
Therefore, the equation can be solved by the numerical method, which is built by substituting (11), (14) and (15) in (16). This method is a bit difficult to calculate for orthogonal constellations, but has a simplified alternation in 8PSK [6]. 5. Iterative estimation with soft decoder In Section 3, p (si | θ (m−1) ) is assumed to be equal for all constellation points. This section studies the turbo-like scenarios. The likelihood of bk, v fed back from the soft decoder is written as b˜ k, v . The probability of which the decision of bk, v is v b is v
lk,bv =
exp(b˜ k, v sgn(2v b − 1)/2) 2 cosh(b˜ k, v /2)
(17)
,
where v b ∈ {0, 1}, sgn(·) is the sign function that the value is one when the variant in the bracket is larger than zero, and zero otherwise. cosh(·) is the hyperbolic cosine function. Under the assumption that transmitted bits are independent to each other, a priori information of rk is
−1 qm = p (rk |si ; θ (m−1) ) p (si ; θ (m−1) ) = exp −|rk − hm−1 si |2 /σm2 −1 k. v
v
lk,bv
siv = v b
=
1 M
exp −
v =1,...,log2 M
|rk − hm−1 si |
σm2 −1
2
+
siv = v b v =1,...,log2 M
b˜ k, v sign(2v b − 1) 2
v =1,...,log2 M
cosh
−1
˜
bk , v 2
.
(18)
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Substitute it into (7), the original probability of constellation point si about rk and θ (m−1) can be expressed as −1 −1 pm = qm k, v k. v
M
−1 qm . k. v
(19)
i =1
Then the other update steps are similar to those in the previous section. Such new algorithm is closely correlated to the extrinsic information, which means that the performance is different for the same modulation with different labels or codes. Therefore, it is necessary to build the model about the extrinsic information which is independent of feedback manner. Fortunately, S. Ten Brink proposed such model [9] as b˜ k, v =
σ w2 2
(2bk, v − 1) + w ,
(20)
where w is the Gaussian noise with zero mean and variance
+∞ I =1−
√
−∞
1 2π σ w
exp −
(ξ − σ w2 /2)2 2 2σ w
σ w2 . And the measurement of extrinsic information is
log2 1 + exp(−ξ ) dξ.
(21)
2 2 According to it, the table of I verse σ w can be built. For given I , σ w is obtained through looking up the table, and then use (16) to finish the extrinsic information model. Besides these, since the ML algorithm has bias, a modification method was proposed in [3]
αˆ =
N −3 N
α˜ −
1 N
(22)
.
This paper also uses it to reduce bias in the following section. 6. Simulation results In this section, simulation results are presented to make performance comparisons of the proposed algorithms in this paper and other typical ones for 8PSK and 16QAM. For the algorithms based on ML, the reduced-bias results are adopted. To differentiate the presented methods, the iterative reduced-complexity one with and without extrinsic information are briefly called MLA and MLI-Mr, respectively. We call ML-HD the ML method based on hard-decision. For the methods based on moment, the first moment and the second moment algorithm (M1M2) is used in 8PSK while the second moment and the fourth moment (M2M4) in 16QAM according to the conclusions in [4]. The algorithm in [6] is named numerical iterative computation (NIC). There are 200 samples in a frame. In particular, a Monte Carlo simulation of 200,000 experiments is used to estimate the root mean squared error (RMSE), which is
β=
[ E ((αˆ − α )2 )]1/2
α
Nt
≈
k=1
(αˆ k − α )2 / N t
α
.
(23)
Also the estimated mean is compared to determine how much bias in these methods. Figs. 2 and 3 show the difference of these algorithms for 8PSK. As for NIC, its convergence is sensitive in the high SNR zone, which means that the optimal point is hard to search. For MMs, it needs to build the table, which corresponds to the relationship between the input SNR and the quotient of different order moment. The quotient changes slowly in high SNR region. It means that the MM resolution deteriorates at such case. It leads to the bumpiness in high SNR region as shown in Figs. 2 and 3, which is the disadvantage of the MMs. For those methods based on ML. It finds that their performance is close to that of the DA estimation when SNR is high, which means that they are asymptotically optimal. Results show that iteration can improve the performance compared with ML-HD. Estimated mean and RMSE of the reduced-complexity version, MLI-4, have the feasible degradation compared with MLI-8 with the same iteration number. In SNR region lower than 3 dB, MLI-8 with 5 iterations approaches M1M2 and NIC, while it is better in other region. In fact, noise influences the reliability updating of (12) which restricts the performance of MLI. When SNR goes beyond some threshold, like 3 dB in 8PSK, the iterative gain is revealed. At the same time, Figs. 2 and 3 provide the results of these estimators with extrinsic information for 8PSK. All the iteration number of MLA is 4. Results show that the extrinsic information can drastically improve the performance by comparing with MLI-8. Besides these, when the extrinsic information is larger than 0.4, MLA has lower bias than M1M2 and NIC. And more, the difference between MLA and DA is almost negligible when the feedback information exceeds 0.7. In practice, BER in turbo-like system is still high when the extrinsic information is 0.4 [10]. It means that MLA can work well in low SNR scenarios with efficient label and code. Similar conclusions are drawn for 16QAM from Figs. 4 and 5. In this case, the moment method, M2M4, is better than MLIs when SNR is smaller than 7 dB. For MLA with 4 iterations, when the extrinsic information is larger than 0.4, its RMSE is better than those of M2M4 and MLIs.
N. Yuanfei et al. / Digital Signal Processing 19 (2009) 278–286
283
Fig. 2. Comparison of estimated mean between the proposed iterative estimators and other estimators for 8PSK.
Fig. 3. Comparison of RMSE between the proposed iterative estimators and other estimators for 8PSK.
7. Applications in the BICM-ID systems Finally, the SNR estimation algorithms are applied in the BICM-ID systems. The realization and analysis can be found in [11] for BICM-ID. Two scenarios are tested. In Fig. 6, the modulation is 8PSK and the label is SSP [11]. The frame length is 150 symbols and the generation polynomial is [7,5]octal with code ratio 1/2. The random interleaver is used. It provides the results of different estimation with six times iterative decoding. In all these SNR estimators, MLA is the best, which approaches the ideal transmission when SNR is higher than 3.5 dB. Even in low SNR region, MLA almost has no degradation. Such result shows that the extrinsic information from the soft decoder can greatly improve the estimated performance. ML-HD is the worst estimator, and degrades about 1 dB. NIC is the second worst. But the performance of this method is the same as the ideal when SNR equals to 6.5 dB. For the other methods, the performance is approximately same. Such results can be explained by the estimators’ RMSE in Fig. 3. When RMSE is smaller, the estimated value is closer to the real one and the system performance is better. But it is noted that there exists error-floor for BICM-ID. So when SNR is very higher, the estimated error becomes obvious-less. In Fig. 7, the modulation is 16QAM and the label is MSP [12]. The result is similar to that in Fig. 6. MLA is close to the ideal and better than the other estimators, especially when SNR is better than 5 dB. Differently, the M2M4 method is always better than the MLI-6 and MLI-16. From Fig. 5, it finds that RMSE of M2M4 is smaller than those of MLIs in the region from
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N. Yuanfei et al. / Digital Signal Processing 19 (2009) 278–286
Fig. 4. Comparison of estimated mean between MLI and other estimators in 16QAM.
Fig. 5. Comparison of RMSE between MLI and other estimators in 16QAM.
−5 dB to 7 dB. When SNR is larger than 7 dB, MLIs are better than M2M4. But this region in the given BICM-ID scenario corresponds to the error-floor, so the advantages of MLIs do not emerge. 8. Conclusions This paper studies the iterative SNR estimation using a priori information. Firstly, the algorithm without feedback information is presented with its reduced-complexity version. Then the algorithm with feedback information is also proposed. The simulation results show that iteration improves estimation accuracy greatly. In middle or high SNR region, the new methods are better than the other foregone NDA ones. Besides it, when feedback soft information is up to some degree, e.g. 0.4, the new method is the best in the NDA ones. Finally, these methods are applied in the given BICM-ID systems to validate these conclusions. Acknowledgments This research was supported by the 863 Plan numbered 2006AA01Z270 and the National Science Foundation of China under grant No. 60496316. Many thanks to Dr. Ai Bo in Beijing Jiaotong University for his constructive comments.
N. Yuanfei et al. / Digital Signal Processing 19 (2009) 278–286
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Fig. 6. The performance of BICM-ID with 8PSK and 6 iterations.
Fig. 7. The performance of BICM-ID with 16QAM and 6 iterations.
References [1] Y. Huang, J.A. Ritcey, EXIT chart analysis of BICM-ID over AWGN channels with SNR mismatch, IEEE Commun. Lett. 8 (8) (2004) 532–534. [2] T.A. Summers, S.G. Wilson, SNR mismatch and online estimation in turbo decoding, IEEE Trans. Commun. 46 (4) (1998) 421–423. [3] David R. Pauluzzi, Norman C. Beaulieu, A comparison of SNR estimation techniques for the AWGN channel, IEEE Trans. Commun. 48 (10) (2000) 1681–1691. [4] Ping Gao, Cihan Tepedelenlio˘glu, SNR Estimation for nonconstant modulus constellations, IEEE Trans. SP 53 (3) (2005) 865–870. [5] Ami Wiesel, Jason Goldberg, Hagit Messer, Non-data-aided signal-to-noise-ratio estimation, in: ICC2002, New York, USA, 2002, pp. 198–201. [6] Bin Li, Robert DiFazio, Ariela Zeira, et al., New results on SNR estimation of MPSK modulated signals, in: Proc. 14th International Symposium on Personal, Indoor and Mobile Radio Communication (PIMRC), Beijing, 2003, pp. 2373–2377. [7] Nader Sheikholeslami Alagha, Cramer–Rao bounds of SNR estimates for BPSK and QPSK modulated signals, IEEE Commun. Lett. 5 (1) (2001) 10–12. [8] D.M. Titterington, A.F.M. Smith, U.E. Makov, Statistical Analysis of Finite Mixture Distributions, Wiley, 1985. [9] S. ten Brink, Convergence behavior of iteratively decoded parallel concatenated codes, IEEE Trans. Commun. 49 (5) (2001) 1727–1737. [10] Thorsten Clevorn, Susanne Godtmann, Peter Vary, BER prediction using EXIT charts for BICM with iterative decoding, IEEE Commun. Lett. 10 (1) (2006) 49–51. [11] X. Li, A. Chindapol, J.A. Ritcey, Bit-interleaved coded modulation with iterative decoding and 8PSK modulation, IEEE Trans. Commun. 50 (8) (2002) 1250–1257. [12] A. Chindapol, J.A. Ritcey, Design analysis and performance evaluation for BICM-ID with square QAM constellations in Rayleigh fading channels, IEEE J. Select. Areas Commun. 19 (5) (2001) 944–957.
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Nie Yuanfei received all the B.Sc., M.S.E. and Ph.D. degrees from Xidian University, China, in 2002, 2005 and 2007, respectively. Now he is engaging in the ASIC design for digital mobile handle TV. His research interests include OFDM, space–time signal processing and iterative decoding in serial concatenated systems. Ge Jianhua received the B.Sc., Master and Ph.D. degrees from Xidian University, China, in 1982, 1985 and 1989, respectively. He is now the professor in both Xidian University and Shanghai JiaoTong University, China. Professor Ge is the senior member of Chinese Electronics Institute. He has won lots of scientific and technical prizes in China. His interests are wireless communications and web security. Wang Yong received the B.Sc., Master and Ph.D. degrees from Xidian University, China, in 1997, 2002 and 2005, respectively. And now Dr. Wang is an associate professor on communications in the Key Lab. of ISN in Xidian University. He has once participated in the key research project on HDTV in TEEG in China and his interests are wireless communications.
Contents lists available at ScienceDirect
Digital Signal Processing www.elsevier.com/locate/dsp
Iterative SNR estimation using a priori information Nie Yuanfei ∗ , Ge Jianhua, Wang Yong State Key Laboratory of Integrated Service Networks, Xidian University, No. 2 Taibainan Street, Xi’an, China
a r t i c l e
i n f o
Article history: Available online 17 July 2008 Keywords: Signal-to-noise ratio (SNR) Expectation maximization (EM) Cramer–Rao bound (CRB) Turbo-like system
a b s t r a c t A class iterative signal-to-noise ratio (SNR) estimation algorithm is proposed in this paper. The data samples are governed by a given distribution with a priori. The expectation maximization (EM) algorithm is applied to iteratively maximize the likelihood function so as to realize the SNR estimation. Cramer–Rao bounds (CRB) with different a priori are compared for binary phase shift keying and orthogonal phase shift keying systems, which show the potential of the SNR estimator in turbo-like systems. In high-order modulations, simulation results show that the reduced-complexity iterative method with equal a priori has better performance in middle or high SNR region than the foregone ones. Moreover, the new method with feedback information is the best when its iteration number is 4 and extrinsic information larger than 0.4. These methods are applied in the bit-interleaved coded modulation with iterative decode (BICM-ID) system to validate the effect of the proposed methods. © 2008 Elsevier Inc. All rights reserved.
1. Introduction Estimation of signal-to-noise ratio (SNR) is essential to the successful of any communications receiver, especially in adaptive coded modulation and turbo-like systems including turbo-like decoder, bit-interleaved coded modulation with iterative decode (BICM-ID) and turbo-equalization. Mismatched SNR parameter degrades the performance of the receiver [1,2]. For the turbo-like detector or decoder, overestimation of SNR is less detrimental than underestimation, and the required SNR accuracy to prevent large performance degradation is in a parochial region. A number of SNR estimators based on non-data-aided (NDA) or data-aided (DA) have been discussed in the literature [2–6]. Numerical evaluation of Crame–Rao bounds (CRBs) was given for BPSK and QPSK [7]. These methods are in two style approaches that recur to maximum-likelihood (ML) or moment method (MM). In NDA algorithms, MM has better performance in the low SNR region, while ML has better performance in the high region. As for the DA, ML seems to be the only choice. In the literature [5], a NDA SNR estimator for BPSK based on expectation maximization (EM) algorithm was proposed to iteratively maximize the NDA likelihood function-offering and improve the performance over a wide range of SNR and data lengths. Another iterative method using numerical computation was proposed in [6]. In this paper, the algorithm in literature [5] is extended to the high-order modulations and a reduced-complexity algorithm presented. Then an iterative SNR estimator is proposed in turbo-like systems. In such systems, extrinsic information fed back from the soft decoder can be used to improve the estimation accuracy, which has been few studied before according to our survey. Such a new algorithm abridges the ML algorithm in the DA mode and that in the NDA. This paper will be organized as follows: in Section 2 we give the system model briefly; in Section 3, the iterative algorithm is presented for high-order modulations with reduced-complexity. Section 4, the CRBs are compared with different a priori information and the iterative algorithm with extrinsic information is put forward. In Section 5 the simulation results are shown for different
*
Corresponding author. E-mail addresses: [email protected] (N. Yuanfei), [email protected] (G. Jianhua), [email protected] (W. Yong).
1051-2004/$ – see front matter doi:10.1016/j.dsp.2008.07.003
© 2008 Elsevier Inc.
All rights reserved.
N. Yuanfei et al. / Digital Signal Processing 19 (2009) 278–286
279
SNR estimators. Finally, we apply these methods in the BICM-ID systems to validate such results. Section 6 concludes this paper. 2. System model In the absence of errors in timing, phase, and frequency offset, symbol spaced samples at the matched filter output (assumed inter-symbol interference free) can be expressed as rk = hk xk + nk ,
k = 1, 2, . . . N ,
(1)
where xk ∈ χ = {s1 , . . . , s M } is a modulated symbol, and hk is the channel fading coefficient. nk is additional white Gaussian noise with zero mean and variance σ 2 . log2 M bits map onto a constellation symbol. The vth bit in a modulated symbol xk is labeled by bk, v , where v = 1, . . . , log2 M. Similarly, siv represents the vth bit in si , where i = 1, . . . , M. The transmitted and received samples in a frame can be conveniently represented in N × 1 column vector form, such as X = [x1 · · · x N ] T and R = [r1 · · · r N ] T respectively, where T is the transpose operator. Since the received samples are assumed with no interference to each other, for rk under the condition of its intrinsic probability pk,i and θ k = [hk σ 2 ] T , the probability density function (PDF) is expressed as
p k ,i
f A (rk | θ k ) =
s i ∈χ
2πσ 2
exp −
|rk − hk si |2
σ2
(2)
,
where pk,i represents the original probability of which xk equals to si . And more, the conditional PDF of the received signal vector is
f A (R | θ ) =
f A (rk | θ).
(3)
k=1,..., N
In flat fading channels, hk can approximate to a constant h in a frame, and θ = [h σ 2 ] T . Without loss of generality, SNR in the received signal can be expressed as α = |h|2 /σ 2 when the constellation is unitary. Also h can be assumed as a real number. 3. CRB for BPSK/QPSK with equiprobable a priori Before proposing the new iterative algorithm, the CRBs of BPSK and QPSK with equiprobable a priori are presented to illustrate the potential in turbo-like scenarios. The CRB is a lower bound for the mean square error (MSE) of any unbiased estimator satisfying a number of regularity conditions. To present the estimated SNR in decibels scale, we use g (α ) = 10 log10 (α ) to replace the decimal one. For the SNR estimation problem, it can be expressed as CRB =
∂ g (α )T −1 ∂ g (α ) , J T θθ ∂θ ∂θ
JθθT = E
∂ 2 f A (R; θ ) ∂θ∂θ T
(4)
,
where J θθ T is the Fisher information matrix. The model of a priori is random process for given signal sequence and channel condition. To make this problem simple, we assume it equal-probability. For BPSK, the original symbol probability is p 1 . Then substitute (3) and the decibel expression into (4), after some operations, the CRB for BPSK is obtained as CRBBPSK (α ; p 1 ) = where
200( α1 − ϕ (α , p 1 ) + 1) N (ln 10)2 (1 − ϕ (α , p 1 ) − 4αϕ (α , p 1 ))
ϕ (α , p 1 ) is a integral function of α and p 1 as +∞
ϕ (α , p 1 ) = A −∞
and
(5)
,
A=2
2
π
ξ 2 exp( −ξ2 ) dξ 2
p 1 exp(ξ
√
2α ) + (1 − p 1 ) exp(−ξ
√
2α )
p 1 (1 − p 1 ) exp(−α ).
Note that in the above equations, α is in linear scale but CRBBPSK is in (dB)2 . In formulation (5), p 1 = 0 and p 1 = 0.5 correspond to the DA and NDA cases, respectively. Similarly, for the QPSK constellation with Gray labeling, if the signal is sampled in the two-dimensional form, not the one-dimensional, the CRB is CRBQPSK (α ; p 1 ) =
100( α2 − ϕ ( α2 , p 1 ) + 1)
N (ln(10))2 (1 − ϕ ( α2 , p 1 ) − 2αϕ ( α2 , p 1 ))
.
(6)
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Fig. 1. CRBs of BPSK and QPSK with different a priori.
For the QPSK with anti-Gray labeling or those higher-order constellations, their CRBs are not easy to calculate. Therefore, only the two CRB in (5) and (6) are drawn in Fig. 1, which illustrate the potential of the estimation methods with some a priori information. When the bit error ratio (BER) arrives 0.1, the bound is much lower than that of 0.5. And when BER is 0.01, it is close to DA. Since low BER can be obtained in turbo-like systems, it is necessary to make use of extrinsic information fed back from the soft decoder in a new SNR estimator. 4. Iterative estimation without soft decoder This section deals with the iterative SNR estimation with no extrinsic information fed back from the soft decoder. Under such condition, the random symbol PDF with no extrinsic information is f N (rk | θ) = f A (rk | θ )| pk (si )=1/ M .
(7)
Similarly, we have PDF of the received vector, f N ( R | θ ). According to the ML principle, a sub-optimal solution of the parameter vector θ is
θ˜ = arg max f N (R | θ).
(8)
θ
But there is no closed-formed analytical solution when f N (rk | θ) is a mixed distribution [7]. As an alternative to potentially costly exhaustive two-dimension search, EM algorithm can be used to obtain suboptimal result through iteratively processing [8]. Define θ (m) = [h(m) σ(2m) ] T the estimated parameter vector after m − 1 iterations. Let Q (θ (m) | θ (m−1) ) denote the expectation of the samples’ logarithmic likelihood PDF, f N (R | θ (m) ), at parameter vector θ (m) with respect to X conditioned on R and θ (m−1) , which is
Q (θ (m) | θ (m−1) ) = E X|R;θ (m−1) ln f N (R | θ (m) ) ,
(9)
where ln(·) is the natural logarithm function and E [·] denotes the expectation operator. Based on R and θ (m−1) , the cost function can be Q (θ (m) | θ (m−1) ) =
M N
−1 pm ln f N (rk | θ (m) ) k ,i
k=1 i =1
= C − N ln σ(2m) −
N |rk |2 − rk∗ h(m) x¯ k,m−1 − rk h∗(m) x¯ k∗,m−1 − |h(m) |2 xˆ k,m−1
σ(2m)
k=1
,
(10)
where ∗ is the complex conjugate operator and C is a constant. x¯ k,m−1 and xˆ k,m−1 stand for the conditional mean and power, respectively, which are x¯ k,m−1 =
M i =1
−1 si pm , k ,i
xˆ k,m−1 =
M i =1
−1 | s i |2 p m . k ,i
(11)
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The original probability of which xk equals to si under known θ (m−1) can be expressed as −1 pm = p (si | rk ; θ (m−1) ) = p (rk | si ; θ (m−1) ) p (si ; θ (m−1) )/ p (rk ; θ (m−1) ). k ,i
(12)
Since no extrinsic information provided by the soft decoder, (12) can be rewritten as M
−1 pm = p (rk | si ; θ (m−1) )/ p (rk | θ m−1 ) = p (rk |si ; θ (m−1) ) k ,i
p (rk | si ; θ (m−1) ).
(13)
i =1
And the conditional probability is
p (rk | si ; θ (m−1) ) ∼ exp −
|rk − h(m−1) si |2
σ(2m−1)
(14)
.
Similarly to [5], straight-forward differentiation yields θ (m) which minimizes Q (θ (m) ; θ (m−1) ), ∂ Q (θ (m) ; θ (m−1) )/∂θ (m) = 0, then
N
∗
¯ k,m−1 rk k=1 x
hm = N
ˆ k,m−1 k−1 x
1
2 m
σ =
N
N
2
|rk | −
k=1
, |
N
∗
¯ k,m−1 rk | k=1 x
N ˆ k,m−1 k=1 x
2
(15)
.
The initial parameter vector can be set as θ (0) = [1 ∞] T , which responds to the ML method based on hard-decision. In this algorithm, the symbol reliability is updated through (13) and (14) and the vector is iteratively obtained by (11) and (15). For high-order modulations, complexity of such processing is direct ratio to the size of the constellation. To simplify the method, a main way is to decrease the searching space. Like the listed sphere algorithm, we only consider part of the constellation. Define βk,i = |rk − hm−1 si |, and the reduced-complexity method is described as follows. (i) Initialization: obtain θ (1) according to the original ML with hard-decision; −1 (ii) Selection: calculate βk,i , order them in ascent, select the first M r value and set the other M − M r infinite, to get pm k ,i through substituting them into (13) and (14); −1 into (11); (iii) Update: achieve the new parameter vector θ (m) by importing pm k ,i (iv) Repetition: repeat (ii) and (iii) until it satisfies such conditions as given iteration number or other stop criterions, and then estimated SNR is found. Different from the previous method, another iterative method was proposed in [6]. Since ML method can converge to a local optimal result, which means lim h(m−1) = h(m) ,
m→∞
lim
m→∞
σ(2m−1) = σ(2m) .
(16)
Therefore, the equation can be solved by the numerical method, which is built by substituting (11), (14) and (15) in (16). This method is a bit difficult to calculate for orthogonal constellations, but has a simplified alternation in 8PSK [6]. 5. Iterative estimation with soft decoder In Section 3, p (si | θ (m−1) ) is assumed to be equal for all constellation points. This section studies the turbo-like scenarios. The likelihood of bk, v fed back from the soft decoder is written as b˜ k, v . The probability of which the decision of bk, v is v b is v
lk,bv =
exp(b˜ k, v sgn(2v b − 1)/2) 2 cosh(b˜ k, v /2)
(17)
,
where v b ∈ {0, 1}, sgn(·) is the sign function that the value is one when the variant in the bracket is larger than zero, and zero otherwise. cosh(·) is the hyperbolic cosine function. Under the assumption that transmitted bits are independent to each other, a priori information of rk is
−1 qm = p (rk |si ; θ (m−1) ) p (si ; θ (m−1) ) = exp −|rk − hm−1 si |2 /σm2 −1 k. v
v
lk,bv
siv = v b
=
1 M
exp −
v =1,...,log2 M
|rk − hm−1 si |
σm2 −1
2
+
siv = v b v =1,...,log2 M
b˜ k, v sign(2v b − 1) 2
v =1,...,log2 M
cosh
−1
˜
bk , v 2
.
(18)
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Substitute it into (7), the original probability of constellation point si about rk and θ (m−1) can be expressed as −1 −1 pm = qm k, v k. v
M
−1 qm . k. v
(19)
i =1
Then the other update steps are similar to those in the previous section. Such new algorithm is closely correlated to the extrinsic information, which means that the performance is different for the same modulation with different labels or codes. Therefore, it is necessary to build the model about the extrinsic information which is independent of feedback manner. Fortunately, S. Ten Brink proposed such model [9] as b˜ k, v =
σ w2 2
(2bk, v − 1) + w ,
(20)
where w is the Gaussian noise with zero mean and variance
+∞ I =1−
√
−∞
1 2π σ w
exp −
(ξ − σ w2 /2)2 2 2σ w
σ w2 . And the measurement of extrinsic information is
log2 1 + exp(−ξ ) dξ.
(21)
2 2 According to it, the table of I verse σ w can be built. For given I , σ w is obtained through looking up the table, and then use (16) to finish the extrinsic information model. Besides these, since the ML algorithm has bias, a modification method was proposed in [3]
αˆ =
N −3 N
α˜ −
1 N
(22)
.
This paper also uses it to reduce bias in the following section. 6. Simulation results In this section, simulation results are presented to make performance comparisons of the proposed algorithms in this paper and other typical ones for 8PSK and 16QAM. For the algorithms based on ML, the reduced-bias results are adopted. To differentiate the presented methods, the iterative reduced-complexity one with and without extrinsic information are briefly called MLA and MLI-Mr, respectively. We call ML-HD the ML method based on hard-decision. For the methods based on moment, the first moment and the second moment algorithm (M1M2) is used in 8PSK while the second moment and the fourth moment (M2M4) in 16QAM according to the conclusions in [4]. The algorithm in [6] is named numerical iterative computation (NIC). There are 200 samples in a frame. In particular, a Monte Carlo simulation of 200,000 experiments is used to estimate the root mean squared error (RMSE), which is
β=
[ E ((αˆ − α )2 )]1/2
α
Nt
≈
k=1
(αˆ k − α )2 / N t
α
.
(23)
Also the estimated mean is compared to determine how much bias in these methods. Figs. 2 and 3 show the difference of these algorithms for 8PSK. As for NIC, its convergence is sensitive in the high SNR zone, which means that the optimal point is hard to search. For MMs, it needs to build the table, which corresponds to the relationship between the input SNR and the quotient of different order moment. The quotient changes slowly in high SNR region. It means that the MM resolution deteriorates at such case. It leads to the bumpiness in high SNR region as shown in Figs. 2 and 3, which is the disadvantage of the MMs. For those methods based on ML. It finds that their performance is close to that of the DA estimation when SNR is high, which means that they are asymptotically optimal. Results show that iteration can improve the performance compared with ML-HD. Estimated mean and RMSE of the reduced-complexity version, MLI-4, have the feasible degradation compared with MLI-8 with the same iteration number. In SNR region lower than 3 dB, MLI-8 with 5 iterations approaches M1M2 and NIC, while it is better in other region. In fact, noise influences the reliability updating of (12) which restricts the performance of MLI. When SNR goes beyond some threshold, like 3 dB in 8PSK, the iterative gain is revealed. At the same time, Figs. 2 and 3 provide the results of these estimators with extrinsic information for 8PSK. All the iteration number of MLA is 4. Results show that the extrinsic information can drastically improve the performance by comparing with MLI-8. Besides these, when the extrinsic information is larger than 0.4, MLA has lower bias than M1M2 and NIC. And more, the difference between MLA and DA is almost negligible when the feedback information exceeds 0.7. In practice, BER in turbo-like system is still high when the extrinsic information is 0.4 [10]. It means that MLA can work well in low SNR scenarios with efficient label and code. Similar conclusions are drawn for 16QAM from Figs. 4 and 5. In this case, the moment method, M2M4, is better than MLIs when SNR is smaller than 7 dB. For MLA with 4 iterations, when the extrinsic information is larger than 0.4, its RMSE is better than those of M2M4 and MLIs.
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283
Fig. 2. Comparison of estimated mean between the proposed iterative estimators and other estimators for 8PSK.
Fig. 3. Comparison of RMSE between the proposed iterative estimators and other estimators for 8PSK.
7. Applications in the BICM-ID systems Finally, the SNR estimation algorithms are applied in the BICM-ID systems. The realization and analysis can be found in [11] for BICM-ID. Two scenarios are tested. In Fig. 6, the modulation is 8PSK and the label is SSP [11]. The frame length is 150 symbols and the generation polynomial is [7,5]octal with code ratio 1/2. The random interleaver is used. It provides the results of different estimation with six times iterative decoding. In all these SNR estimators, MLA is the best, which approaches the ideal transmission when SNR is higher than 3.5 dB. Even in low SNR region, MLA almost has no degradation. Such result shows that the extrinsic information from the soft decoder can greatly improve the estimated performance. ML-HD is the worst estimator, and degrades about 1 dB. NIC is the second worst. But the performance of this method is the same as the ideal when SNR equals to 6.5 dB. For the other methods, the performance is approximately same. Such results can be explained by the estimators’ RMSE in Fig. 3. When RMSE is smaller, the estimated value is closer to the real one and the system performance is better. But it is noted that there exists error-floor for BICM-ID. So when SNR is very higher, the estimated error becomes obvious-less. In Fig. 7, the modulation is 16QAM and the label is MSP [12]. The result is similar to that in Fig. 6. MLA is close to the ideal and better than the other estimators, especially when SNR is better than 5 dB. Differently, the M2M4 method is always better than the MLI-6 and MLI-16. From Fig. 5, it finds that RMSE of M2M4 is smaller than those of MLIs in the region from
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Fig. 4. Comparison of estimated mean between MLI and other estimators in 16QAM.
Fig. 5. Comparison of RMSE between MLI and other estimators in 16QAM.
−5 dB to 7 dB. When SNR is larger than 7 dB, MLIs are better than M2M4. But this region in the given BICM-ID scenario corresponds to the error-floor, so the advantages of MLIs do not emerge. 8. Conclusions This paper studies the iterative SNR estimation using a priori information. Firstly, the algorithm without feedback information is presented with its reduced-complexity version. Then the algorithm with feedback information is also proposed. The simulation results show that iteration improves estimation accuracy greatly. In middle or high SNR region, the new methods are better than the other foregone NDA ones. Besides it, when feedback soft information is up to some degree, e.g. 0.4, the new method is the best in the NDA ones. Finally, these methods are applied in the given BICM-ID systems to validate these conclusions. Acknowledgments This research was supported by the 863 Plan numbered 2006AA01Z270 and the National Science Foundation of China under grant No. 60496316. Many thanks to Dr. Ai Bo in Beijing Jiaotong University for his constructive comments.
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Fig. 6. The performance of BICM-ID with 8PSK and 6 iterations.
Fig. 7. The performance of BICM-ID with 16QAM and 6 iterations.
References [1] Y. Huang, J.A. Ritcey, EXIT chart analysis of BICM-ID over AWGN channels with SNR mismatch, IEEE Commun. Lett. 8 (8) (2004) 532–534. [2] T.A. Summers, S.G. Wilson, SNR mismatch and online estimation in turbo decoding, IEEE Trans. Commun. 46 (4) (1998) 421–423. [3] David R. Pauluzzi, Norman C. Beaulieu, A comparison of SNR estimation techniques for the AWGN channel, IEEE Trans. Commun. 48 (10) (2000) 1681–1691. [4] Ping Gao, Cihan Tepedelenlio˘glu, SNR Estimation for nonconstant modulus constellations, IEEE Trans. SP 53 (3) (2005) 865–870. [5] Ami Wiesel, Jason Goldberg, Hagit Messer, Non-data-aided signal-to-noise-ratio estimation, in: ICC2002, New York, USA, 2002, pp. 198–201. [6] Bin Li, Robert DiFazio, Ariela Zeira, et al., New results on SNR estimation of MPSK modulated signals, in: Proc. 14th International Symposium on Personal, Indoor and Mobile Radio Communication (PIMRC), Beijing, 2003, pp. 2373–2377. [7] Nader Sheikholeslami Alagha, Cramer–Rao bounds of SNR estimates for BPSK and QPSK modulated signals, IEEE Commun. Lett. 5 (1) (2001) 10–12. [8] D.M. Titterington, A.F.M. Smith, U.E. Makov, Statistical Analysis of Finite Mixture Distributions, Wiley, 1985. [9] S. ten Brink, Convergence behavior of iteratively decoded parallel concatenated codes, IEEE Trans. Commun. 49 (5) (2001) 1727–1737. [10] Thorsten Clevorn, Susanne Godtmann, Peter Vary, BER prediction using EXIT charts for BICM with iterative decoding, IEEE Commun. Lett. 10 (1) (2006) 49–51. [11] X. Li, A. Chindapol, J.A. Ritcey, Bit-interleaved coded modulation with iterative decoding and 8PSK modulation, IEEE Trans. Commun. 50 (8) (2002) 1250–1257. [12] A. Chindapol, J.A. Ritcey, Design analysis and performance evaluation for BICM-ID with square QAM constellations in Rayleigh fading channels, IEEE J. Select. Areas Commun. 19 (5) (2001) 944–957.
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Nie Yuanfei received all the B.Sc., M.S.E. and Ph.D. degrees from Xidian University, China, in 2002, 2005 and 2007, respectively. Now he is engaging in the ASIC design for digital mobile handle TV. His research interests include OFDM, space–time signal processing and iterative decoding in serial concatenated systems. Ge Jianhua received the B.Sc., Master and Ph.D. degrees from Xidian University, China, in 1982, 1985 and 1989, respectively. He is now the professor in both Xidian University and Shanghai JiaoTong University, China. Professor Ge is the senior member of Chinese Electronics Institute. He has won lots of scientific and technical prizes in China. His interests are wireless communications and web security. Wang Yong received the B.Sc., Master and Ph.D. degrees from Xidian University, China, in 1997, 2002 and 2005, respectively. And now Dr. Wang is an associate professor on communications in the Key Lab. of ISN in Xidian University. He has once participated in the key research project on HDTV in TEEG in China and his interests are wireless communications.