A Systematic Approach for Calculating the Symbol Error Rate for the Entire Range of Eb/N0 above and below the Threshold Point for the CE-OFDM System

Recently, the performance of the constant envelope OFDM (CE-OFDM) was analyzed in additive white Gaussian noise (AWGN) with the help of a closed-form approximated expression for the symbol error rate (SER). This expression was obtained with the assumption of having a high carrier-to-noise ratio (CNR)which, in effect, linearized the phase demodulator (the phase demodulator was implemented with an arctangent calculator) and simplified the analysis. Thus, this expression is not accurate for the lower range of CNR. As a matter of fact, it was already observed that there is a threshold point from which the simulated SER result vanishes from the theoretically obtained expression. In this paper, we present a systematic approach for calculating the SERwithout assuming having the high CNR case or using linearization techniques. In other words, we derive the SER for the nonlinear case. As a byproduct, we obtain a new closed-form approximated expression for the SER based on the Laplace integral method and the Edgeworth expansion. Simulation results indicate that the simulated results and those obtained from the new derived expression are very close for the entire range of bit energy-to-noise density ratio (Eb/N0) above and below the threshold point.


Introduction
The orthogonal multicarrier modulation technique, also known as "orthogonal frequency division multiplexing" (OFDM) technique, is the standard modulation scheme used in Europe for Digital Audio Broadcasting (DAB) and Digital Video Broadcasting (DVB) [1][2][3].In addition, local area networks (LANs) such as IEEE 802.11a are OFDM based [3].OFDM is an excellent solution to the frequency selective fading problem in wireless applications [3].However, spurious high amplitude peaks in the composite time signal occur when the signals from the different tones add constructively [4].Compared to the average signal power, the instantaneous power of these peaks is high and, consequently, so is the peak-to-average power ratio (PAPR) [4].The occurrence of these peaks seriously hampers practical implementations and is generally considered as one of the major drawbacks of OFDM [4].Recently [5][6][7][8], the constant envelope OFDM (CE-OFDM) was proposed for solving the PAPR problem associated with OFDM.CE-OFDM transforms the OFDM signal, by way of phase modulation, to a signal designed for efficient power amplification [5].At the receiver, the inverse transformation-phase demodulation-is applied prior to the conventional OFDM demodulator [5].The performance of CE-OFDM was analyzed in [5] where a closed-form approximated expression was given for the SER for the AWGN channel case.However, this proposed expression for the SER [5] was obtained by assuming having high values for   / 0 and by using linearization techniques to avoid the nonlinear problem caused by the phase demodulator (the phase demodulator was implemented with an arctangent calculator).Thus, it is accurate only for the higher range of   / 0 .It should be pointed out that it was already observed in [5] that the simulated SER results vanish from those obtained by the proposed expression for the SER for the lower range of   / 0 .
In the engineering field, it is very common to avoid the nonlinear problem caused by the phase demodulator Mathematical Problems in Engineering by assuming high   / 0 and using linearization techniques [9,10].The main purpose of this paper is to show a systematic approach for calculating the SER without avoiding the nonlinear problem caused by the phase demodulator.As a byproduct, we obtain a new closed-form approximated expression for the SER that is valid for the entire range of   / 0 and AWGN case.Namely, the new expression is valid also for the lower range of   / 0 .It is based on the Edgeworth expansion up to order four [11,12], Laplace integral method (see [13, page. 261-274], [3,12,14]), and is obtained without using linearization techniques.
The paper is organized as follows: after having described the system under consideration in Section 2, we introduce in Section 3 our systematic approach for calculating the SER.In Section 4 we present simulation results, and Section 5 is our conclusion.

System Description
The system under consideration is described in Figure 1 which is similar to that described in [5] for the AWGN case.
Next, a cyclic prefix (CP) is appended to the sequence  [𝑛].The discrete-time samples are then passed through a digital-to-analog (D/A) converter, and the result is amplified and transmitted into the channel where it is corrupted by an additive white Gaussian noise ().The received signal is () = () + () where () =   exp(2ℎ()), − CP ≤  < ,  CP is the cyclic prefix duration and   is the signal amplitude.This signal (()) is then passed through an analog-to-digital (A/D) converter and after that sent to the block that discards the cyclic prefix samples.The samples that remain, ŝ[], are sent to the phase demodulator.The phase demodulator is implemented with an arctangent calculator followed by a phase unwrapper.
where Ñ[] is defined by Next, the vector ] is sent to the symbol demapper that yields the received bits.Since the vector ] is sent to the symbol demapper, the noise component we are interested in is defined by

The Systematic Approach for Calculating the SER
Recently [5], the performance of CE-OFDM was analyzed in an AWGN channel where a closed-form approximated expression was given for the SER given by where As already mentioned earlier, this expression (7) was obtained by assuming the high   / 0 case and using linearization  techniques.Thus, avoiding the nonlinear problem caused by the phase demodulator.In this section, we show a systematic approach for calculating the SER applicable for CE-OFDM system valid for the entire range of   / 0 without using linearization techniques or avoiding the nonlinear problem caused by the phase demodulator.According to Figure 1, the CE-OFDM demodulator operates in the discrete-time domain.It is convenient, however, to consider the continuous-time model for analysis [5].When no phase offset is caused by the channel, the output of the phase demodulator for the simple AWGN case is according to [5] where is a nonlinear noise component,   () = |()| (where | ⋅ | is the absolute function of (⋅)), and () is the phase of ().
Theorem 1.For the following assumptions, where (⋅) * is the conjugate operation on (⋅) and [⋅] is the expectation operator, where Proof.According to [15], the symbol error rate is computed by determining the probability of error for each signal point in the M-PAM constellation.For the M-2 inner points, the probability of error is where  = 2ℎ√1/2 2  as given in (14).For the two outer points, the probability of error is [15] Therefore, the overall symbol error rate is [15] Now, in order to calculate the probability that [] >  (15), the probability density function (pdf) of [] is needed.Since we do not know the pdf of [], we use the Edgeworth expansion up to order four [11,12] for approximating it: where [  4 ] may be given by where Proof.Let us rewrite (9) as where  =   ,  =   () sin(() − ()) and  =   ()(cos(() − ())).According to [3,16], the pdf of the random variable () is given for |()| < /2 by where Next, we recall from [3] the derived expression for [(()) 2 ] based on (22): which we split into two integrals: where Mathematical Problems in Engineering 5 and  is a small fixed positive value.Now, if we define we obtain, by using ( 28) and ( 27): In order to solve (29), we use the Laplace integral method following [3].According to [3], the Laplace method [13, page.261-274] is a very general technique for obtaining the asymptotic behavior as  → 0 of integrals in which the large parameter 1/ appears in the exponent.The main idea of Laplace's method [13, page.261-274] is to use the fact that if the real continuous function Ψ(()) has its minimum at  0 (), which is between infinity and minus infinity, then it is only the immediate neighborhood of () =  0 () that contributes to the full asymptotic expansion of the integral (29) for large 1/.The function Ψ(()) has its minimum at () = 0. Therefore, for  → 0 and for the immediate neighborhood of () = 0, we may write, as was done in [3], Using the Laplace Integral method [13, page.261-274] and (30) we may approximate (29) in the following way where 0 () is the point where the function Ψ(()) has its minimum and  1 is a small fixed positive value.According to the Laplace Integral method [13, page.261-274] and [3], we may write (31) in the following way where  () 1 ( 0 ()) and Ψ () ( 0 ()) are the th derivation of  1 and Ψ, respectively, at the minimum point  0 () = 0 and () is defined as lim  → 0 (()/) = , where "" is a constant value.Next, we derive the second, fourth, and sixth derivative of  1 (()) at the minimum point  0 () = 0: (24 √ 2 cos  () − 72 √ 2( ()) ( Next, we calculate  1 : We have already mentioned earlier that  =  2  / 2 .Thus, our next task is to find a closed expression for  2  / 2 as a function of   / 0 and the various system parameters used such as the number of useful carriers (), number of total carriers ( DFT ), and the order of the PAM modulation used ().For our case, the expression for   / 0 can be written by where 1/  is the frequency sampling rate and  =  DFT   .Now, by using (37), we may define  2  as Next, by dividing both sides of (38) by  2 , the expression of ( 20) is obtained.By using now  =  2  / 2 , (38), (36), (35), and (23), we obtain (19).This completes the proof of Theorem 2. Now, we turn to calculate [ 2 []] and [ 4 []].By using (6)

Article expression
Figure 2: SER comparison as a function of   / 0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). DFT = 64, ℎ = 0.1/2,   = 2, modulation type:  = 8.The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.

Article expression
Figure 3: SER comparison as a function of   / 0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). DFT = 64, ℎ = 0.05/2,   = 2, modulation type:  = 8.The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.

Simulation
In this section, we test the new obtained expression for the SER (10) for the  = 2 and  = 8 case with DFT length of  DFT = 8, 32, 64 for a wide range of   / 0 values.The results are compared with those obtained from (7) and with

Symbol error rate
Figure 4: SER comparison as a function of   / 0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). DFT = 64, ℎ = 0.5/2,   = 2, modulation type:  = 8.The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.

Article expression
Figure 5: SER comparison as a function of   / 0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). DFT = 32, ℎ = 0.1/2,   = 2, modulation type:  = 8.The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.
those obtained by simulation.In the following, we denote the expression given in (7) as "Article Expression" and our new obtained expression for the SER (10) as "New Expression." Figures 2,3,4,5,and 6 show the performance comparison of the SER as a function of   / 0 between our expression for the SER (10), the expression given in (7), and the simulated    (10) and the expression proposed by [5] given in (7). DFT = 8, ℎ = 0.1/2,   = 2, modulation type:  = 8.The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.

Symbol error rate
Figure 7: SER comparison as a function of   / 0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7).

Symbol error rate
Figure 8: SER comparison as a function of   / 0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). DFT = 64, ℎ = 0.05/2,   = 2, modulation type:  = 2.The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.

Symbol error rate
Figure 9: SER comparison as a function of   / 0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). DFT = 64, ℎ = 0.5/2,   = 2, modulation type:  = 2.The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.
for the SER (10) describes better the simulated results for the entire range of   / 0 values compared with the expression given in (7).Note that our obtained expression for the SER (10) was derived under the following condition of  → 0 which is equivalent to  DFT /log 2 ()(  / 0 ) → 0 or

Symbol error rate
Figure 10: SER comparison as a function of   / 0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). DFT = 32, ℎ = 0.1/2,   = 2, modulation type:  = 2.The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.
log 2 ()(  / 0 )/ DFT → ∞.The reason for having good results may be that, in practice, analysis based on low noise, which makes the Laplace integral and singular perturbation method feasible, can be extended to the region where the noise is not low.Note, for example, the papers [17,18] where the Laplace integral and the singular perturbation method were applied under low noise assumption, and the results could be very well extended to the medium and high noise ranges.(As a matter of fact, these methods were rather successful, even in calculating the threshold region.)This is also the case in this paper, where good results are obtained even under the very low   / 0 condition (Figures 2 to 11).In addition, it should not be forgotten that we have not approximated the pdf of the random variable [] as a Gaussian pdf as was done in [5,15] but rather used the exact expression (22).Although we used the Edgeworth expansion up to order four for approximating the pdf of [] where satisfying results were obtained, it is possible that better results might be obtained when a higher order of the Edgeworth expansion is used.

Conclusion
In this paper, we presented a systematic approach for calculating the SER, applicable for CE-OFDM and for the AWGN case, which does not avoid the nonlinear problem caused by the phase demodulator as was done in [5,15].As a byproduct, a new expression was obtained for the SER that outperforms the recently proposed expression by [5,15].Although the SER performance difference between the new derived expression and the one proposed in [5,15] is not very

Symbol error rate
Figure 11: SER comparison as a function of   / 0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). DFT = 8, ℎ = 0.1/2,   = 2, modulation type:  = 2.The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.
high, it is still important from the mathematical point of view to have a systematic approach showing how to deal with the nonlinear problem.CE-OFDM was recently considered also for the power line communication channel where impulse noise is present.In future work, we intend to calculate the SER, applicable for CE-OFDM for the impulse noise case where the Symmetric Alpha Stable (SS) Model is going to be used for statistical modelling of impulsive noise.Please note, the Cauchy distribution is a special case of the SS distribution.It should be pointed out that the probability density function (pdf) of a SS process in the presence of zero-mean Gaussian noise (designated SS+ G) is a close approximation to the pdf of class B noise where by class B noise we mean the Middleton Class B Model which refers to impulsive noise with a spectrum that is broad compared to the receiver bandwidth [19].Class B noise impulses produce transients in the receiver [19].In addition, we also plan to use fractional Gaussian noise [20][21][22] instead of the additive white Gaussian noise we used in this paper for calculating the SER applicable for CE-OFDM.When using fractional Gaussian noise, we may consider the fluctuation range of   / 0 and find if possible trivial fluctuations and nontrivial ones, methodologically referring to [21,22] for references.