Non-data-aided SNR estimation method for APSK exploiting rank discrimination test A.W. Azim, S.S. Khalid and S. Abrar Presented is a non-data-aided (NDA) signal-to-noise ratio (SNR) estimation method for amplitude-phase shift-keying (APSK) in the presence of additive white Gaussian noise over a quasi-static channel. The proposed estimator is based on a rank discrimination test and uses a look-up table for SNR estimation. The performance of the proposed estimator is evaluated for 8APSK and 16APSK. It is shown through Monte-Carlo simulations that the proposed estimator outperforms existing NDA moment-based SNR estimators in terms of normalised mean-square error.
Introduction: In this Letter, we look at the SNR estimation problem in the perspective of a hypothesis testing problem, or more precisely, an optimum scale-invariant rank discrimination testing problem. Assuming a quasi-static channel model, the sampled (at√ the symbol √ rate) matched filter output is given by yk = S ak + N wk , k = 1, · · · , N ,√ where ak are the complex-valued transmitted APSK symbols, S is the unknown channel gain, and the complex-valued noise samples wk are independent, zero-mean, unit-variant and circular Gaussian. The energy normalisation is assumed, i.e. E[|a|2 ] = 1. Given N received signal samples in vector y = [y1 , · · · , yN ], we define a rank discriminatory hypotheses as follows: √ H0 : y = S a (1) H1 : y = N w A test statistic T (y) and a detection threshold g jointly describe a rank discrimination test such that if T (y) . g . 0, then the hypothesis H1 is rejected (H0 is selected). Otherwise, the null hypothesis is rejected (H1 is selected). The Neyman-Pearson lemma provides an optimum solution to this problem and suggests the most powerful test which leads to the following likelihood ratio: TNP (y) =
pY (y : H0 ) pY (y : H1 )
(2)
Proposed SNR estimator: In the context of our application, however, √ the hypotheses are parameterised by unknown gains (or scales) S and √ N . An additional requirement is needed so that an optimal test can be derived with respect to the Neyman-Pearson criterion over all values of unknown parameters; such a test is called the uniformly most powerful test (UMP). From [1], we obtain a monotonic scaleinvariant UMP test as given by 1 pY (ly : H0 )lN −1 dl .g (3) TUMP (y) = 01 p (ly : H1 )lN −1 dl 0 Y We intend to exploit (3) to design a test for the purpose of SNR estimation of APSK; the SNR is given as r = S/N . Consider a continuous APSK, where alphabets {aR + ȷaI } are assumed to be uniformly distributed in a ring-shaped circular region; where Ri and Ro are the inner and the outer radii, respectively. The joint PDF of aR and aI is given by 2 2 −1 2 2 pA (aR + ȷaI ) = (p(Ro − Ri )) , Ri ≤ aR + aI ≤ Ro (4) 0, otherwise Considering Y = |A|= a2R + a2I and after some manipulation, we obtain pY ( y) = 2 y/(R2o − R2i ). Let Y 1 , Y 2 , · · · , Y N be a sequence of N y) = pY ( yk ) denotes N-variate continuous density. size N, and pY ( k=1
yk = |yk |): We establish scale-invariancy under H0 as follows (below 1 y : H0 )lN −1 dl pY (l 0
(Ro −Ri )/( zN − z1 ) N 2N l2N −1 dl y k N 2 2 (Ro − Ri ) k=1 0
N 2N −1 Ro − Ri N = yk N ( zN − z1 )2N Ro + Ri k=1 =
(5)
where z1 , z2 , · · · , zN are the order statistic of y, so that z1 = min{ y}, zN = max{ y}, and z1 , z2 , · · · , zN−1 , zN . Now consider the next hypothesis H1 that we receive only complex-valued Gaussian noise. The modulus of the complex noise follows Rayleigh distribution, ⎛ ⎞ 2 yk y k yk : H1 ) = 2 exp⎝− 2 ⎠, .0 (6) yk ≥ 0, s pY ( y s 2s y y Establishing scale-invariancy under H1 , we obtain 1 y : H1 )lN −1 dl pY (l 0 N
=
⎞ N 2 2 l y ˆ ⎜ k⎟ ˆ ⎟ 2N −1 ⎜ k=1 exp⎜− dl ⎟l 2 ⎝ 2s ⎠ y ⎛
yk 1
k=1 s2N y
0
(7)
N 2N −1 G(N ) = ykˆ
N N 2 k=1 ˆ yk k=1
Combining the results (5) and (7), it gives (below K . 0) 1 01 0
y : H0 )l pY (l
N −1
dl
y : H1 )lN −1 dl pY (l
=K
N
1/N
k=1
y2k
( zN − z1 ) 2
N H0
(8)
.g , H1
Dropping the constant K and taking Nth root, we obtain a test: 1/N TUMP (y) =
N
|yk |2
k=1
(max {|yk |} − min {|yk |})2
=
ˆ2 M ˆ +1 − M ˆ −1 )2 (M
(9)
ˆ −1 := min {|yk |}. Now ˆ +1 := max {|yk |} and M where we denote M √ consider receiving a mixture of signal and noise, i.e. yk = S ak + √ N wk . Under this scenario, the test (9) becomes a monotonically non-decreasing function of SNR, r = S/N . However, for r ≥ 50dB, the test is found to exhibit squashing behaviour and consequently becomes less sensitive to the higher values of r. To overcome this problem, we suggest the following non-squashing UMP test: T NUMP (y) =
ˆ2 M ˆ ˆ −1 )2 (M+1 − b M
(10)
where b = max{|ak |}/ min{|ak |}. Note that, for r 1, the expected value of T NUMP (y) satisfies E[TNUMP (y)] 1. Our method suggests performing this test (10) a priori for a range of SNR values and given APSK signal (a number of times), and store the estimated expected ˆ NUMP (y)], in a look-up table (LUT). To estimate the value, E[T unknown SNR of a new data y, we first calculate the test T NUMP (y) and then interpolate LUT entries in the vicinity of T NUMP (y) to obtain r, r LUT[T NUMP (y)]
(11)
We denote the proposed SNR estimator as bM2 M1 (APSK). Note that the LUT-based SNR estimation is widely used in practice, refer to [2 – 4]. Simulation results: Simulation results of our proposed estimator bM2 M1 (APSK) are provided for 8APSK and 16APSK signals over additive white Gaussian noise quasi-static channels and are compared with two moment-based estimators, M2 M4 [5] and M6 [6] using Monte-Carlo simulations which are averaged over NR = 10000 runs. The performance metrics are the square-root normalised mean-square error (SNMSE), viz
NR 1 ri − r 2 (12) SNMSE = NR i=1 r N R ri . and the mean value of the estimated SNR, i.e. MSNR = NR−1 i=1
Fig. 1 provides the SNMSE of the proposed estimator M2 M1 (APSK) and is compared with those of M2 M4 and M6 . For both signals, it is shown that bM2 M1 (APSK) outperforms M2 M4 and M6 (especially at higher SNR values), while the M2 M4 and the M6 diverge for r ≥ 15 dB. Fig. 2 provides the MSNR; it is evident that
ELECTRONICS LETTERS 5th July 2012 Vol. 48 No. 14
bM2 M1 (APSK) is noticeably unbiased for higher SNR values. Also note that M2 M4 and M6 exhibit low success rate owing to the occurrence of negative radicand; however, our estimator has 100% success rate as it does not involve the calculations of square-root.
8APSK: SNR estimation with N=1000 samples
2
10 estimated sq. norm. MSE
# The Institution of Engineering and Technology 2012 23 April 2012 doi: 10.1049/el.2012.1327
8APSK(4,4)
βM2M∞(APSK) M2M4 1
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M6
A.W. Azim, S.S. Khalid and S. Abrar (COMSATS Institute of Information Technology, Islamabad 44000, Pakistan)
0
10
E-mail:
[email protected] References
−1
10
a 16APSK: SNR estimation with N=1000 samples
2
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16APSK(8,8)
βM M (APSK) ∞
2
estimated sq. norm. MSE
Conclusion: In this Letter, we have studied the problem of estimating the SNR of APSK signals. It is shown that the proposed method improves the performance of average SNR estimation in the high SNR regime and considerably outperforms the existing moment-based estimators.
M M 2
1
4
M6
10
0
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−1
10 −20
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0
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20
30
40
b
Fig. 1 Plots of square-root normalised mean-square error in rˆ obtained from b M2M1(APSK), M2M4 and M6
1 Sidak, Z., Sen, P., and Hajek, J.: ‘Theory of rank tests’ (Academic Press, 1999, 2nd edn) 2 Summers, T.A., and Wilson, S.G.: ‘SNR mismatch and online estimation in turbo decoding’, IEEE Trans. Commun., 1998, 46, pp. 421–423 3 Im, S., and Power, E.J.: ‘An algorithm for estimating signal-to-noise ratio of UWB signals’, IEEE Trans. Veh. Technol., 2005, 54, pp. 1905– 1908 4 Suhadi, S., Last, C., and Fingscheidt, T.: ‘A data-driven approach to a priori SNR estimation’, IEEE Trans. Audio Speech Lang. Process., 2011, 19, pp. 186– 195 5 Pauluzzi, D.R., and Beaulieu, N.C.: ‘A comparison of SNR estimation techniques for AWGN channel’, IEEE Trans. Commun., 2000, 48, pp. 1681–1691 6 Lopez-Valcarce, R., and Mosquera, C.: ‘Sixth-order satistics-based nondata-aided SNR estimation’, IEEE Commun. Lett., 2007, 11, (4), pp. 351– 353
a 8APSK b 16APSK
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8APSK: N = 1000, MC = 10000 βM M (APSK) ∞
estimated mean SNR, dB
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M2M4 M6 20
unbiased
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16APSK: N = 1000, MC = 10000 βM2M∞(APSK) M M 2
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Fig. 2 Plots of mean rˆ obtained from bM2M1(APSK), M2M4 and M6 a 8APSK b 16APSK
ELECTRONICS LETTERS 5th July 2012 Vol. 48 No. 14
