PSK Error Performance with Tikhonov Distributed Phase Error over Nakagami Fading Channel Ananya Patra
Aniruddha Chandra
ECE Department NIT Durgapur WB, India-713209
ECE Department NIT Durgapur WB, India-713209
[email protected]
[email protected]
Abstract—In this paper average bit error rate (BER) of binary phase shift keying (BPSK) modulation with improper phase estimation has been derived. A simple slow flat wireless fading channel obeying Nakagami-m distribution is assumed for channel modelling. In addition the channel is also perturbed by additive white Gaussian noise (AWGN). The phase distortions considered in the paper are random, unbiased, i.e. having zeromean and follows Tikhonov distribution. Analytical BER is calculated through moment generating function (MGF) approach and plotted as a function of signal to noise ratio (SNR) per bit for various values of fading parameter m as well as for different values of phase error parameter α. Extensive Monte Carlo simulations were performed to validate the theoretical results. Also, a graphical comparison of the derived BER with earlier works has been included. The figure clearly shows significant improvement in accuracy of the calculated theoretical BER. Keywords - PSK systems; carrier synchronization; phase error; Nakagami-m fading; moment generating function; Tikhonov distribution.
I.
INTRODUCTION
Synchronization is a critical function in digital communication systems and its failure may have catastrophic effects on the system performance. For coherent communication systems, error performance is usually evaluated under the assumption that a perfect phase reference is available in the receiver for demodulation [1]. In practice, however, this local phase reference is reconstructed from a noise-corrupted version of a received signal, resulting in a phase error that is normally modeled with either Gaussian [2] or Tikhonov [2-3] densities. The immediate effect of the phase error is to degrade the detection performance of coherent systems. This degradation, to some extent, may be mitigated by sending a pilot tone with the transmitted signal. But, this calls for additional transmitter power. Also to make room for the pilot tone the effective bandwidth will decrease, which is undesirable. During the demodulation with practical filters the signal power associated with the pilot tone may result in undesired DC component. This may further degrade the decision process or pose problems to the receiver amplifier. A better way to generate phase reference is to use carrier recovery techniques such as phase locked loops (PLLs). This method generally works well when the mobility of the receiver is limited and the phase distortions are not
severe. It has been shown [4] that Tikhonov distribution matches with the probability density function (PDF) of the carrier phase error for first-order PLLs, and approximates the PDF for second-order loop. Apart from synchronization errors, the wireless channels are subjected to random multipath fading. Nakagami-m distribution serves as the most general distribution to characterize such fading effects. It also incorporates the poplar Rayleigh model as a special case. In this paper we discuss the BER performance of BPSK over Nakagami-m fading channel with Tikhonov distributed phase error and thus address a generalized form of the phase jitter problem. The performance of PSK systems in presence of imperfect carrier phase recovery has been studied extensively in the last five decades. In 1960, Viterbi [5] was first to provide a statistical model for the carrier phase error of a first-order PLL and Lindsey [6], in 1966, addressed the performance of BPSK with phase error and AWGN. Later a usable but approximated expression was provided by Kam et al. [7]. In all these works carrier recovery through a first order PLL was considered, while Tikhonov distribution was used for statistical modelling of phase error. Weber [8] extended the calculations with Tikhonov model to fading channels and analyzed the performance of PLLs in Rayleigh, Rician and log-normal fading environments. For Nakagami-m fading channel, Lo and Lam [2] and Simon and Alouini [3] derived BER of BPSK with phase error. Lo and Lam [2] used an infinite series for error function whereas Simon and Alouini [3] approached the problem by truncating the conditional error probability (CEP) expression (a function of both instantaneous SNR and phase error) with McLaurin’s series. In both papers the carrier recovery loop SNR α is assumed to be proportional to instantaneous channel SNR γ, i.e. α = Kγ . A more general case, i.e. α ≠ Kγ , was studied by Falujah and Prabhu [9]. The authors have calculated BER of BPSK and QPSK in the presence of three system imperfections namely phase recovery error, slow multipath fading, and AWGN, using an infinite series expression to tackle the error function present in the CEP. However their end expressions involved infinite series and complex mathematical functions, were difficult to evaluate, and not guaranteed to converge. In the current paper we propose a solution in terms of finite range integral
instead. The result is derived through simple MGF method and it converges rapidly. The rest of the paper is organized as follows. The model of a system using BPSK with imperfect phase estimation and operating over a wireless fading channel is detailed in Section II. In Section III the CEP, conditioned on the phase error and channel fading, is averaged over the fading and phase error PDF using the MGF method which yields the final BER expression. Section IV contains performance analysis of BER plots and related discussions. The paper finally ends with some concluding remarks in Section V. II.
As seen from Fig. 1, the decision variable in presence of phase error is no longer the received signal (r) itself, rather its vertical projection r cos θ . The corresponding BER expression of BPSK (with phase error) is thus a function of two variables η and θ [7] 1 (2) Pe (η, θ) = erfc η cos θ 2
[
B. Wireless Fading Channel A slow and flat fading channel is assumed here, and the effect of such kind of fading on the signal amplitude is multiplicative distortion characterized by attenuation factor α. Besides fading, the transmitted signal is also perturbed by real bandpass noise having a two-sided power spectral density (PSD) N0/2 which is typically assumed to be independent of the fading process. The equivalent baseband AWGN is therefore complex circular with PSD N0 per quadrature. The overall fading channel model at baseband can be expressed as, ~ r (t ) = α exp(− jφ)~ s (t ) + n~ (t ) (3) where α and φ denote random amplitude and phase variation and n~ (t ) is a zero-mean circularly symmetric complex Gaussian process.
SYSTEM AND CHANNEL MODEL
A. BPSK Modulation Phase-shift keying is a digital modulation scheme that conveys data by changing or modulating the phase of a reference signal (the carrier wave). PSK uses a finite number of phases; each assigned a unique pattern of binary bits. BPSK (also sometimes called PRK, Phase Reversal Keying) is the simplest form of PSK. It uses two phases which are separated by 180° and so can also be termed as 2PSK. The BER of BPSK in AWGN is given by Pe (η ) =
1 erfc 2
[ η]
For Nakagami-m fading channel the PDF of α is given
(1)
by,
where η = E b N 0 .
φ2
f α (α ) =
r
θ − Eb
m
m 2 m 2 m−1 exp − α 2 ; α > 0 α Γ(m ) Ω Ω
(4)
where Γ(.) is the Gamma function, E {α 2 } = Ω , and m is the Nakagami-m fading parameter, which ranges from 0.5 to ∞ . For m=1, Nakagami-m distribution reduces to Rayleigh. As α2 denotes instantaneous attenuation in received signal power, the instantaneous SNR per bit can be defined as γ = α 2 η with an average value γ = E{α 2 }η .
φ1
r cos θ
]
Eb
Figure 1. Constellation diagram of BPSK with phase error (where r = received signal vector ).
_________________________________________________________________________________________________________ s(t ) = ±
Input Signal an
BPSK Modulator
2 Es cos(2πfct ) Ts Tx Signal
s (t )
r (t ) = α exp(− jφ )s (t ) + n(t )
Rx Signal
Faded Signal
∫ (⋅)dt Ts
r
Decision Device
0
Binary Information
α exp (− jφ )
Transmitter
Demodulator
r (t )
n (t ) AWGN
Channel
±
2 Es cos (2 πf c t + θ ) Ts
1 if ...r ≥ 0 aˆ n = 0 if ...r ≤ 0
Th = 0
Receiver
Figure 2. Transmitter–Receiver block diagram. __________________________________________________________________________________________________________________________________
The mean square value of α is generally normalized to one. The PDF of γ is given by 1 m f γ (γ ) = Γ (m ) γ
m
m −1 mγ γ exp − γ
III. (5)
Accordingly, in fading channel, the BER expression of BPSK as given in (2) may be modified as 1 (6) Pe (γ , θ ) = erfc γ cos θ 2 to incorporate the effect of both fading and phase error.
[
]
C. Phase Error Modeling Phase error in this paper is modeled with Tikhonov distribution. In probability theory and statistics, the Tikhonov or von Mises distribution is a continuous probability distribution on the circle. It may be thought of as a circular analogue of the normal distribution.
Fig. 2 depicts the system model employing BPSK modulation. The effect of fading, AWGN and phase error is also shown. At the receiver side due to the imperfect synchronization estimated phase reference at the receiver (φe) differs from the carrier phase (φr) actually received. The phase error θ=φr-φe is a random variable and if a Tikhonov model is assumed for it then the corresponding PDF may be written as,
f θ (θ) =
distribution approaches a normal distribution in θ , N (µ, 1 α ) , with mean µ and variance 1/ α .
1 exp[α cos(θ − µ )] 2πI 0 (α )
(7)
where I 0 (.) is the modified Bessel function of first kind and order 0, µ is a measure of location and α is a measure of concentration.
ANALYTICAL DERIVATION
In this section we will be deducing the mathematical expressions for BER of BPSK when affected by Tikhonov distributed phase error. For the purpose, a simpler approach to evaluate the average BEP is adopted which involves MGF of the fading SNR γ, Mγ(s). The CEP of BPSK with phase error and fading from (6) can be expressed by utilizing alternate representation of erfc(⋅) as
Pe (γ , θ ) =
π/2 γ cos 2 θ 1 dφ exp − ∫ π φ= 0 sin 2 φ
(8)
In presence of fading and phase error the CEP for BPSK becomes a function of two variables γ and θ and needs to be averaged over both of them. The average BER over AWGN channel may be found by simply averaging this CEP over phase error PDF,
Pe =
∞
∫ P (η, θ) f (θ)dθ
(9)
θ
e
−∞
where f θ (θ) is the PDF as mentioned in (7) whereas Pe (η, θ ) denotes the CEP given in (2). In a fading channel, however, SNR γ itself becomes a random variable and to obtain the average BER one has to perform a two-fold integration, the first one over the phase error PDF and the second over the fading SNR PDF. We may thus formulate a general mathematical equation to calculate the average probability of error as
Pe =
π
∞
∫ ∫ P (γ, θ) f (γ ) f (θ)dθdγ γ
e
(10)
θ
θ= − π γ = 0
where fγ(γ) is given by (5).We proceed by substituting (8) in (10) and interchanging the order of integrals γ cos 2 θ 1 π π/ 2 ∞ f γ (γ )dγ f θ (θ )dφdθ (11) Pe = ∫ exp − 2 ∫ ∫ π θ=− π φ=0 γ =0 sin φ This is possible as we have considered α ≠ Kγ , i.e. the loop SNR α is independent of the channel SNR γ . From the
M γ (s ) = ∫ f γ (γ ) exp(− sγ )dγ ∞
definition of MGF replace the term
(
∫γ=0 exp(− γ cos ∞
)
0
2
we may
)
θ sin 2 φ f γ (γ )dγ in (11) by
M γ cos 2 θ sin 2 φ . Further substituting f θ (θ ) from (7) in (11) Figure 3. PDF of Tikhonov distributed phase error.
Fig. 3 portrays the PDF of Tikhonov distribution given by (7) for different values of α . For small α values the distribution f θ (θ ) approaches uniform distribution U (⋅,⋅) and
at α = 0 , f θ (θ) ≡ U (− π, π) . When α is large, the distribution becomes concentrated about the angle µ with α being a measure of the concentration. In fact, as α increases, the
we have
Pe =
π π/ 2 cos 2 θ 1 exp(α cos θ )dθdφ M γ ∫ ∫ 2π I 0 (α ) θ=− π φ=0 sin 2 φ 2
(12)
Now using the expression of MGF of SNR γ Nakagami-m fading channel from (2.22) [10]
m M γ (s ) = m + sγ
in
m
(13)
the final expression for average BER of BPSK becomes, m
π π/2 1 m Pe = exp (α cos θ )dθdφ 2 ∫ ∫ 2 π I 0 (α ) θ=− π φ=0 g (m, γ , θ, φ)
(14)
where g (m, γ, θ, φ) = m + γ (cos 2 θ sin 2 φ ) . One may argue that instead of adopting MGF method a direct numerical integration of (10) through powerful tools like MATHEMATICATM would yield the same result and that too with lesser complexity. But interestingly this is not true. The infinite limits of the integrals lead to numerical instability. It is verified that this instability results in considerable inaccuracy when compared to values that are found by simulation runs. IV.
RESULTS AND DISCUSSIONS
Monte Carlo simulations were performed to validate the analytical results in the paper. The stochastic simulation process is used to estimate the BER by counting the erroneous bits at the receiver and then dividing the count by the total number of bits passed through the system. Usually the number of bits examined at a SNR point is at least 10 times higher than the inverse of the expected error rate, i.e. to test a BER of 10-4, 105 bits were examined. Further, an average of 30 individual runs was taken to smooth the variation about the mean. The simulation and analytical values almost coincide with each other. In Fig. 4 and Fig. 5, bit error probabilities for BPSK with Tikhonov distributed phase error in Nakagami-m fading channel are depicted. The BER variation as a function of SNR per bit is shown for various values of the fading parameter m in Fig. 4 whereas Fig. 5 displays the effect of changing the carrier recovery loop SNR α. All the curves show that the error probabilities rise significantly, when fading severity (m) decreases or loop SNR (α) increase. The simulated points are superimposed (shown by asterisk mark) on the analytical values obtained in the paper.
Figure. 4. Average BER of BPSK in Nakagami-m fading channel with phase error (plotted for different values of fading parameter m).
Figure.5. Average BER of BPSK in Nakagami-m fading channel with phase error (plotted for different values of carrier recovery loop SNR α). Fig. 6 and Fig. 7 portrays the theoretical comparison of MGF method with earlier works. In Fig. 6 it has been shown that results for MGF method almost coincides with FalujahPrabhu’s theoretical result [9]. So, we can say that MGF method is an efficient alternative method where without having a close-form expression we can achieve the same accuracy and it also validates our analysis. For plotting Fig. 6 and Fig. 7 we assumed α = 10dB and m = 2. Also comparing the plots of percentage of error in Fig. 7 we can say that MGF method provides better accuracy. For 3dB ≤ γ ≤ 12dB the percentage error is zero and it slowly increases whereas the for the earlier result given in [9], the percentage of error increases exponentially. So we can summarize that both in terms of accuracy and numerical complexity, MGF method is the most suitable method for error performance analysis over fading channel in presence of phase error.
Figure. 6. Comparison of MGF method and numerical integration with Falujah-Prabhu’s [9] theoretical result.
extended to the encompass M-PSK in general, i.e. analysis for arbitrary M values. Another scope of future research is to see how coherent PSK schemes with phase error will perform against differential schemes in a multi-antenna system. REFERENCES [1]
Figure.7. Percentage of error graph for MGF method and Falujah-Prabhu’s [9] theoretical result.
V.
CONCLUSIONS
In this article we have provided an alternative method for the analysis of BER for BPSK operating over Nakagami-m fading channel, when perturbed with erroneous phase estimate. Using MGF method, the bit error probabilities of BPSK for Tikhonov distributed phase error are calculated. Numerical evaluation of these formulas provides excellent accuracy when compared with the exact evaluation. Extensive simulations were also performed to authenticate the theoretical values. A graphical comparison (with earlier works) of BER curves of BPSK suffering from phase synchronization problems is displayed. The current work may be easily extended to other wireless fading models (Rayleigh, Rician, Hoyt, Weibull or, generalized Gamma). Also, the analysis for BPSK may be
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