Computation of Bit-Error Rate of Coherent and Non-Coherent Detection M-Ary PSK With Gray Code in BFWA Systems

Transcription

1 Computation of Bit-Error Rate of Coherent and Non-Coherent Detection M-Ary PSK With Gray Code in BFWA Systems Department of Electrical Engineering, College of Engineering, Basrah University Basrah Iraq, doi: /ijact.vol3. issue1.13 Abstract In this paper we examined the broadband fixed wireless access (BFWA) systems by coherent and non-coherent detection using M-ary PSK with Gray code. A closed form for the exact symbol error rate and bit error rate (BER) of M-ary PSK is presented. We show through analysis the physical limitations of the BFWA channels over different values of M (2,4,8,16,32) and the performance difference between the two type of detection (coherent and non-coherent). 1. Introduction Keywords: BER, BFWA, M-PSK One of the simplest forms of digital modulation is binary or Bi-Phase Shift Keying (BPSK). One application where this is used is for deep space telemetry. A more common types of phase modulation is Quadrature Phase Shift Keying (QPSK) and Octal Phase shift Keying (OPSK). They are used extensively in applications including CDMA (Code Division Multiple Access) cellular service, wireless local loop, Iridium (a voice/data satellite system) and DVB-S (Digital Video Broadcasting -Satellite). Digital modulation techniques are classified into coherent and non-coherent techniques, depending on whether the receiver is equipped with a phase recovery circuit or not. BFWA enables operators in a competitive environment to roll-out wideband services in a rapid and cost efficient manner [1] [2]. The IEEE working group on broadband wireless access standards developed IEEE [3]. The important limiting factor in outdoor wireless transmission is the multipath channel between the transmitter and receiver causing inter-symbol interference (ISI) which degrades the system performance and limits the maximum achievable data rate [1] [4]. The application of wavelet transform is used in [5] to identify a digitally modulated signal. In this paper we provide a general theoretical approach to analyze the effect of ISI on the performance of the BFWA systems for M-ary PSK. The multipath channel can be modeled by an equivalent baseband system where the transmit filter, the channel and the receive filter, are represented by a discrete-time L-tap transversal filter with finite-length impulse response h = h δ, where hl denotes the complex channel coefficients. Tailored for different terrain conditions, a set of 6 typical channel models called Standford University Interim (SUI) channel models used for simulation, design, development and testing of technologies suitable for fixed broadband wireless applications where proposed in [6]. The method in [1] [4] is only applicable to QPSK (M=4), while in this paper we develop this method for M=2,4,8, 16 and 32. The methods in [7] [8] are only for coherent method without including the ISI effect, while in this paper, the both types of modulation (coherent and non-coherent) are presented. Also, the effect of ISI is introduced in all calculations. In section 2, we propose new method to obtain the average bit-error rate as a function of signal to noise ratio for M-ary PSK signal modulated in coherent and non-coherent methods. Section 3, compares the simulated results for coherent and non-coherent modulation of M-ary PSK and discusses the performance comparison of M values. Finally, we draw the conclusion in section

2 2. The Analytical Model 2.1. Coherent Detection for M-ary PSK Modulation Denote h as an estimate of h, and assume it is an accurate estimate of h, i.e., h = h, [1]. To detect the transmitted symbol coherently, we multiply the received signal with the conjugate of h, i.e., = h (h + h + h + ) = h h + h (h + h + ) = h h + h + (1) Where the combined ISI and noise = + is a complex Gaussian random variable with PDF~(, ) and variance = h ([ h ] + [ h ] + ) = h ( + + ), where and are the powers of the second and third taps, respectively, and is the noise power. The constellation for an M-ary PSK signaling is shown in figure-1-. In this constellation the decision region D o is also shown. Due to the symmetry of the M-ary PSK constellation, the symbol error probability P e equals the conditional error probability [1]. Suppose s o (1,0) is transmitted, the probability of making correct decision P(c s n =s o ) is the probability of r n falling in the correct decision region D o ( 1 < < 1), i.e., ( = ) = { = h + } = ( h ) + > 0 1 < tan < 1 (2) Because of the value ( h ) + always grater than zero, so the probability is equal to one, then Where k1=tan(-θ1) and k2=tan(θ1), ( = ) = 1 < tan h < 1 + = tan > 1 tan < 1 = h > 1 + h < 2 + = h 1 > + 1 h 2 < + 2 (3) Figure 1. Space diagrams of several signals for M- ary PSK

3 and are gaussian random variables, ~ 0, and ~ 0, are statistically independent. Normalizing and to unit variance, yield h 1 ( ) = (1 + 1 ) > + 1 h 2 ( ) 2 (1 + 2 ) < + 2 ( ) 2 = (4) ( )( ) ( )( ) Where () = exp = ( ) = 1 ( ) is the complementary aussian cumulative distribution function. From [ ], = h is Ricean distribution with PDF = 1 (5) ( )( ) ( )( ) () = exp, 0 (6) Where () is the 0 th order modified Bessel function of the first kind [9]. For gray coded [9],, where k is the number of bits per symbol, therefore () ()/ 1 ( )( ) The error probability when r is random is the average of (), i.e., = 1 ( )( ) = ()() ( )( ) 2.2. Non-Coherent Detection for M-ary PSK Modulation ( )( ) (7) exp (8) The original signal,, received from the channel is directly passed to the PSK demodulator without correcting the phase shift [1] [4]. The received signal from the channel is given in [4] as = h + h + h + = h + (9) Where = h + h + ~(0, ) stands for combined ISI and noise, which is Gaussian distributed, and = [ h ] + [ h ] + = Non-Coherent Detection for BPSK Modulation (M=2) The probability of a symbol error occurring, for example, = given is transmitted can be computed as [1] [4], where = (1,0) and = ( 1,0) : ( = = ) = {h + } = (h + h ) + = {h + < 0} = (10)

4 Let = h is Gaussian distributed random variable with PDF () = ( ) (11) According to the signal constellation plotted in figure 2-a, the error event ( = = ) results in one bit error. The relationship between bit error probability and the conditional symbol error probabilities is: And the bit error probability is = [1 ( = = )] = ( = = ) (12) () = (13) According to [10], if z is a zero-mean, unit-variance, normal random variable, then [( + )] =. Assign = ( )/ such that z is a zero-mean, unit-variance, normal random variable, so : = ( ) = (14) Where =, and =. Figure 2. Signal space diagram of : a- DPSK, b- QPSK, c- OPSK Non-Coherent Detection for QPSK Modulation (M=4) The average bit error probability can be derived as [1] [4] as:

5 Where = = =, =, = , and = Non-Coherent Detection for OPSK Modulation (M=8) (15) The probability of a symbol error occurring, for example, = given is transmitted can be computed as before, where = (1,0) and =, as shown in figure 2 -c : ( = = ) = {h + } = (h + h ) + = < tan h + h + = tan h + h + = h + h + < > tan h + h + > h + h + < < Where = and =, = (h h ) > ( ) (h h ) > ( ) and are Gaussian random variables, ~ 0, and ~ 0, are statistically independent. Normalizing and to unit variance, yield (h h ) = (1 + ) > ( ) (h h ) (1 + 2 ) 2 (1 + ) > ( ) (1 + 2 ) 2 = ( ) ( ) (16) Similarly, other conditional error probabilities can be obtained as : ( = = ) = {h + } = ( ) ( ) (17) ( = = ) = {h + } = ( ) ( ) (18)

6 ( = = ) = {h + } = ( ) ( ) (19) Denote = h h, = h h,. = h h. All of them are Gaussian distributed random variable with PDFs : ( ) = (20) ( ) = (21) ( ) = (22) According to the signal constellation in fig 2-c, all error events ( = = ), ( = = ), ( = = ), ( = = ), ( = = ), and ( = = ) result in two bits error, while ( = = ) results in one bit error. The relation between bit error probability and the conditional symbol error probabilities is : = [2 ( = = ) + 2 ( = = ) + 2 ( = = ) + 2 ( = = ) + 2 ( = = ) + 2 ( = = ) + 1 ( = = )] (23) For OPSK (M=8) modulation, k=3. And the bit error probability will be as : (,,,,,,, ) = 2 ( ) ( ) + 2 ( ) ( ) + 2 ( ) ( ) ( ) + 1 ( ) ( ) + 2 And the average bit error probability is : ( ) ( ) + 2 ( ) ( ) ( ) + 2 (24) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + 2 (25) ( ) ( ) ( ) ( )

7 Wherer,,, are zero-mean, unit variance, normal random variable. and = +, where n=1,2,..,8. And = (26) Where = 3. Simulation Results, and =, n=1,2,..,8. This section gives a comparison between the average BER of M-ary PSK as a function of signal to noise ratio for coherent and non-coherent detection. The statistics of the SUI-3 channel coefficients are adopted from [1] and can be summarized as : [h ] = , = [ h ] = 1.5, [h ] = 0.5, [h ] = [h ] = 0, = , = 0.1, = 0.25, = 1, in coherent method and = 0.5, = = 0.5 in non-coherent method. In figure 3 and 4 the symbol error rate and bit error rate are evaluated for M=2, 4, 8, 16, and 32 using coherent detection, the result show that the symbol error rate increase as M increase, also the bit error rate increase too until M=8 and above it will decrease because the large value of M. in figure 5 and 6 we have the same relation between the symbol error rate and bit error rate with signal to noise ration but in non-coherent detection, the results show that for non-coherent detection when M is greater than 8, the bit error rate will be so much and the power needs become excessive that make 16PSK and above are not widely used in communication PSK 4-PSK 8-PSK 16-PSK 32-PSK Symbol Error Rate SNR(dB) Figure 3. Symbol error rate as a function of SNR for multi levels of PSK: 2, 4, 8, 16, and 32 over the BFWA channel coherent detection

8 PSK 4-PSK 8-PSK 16-PSK 32-PSK Bit Error Rate SNR(dB) Figure 4. Bit error rate as a function of SNR for multi levels of PSK: 2, 4, 8, 16, and 32 over the BFWA channel coherent detection PSK 4-PSK 8-PSK Symbol Error Rate SNR(dB) Figure 5. Symbol error rate as a function of SNR for multi levels of PSK: 2, 4, and 8 over the BFWA channel for non-coherent detection PSK 4-PSK 8-PSK Bit Error Rate SNR(dB) Figure 6. Bit error rate as a function of SNR for multi levels of PSK: 2, 4, and 8 over the BFWA channel for non-coherent detection

9 4. Conclusions In this paper, we have analyzed M-ary PSK modulated BFWA system for both detection methods (coherent and non-coherent). On SUI channel model, many values of M-ary of PSK are examined to evaluate conditional probability of error, symbol error rate, and bit error rate. The performance show that for any value of SNR, coherent M-ary PSK produce a smaller and faster decay of bit error rate than the non-coherent one. Also, the increasing of M results in a reduced the channel bandwidth. However, this reduction in bandwidth is achieved at the cost of increased the power requirement. 5. References [1] P. Xiao, R. Carrasco, and I. Wassell, Performance Analysis of Conventional Detection in BFWA Systems, IEEE Proc. Second IFIP Int. Conf. Wireless and Optical Communication Network, WOCN 2005, pp , Dubai, UAE, March, [2] I. Tardy, and O. Groudalen, On the Role of Future High-Frequency BFWA Systems in Broadband Communication Networks, IEEE Communication Magazine, pp , February, [3] IEEE Standard, Part 16: air interface for fixed broadband wireless access systems-amendment 2: Media access control modifications and additional physical layer specifications for 2-11 GHz, Standard IEEE802.16a, January, [4] M. K. Khan, R. A. Carrasco, I. J. Wessell, and J. A. Neasham, Performance Comparison of Low Density Parity Check Codes Using Square Root Kalman Equalization and Orthogonal Frequency Division Multiplexing Techniques for Broadband Fixed Wireless Access Systems, IET Communication, vol. 2, no. 2, pp , February, [5] S. B. Sadkhan, and N. A. Abbas, Proposed Simulation of Modulation Identification Based On Wavelet Transform, International Journal of Advancements in Computing Technology (IJACT),vol. 1,no. 1, September, [6] V. Ereceg. An Empirically Based Path Loss Model for Wireless Channels in Suburban Environments. IEEE JSAC, vol. 17, no. 7, pp , July [7] P. J. Lee, Computation of the Bit-Error Rate of Coherent M-ary PSK with Gray Code Bit Mapping, IEEE Trans. Commun., vol. COM-34, pp , May, [8] J. Lassing, E. G. Strom, E. Agrell, and T. Ottosson, Computation of the Exact Bit-Error Rate of Coherent M-ary PSK with Gray Code Bit Mapping, IEEE Transaction on Communications, vol. 51, no. 11, pp , November, [9] J. Proakis, and M. Salehi, Digital Communications, Fifth Edition, McGraw-Hill, [10] S. Verdu, Multiuser Detection, 1st edition, Cambridge University Press,