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

Abstract

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 SER without 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 (𝐸_𝑏/𝑁₀) above and below the threshold point.

1. 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) [1–3]. 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 [5–8], 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 𝐸_𝑏/𝑁₀ 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 𝐸_𝑏/𝑁₀. 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 𝐸_𝑏/𝑁₀.

In the engineering field, it is very common to avoid the nonlinear problem caused by the phase demodulator by assuming high 𝐸_𝑏/𝑁₀ 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 𝐸_𝑏/𝑁₀ and AWGN case. Namely, the new expression is valid also for the lower range of 𝐸_𝑏/𝑁₀. 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.

2. System Description

The system under consideration is described in Figure 1 which is similar to that described in [5] for the AWGN case.

Figure 1 

System description.

During each 𝑇-second block interval, an 𝑁_DFT-point inverse discrete Fourier transform (IDFT) calculates a block of time samples 𝑥[𝑛] where the input to the IDFT is a conjugate symmetric data vector [0,𝑋[1],𝑋[2],…,𝑋[𝑁_QAM], 0,𝑋^∗[𝑁_QAM],…,𝑋^∗[2],𝑋^∗[1]] where {𝑋[𝑘]}^𝑁QAM_𝑘=1 are 𝑀_QAM-QAM data symbols and (⋅)^∗ is the conjugate operator on (⋅). The total length of the IDFT is equal to 2𝑁_QAM+2 and is defined in the following as 𝑁_DFT. In addition, we denote 𝑁=2𝑁_QAM. The samples 𝑥[𝑛]𝑥[𝑛]=∑^𝑁DFT−1_𝑘=0𝑋[𝑘]exp𝑗2𝜋𝑘𝑛𝑁_DFT=2∑^𝑁QAM_𝑘=1Re(𝑋[𝑘])cos2𝜋𝑘𝑛𝑁_DFT−Im(𝑋[𝑘])sin2𝜋𝑘𝑛𝑁_DFTfor𝑛=0,1,2,…,𝑁_DFT−1 (1) are sent to the phase modulator to obtain the output signal 𝑠[𝑛] defined by 𝑠[𝑛]=exp𝑗2𝜋ℎ𝐶𝑥[𝑛], (2) where ℎ is the modulation index, 𝑗=√−1, Re(⋅) and Im(⋅) are the real and imaginary parts of (⋅), respectively, and 𝐶 is defined by 𝐶=12𝑁𝜎²_𝐼, (3) where 𝜎²_𝐼=(𝑀²−1)/3 and 𝑀=𝑀_QAM. According to [5], the phase modulating OFDM signal is comprised of 𝑁=2𝑁_QAM subcarriers, each modulated by 𝑀=𝑀_QAM-ary pulse amplitude modulation (PAM) data symbols (±1,±3,…,±(𝑀−1)).

Next, a cyclic prefix (CP) is appended to the sequence 𝑠[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 𝑛(𝑡). The received signal is 𝑟(𝑡)=𝑠(𝑡)+𝑛(𝑡) where 𝑠(𝑡)=𝐴_𝑠exp(𝑗2𝜋ℎ𝐶𝑥(𝑡)), −𝑇_CP≤𝑡<𝑇, 𝑇_CP is the cyclic prefix duration and 𝐴_𝑠 is the signal amplitude. This signal (𝑟(𝑡)) 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, 𝑠[̂𝑛], are sent to the phase demodulator. The phase demodulator is implemented with an arctangent calculator followed by a phase unwrapper. The output of the phase demodulator is given by 𝜙[𝑛]=𝜙[𝑛]+𝜉[𝑛] where 𝜉[𝑛] is a nonlinear noise component and 𝜙[𝑛]=2𝜋ℎ𝐶𝑥[𝑛]. Next, the sequence 𝜙[𝑛] is sent to the 𝑁_DFT-point DFT block to obtain 𝑄^∼[𝑘]=1𝑁_DFT^𝑁DFT−1_𝑛=0𝜙[𝑛]exp−𝑗2𝜋𝑘𝑛𝑁_DFT=1𝑁_DFT^𝑁DFT−1_𝑛=0𝜙[𝑛]exp−𝑗2𝜋𝑘𝑛𝑁_DFT+𝑁^∼[𝑘]=2𝜋ℎ𝐶𝑋[𝑘]+𝑁^∼[𝑘], (4) where 𝑁^∼[𝑘] is defined by 𝑁^∼[𝑘]=1𝑁_DFT^𝑁DFT−1_𝑛=0𝜉[𝑛]exp−𝑗2𝜋𝑘𝑛𝑁_DFT. (5) Next, the vector [Re(𝑄^∼[𝑘])Im(𝑄^∼[𝑘])] is sent to the symbol demapper that yields the received bits. Since the vector [Re(𝑄^∼[𝑘])Im(𝑄^∼[𝑘])] is sent to the symbol demapper, the noise component we are interested in is defined by 𝑁[𝑘]=1𝑁_DFT^𝑁DFT−1_𝑛=0𝜉[𝑛]√2exp−𝑗2𝜋𝑘𝑛𝑁_DFT. (6)

3. 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 SER≃2𝑀−1𝑀𝑄2𝜋ℎ6log₂𝑀𝑀²𝐸−1𝑁^𝑏₀,where𝑄(𝑥)=1√2𝜋^∞𝑥𝑒^−𝑢2/2𝑑𝑢. (7) As already mentioned earlier, this expression (7) was obtained by assuming the high 𝐸_𝑏/𝑁₀ 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 𝐸_𝑏/𝑁₀ 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] 𝜙(𝑡)=𝜙(𝑡)+𝜉(𝑡), (8) where 𝜉(𝑡)𝐴=arctan^𝑛(𝑡)sin𝜓(𝑡)−𝜙(𝑡)𝐴_𝑠+𝐴_𝑛(𝑡)cos𝜓(𝑡)−𝜙(𝑡) (9) is a nonlinear noise component, 𝐴_𝑛(𝑡)=|𝑛(𝑡)| (where |⋅| is the absolute function of (⋅)), and 𝜓(𝑡) is the phase of 𝑛(𝑡).

Theorem 1. For the following assumptions,(1)𝐸[𝑛(𝑡)]=0, 𝐸[𝑛(𝑡)(𝑛^∗(𝑡))]=𝜎²_𝑛, where (⋅)^∗ is the conjugate operation on (⋅) and 𝐸[⋅] is the expectation operator,(2)𝐸[𝐴_𝑛(𝑡)sin(𝜓(𝑡)−𝜙(𝑡))] = 𝐸[𝐴_𝑛(𝑡)cos(𝜓(𝑡)−𝜙(𝑡))]=0,(3)𝐸[(𝐴_𝑛(𝑡)sin(𝜓(𝑡)−𝜙(𝑡)))²] = 𝐸[(𝐴_𝑛(𝑡)cos(𝜓(𝑡)−𝜙(𝑡)))²]=𝜎²_𝑛/2,(4)𝐴_𝑛(𝑡)sin(𝜓(𝑡)−𝜙(𝑡)) and 𝐴_𝑛(𝑡)cos(𝜓(𝑡)−𝜙(𝑡)) are Gaussian processes,
the SER is approximately given by 𝑆𝐸𝑅≃2𝑀−1𝑀𝑄𝑑𝜎_𝑁+2𝑀−1𝑀𝐸𝑁×3⁴[𝑘]−3𝜎²_𝑁²4!𝜎²_𝑁²𝑄𝑑𝜎_𝑁+2𝑀−1𝑀𝐸𝑁⁴[𝑘]−3𝜎²_𝑁²4!𝜎²_𝑁⁴×14√2√𝜋𝜎_𝑁2𝑑³𝑒^{−(1/2)(𝑑2/𝜎2𝑁)}+6𝑑𝜎²_𝑁𝑒^{−(1/2)(𝑑2/𝜎2𝑁)}+3𝜎⁴_𝑁𝑄𝑑𝜎_𝑁−2𝑀−1𝑀6𝐸𝑁⁴[𝑘]−3𝜎²_𝑁²4!𝜎²_𝑁²×1√𝑑2𝜋𝜎_𝑁𝑒^{−(𝑑/𝜎𝑁)2/2}+√𝑑2𝜋𝑄𝜎_𝑁, (10) where 𝜎²_𝑁≃14𝑁log₂𝑀𝐸_𝑏/𝑁₀𝑁1+^𝐷𝐹𝑇2𝑁log₂𝑀𝐸_𝑏/𝑁₀+124𝑁_𝐷𝐹𝑇𝜋²𝑒^{−(𝑁/𝑁𝐷𝐹𝑇)(log2𝑀)(𝐸𝑏/𝑁0)},𝐸𝑁⁴[𝑘≃3]116𝑁log₂𝑀𝐸_𝑏/𝑁₀²×51+3𝑁_DFT𝑁log₂𝑀𝐸_𝑏/𝑁₀+1320𝑁²_DFT𝜋⁴𝑒^{−(𝑁/𝑁DFT)(log2𝑀)(𝐸𝑏/𝑁0)},𝑄𝑑𝜎_𝑁=1√2𝜋^∞𝑑/𝜎_𝑁𝑒^−𝑢2/2𝑑𝑢,𝑑=2𝜋ℎ12𝑁𝜎²_𝐼 (11)(12)(13)(14) and !=1⋅2⋅3⋅⋯⋅(𝑛−1),0!=1.

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 𝑃_inner=𝑃(|𝑁[𝑘]|>𝑑)=2𝑃(𝑁[𝑘]>𝑑), (15) where 𝑑=2𝜋ℎ1/2𝑁𝜎²_𝐼 as given in (14). For the two outer points, the probability of error is [15] 𝑃_outer(𝑁[𝑘])=𝑃=𝑃>𝑑^inner2. (16) Therefore, the overall symbol error rate is [15] SER=𝑀−2𝑀𝑃_inner+2𝑀𝑃_outer=𝑀−2𝑀𝑃_inner+1𝑀𝑃_inner=𝑀−1𝑀𝑃_inner. (17) Now, in order to calculate the probability that 𝑁[𝑘]>𝑑 (15), the probability density function (pdf) of 𝑁[𝑘] is needed. Since we do not know the pdf of 𝑁[𝑘], we use the Edgeworth expansion up to order four [11, 12] for approximating it: 𝑓(𝑁[𝑘≃1])√2𝜋𝜎_𝑁𝑒^{−𝑁2[𝑘]/2𝜎2𝑁}×𝐸𝑁1+⁴[𝑘]−3𝜎²_𝑁²4!𝜎²_𝑁²×𝑁⁴[𝑘]𝜎²_𝑁²−6𝑁²[𝑘]𝜎²_𝑁+3, (18) where 𝐸[𝑁²[𝑘]]=𝜎²_𝑁. According to (6), 𝑁[𝑘] is a function of the random variable 𝜉[𝑛]. Thus, having the statistics of 𝜉(𝑡), namely, by knowing the following expressions 𝐸[(𝜉(𝑡))²] and 𝐸[(𝜉(𝑡))⁴], the approximated pdf for 𝑁[𝑘] (18) is well defined.
Theorem2. 𝐸[(𝜉(𝑡))²] and 𝐸[(𝜉(𝑡))⁴] may be given by𝐸(𝜉(𝑡))²≃𝜎²_𝑛2𝑎²𝜎1+^2𝑛2𝑎²+1𝜋12²𝑒^{−(𝑎2/𝜎2𝑛)},𝐸(𝜉(𝑡))⁴≃54𝜎²_𝑛𝑎²³+34𝜎²_𝑛𝑎²²+1𝜋80⁴𝑒^{−(𝑎2/𝜎2𝑛)}, (19)where 𝜎²_𝑛𝑎²=𝑁_𝐷𝐹𝑇𝑁log₂(𝑀)𝐸_𝑏/𝑁₀. (20)Proof. Let us rewrite (9) as 𝜉(𝑡)𝑥=arctan,𝑎+𝑦 (21) where 𝑎=𝐴_𝑠, 𝑥=𝐴_𝑤(𝑡)sin(𝜓(𝑡)−𝜙(𝑡)) and 𝑦=𝐴_𝑤(𝑡)(cos(𝜓(𝑡)−𝜙(𝑡))). According to [3, 16], the pdf of the random variable 𝜉(𝑡) is given for |𝜉(𝑡)|<𝜋/2 by 𝑓(𝜉(𝑡))=exp−1/𝜌𝜋+2/𝜌cos(𝜉(𝑡))exp−sin²(𝜉(𝑡))/𝜌𝜋/2×erf2𝜌cos(𝜉(𝑡)), (22) where 𝜌=2(𝜎²_𝑛/2)/𝑎²=𝜎²_𝑛/𝑎² and erf(𝑥)=(1/√2𝜋)∫^𝑥₀exp(−𝑦²/2)𝑑𝑦. Next, we recall from [3] the derived expression for 𝐸[(𝜉(𝑡))²] based on (22): 𝐸(𝜉(𝑡))²≃𝜎²_𝑛2𝑎²𝜎1+^2𝑛2𝑎²+1𝜋12²𝑒^{−(𝑎2/𝜎2𝑛)}, (23) thus leaving us now only with the derivation of 𝐸[(𝜉(𝑡))⁴] defined by 𝐸(𝜉(𝑡))⁴=^{𝜋/2−𝜀}_{−𝜋/2+𝜀}𝜉⁴(𝑡)𝑓(𝜉(𝑡(𝑡)=))𝑑𝜉_{𝜋/2−𝜀−𝜋/2+𝜀}𝜉⁴(𝑡)exp−1/𝜌𝜋+2/𝜌cos(𝜉(𝑡))exp−sin²(𝜉(𝑡))/𝜌𝜋/2×erf2/𝜌cos(𝜉(𝑡))𝑑𝜉(𝑡), (24) which we split into two integrals: 𝐸(𝜉(𝑡))⁴=𝐼₁+𝐼₂, (25) where 𝐼₁=^{𝜋/2−𝜀}_{−𝜋/2+𝜀}𝜉⁴(𝑡)exp−1/𝜌𝜋𝑑𝜉(𝑡),𝐼₂=₋₍₎^{(𝜋/2)−𝜀}_𝜋/2+𝜀𝜉⁴(𝑡)2/𝜌cos(𝜉(𝑡))exp−sin²(𝜉(𝑡))/𝜌𝜋/2×erf2𝜌cos(𝜉(𝑡))𝑑𝜉(𝑡), (26)(27) and 𝜀 is a small fixed positive value. Now, if we define Ψ(𝜉(𝑡))=sin²(𝜉(𝑡,𝑓(𝜉(𝑡=))))2/𝜌cos(𝜉(𝑡))𝜉⁴(𝑡)𝜋/2erf2𝜌cos(𝜉(𝑡)), (28) we obtain, by using (28) and (27): 𝐼₂=^{𝜋/2−𝜀}_{−𝜋/2+𝜀}𝑓(𝜉(𝑡−Ψ(𝜉(𝑡))exp))𝜌𝑑𝜉(𝑡). (29) 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 𝜌→0 of integrals in which the large parameter 1/𝜌 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(𝜉(𝑡)) has its minimum at 𝜉₀(𝑡), which is between infinity and minus infinity, then it is only the immediate neighborhood of 𝜉(𝑡)=𝜉₀(𝑡) that contributes to the full asymptotic expansion of the integral (29) for large 1/𝜌. The function Ψ(𝜉(𝑡)) has its minimum at 𝜉(𝑡)=0. Therefore, for 𝜌→0 and for the immediate neighborhood of 𝜉(𝑡)=0, we may write, as was done in [3], erf2𝜌cos(𝜉(𝑡))≅erf2𝜌≅0.5. (30) Using the Laplace Integral method [13, page. 261–274] and (30) we may approximate (29) in the following way 𝐼₂≃2𝜌^{𝜉0(𝑡)+𝜀1}_{𝜉0(𝑡)−𝜀1}𝑓₁(𝜉(𝑡−Ψ(𝜉(𝑡))exp))𝜌𝑑𝜉(𝑡), (31) where 𝑓₁(𝜉(𝑡=(𝜉(𝑡𝜉))cos))⁴(𝑡)2,𝜋/2 (32)𝜉₀(𝑡) is the point where the function Ψ(𝜉(𝑡)) has its minimum and 𝜀₁ 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 𝐼₂≃2𝜌−Ψ𝜉exp⁰(𝑡)𝜌2𝜋𝜌Ψ^′′𝜉₀(𝑡)⋅𝑓₁𝜉₀(𝑡)+𝑓₁^′′𝜉₀(𝑡)2𝜌Ψ^′′𝜉₀(𝑡)+𝑓₁^𝐼𝑉𝜉₀(𝑡)8𝜌Ψ^′′𝜉₀(𝑡)²+−Ψ^𝐼𝑉𝜉₀(𝑡)𝑓48𝜌^1′′𝜉₀(𝑡𝜌)15Ψ^′′𝜉₀(𝑡)³+−Ψ^𝐼𝑉𝜉₀(𝑡)(24)²𝜌𝑓₁^𝐼𝑉𝜉₀(𝑡)𝜌105Ψ^′′𝜉₀(𝑡)⁴+𝑓₁^𝑉𝐼𝜉₀(𝑡)𝜌72015Ψ^′′𝜉₀(𝑡)³, (33) where 𝑓₁^(𝑛)(𝜉₀(𝑡)) and Ψ^(𝑛)(𝜉₀(𝑡)) are the 𝑛th derivation of 𝑓₁ and Ψ, respectively, at the minimum point 𝜉₀(𝑡)=0 and 𝑂(𝑥) is defined as lim_𝑥→0(𝑂(𝑥)/𝑥)=𝑟, where “𝑟” is a constant value. Next, we derive the second, fourth, and sixth derivative of 𝑓₁(𝜉(𝑡)) at the minimum point 𝜉₀(𝑡)=0: 𝑓₁^′′(𝜉(𝑡1))=−2√𝜋√2(𝜉(𝑡))⁴(𝑡)√cos𝜉−122(𝜉(𝑡))²(𝑡)√cos𝜉+82(𝜉(𝑡))³(𝑡)⇓𝑓sin𝜉_1′′𝜉₀(𝑡)𝑓=0,_1𝐼𝑉(𝜉(𝑡=1))2√𝜋24√2cos𝜉(𝑡)−72√2(𝜉(𝑡))²(𝑡)+√cos𝜉2(𝜉(𝑡))⁴cos𝜉(𝑡)+16√2(𝜉(𝑡))³(𝑡)√sin𝜉−96(𝑡)(𝑡)⇓𝑓2𝜉sin𝜉_1𝐼𝑉𝜉₀(𝑡)=12√𝜋24√𝑓2,_1𝑉𝐼(𝜉(𝑡1))=−2√𝜋360√2cos𝜉(𝑡)−180√2(𝜉(𝑡))²(𝑡)+√cos𝜉2(𝜉(𝑡))⁴cos𝜉(𝑡)+24√2(𝜉(𝑡))³(𝑡)√sin𝜉−480(𝑡)(𝑡)⇓𝑓2𝜉sin𝜉_1𝑉𝐼𝜉₀(𝑡)1=−2√𝜋360√2. (34) By substituting (34), Ψ^′′(𝜉₀(𝑡))=2, Ψ^𝐼𝑉(𝜉₀(𝑡))=−8, and 𝑓₁(𝜉₀(𝑡))=0 into (33), we obtain 𝐼₂≃√√2𝜋1/2𝜋24√28𝜌2²+8(24)²𝜌12√𝜋24√2𝜌1052⁴−12√𝜋360√21𝜌720152³⇓𝐼₂≃54𝜌³+34𝜌². (35) Next, we calculate 𝐼₁: 𝐼₁=^{𝜋/2−𝜀}_{−𝜋/2+𝜀}𝜉⁴(𝑡)exp−1/𝜌𝜋(𝑡)=1𝑑𝜉𝜋80⁴𝑒^−1/𝜌. (36) We have already mentioned earlier that 𝜌=𝜎²_𝑛/𝑎². Thus, our next task is to find a closed expression for 𝜎²_𝑛/𝑎² as a function of 𝐸_𝑏/𝑁₀ and the various system parameters used such as the number of useful carriers (𝑁), number of total carriers (𝑁_DFT), and the order of the PAM modulation used (𝑀). For our case, the expression for 𝐸_𝑏/𝑁₀ can be written by 𝐸_𝑏𝑁₀=𝐸_𝑠/𝑁₀𝑁log₂(𝑀)=𝑎²𝑁_DFT𝑇_𝑠𝑁₀𝑁log₂(𝑀)⇓𝑁₀=𝑎²𝑁_DFT𝑇_𝑠𝑁log₂(𝑀)𝐸_𝑏/𝑁₀, (37) where 1/𝑇_𝑠 is the frequency sampling rate and 𝑇=𝑁_DFT𝑇_𝑠. Now, by using (37), we may define 𝜎²_𝑛 as 𝜎²_𝑛=𝑁₀𝑇_𝑠=𝑎²𝑁_DFT𝑁log₂(𝑀)𝐸_𝑏/𝑁₀. (38) Next, by dividing both sides of (38) by 𝑎², the expression of (20) is obtained. By using now 𝜌=𝜎²_𝑛/𝑎², (38), (36), (35), and (23), we obtain (19). This completes the proof of Theorem 2.
Now, we turn to calculate 𝐸[𝑁²[𝑘]] and 𝐸[𝑁⁴[𝑘]]. By using (6) and the property of 𝐸[𝜉[𝑛]𝜉[𝑚]]=𝐸[𝜉²[𝑛]]𝛿[𝑛−𝑚], we obtain 𝐸𝑁²[𝑘=1]2𝑁²_DFT^{𝑁DFT−1𝑁}_𝑛=0^DFT−1_𝑚=0𝐸𝜉[𝑛]𝜉[𝑚]×exp−𝑗2𝜋𝑘𝑛𝑁_DFTexp−𝑗2𝜋𝑘𝑚𝑁_DFT=12𝑁_DFT𝐸𝜉²[𝑛,𝐸𝑁₄][𝑘=1]4𝑁⁴_DFT^{𝑁DFT−1𝑁}_𝑛=0^DFT−1𝑁_𝑚=0^DFT−1_̆^𝑁_𝑛=0^DFT−1_̆𝑚=0⋅𝐸𝜉[𝑛]𝜉[𝑚]𝜉[𝑛]̆𝜉[𝑚̆]exp−𝑗2𝜋𝑘𝑛𝑁_DFT×exp−𝑗2𝜋𝑘𝑚𝑁_DFT𝑛̆exp−𝑗2𝜋𝑘𝑁_DFT𝑚̆×exp−𝑗2𝜋𝑘𝑁_DFT≅14𝑁²_DFT𝐸𝜉⁴[𝑛]. (39) By substituting (20) and (19) into (39), the expressions for 𝐸[𝑁²[𝑘]] and 𝐸[𝑁⁴[𝑘]] given in (11) and (12), respectively, are obtained. Now, we are ready to carry out the calculation of the probability that 𝑁[𝑘]>𝑑 which is needed in (15). By using the estimated pdf of 𝑁[𝑘] (18), the probability that 𝑁[𝑘]>𝑑 is defined by 𝑃(𝑁[𝑘])≃>𝑑_∞𝑑1√2𝜋𝜎_𝑁𝑒^{−(𝑁2[𝑘]/2𝜎2𝑁)}×𝑁1+𝐻⁴[𝑘]𝜎²_𝑁²−6𝑁²[𝑘]𝜎²_𝑁+3𝑑𝑁,where,𝐸𝑁𝐻=⁴[𝑘]−3𝜎²_𝑁²4!𝜎²_𝑁². (40) In order to find a solution for (40), the following integrals are first carried out: ∫^∞_𝑑(1/√2𝜋𝜎_𝑁)𝑒^{−(𝑁2[𝑘]/2𝜎2𝑁)}𝑑𝑁[𝑘], ∫^∞_𝑑𝑁²[𝑘](1/√2𝜋𝜎_𝑁)𝑒^{−(𝑁2[𝑘]/2𝜎2𝑁)}𝑑𝑁[𝑘], and ∫^∞_𝑑𝑁⁴[𝑘](1/√2𝜋𝜎_𝑁)𝑒^{−(𝑁2[𝑘]/2𝜎2𝑁)}𝑑𝑁[𝑘].
By using 𝑁[𝑘]/𝜎_𝑁=𝑢 and 𝑑𝑁[𝑘]/𝜎_𝑁=𝑑𝑢, we may write ^∞_𝑑1√2𝜋𝜎_𝑁𝑒^{−𝑁2[𝑘]/2𝜎2𝑁}[𝑘]=1𝑑𝑁√2𝜋^∞𝑑/𝜎_𝑁𝑒^−𝑢2/2𝑑𝑑𝑢=𝑄𝜎_𝑁∞𝑑,𝑁²[𝑘]1√2𝜋𝜎_𝑁𝑒^{−𝑁2[𝑘]/2𝜎2𝑁}[𝑘]=𝜎𝑑𝑁_2𝑁√2𝜋^∞𝑑/𝜎_𝑁𝑢²𝑒^−𝑢2/2=𝜎𝑑𝑢_2𝑁√2𝜋^∞0𝑢²𝑒^−𝑢2/2𝑑𝑢−^{𝑑/𝜎𝑁}₀𝑢²𝑒^−𝑢2/2=𝜎𝑑𝑢_2𝑁√𝑑2𝜋𝜎_𝑁𝑒^{−(𝑑/𝜎𝑁)2/2}+√𝑑2𝜋𝑄𝜎_𝑁∞𝑑,𝑁⁴[𝑘]1√2𝜋𝜎_𝑁𝑒^{−𝑁2[𝑘]/2𝜎2𝑁}[𝑘]=𝜎𝑑𝑁_4𝑁√2𝜋^∞𝑑/𝜎_𝑁𝑢⁴𝑒^−𝑢2/2=1𝑑𝑢4√2√𝜋𝜎_𝑁2𝑑³𝑒^{−(1/2)(𝑑2/𝜎2𝑁)}+6𝑑𝜎²_𝑁𝑒^{−(1/2)(𝑑2/𝜎2𝑁)}+3𝜎⁴_𝑁𝑄𝑑𝜎_𝑁. (41) Now, by substituting (41) into (40), we obtain 𝑃(𝑁[𝑘])≃𝐸𝑁>𝑑1+3⁴[𝑘]−3𝜎²_𝑁²4!𝜎²_𝑁²𝑄𝑑𝜎_𝑁+𝐸𝑁⁴[𝑘]−3𝜎²_𝑁²4!𝜎²_𝑁⁴×14√2√𝜋𝜎_𝑁2𝑑³𝑒^{−(1/2)(𝑑2/𝜎2𝑁)}+6𝑑𝜎²_𝑁𝑒^{−(1/2)(𝑑2/𝜎2𝑁)}+3𝜎⁴_𝑁𝑄𝑑𝜎_𝑁𝐸𝑁−6⁴[𝑘]−3𝜎²_𝑁²4!𝜎²_𝑁²1√×𝑑2𝜋𝜎_𝑁𝑒^{−(𝑑/𝜎𝑁)2/2}+√𝑑2𝜋𝑄𝜎_𝑁. (42) Next, by using (42), (17), and (15), the expression of (10) is obtained. This completes the proof of Theorem 1.

4. Simulation

In this section, we test the new obtained expression for the SER (10) for the 𝑀=2 and 𝑀=8 case with DFT length of 𝑁_DFT=8,32,64 for a wide range of 𝐸_𝑏/𝑁₀ values. The results are compared with those obtained from (7) and with 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 𝐸_𝑏/𝑁₀ between our expression for the SER (10), the expression given in (7), and the simulated results for the 𝑀=8 case with various values for the parameter ℎ and DFT length 𝑁_DFT. Figures 7, 8, 9, 10, and 11 show the performance comparison of the SER as a function of 𝐸_𝑏/𝑁₀ between our expression for the SER (10), the expression given in (7), and the simulated results for the 𝑀=2 case with various values for the parameter ℎ and DFT length 𝑁_DFT. According to Figures 2 to 11, our expression for the SER (10) describes better the simulated results for the entire range of 𝐸_𝑏/𝑁₀ values compared with the expression given in (7).

Figure 2 

SER comparison as a function of 𝐸𝑏/𝑁0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). 𝑁DFT=64, ℎ=0.1/2𝜋, 𝐴𝑠=2, modulation type: 𝑀=8. The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.

Figure 3 

SER comparison as a function of 𝐸𝑏/𝑁0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). 𝑁DFT=64, ℎ=0.05/2𝜋, 𝐴𝑠=2, modulation type: 𝑀=8. The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.

Figure 4 

SER comparison as a function of 𝐸𝑏/𝑁0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). 𝑁DFT=64, ℎ=0.5/2𝜋, 𝐴𝑠=2, modulation type: 𝑀=8. The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.

Figure 5 

SER comparison as a function of 𝐸𝑏/𝑁0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). 𝑁DFT=32, ℎ=0.1/2𝜋, 𝐴𝑠=2, modulation type: 𝑀=8. The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.

Figure 6 

SER comparison as a function of 𝐸𝑏/𝑁0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). 𝑁DFT=8, ℎ=0.1/2𝜋, 𝐴𝑠=2, modulation type: 𝑀=8. The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.

Figure 7 

SER comparison as a function of 𝐸𝑏/𝑁0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). 𝑁DFT=64, ℎ=0.1/2𝜋, 𝐴𝑠=2, modulation type: 𝑀=2. The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.

Figure 8 

SER comparison as a function of 𝐸𝑏/𝑁0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). 𝑁DFT=64, ℎ=0.05/2𝜋, 𝐴𝑠=2, modulation type: 𝑀=2. The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.

Figure 9 

SER comparison as a function of 𝐸𝑏/𝑁0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). 𝑁DFT=64, ℎ=0.5/2𝜋, 𝐴𝑠=2, modulation type: 𝑀=2. The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.

Figure 10 

SER comparison as a function of 𝐸𝑏/𝑁0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). 𝑁DFT=32, ℎ=0.1/2𝜋, 𝐴𝑠=2, modulation type: 𝑀=2. The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.

Figure 11 

SER comparison as a function of 𝐸𝑏/𝑁0 between the simulated, new proposed expression (10) and the expression proposed by [5] given in (7). 𝑁DFT=8, ℎ=0.1/2𝜋, 𝐴𝑠=2, modulation type: 𝑀=2. The averaged results were obtained in 100 Monte Carlo trials where 500 CE-OFDM signals were produced for each trial.

Note that our obtained expression for the SER (10) was derived under the following condition of 𝜌→0 which is equivalent to 𝑁_DFT/𝑁log₂(𝑀)(𝐸_𝑏/𝑁₀)→0 or 𝑁log₂(𝑀)(𝐸_𝑏/𝑁₀)/𝑁_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 𝐸_𝑏/𝑁₀ condition (Figures 2 to 11). In addition, it should not be forgotten that we have not approximated the pdf of the random variable 𝜉[𝑛] 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 𝑁[𝑘] where satisfying results were obtained, it is possible that better results might be obtained when a higher order of the Edgeworth expansion is used.

5. 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 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 (S𝛼S) Model is going to be used for statistical modelling of impulsive noise. Please note, the Cauchy distribution is a special case of the S𝛼S distribution. It should be pointed out that the probability density function (pdf) of a S𝛼S process in the presence of zero-mean Gaussian noise (designated S𝛼S+ 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 [20–22] 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 𝐸_𝑏/𝑁₀ and find if possible trivial fluctuations and nontrivial ones, methodologically referring to [21, 22] for references.

Acknowledgment

The authors would like to thank the anonymous reviewers for their fruitful and detail comments.