IMPROVED SNR ESTIMATION FOR BPSK AND QPSK SIGNALS

Transcription

1 MPROVD NR TMATON FOR BPK AND QPK GNA Ayesha jaz, Adegbenga B. Awoseyila and Barry G. vans Centre for Communication ystems Research (CCR), University of urrey, Guildford GU 7XH, U.K. A novel non-data-aided NR estimation technique for BPK and QPK modulations in complex additive white Gaussian noise (AWGN) is proposed. t uses the constant amplitude property of in-phase and quadrature components of these modulations to achieve an improved performance. ts complexity is shown to be lower while its accuracy supersedes the popular moments-based estimator, approaching closely to the CRB. ntroduction: ignal-to-noise ratio (NR) is an important measure of the channel quality in many modern wireless communication systems. An accurate estimate of NR is required for various applications such as lin adaptation [1] and iterative decoding []. NR estimation techniques can broadly be divided into two categories: data-aided (DA) and non-data-aided (NDA). Although DA estimators tend to be more accurate, NDA estimators have increased relevance when transmitted data are unnown, as is usually the case for applications which require frequent NR estimates. Moreover, they are more bandwidth efficient since they do not require training symbols (pilots). Various NDA NR estimation methods have been proposed for PK signals in complex AWGN. n [3], the moments-based estimator (second- and fourth-order: M M ) is reviewed and shown to achieve the Cramer-Rao lower 1

2 bound (CRB) [] for a large sample size and at very high NR. However, its performance degrades with respect to the CRB as the sample size reduces. This constitutes a disadvantage for fast lin adaptation where small observation windows are more appropriate. n [5], four NR estimation algorithms using the absolute values of in-phase and quadrature components of the received QPK signal are proposed, wherein the best performance shows some improvement over M M for a moderate sample size and at NR > 5dB. The method proposed in [6] is based on the data-aided M approach [3], wherein modulation is removed by taing the M th power of the received signal. However, this process introduces a noise penalty (~1dB for QPK) which degrades accuracy. The envelope-based estimator in [7] achieves the same accuracy as M M while the iterative bias compensation estimator in [8] achieves a reduced variance in the low NR region. However, it is computationally intensive due to many iterations needed to achieve such accuracy. We present a novel NDA NR estimation technique for BPK and QPK based on the constant amplitude property of their in-phase and quadrature components. ts improved performance over the M M estimator at moderate/high NR is explained by mathematical derivations and verified by computer simulations. ystem model: et and Q, 1,,..., be the in-phase and quadrature components of QPK symbols transmitted over an AWGN channel. The signal components are assumed to be independent and identically distributed (i.i.d) discrete random variables, taing values in the set {a,-a} with equal

3 probability. Assuming one complex sample is taen for each of the symbols transmitted and that carrier synchronization exists, the th received signal Z can be described as: X n (1) Y n () Q Q where X and Y represent the in-phase and quadrature components of Z respectively, while n and Q n represent the in-phase and quadrature components of noise which are taen to be zero-mean i.i.d. Gaussian random variables, with variances Q, respectively. The NR of the received signal is given by: Q a a a (3) N { n nq } Q where is the signal power and N is the noise power. Assuming the same conditions as above for BPK modulation with { a, a} and 0, signal power equal to Q a is employed in (3). The second order moment ( M ) of the received signal is is given as: M Z X Y n n n n () Q Q Q Q 3

4 ince the noise and signal components are independent, M is shown to be equal to the sum of signal and noise power as follows: Q Q n n M N a a a Q (5) Therefore, the NR can be estimated as follows: M ˆ ˆ ˆ ˆ (6) where Y X Z M ˆ Proposed Method: For BPK and QPK modulations in complex AWGN, it is observed that the absolute values of the in-phase and quadrature components of the received signal have a close relationship with the signal power, since these components have a constant amplitude in the transmit signal. Consider a received QPK signal as presented in the system model. ince, a Q =1,,,, a Q (7) The absolute values of the in-phase and quadrature components of the received signal can be expressed as: n X (8)

5 Y n (9) Q Q Assuming that n (which is usually valid at moderate/high NR): X n ( X 0) (10) X n ( X 0) (11) Y n ( Y 0) (1) Q Q Y n ( Y 0) (13) Q Q Using (7) and (10) -(13) and given that the noise components are zero-mean, X n a (1) Y n a (15) Q Q Q Hence, the transmitted signal power can be determined using the mean of the absolute values of in-phase and quadrature components of the received signal, i.e. X Y a a a (16) Consequently, our proposed method estimates the transmitted signal power using the samples of the received signal as follows: 5

6 1 ˆ proposed X Y (17) 1 1 The moments-based method estimates the transmitted signal power as follows [3]: ˆ M M Mˆ Mˆ (18) where Mˆ 1 Z 1 n contrast to the M M estimator which maes use of the constant envelope property of the transmitted signal, our proposed estimator taes full advantage of the constant amplitude property of in-phase and quadrature components of the signal, and is therefore able to achieve improved accuracy when the estimator assumptions are valid. The proposed method also has a lower complexity than M M as shown in Table 1, wherein it is seen that only three real multiplications are required in the computation of Ŝ proposed while the number of real multiplications needed to compute ˆM increases linearly with the observation window. For a moderate sample size of =6, this translates into 193 real multiplications. Computer simulations: Computer simulations (10,000 trials) were performed to verify the performance of the proposed method in comparison to M M and the best estimator in [5], using =6 QPK/BPK symbols in complex 6

7 AWGN. Fig. 1 shows the bias of the estimators in terms of mean NR estimate, wherein it can be seen that all the estimators exhibit very low bias at moderate/high NR. The proposed method has a low bias for BPK at low NR while M M maintains this for both BPK and QPK. The increasing bias noticed for the proposed method at low NR is due mainly to an increasing departure from the assumptions used to derive it. Fig. shows the accuracy of the estimators in terms of mean square error (M), normalised to the true NR as defined in [3], wherein the CRB (DA) is also shown for comparison purposes. For BPK, the proposed method outperforms M M at all values of NR, with its M approaching closely to the CRB at NR > 3dB. n the case of QPK, the proposed method maintains its superiority over M M for NR > 3dB, while approaching closely to the CRB at NR > 7dB. Conclusions: We have proposed a novel NDA NR estimator for BPK and QPK modulations and it is shown to be better than the popular momentsbased estimator in terms of lower computational complexity and improved accuracy at NR regions of interest. Furthermore, its accuracy approaches closely to the CRB at moderate NR. Acnowledgment: This wor was supported in part by the C FP6 atnx- Project (T-07393). 7

8 References 1. Chung T.., and Goldsmith A. J., Degrees of freedom in adaptive modulation: a unified view, Trans. Commun., vol. 9, no. 9, pp , ep ummers T. A., and Wilson. G., NR mismatch and online estimation in turbo decoding, Trans. Commun., vol. 6, no., pp. 1-3, Apr Pauluzzi D. R., and Beaulieu N. C., A comparison of NR estimation techniques for the AWGN channel, Trans. Commun., vol. 8, no. 10, pp , Oct Alagha, N.., Cramer-Rao bounds of NR estimates for BPK and QPK modulated signals, Commun. ett., vol. 5, no. 1, pp. 10-1, Jan Beaulieu N. C., Toms A.., and Pauluzzi D. R., Comparison of four NR estimators for QPK modulations, Commun. ett., vol., no., pp. 3-5, Feb Xu H., and Zheng H., The simple NR estimation algorithm for MPK signals, Proc. ntl. Conf. ig. Process. (CP), vol., pp , Aug. - ep Pei-jun Y. and Jian-ping A., NR estimation for constant envelope signals in AWGN channel, Proc. ntl. Conf. on Wireless Commun., Net. and Mobile Computing, vol. 1, pp , ep Baali M., téphenne A., and Affes., terative NR estimation for MPK modulation over AWGN channels, Proc. 6th Veh. Tech. Conf. (VTC), pp. 1-5, ep

9 Figure/Table captions: Table 1 Complexity comparison between proposed method and M M Fig. 1 Mean of estimated NR, =6 Fig. Normalised M of different estimators, =6 9

10 Table 1 Computation Ŝ proposed Real Multiplications Real Additions -1-1 ˆM 10

11 Figure 1 Mean of estimated NR, db M M (BPK) M M (QPK) Best in [5] Proposed (BPK) Proposed (QPK) Unbiased NR, db 11

12 Figure Normalized M M M (BPK) M M (QPK) Best in [5] Proposed (BPK) Proposed (QPK) CRB NR, db 1