# Bit error performance of BPSK and QPSK receiver in the presence of [eta]-[mu] fading and carrier phase error.

1 INTRODUCTIONPerformance of coherent phase shift keying (PSK) signal receiver strongly depends on the quality of the reference carrier extraction. Impairments that occur in the channel and receiver affect the extraction of the reference carrier, so that an ideal reference carrier recovery in the fading channel is not possible [1]-[2]. The influence of the imperfect carrier extraction on performance of the receiver has long been of interest, starting from the early 1960s until the most recent papers [3]-[5]. In [3], under assumption of uniformly distributed phase error, the error probability has been determined in the case of digital binary and quaternary phase shift keying (BPSK and QPSK) signal detection in the presence of Gaussian noise, Nakagami-m fading and imperfect reference signal carrier extraction. In [4] the derivation procedure of the general explicit analytical expression for average symbol error probability for MPSK signal detection in the presence of Nakagami-m fading, Gaussian noise and imperfect phase reference has been presented. It was assumed that reference signal recovery is performed by phase-locked loop (PLL) from unmodulated signal. The effects of carrier phase error on equal gain combining (EGC) receivers in spatially correlated Nakagami-m fading have been considered in [5]. Taking the effects of imperfect phase estimation, performance of BPSK and QPSK receivers over Rayleigh fading channel was determined in [6].

In this paper we consider coherent detection of BPSK and QPSK signals in [eta]-[mu] fading channel. The [eta]-[mu] distribution is quite general and is used to describe small variations of the signal affected by fading in cases where there is no line of sight between the transmitter and receiver. This distribution includes the Hoyt, Nakagami-m, one-sided Gaussian and Rayleigh distribution as special cases. The extraction of the reference carrier signal is assumed imperfect and the difference between the estimated phase and the received signal phase is defined as a phase error that follows Tikhonov distribution. Using expansion in Maclaurin series, we obtain appoximative expressions in a form of infinitive series for average bit-error rate (BER) that converge quickly. A very good accuracy of these results over a large range of average signal-to-noise ratios (SNRs) is demonstrated by comparison with exact results that require two-fold numerical integration for their evaluation. By using the expressions derived in the paper, we determine in which measure the imperfect reference signal recovery and fading severity affect that accuracy and how much they impair performances of the receiver.

2 CHANNEL MODEL

We observe the transmission of BPSK and QPSK signals over [eta]-[mu] fading channel. This fading model is proposed in [7]. As shown, measured fading values can be better approximated by using this model than using the standard models of short-term fading. The generalized distribution is quite general and is used to describe small variations of the signal affected by fading in cases where there is no line of sight between the transmitter and receiver. In [7] two formats of this distribution are presented and they are related to two physical models. In this paper, the format 1 is used. It refers to the multipath signal that is spread through the inhomogeneous environment. Phases of fading components in one cluster are random. Components in phase and quadrature are independent and have different strengths. The probability density function of instantaneous symbol SNR [[gamma].sub.s] in the [eta]-[mu] fading channel is given by [7]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where it is h = 1 / 4 (2 + [[eta].sup.-1] + [eta]) and H = 1 / 4 [[eta].sup.-1] + [eta]). Average symbol SNR is denoted by [[??].sub.s]. Average symbol and bit SNR are related as [[??].sub.s] = [log.sub.2]M x [??], where [??] is the average bit SNR, and M is the number of phase levels. (The same relation exists between instantaneous symbol and bit SNR [[gamma].sub.s] = [gamma][log.sub.2]M). The gamma function [10, Eqn. 06.05.02.0001. 01] is indicated by [GAMMA](.) and [l.sub.v] represents the modified Bessel function of the first kind and an arbitrary order v [10, Eqn. 03.02.02.0001.01]. The parameter [eta] (power ratio between the components in phase and quadrature in each cluster) can take the values from the range 0<[eta]<+[infinity]. However, since this distribution is symmetric around [eta]=1, it is enough to observe only one of the bands [0,1] and [1,+[infinity]). Parameter [mu]>0 is defined as [mu] = [E.sup.2][[r.sup.2]] / 2V[[r.sup.2]] (1 + [(H /h).sup.2]) where E[.] and V[.] denote the operators of mathematical expectation and variance, and r is the fading envelope. Special cases of [eta]-[mu] distribution are Hoyt and Nakagami-m distribution. In format 1, by setting the parameter [mu] = 0.5, one can obtain Hoyt distribution. By setting [mu] = m for [eta]>0 or [eta]>+[infinity] in the same format, (1) yields Nakagami-m distribution (m is the parameter of Nakagami-m distribution). Nakagami-m distribution can be also obtained by setting [eta] = 1 and [mu] = m/2. From these two distributions, as special cases, one-sided Gaussian and Rayleigh distribution can be obtained.

3 MODEL OF THE RECEIVER AND BIT-ERROR RATE

The conditional BER in the case of nonideal coherent detection of BPSK signal can be written as

[P.sub.b][([[phi].sub.c],[gamma]).sub.BPSK] = 1 / 2 erfc([square root of [gamma]]cos[[phi].sub.c]). (2)

and in the case of QPSK signal detection as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where the complementary error function is denoted as erfc (*) [8, Eqn. (7.1.2)]. Difference between the estimated phase [[??].sub.i] and the received signal phase [[delta].sub.i] in every moment of sampling is a phase error and denoted with [PHI]c =[[delta].sub.i] - [[??].sub.i]. When the phase estimation is done using PLL from unmodulated carrier, the PDF of this phase error corresponds to Tikhonov distribution [4]-[5], [9]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Where [[rho].sub.pll] is the SNR in the PLL circuit (loop SNR). Under the assumption that the ratio of carrier-to-data power is constant (fixed), loop SNR depends linearly on bit SNR [9]

[[rho].sub.pll] = C[gamma]. (5)

where C is a constant of proportionality. For this reason, it is not likely to expect the appearance of an irreducible error probability [9].

Using (1)-(4), the average error probability can be evaluated from

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

which requires two-fold numerical integration. In order to simplify this calculation, the approximation of conditional BER (2) and (3) will be introduced.

By expanding conditional BER in a Maclaurin series in [[phi].sub.c] (2) and (3) become

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

respectively.

Knowing the [P.sub.b]([[phi].sub.c,[gamma]) for BPSK and QPSK signal detection and the distribution of the phase error (given by (4)), the conditional error probability on [gamma] as a function of average bit SNR [[??].sub.b] can be calculated from

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

By maintaining only first two terms in (7) and (8) for BPSK and QPSK detection, respectively, and using (9), one can get simple expressions, which lead to infinitive series-form evaluation of BER performance for wide range of propagation scenarios. These results approximate very well the exact values that require two-fold numerical integration for their evaluation.

In the case of BPSK the conditional error probability on the instantaneous SNR [gamma] becomes

[P.sub.b][([gamma]).sub.BPSK] = 1 / 2erfc([square root of [gamma]]) + [[sigma].sup.2.sub.[phi]] / 2 [square root of [gamma] / [pi] exp(-[gamma]) (10)

and in the case of QPSK signal detection it is

[P.sub.b][([gamma]).sub.|QPSK] = 1 / 2 erfc([square root of [gamma]]) + [[sigma].sup.2.sub.[phi]] / 2 [square root of [gamma] / [pi]] (1 + 2[gamma] exp(-[gamma] (11)

In (10) and (11) the phase error standard deviation is denoted as [[sigma].sub.[PHI]] and can be evaluated as [4]-[5], [9]

[[sigma].sup.2.sub.[phi]] = 1 / [[rho].sub.pll] = [(C[gamma]).sup.-1] (12)

The average error probability as a function of [[??].sub.b] can be calculated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

In order to obtain an infinitive series-form expression for average BER in [eta]-[mu] fading channel it was necessary to use a series representation of the modified Bessel function [10, Eqn. 03.02.06.0002.01] that appears in (1),

[I.sub.p](u) = [+[infinity].summation over (k=0)] 1 / [GAMMA](k + 1)[GAMMA](p + k + 1) [(u / 2).sup.2k+p]. (14)

In the case of BPSK the average error probability becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

and in the case of QPSK signal detection it is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

where [sub.2][[??].sub.1](*,*;*;*) is regularized hypergeometric function [10, Eqn. 07.24.02.0001.01].

4 NUMERICAL RESULTS

The following examples illustrate the matching between introduced approximations and exact values, obtained by two-fold numerical integration, while the phase error follows Tikhonov distribution.

In figure 1 the influence of fading parameter [eta] on the performance of the receiver with an imperfect reference carrier extraction can be traced for the case of QPSK detection. It may be noted that [eta] very little affects BER values. For the purpose of comparison, performance of a receiver with an ideal reference carrier extraction is presented in the figure. In all cases the introduced approximations show good agreement with the exact values of BER, while fading parameter [eta] doesn't show a significant impact on the degree of this agreement.

[FIGURE 1 OMITTED]

In figures 2 and 3 the effect of fading parameter [mu] on the performance of the receiver with an imperfect reference carrier extraction is shown in the case of BPSK and QPSK detection, respectively. Decrease of [mu] value impairs system performance. On the other hand, in the case of QPSK detection smaller [mu] value causes a better match between the approximations (asterisk) and exact values of BER (open marks).

In figures 4 and 5 the impact of the loop parameter C on the performance of BPSK and QPSK receiver is presented, respectively. The constant C (in dB) represents the amount by which the loop SNR (in dB) exceeds the instantaneous incoming SNR (in dB). Parameter C influents the BER significantly, especially in the case of QPSK detection and low fading severity (larger [mu]). The increase of this parameter's value reduces the BER. For example, in the case of QPSK detection, in order to obtain the same value of BER= [10.sup.-6], for fading parameters' values [eta]=2 and [mu]=1.5, it is necessary for average SNR to rise approximately 2.57dB, when loop parameter C changes from 15dB to 5dB.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

5 CONCLUSION

In this paper we considered coherent detection of BPSK and QPSK signals in [eta]-[mu] fading channel. The imperfect reference signal extraction is taken into account. Using expansion in Maclaurin series, we obtained appoximative expressions in a form of infinitive series for average BER that converge quickly. In this way a cumbersome exact evaluation that includes two-fold numerical integration has been avoided, while introduced approximations show very good fit to the exact values over a large range of average SNRs.

Decrease of fading parameter [mu] impairs system performance, while parameter [eta] very little affects BER values. Loop parameter c has a strong influence on BER, especially in the case of QPSK detection.

As for the impact on the agreement between the introduced approximations and exact values, fading parameter [eta] doesn't affect significantly a degree of this agreement, while, in the case of QPSK detection, smaller [mu] value causes a better fit.

ACKNOWLEDGEMENT

This work was supported in part by the Ministry of Sciences and Technological Development of Serbia within the Project TR-33008.

REFERENCES

[1.] Simon. M, Alouini. M, "Digital Communications over Fading Channels", 2nd ed., Wiley, New York, 2005.

[2.] Lindsey. W, Simon. M, "Telecommunication Systems Engineering", Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1973.

[3.] Lo. C, Lam. W, "Average BER of BPSK and QPSK systems with noisy phase reference over Nakagami-m fading channels", IEICE Transactions on Communications 2001; 84 (6):1687-1689.

[4.] Kostic. I, "Average SEP for M-ary CPSK with noisy phase reference in Nakagami fading and Gaussian noise", European Transactions on Telecommunications 2007; 18(2):109-113.

[5.] Sagias. N, Karagiannidis. G, "Effects of carrier phase error on EGC receivers in correlated Nakagami-m fading", IEEE Communications Letters 2005; 9(7):580-582.

[6.] Chandra. A, Biswas. D, Bose. C, "BER Performance of Coherent PSK in Rayleigh Fading Channel with Imperfect Phase Estimation", in Proc. International Conf. on Recent Trends in Information, Telecommunication and Computing, 2010.

[7.] Yacoub. M, "The k-m distribution and the [eta]-[mu] distribution", IEEE Antennas and Propagation Magazine 2007; 49(1):68-81.

[8.] Abramowitz. M, Stegun. I, "Handbook of Mathematical Functions", Dover publications, INC., New York, 1972.

[9.] Simon. M, Alouini. M, "Simplified noisy reference loss evaluation for digital communication in the presence of slow fading and carrier phase error", IEEE Transactions On Vehicular Technology 2001; 50(2):480-486.

[10.] http://functions.wolfram.com

Bojana Nikolic, Goran Djordjevic, Mihajlo Stefanovic, Dragan Denic

Faculty of Electronic Engineering, University of Nis

bojana.nikolic@elfak.ni.ac.rs, goran@elfak.ni.ac.rs, mihajlo.stefanovic@elfak.ni.ac.rs, dragan.denic@elfak.ni.ac.rs

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Author: | Nikolic, Bojana; Djordjevic, Goran; Stefanovic, Mihajlo; Denic, Dragan |
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Publication: | International Journal of Emerging Sciences |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Dec 1, 2011 |

Words: | 2235 |

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