Semi-analytical approach for performance evaluation of direct-detection OFDM optical communication systems

Tiago M. F. Alves; Adolfo V. T. Cartaxo

doi:10.1364/OE.17.018714

1. Introduction

Transmission of orthogonal frequency-division multiplexing (OFDM) signals in optical communication systems has received an enhanced interest in the past few years due to its robustness to intersymbol interference (ISI) [1]. Transmission of OFDM signals over optical fibre has been experimentally proposed for different systems such as radio-over-fibre for access networks [2, 3], metro networks [4] and long-haul systems [5]. In addition to experimental work, some important aspects of the transmission of OFDM signals over optical fibre have been also analyzed through numerical simulation [6, 7].

The bit error ratio (BER) is the figure of merit commonly used to assess the transmission performance of OFDM signals over optical fibre. Three approaches have been used to calculate the BER of direct-detection (DD) OFDM systems: (i) Monte Carlo (MC) simulation, where the signal and the noise introduced in the system are generated and propagated together along the transmission system and the BER is obtained by direct error counting [6, 8, 9]; (ii) semi-analytical approaches, where the signal is generated and propagated along the transmission system to assess the channel induced distortions on the signal waveform, whereas the noise and the BER are characterized by analytical means using the information of the noise statistical properties [10]; and (iii) analytical methods, where the signal and the noise at the system output are characterized analytically [11]. Though accurate BER estimation is achieved byMCsimulation, low BER values are remarkably difficult to simulate as they require unacceptable computation time. Instead, the assessment of the system performance by using analytical methods provide fast results independently of the BER range. However, this is achieved at the expense of the performance estimation accuracy as closed-form expressions are usually obtained by neglecting a few sources of signal distortion. Semi-analytical approaches provide a compromise between the other approaches. The semi-analytical method reported in [10] allows quickly evaluating the BER in each subcarrier from the signal-to-noise ratio. However, the bandwidth limitation and the distortion induced by the OFDM receiver on the noise and on the OFDM signal are neglected and it is not clear the way how the distortions induced on each OFDM subcarrier by the optical link are considered as the formulation used to evaluate the BER considers absence of ISI and intercarrier interference (ICI).

In this work, a semi-analytical Gaussian approach (SAGA) to evaluate the BER through numerical simulation in OFDM optical transmission systems for optically pre-amplified/DD receivers is proposed. Although the SAGA has been initially developed to evaluate the transmission performance of DD-OFDM ultra-wideband (UWB) radio signals over optical fiber, it is shown that it applies also to other DD-OFDM systems. The SAGA presents several novel contributions relative to the previous semi-analytical approach reported in [10]: (i) it allows obtaining the variance of the I and Q components of each subcarrier at OFDM receiver output; (ii) the dependence of the signal-noise and noise-noise beat terms contribution to the variance of each subcarrier on the optical and electrical filters shape and on the FFT block are considered; (iii) it takes into account eventual phase differences of the local oscillator in the I and Q branches of the electrical receiver; (iv) it considers the influence of the equalizer transfer function on the signal-noise and noise-noise beat terms; and (v) the statistical distribution of the ISI between consecutive OFDM symbols and of the ICI are correctly taken into account.

2. Theory

In this section, the optically pre-amplified DD receiver of the OFDM optical transmission system is presented. Afterwards, closed-form expressions for the mean and the variance of each subcarrier at system output are derived and the SAGA used to evaluate the system performance is presented.

2.1. System modeling

Fig. 1. Block diagram of the optically pre-amplified DD-OFDM receiver.

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Fig. 1 shows a scheme of the optically pre-amplified DD-OFDM receiver. The receiver includes an optical amplifier (OA), an optical filter (OF) and the PIN photodetector. The OF is characterized by its impulse response, h_o(t), and transfer function, H_o(ν), and the PIN photodetector by a quadratic detection characteristic with responsivity R_λ. The electrical part of the receiver consists in the I and Q demodulator branches, the FFT block, the equalizer and the symbol demapping block. The phase of the local oscillator for the I and Q branches, identified in Fig. 1 as ϕ ₁ and ϕ ₂, respectively, can represent either the inaccuracy of the local oscillator or different propagation delays between the I and Q branches of the electrical receiver. In the FFT block, N represents the total number of subcarriers transmitted in each OFDM symbol.

2.2. Gaussian distribution characterization

To characterize analytically the signal as a Gaussian distribution, closed-form expressions for the mean and variance of the signal at the equalizer output are derived in this subsection.

Considering that the signal field at OF input is polarized and the noise field is completely unpolarized, the Jones vectors of the OFDM signal field, r _s(t), and the amplified spontaneous emission (ASE) noise field, r_n(t), at OF input can be written as r _s(t)=r _s,‖(t)u _‖ and r_n(t)=r _n,‖(t)u _‖+r _n,⊥(t)_u⊥, respectively, where r _s,‖(t), r _n,‖(t) and r _n,⊥(t) are the OFDM signal and the ASE noise field components in the parallel (‖) and perpendicular (⊥) polarizations directions defined by u _‖ and u _⊥, respectively. The derivation of the SAGA assumes that the received signal can be demodulated by the DD-OFDM receiver shown in Fig. 1, independently of the OFDM transmitter employed. OFDM-UWB radio-over-fiber and single-sideband (SSB) OFDM signals used in access [2] and in long-haul DD systems [14, 15], respectively, are examples of different OFDM transmitters that can use the receiver of Fig. 1 to demodulate the received signal. The ASE noise field due to the OA is assumed zero mean additive white Gaussian noise with power spectral density (PSD) in each polarization direction (parallel and perpendicular to the signal) given by S _{ASE,(‖,⊥)}=hν ₀ n_sp(g-1), where h is the Planck’s constant, n_sp is the spontaneous emission noise factor, ν ₀ is the optical carrier frequency and g is the OA gain. Considering that the OF operation is independent of the polarization direction, the field at the OF output can be expressed as a function of the signal and noise field components as e(t)=s(t)+n(t), where s(t)=r _s(t) * h_o(t), n(t)=r_n(t)*h_o(t). The low-pass equivalent (LPE) of the field at OF output, e_l(t), is defined as e(t)=√2ℜ{e_l(t)exp[j2πν ₀ t]}=√2ℜ{[s_l(t)+n_l(t)]exp[j2πν ₀ t]}, where s_l(t) and n_l(t) are the LPE of s(t) and n(t), respectively, and ℜ{z} is the real part of z. The LPE of the OF impulse response, h_o,l(t), and transfer function, H_o,l(f), are defined as h_o(t)=2ℜ{h_o,l(t)exp[j2πν ₀ t]} and by H_o(ν)=H_o,l(ν-ν ₀)+H*_o,l(-ν-ν ₀), respectively. Considering the representation of the LPE of each noise polarization in their I and Q components, the LPE of the field at the OF output is given by:

e_{1} (t) = [s_{l, ∥} (t) + n_{l, I, ∥} (t) + j n_{l, Q, ∥} (t)] u_{∥} + [n_{l, I, ⊥} (t) + j n_{l, Q, ⊥} (t)] u ⊥

where s _l,‖(t), n _l,‖(t) and n _l,⊥(t) are the ‖ and ⊥ components of s_l(t) and n_l(t). Hereinafter, the subscript l is dropped for the sake of notation simplicity, as the analytical derivations are performed with the LPE of the signals. As the OFDM signal field is completely polarized on the parallel direction, the subscript ‖ in the signal description is also dropped.

The derivation of the mean of the I and Q electrical components of the k-th subcarrier of the γ-th OFDM symbol at equalizer output from the field at OF output is performed in Appendix A. The mean of those components can be written, respectively, as:

\begin{array}{c} m_{e, I}^{(γ)} [k] = ℜ {E [s_{e}^{(γ)} [k]]} & m_{e, Q}^{(γ)} [k] = ℑ {E [s_{e}^{(γ)} [k]]} \end{array}

where ℑ{z} is the imaginary part of z and E[s ^(γ) _e[k]] is the mean of the signal at equalizer output given by Eq. 17 in Appendix A.

The variance of the k-th subcarrier of the γ-th OFDM symbol for the I and Q components at the FFT block output is also derived in Appendix A and it can be written as a function of the signal-noise beat terms variance and the noise-noise beat terms variance of each polarization:

{(σ_{FFT, (I, Q)}^{(γ)} [k])}^{2} = \sum_{y = 1}^{4} {(σ_{s - ASE, (I, Q), y}^{(γ)} [k])}^{2} + {(σ_{ASE - ASE, ∥, (I, Q), y}^{(γ)} [k])}^{2} + {(σ_{ASE - ASE, ⊥, (I, Q), y}^{(γ)} [k])}^{2}

where the first and the second subscripts in (I;Q) should be used for either the I or the Q component of each subcarrier. Each one of the four signal-noise beat terms are the variances of the beats between the I and Q components of the signal and the noise fields while the four noise-noise beat terms are the variances of the beats between the I and Q components of the noise field in each polarization direction. As shown in Appendix B, the variance of each signal-noise beat term of the k-th subcarrier of the γ-th OFDM symbol at FFT output is given by:

{(σ_{s - ASE, (I, Q), y}^{(γ)} [k])}^{2} = \frac{R_{λ}^{2}}{4} \int_{- \infty}^{+ \infty} S_{y} (f_{1}) {\frac{\exp (j 2 π f_{RF} t_{0}^{(γ)})}{2} H_{1, m, (I, Q)}^{(γ)} (f_{1} + f_{RF}, k) +

where the function H ^(γ) _i,m,(I,Q)(f,k) represents the effects of the different phases of the local oscillator in the I and Q branches of the OFDM receiver and the bandwidth limitation imposed by the electrical low-pass filter (LPF) and the FFT block on the signal-noise beat terms. H ^(γ) _i,m,(I,Q)(f,k) is defined by Eq. 33 in Appendix B. t ^(γ) ₀ is the first sampling time instant of the γ-th OFDM symbol (see Appendix A) and S_y(f) is the PSD (or cross-PSD, depending on which signal-noise beat term is being evaluated) of the I and the Q noise components at PIN input.

Considering the same PSD for the I and Q noise components of each polarization direction and the same PSD for the noise in each polarization direction, we can drop the noise-noise beat terms dependence on the polarization in Eq. 3. As shown in Appendix C, the variance of each noise-noise beat term of each subcarrier at FFT block output is given by:

{(σ_{ASE - ASE, (I, Q), y}^{(γ)} [k])}^{2} = \frac{1}{8} \int_{- \infty}^{+ \infty} [S_{y} (f_{1}) * S_{y} (f_{1})] {∣ H_{eq, (I, Q)}^{(γ)} (f_{1}, k) ∣}^{2} d f_{1}

where H ^(γ) _eq,(I,Q)(f ₁,k) represents the equivalent transfer function of the OFDM receiver and describes the influence of the phase of the local oscillator and the bandwidth limitation imposed by the electrical LPF and the FFT block on the noise-noise beat term.

The variance of the I and Q components of each subcarrier at equalizer output can be expressed in terms of the variance of the signal at FFT output (as shown in Appendix D) as:

{(σ_{e, (I, Q)}^{(γ)} [k])}^{2} = {(A_{e}^{(γ)} [k])}^{2} {(σ_{FFT, (I, Q)}^{(γ)} [k]))}^{2}

where A ^(γ) _e [k] is the amplitude response of the equalizer. Eq. 6 shows that the variance of the I and Q components of the signal at equalizer output is independent of the equalizer phase.

2.3. Performance evaluation

In order to take into account correctly the statistical distribution of the ISI and the ICI, the BER can be evaluated using an exhaustive Gaussian approach (GA) [12] for the I and Q components of each OFDM subcarrier at the equalizer output:

{BER}_{(I, Q)} [k] = \frac{1}{N_{s}} [\sum_{\underset{a_{(I, Q)}^{(γ)} = 0}{γ} = 1}^{N_{s}} Q (\frac{F_{(I, Q)} [k] - m_{e, (I, Q), 0}^{(γ)} [k]}{σ_{e, (I, Q)}^{(γ)} [k]}) + \sum_{\underset{a_{(I, Q)}^{(γ)} = 1}{γ} = 1}^{N_{s}} Q (\frac{m_{e, (I, Q), 1}^{(γ)} [k] - F_{(I, Q)} - [k])}{σ_{e, (I, Q)}^{(γ)} [k]})]

where m ^(γ) _e,(I,Q),0[k] and m ^(γ) _e,(I,Q),1[k] are the mean and σ ^(γ) _e,(I,Q)[k] is the standard deviation of the k-th subcarrier of the γ-th OFDM symbol at the equalizer output when the bit transmitted in the (I,Q) component, a ^(γ) _(I,Q), is 0 or 1, respectively, and F _(I,Q)[k] is the decision threshold level. Although Eq. 7 considers binary phase shift keying (BPSK) or quadrature phase shift keying (QPSK), it can be easily generalized to other more efficient mappings. Notice that k∈Ω, where Ω is the subset of the subcarriers indexes corresponding only to the information subcarriers, and 1≤γ≤N_s, γ∈ℕ, where N_s is the number of simulated OFDM symbols.

If BPSK symbol mapping is used (with symbols j and -j), the means and the standard deviations considered in Eq. 7 are relative to the Q component only. Instead, if QPSK is used and assuming that the I and Q signal components at the equalizer output are uncorrelated and that Gray mapping is used, the BER of each OFDM subcarrier is given by $BER [k] = \frac{1}{2} [1 - (1 - {BER}_{I} [k]) (1 - {BER}_{Q} [k])]$ , where BER_I [k] and BER_Q[k] are the BER of the I and Q components of each subcarrier at the equalizer output given by Eq. 7. The overall BER is evaluated averaging the BER of each subcarrier over all information subcarriers, N_i:

BER = \sum_{\underset{(k \in Ω)}{k = 1}}^{N} \times \frac{BER [k]}{N_{i}}

3. Results and analysis for OFDM-UWB radio signals

In this section, the accuracy of the SAGA is assessed and discussed considering the transmission of OFDM-UWB radio signals [13]. The OFDM-UWB baseband signal is composed of 128 subcarriers: 100 are used as information subcarriers, 12 are used as pilot subcarriers, 10 are used as guard subcarriers, and 6 are used as null subcarriers. The OFDM-UWB signal bandwidth is 528 MHz and the time interval of each OFDM-UWB symbol is 312.5 ns. The I and Q components of the generated OFDM-UWB signal are filtered by 6^th order Bessel LPFs with a -3 dB bandwidth of 400 MHz to reduce the aliasing components. The electro-optic conversion is modeled by a Mach-Zehnder modulator (MZM) biased at quadrature point and the analysis performed considers OFs tuned to the optical carrier. Fig. 2(a) shows the spectra of the double sideband OFDM-UWB signal with f_RF=3.4 GHz applied to MZM arms and at MZM output.

Null phases for the I and Q branches of the OFDM receiver are considered (ϕ ₁=ϕ ₂=0). The results are obtained in back-to-back, with R _λ=1 A/W. In almost all the results presented, BPSK symbol mapping is used. In other cases, an explicit reference to the symbol mapping used is presented. The transfer function of the equalizer is estimated by using the information provided by the channel distortion induced on the OFDM-UWB pilots subcarriers (spread along the OFDM signal band). A linear regression method is implemented to estimate the equalizer transfer function along the band of the transmitted subcarriers.

3.1. Probability density function of the signal at equalizer output

In order to assess the accuracy of the proposed GA, the probability density function (PDF) of the I and Q components of each subcarrier at the equalizer output is evaluated by MC simulation and compared with the one provided by the GA. 32 OFDM-UWB symbols are considered in numerical results and f_RF=3.4 GHz is used to generate the OFDM-UWB radio signal. Two 6^th order Bessel LPFs with a -3 dB bandwidth of 400 MHz are used at the OFDM-UWB electrical receiver. The OF is modeled by a 2^nd order super-Gaussian filter with -3 dB bandwidth of 30 GHz. The estimation of the PDF for each subcarrier is performed over 3.7×10⁶ noise runs.

Fig. 2(b) and 2(c) show the PDFs of the Q component of the signal at the equalizer output obtained by MC and by the GA, for modulation indexes of 19% and 64%, respectively, and for an optical signal-to-noise ratio (OSNR), defined in a reference bandwidth of 0.1 nm, of 7 dB. The modulation index is defined as m=V_RMS/V_b, where V_RMS is the root-mean-square (rms) voltage of the OFDM-UWB signal applied to the MZM arms and V_b is the MZM bias point.

Fig. 2. a) Spectra of the OFDM-UWB radio signal applied to MZM arms and at the MZM output, with ν0 the optical carrier frequency. PDF of the Q component of two subcarriers of the signal at the equalizer output for a) a modulation index of 19% and b) a modulation index of 64%. PDF obtained by using the GA (lines) and MC simulation (marks).

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Fig. 2(b) and 2(c) show the PDFs of two subcarriers of the first OFDM-UWB received symbol that carry a negative and a positive BPSK symbol. The PDF of the OFDM-UWB signal is well approximated by the proposed GA. Particularly, a very good accuracy can be observed in the tail regions for both modulation indexes, confirming that the statistics of the signal at equalizer output is adequately modeled by a Gaussian distribution.

3.2. Mean and variance of the signal at equalizer output

In this section, the mean and the variance of each subcarrier at the equalizer output is evaluated by MC simulation and compared with the mean and the variance given by the GA. The OFDM-UWB signal is generated with the same features as the ones described in the previous section and 2×10⁴ runs are used to estimate the mean and the variance of each subcarrier. The results are evaluated for the first OFDM-UWB symbol and for the OFDM-UWB signal located into the UWB sub-band 1 (f_RF=3.4 GHz) or the UWB sub-band 14 (f_RF=10.3 GHz).

Fig. 3. a) Mean of the Q component of the first OFDM symbol at equalizer output as a function of the transmitted information subcarrier. Results obtained by using the GA (×) and MC simulation (circles). b) Variance contributions to the total variance of the Q component of the first OFDM symbol at the equalizer output as a function of the transmitted information subcarrier. Results obtained by using the GA (lines) and MC simulation (marks).

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Fig. 3(a) shows the mean of the Q component of the first OFDM symbol at the equalizer output as a function of the information subcarriers index. The subcarriers with indexes from 1 to 50 correspond to subcarriers transmitted in the frequencies below f_RF while subcarriers with indexes from 51 to 100 correspond to subcarriers transmitted in frequencies above f_RF (indexes 1 and 100 correspond to the information subcarriers transmitted on the edges of the OFDM spectrum). An excellent agreement between the results obtained by MC and by the GA can be observed, confirming that the mean at the equalizer output is well described by Eq. 2. Fig. 3(b) depicts the variance of the first OFDM symbol and the different beat terms contributions at the equalizer output obtained by MC simulation and by the GA as a function of the information subcarriers. In the GA, the noise-noise beat term (in both polarizations) at equalizer output is obtained by using Eq. 3, 5 and 6. The optical carrier-noise beat term is obtained by switching-off the voltage signal applied to the MZM arms and by subtracting the noise-noise beat contributions from the total variance. The comparison between the MC and the GA results shows that the beat terms contributions to the total variance of each subcarrier at equalizer output are very well estimated by the closed-form expressions given by the GA. Fig. 3(b) shows also that the main contribution to the total variance of each subcarrier comes from the optical carrier-noise beat term. Fig. 3(b) does not show the OFDM signal-noise beat as its contribution to the total variance is negligible.

Fig. 3(b) shows that the variance at the equalizer output is higher for the subcarriers with lower and higher indexes (subcarriers located on the edges of the OFDM-UWB spectrum). The filters shape reduces the amplitude of the OFDM subcarriers located at the edges of the spectrum requiring an higher amplitude response of the equalizer transfer function on those frequencies. Therefore, though lower noise PSD is observed for the subcarriers located on the edges of the OFDM-UWB spectrum at the equalizer input due to the non-rectangular frequency response of the optical and, mainly, of the electrical filters, the noise PSD at the equalizer output is higher for those subcarriers leading to lower signal-to-noise ratio. This is in agreement with the experimental results shown in [5]. The results of Fig. 3(b) are different from the ones presented in [10] (the PSD of the noise decreases with the increase of the subcarrier frequency) as, in that case, the noise PSD is affected only by the optical filter and the influence of the equalizer transfer function on the noise PSD has not been considered.

Fig. 4. Variance of the Q component of the first OFDM symbol at the equalizer output as a function of the transmitted information subcarrier for a) different types of LPF and modulation indexes values and b) different types of optical filters. Variance obtained by the GA (lines) and the MC simulation (marks).

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Fig. 4(a) shows the variance of the Q component of the first OFDM symbol at the equalizer output as a function of the transmitted information subcarrier considering the OF modeled by a rectangular filter with a -3dB bandwidth of 30 GHz and OSNR=7 dB. Fig. 4(a) shows that the influence of the electrical LPF of the OFDM-UWB receiver on the variance of each subcarrier at the equalizer output given by Eq. 6 is correctly taken into account. In addition, Fig. 4(a) shows also that the variance changes considerably from subcarrier to subcarrier when different types of LPF are used. This change is due to the combined effect of two main features: i) different LPF shapes lead to different gains of each OFDM subcarrier. These gains are compensated by the equalizer transfer function affecting the noise variance at the equalizer output, as indicated by Eq. 6. ii) As shown in Appendix B, the noise power at the equalizer output is imposed by an equivalent filter given by the magnitude of the LPF transfer function and by the magnitude of the equivalent FFT transfer function, T ^(γ) _(1,2)(f,k).

Fig. 4(b) shows results similar to Fig. 4(a), but considering different types of OFs and 6^th order Bessel LPFs with a -3 dB bandwidth of 400 MHz at the electrical receiver. These results show that the impact of the OF shape on the variance of each subcarrier at the equalizer output is also correctly described by Eq. 6. The different variance levels obtained for the different OF types is due the influence of the OF shape on the noise power and on the subcarriers amplitude.

3.3. BER

Fig. 5. BER at the equalizer output as a function of a) the information subcarriers and b) the modulation index. Results obtained by using the GA (lines) and MC simulation (marks).

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Fig. 5(a) depicts the BER evaluated by MC and by the GA as a function of the information subcarriers for different optical and electrical filters. In MC simulation, the BER is computed when 100 errors occur in the subcarrier with better performance. The BER changes considerably from subcarrier to subcarrier mainly due to the dependence of the variance of each subcarrier at equalizer output on the amplitude of the frequency response of the optical and electrical filters. As shown in [5], the subcarriers more affected by the amplitude reduction induced by the channel response present higher BER levels. Fig. 5(b) shows the BER obtained by MC and by Eq. 8, as a function of the modulation index for BPSK and QPSK symbol mappings. 2nd order super-Gaussian OF with a -3 dB bandwidth of 30 GHz and 6^th order Bessel LPFs with a -3 dB bandwidth of 400 MHz are considered. The results are presented for two different OSNR values. Fig. 5(b) indicates that the SAGA provides excellent BER estimates for all modulation indexes, i. e., in case of performance dominantly impaired by noise or by interference, or simultaneously by both effects.

4. Results and analysis of OFDM signals proposed for long-haul networks

In this section, the accuracy of the SAGA is studied for two different OFDM transmitters (both using DD in the receiver) recently proposed for long-haul networks: the RF tone assisted gapped OFDM system (referred henceforth as A-OFDM transmitter) [14] and ii) the OFDM system proposed in [15] (referred henceforth as B-OFDM transmitter).

QPSK symbol mapping and 10 Gbit/s OFDM signals are generated in both transmitters. A-OFDM and B-OFDM transmitters use 128 and 512 data subcarriers, respectively, with 2-times oversampling used to obtain brick-wall spectra. The electrical OFDM signal is converted to the optical domain by a linear IQ modulator and back-to-back operation with R _λ=1 A/W are considered in the analysis. It should be stressed that SSB-OFDM spectra are obtained at the IQ modulator output in both implementations. The receiver used for both systems is the one presented in Fig. 1. Rectangular optical and electrical filters with a bandwidth of 15 GHz and 12 GHz, respectively, are considered in the receiver of the system employing the A-OFDM transmitter, while the receiver of the system employing the B-OFDM transmitter uses rectangular optical and electrical filters with a bandwidth of 25 GHz and 3 GHz. In both cases, the OF is centered at ν ₀=193.1 THz. OFDM signals with 5 GHz bandwidth (centered around 7.5 GHz) at the PIN output are obtained in both configurations with a 5 GHz gap to accommodate for the distortion induced by the subcarriers-subcarriers beat term [14, 15].

For the system employing the B-OFDM transmitter, the down conversion process at the electrical receiver is performed using f_RF=7.5 GHz. For the A-OFDM transmitter, the down conversion process is performed using two different approaches: i) using f_RF=5 GHz and ii) using an FFT implemented over twice the number of points used in the IFFT of the B-OFDM transmitter. The latter approach avoids the I and Q down conversion process presented in Fig. 1. In the proposed GA, this is equivalent to set f_RF=0. The equalizer transfer function is estimated using training symbols, and in both systems, the BER given by the GA and DEC is evaluated as a function of the carrier-to-signal power ratio (CSPR) considering ϕ ₁=ϕ ₂=0.

Fig. 6. a) Spectra of the OFDM signal at the IQ modulator output (I and III) and at the PIN output (II and IV) considering the A-OFDM and the B-OFDM transmitters. b) BER at the equalizer output as a function of CSPR. Results obtained by using the GA (lines) and MC simulation (marks), with the approach i) of OFDM-A (diamond marks and dashed lines) and the approach ii) of OFDM-A (square marks and dashed-dotted lines).

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Fig. 6(a) depicts the spectra of the OFDM signals at the IQ modulator output (I and III) and at the PIN output (II and IV) for the A-OFDM and B-OFDM transmitters. The spectra at the IQ modulator output confirm that both transmitters generate SSB-OFDM signals while the spectra at the PIN output show the unwanted subcarriers-subcarriers components falling close to DC and the useful OFDM signal located around 7.5 GHz [14, 15]. Fig. 6(b) shows the BER obtained using Eq. 8 as a function of the CSPR for two OSNR levels and for systems employing the A-OFDM and B-OFDM transmitters. The comparison between the results given by the GA and DEC clearly shows that the system performance is very well estimated by the GA, independently of the OFDM transmitter considered. Furthermore, Fig. 6(b) shows that for low OSNR levels the optimum CSPR is around 0 dB, as presented in [16].

5. Conclusion

A SAGA for each subcarrier of each OFDM symbol at the equalizer output, that allows evaluating the BER of each OFDM subcarrier in optically pre-amplified and DD receivers has been proposed. The approach is valid for arbitrary OFDM transmitters as long as the OFDM signal can be demodulated by the considered receiver structure. Closed-form analytical expressions for the mean and variance of the signal-noise and noise-noise beat terms of each subcarrier at equalizer output have been derived. These expressions take into account the distortions induced by the frequency response of the optical and electrical filters, the different phases of the local oscillator in the I and Q branches of the OFDM receiver and the equalizer transfer function.

The results obtained by the proposed method have shown excellent agreement with MC estimates for OFDM-UWB radio signals used in access networks and for two SSB-OFDM signals proposed for long-haul systems.

Acknowledgments

The work of Tiago Alves was supported by Fundação para a Ciência e a Tecnologia from Portugal under contract SFRH/BD/29871/2006 and the projects TURBO-PTDC/EEA-TEL/104358/2008 and SAMURAI-PTDC/EEA-TEL/108165/2008. This work was also supported in part by the UCELLS-FP7-IST-1-216785 project.

A. Mean and variance of the signal at the equalizer output

Considering a quadratic characteristic for the PIN, the current at the PIN output is given by:

s_{PIN} (t) = R_{λ} e_{l} (t) \cdot e_{l}^{*} (t) = R_{λ} {{∣ s (t) ∣}^{2} + {2 s}_{I} (t) n_{I, ∥} (t) + {2 s}_{Q} (t) n_{Q, ∥} (t) + n_{I, ∥}^{2} (t) + n_{Q, ∥}^{2} (t) +

with s_I(t)=ℜ{s(t)}, s_Q(t)=ℑ{s(t)} and |s(t)|²=s ² _I(t)+s ² _Q(t). Considering the scheme of the OFDM receiver depicted in Fig. 1, the current at LPF output in the I and Q branches of the electrical OFDM receiver can be expressed as:

s_{LPF, I} (t) = [\frac{s_{PIN} (t)}{2} \cos (2 π f_{R F^{t}} + ϕ_{1})] * h_{r} (t) = \frac{R_{λ}}{2} \int_{- \infty}^{+ \infty} [{∣ s (τ) ∣}^{2} \times + {2 s}_{I} (τ) n_{I, ∥} (τ) +

where h_r(t) is the impulse response of the LPF of the electrical receiver and the 1/2 factor is due to the current division by the two branches of the electrical receiver.

Prior to the FFT block, the signal is sampled and the guard time is removed. Thus, the continuous time, t, is replaced by t ^(γ) ₀+(i-1)T_c, where t ^(γ) ₀=t ₀+(γ-1)T is the first sampling time instant of the γ-th OFDM symbol, T is the OFDM symbol time, T_c=(T-T_g)/N is the chip time and T_g is the guard time. The index i stands for the i-th chip and thus, 1≤i≤N, i∈ℕ. The signal at FFT block input can be expressed as:

s_{LPF} (t_{0}^{(γ)} + (i - 1) T_{c}) = \int_{- \infty}^{+ \infty} \frac{s_{PIN} (τ)}{2} [\cos (2 π f_{RF} τ + ϕ_{1}) - j \sin (2 π f_{RF} τ + ϕ_{2})] \times

Hereinafter, we define s ^(γ) _LPF[i]=s_LPF(t ^(γ) ₀+(i-1)T_c) for the sake of notation simplicity. The signal at FFT block output corresponding to the k-th subcarrier of the γ-th symbol of the received signal can then be written as:

s_{FFT}^{(γ)} [k] = s_{FFT, I}^{(γ)} [k] + j s_{FFT, Q}^{(γ)} [k] = \sum_{i = 1}^{N} s_{LPF}^{(γ)} [i] \exp [- j W_{N} (k, i)]

where W_N(k,i)=2π(k-1)(i-1)/N. From Eq. 12, the I and Q components of the signal at the FFT output are given by:

s_{FFT, (I, Q)}^{(γ)} [k] = \frac{1}{2} \sum_{i = 1}^{N} \int_{- \infty}^{+ \infty} s_{PIN} (τ) f_{(I, Q)} (τ, k, i) h_{r} (t_{0}^{(γ)} + (i - 1) T_{c} - τ) dτ

where

f_{I} (τ, k, i) = \cos (2 π f_{RF} τ + ϕ_{1}) \cos [W_{N} (k, i)] - \sin (2 π f_{RF} τ + ϕ_{2}) \sin [W_{N} (k, i)]

f_{Q} (τ, k, i) = - \cos (2 π f_{RF} τ + ϕ_{1}) \sin [W_{N} (k, i)] - \sin (2 π f_{RF} τ + ϕ_{2}) \cos [W_{N} (k, i)]

From Eq. 12 and by evaluating the mean of the current at PIN output using Eq. 9, the mean of the signal at FFT block output is given by:

E [s_{FFT}^{(γ)} [k]] = \frac{R_{λ}}{2} \sum_{i = 1}^{N} {{{∣ s (t_{0}^{(γ)} + (i - 1) T_{c}) ∣}^{2} [\cos (2 π f_{RF} (t_{0}^{(γ)} + (i - 1) T_{c}) + ϕ_{1}) -

where P_n is the optical noise power at the OF output. Eq. 16 shows that if the LPF bandwidth is narrow when compared with the RF carrier frequency the contribution of the noise-noise beat term to the mean of the signal at FFT block output can be neglected.

Considering the equalizer transfer function of the k-th subcarrier of the γ-th OFDM symbol, H ^(γ) _e[k]=A ^(γ) _e[k]exp(-jΦ^(γ) _e [k]), the mean of the signal at the equalizer output can be expressed as:

E [s_{e}^{(γ)} [k]] = E [s_{FFT}^{(γ)} [k]] A_{e}^{(γ)} [k] \exp (- j Φ_{e}^{(γ)} [k])

Hence, the mean of the I and Q components of each subcarrier at equalizer output are given by:

m_{e, I}^{(γ)} [k] = ℜ {E [s_{e}^{(γ)} [k]]} m_{e, Q}^{(γ)} [k] = ℑ {E [s_{e}^{(γ)} [k]]}

The variance of the k-th subcarrier of the γ-th OFDM symbol for the I and Q components at the FFT block output can be evaluated as:

{(σ_{FFT, (I, Q)}^{(γ)} [k])}^{2} = E [{(s_{FFT, (I, Q)}^{(γ)} [k])}^{2}] - {E [s_{FFT, (I, Q)}^{(γ)} [k]]}^{2}

From Eq. 16, the mean of the I and Q components of the k-th subcarrier of the γ-th OFDM symbol at FFT block output can be expressed as:

E [s_{FFT, (I, Q)}^{(γ)} [k]] = \frac{1}{2} \sum_{i = 1}^{N} \int_{- \infty}^{+ \infty} E [s_{PIN} (τ)] f_{(I, Q)} (τ, k, i) h_{r} (t_{0}^{(γ)} + (i - 1) T_{c} - τ) dτ

By using Eq. 9, the mean of the current at PIN output is given by:

E = [s_{PIN} (τ)] = R_{λ} {{∣ s (τ) ∣}^{2} + E [n_{I, ∥}^{2} (τ)] + E [n_{Q, ∥}^{2} (τ)] + E [n_{I, ⊥}^{2} (τ)] + E [n_{Q, ⊥}^{2} (τ)]}

Considering the statistical independence of the noise over both polarization directions, i. e., E{n _x,‖ n _y,⊥]=E[n _x,‖] E[n _y,⊥] where x and y stands for the I and Q components, the odd order moments of Gaussian processes with zero mean are null and that the fourth order moments of Gaussian processes with zero mean are given by E[X ₁ X ₂ X ₃ X ₄]=E[X ₁ X ₂] E[X ₃ X ₄]+E[X ₁ X ₃] E[X ₂ X ₄]+E[X ₁ X ₄] E[X ₂ X ₃], the variance of the I and Q components of each subcarrier at FFT output can be written as:

{(σ_{FFT, (I, Q)}^{(γ)} [k])}^{2} = R_{λ}^{2} \sum_{i = 1}^{N} \sum_{n = 1}^{N} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} {\int_{- \infty}^{+ \infty} [s_{I} (τ_{1}) s_{I} (τ_{2}) + S_{n_{I, ∥} n_{I, ∥}} (f_{1}) + s_{I} (τ_{1}) \times

where $S_{n_{I, y} n_{I, y}} (f)$ , $S_{n_{Q, y} n_{Q, y}} (f)$ , $S_{n_{I, y} n_{Q, y}} (f)$ and $S_{n_{Q, y} n_{I, y}} (f)$ are the PSD of n_I,y(t) and n_Q,y(t), and the cross PSD between n_I,y(t) and n_Q,y(t) and between n_Q,y(t) and n_I,y(t), respectively, at the OF output. y represents either the ‖ or the ⊥ polarization direction. The derivation of Eq. 22 used the Wiener-Kintchine theorem. For tuned OF, the PSD of n_I(t) or n_Q(t) can be evaluated from the PSD at the OF input as $S_{n_{(I, Q)} n_{(I, Q)}, y} (f) = 0.5 \times S_{ASE, y} {∣ H_{o, l} (f) ∣}^{2}$ , while the cross PSD between both noise field components at the OF output is null. If detuned OFs are used, the relation between the noise PSD at the OF output and at the OF input is described in [17]. The functions g _(I,Q)(τ ₁,τ ₂,k, i,n) are given by:

g_{(I, Q)} (τ_{1}, τ_{2}, k, i, n) = f_{(I, Q)} (τ_{1}, k, i) f_{(I, Q)} (τ_{2}, k, n) h_{r} (t_{0}^{(γ)} + (i - 1) T_{c} - τ_{1}) \times

Eq. 22 can still be written as shown in Eq. 3.

B. Signal-noise beat contribution

From the variance of the I and Q components of the k-th subcarrier of the γ-th OFDM symbol at FFT block output given by Eq. 22 of Appendix A, the contribution of the I and Q signal-noise beat to the variance are given by:

{(σ_{s — ASE, (I, Q), y}^{(γ)} [k])}^{2} = R_{λ}^{2} \sum_{i = 1}^{N} \sum_{n = 1}^{N} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} s_{m} (τ_{1}) s_{p} (τ_{2}) S_{y} (f_{1}) \exp [j 2 π f_{1} (τ_{1} - τ_{2})] d f_{1} \times

where

s_{m} (τ_{1}) = {\begin{array}{c} s_{I} (τ_{1}) & if y = 1 or y = 2 \\ s_{Q} (τ_{1}) & if y = 3 or y = 4 \end{array} s_{p} (τ_{2}) = {\begin{array}{c} s_{I} (τ_{2}) & if y = 1 or y = 3 \\ s_{Q} (τ_{2}) & if y = 2 or y = 4 \end{array}

S_{y} (f_{1}) = {\begin{array}{c} S_{n_{I, ∥} n_{I, ∥}} (f_{1}) & if y = 1 \\ S_{n_{I, ∥} n_{Q, ∥}} (f_{1}) & if y = 2 \\ S_{n_{Q, ∥} n_{I, ∥}} (f_{1}) & if y = 3 \\ S_{n_{Q, ∥} n_{Q, ∥}} (f_{1}) & if y = 4 \end{array}

Using Eq. 23, ${(σ_{s - ASE, (I, Q), y}^{(γ)} [k])}^{2}$ can be written as:

{(σ_{s - ASE, (I, Q), y}^{(γ)} [k])}^{2} = R_{λ}^{2} \sum_{i = 1}^{N} \sum_{n = 1}^{N} \int_{- \infty}^{+ \infty} \dots \int_{- \infty}^{+ \infty} S_{y} (f_{1}) S_{m} (f_{2}) F_{(I, Q)} (f_{3}, k, i) H_{r} (f_{4}) S_{p} (f_{5}) \times

where S _(m,p)(f)=𝓕{s _(m,p)(τ)}, F _(I,Q)(f,k,i)=𝓕{f _(I,Q)(τ,k,i)}and H_r(f)=𝓕 {h_r(τ)g. From Eq. 14 and 15, F _(I,Q)(f,k,i) are given by:

F_{I} (f, k, i) = 𝓕 {f_{1} (f, k, i)} = \frac{1}{2} [A (k, i) \exp ({jϕ}_{1}) + jB (k, i) \exp ({jϕ}_{2})] δ (f - f_{RF}) +

F_{Q} (f, k, i) = 𝓕 {f_{Q} (f, k, i)} \frac{1}{2} [- B (k, i) \exp ({jϕ}_{1}) + jA (k, i) \exp ({jϕ}_{2})] δ (f - f_{RF}) +

where A(k,i)=cos[W_N(k,i)] and B(k,i)=sin [W_N(k,i)]. Thus, after performing some algebra, ${(σ_{s - ASE, (I, Q), y}^{(γ)} [k])}^{2}$ can be written as:

{(σ_{s - ASE, (I, Q), y}^{(γ)} [k])}^{2} \frac{R_{λ}^{2}}{4} \int_{- \infty}^{+ \infty} S_{y} (f_{1}) {R_{l, m, (I, Q)}^{(γ)} (- f_{1} - f_{RF}, k) + R_{2, m, (I, Q)}^{(γ)} (- f_{1} + f_{RF}, k)} \times

where (z)* represents the complex conjugate of z. The functions R ^(γ) _l,m,(I,Q)(-f ₁-f_RF,k) and R ^(γ) _2,m,(I,Q)(-f ₁+f_RF,k) are given by:

R_{1, m, (I, Q)}^{(γ)} (- f_{1} - f_{RF}, k) = \frac{\exp (j 2 π f_{RF} t_{0}^{(γ)})}{2} H_{1, m, (I, Q)}^{(γ)} (f_{1} + f_{RF}, k)

R_{2, m, (I, Q)}^{(γ)} (- f_{1} + f_{RF}, k) = \frac{\exp (- j 2 π f_{RF} t_{0}^{(γ)})}{2} H_{2, m, (I, Q)}^{(γ)} (f_{1} - f_{RF}, k)

where the function H ^(γ) _i,m,(I,Q)(f,k) is given by:

H_{i, m, (I, Q)}^{(γ)} (f, k) = 𝓕 {ν_{m}^{(γ)} (- t) g_{i, (I, Q)}^{(γ)} (t, k)} g_{i, (I, Q)}^{(γ)} (t, k) = 𝓕^{- 1} {G_{i, (I, Q)}^{(γ)} (f, k)}

ν_{m}^{(γ)} = 𝓕^{- 1} {V_{m}^{(γ)} (f)} = 𝓕^{- 1} {\exp (j 2 π {f t}_{0}^{(γ)}) s_{m} (f)}

The same procedure can be performed to obtain R ^(γ) _1,p,(I,Q)(-f ₁-f_RF,k) and R ^(γ) _2,p,(I,Q)(-f ₁+f_RF,k), respectively, by replacing m by p. The transfer functions G ^(γ) _i,(I,Q)(f,k) can be represented as the elements of a column vector, G ^(γ) _(I,Q), given by:

G_{(I, Q)}^{(γ)} = [\begin{array}{c} G_{1, (I, Q)}^{(γ)} (f, k) \\ G_{2, (I, Q)}^{(γ)} (f, k) \end{array}] = H_{r} (- f) P_{(I, Q)} T^{(γ)}

where the vector T ^(γ)=[T ^(γ) ₁(f,k) T ^(γ) ₂ (f,k)]^T represents the effects of the FFT operation and is given by:

T_{1}^{(γ)} (f, k) = \exp {j 2 π \frac{N - 1}{2} [{fT}_{c} + \frac{(k - 1)}{N}]} [\frac{\sin {2 π [{fT}_{c} + \frac{(k - 1)}{N}] \frac{N}{2}}}{\sin {π [{fT}_{c} + \frac{(k - 1)}{N}]}}]

T_{2}^{(γ)} (f, k) = \exp {j 2 π \frac{N - 1}{2} [{fT}_{c} - \frac{(k - 1)}{N}]} [\frac{\sin {2 π [{fT}_{c} - \frac{(k - 1)}{N}] \frac{N}{2}}}{\sin {π [{fT}_{c} - \frac{(k - 1)}{N}]}}]

P_I and P _Q are the I and Q matrixes that describe the mismatch of the phases of the local oscillator:

P_{I} = [\begin{array}{c} \exp ({jϕ}_{1}) + \exp ({jϕ}_{2}) & \exp ({jϕ}_{1}) - \exp ({jϕ}_{2}) \\ \exp (- {jϕ}_{1}) - \exp (- {jϕ}_{2}) & \exp (- {jϕ}_{1}) + \exp (- {jϕ}_{2}) \end{array}]

P_{Q} = j [\begin{array}{c} \exp ({jϕ}_{1}) + \exp ({jϕ}_{2}) & - \exp ({jϕ}_{1}) + \exp ({jϕ}_{2}) \\ \exp (- {jϕ}_{1}) - \exp (- {jϕ}_{2}) & - \exp (- {jϕ}_{1}) - \exp (- {jϕ}_{2}) \end{array}]

The P_I and P _Q matrixes allow assessing the orthogonality between the I and Q branches of the OFDM receiver and, consequently, the orthogonality between the OFDM subcarriers at FFT output. P_I and P _Q become diagonal matrixes if there is not phase difference between the I and Q branches (ϕ ₁=ϕ ₂).

The transfer functions G ^(γ) _i,(I,Q)(f,k) describe the effects of the different phases of the local oscillator in the I and Q branches, of the electrical filtering and of the FFT operation of the OFDM receiver on the signal-noise beat terms. It should be stressed that Eq. 35, 36 and 37 indicate that the noise at the equalizer output is limited by the LPF and also by the FFT block due to the magnitude response of terms T ^(γ) ₁(f,k) and T ^(γ) ₂(f,k).

Substitution of Eq. 31 and 32 in Eq. 30 allow writing the signal-noise beat terms variance of the k-th subcarrier of the γ-th OFDM symbol at FFT block output as shown in Eq. 4.

C. Noise-noise beat contribution

From Eq. 3 and 22, the I and Q noise-noise beat contribution to the variance of the k-th subcarrier of the γ-th OFDM symbol at FFT output are given by:

{(σ_{ASE - ASE, (I, Q), y}^{(γ)} [k])}^{2} = \frac{R_{λ}^{2}}{2} \sum_{i = 1}^{N} \sum_{n = 1}^{N} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} S_{y} (f_{2}) S_{y} (f_{3}) \times

The simplification of Eq. 40 follows a similar procedure to the one used in Appendix B to simplify the signal-noise beat contribution. Each noise-noise beat term can be written as:

(σ_{ASE - ASE, (I, Q), y}^{(γ)} [k]) = \frac{1}{8} \int_{- \infty}^{+ \infty} [S_{y} (f_{1}) * S_{y} (f_{1})] {∣ H_{eq, (I, Q)}^{(γ)} (f_{1}, k) ∣}^{2} d f_{1}

where H ^(γ) _eq,(I,Q)(f ₁,k)=H ^(γ) _eq,1,(I,Q)(f ₁+f_RF,k)+H ^(γ) _eq,2,(I,Q)(f ₁-f_RF,k) and with:

[\begin{array}{c} H_{eq, 1, (I, Q)}^{(γ)} (f, k) \\ H_{eq, 2, (I, Q)}^{(γ)} (f, k) \end{array}] = \frac{H_{r} (f)}{2} \exp [j 2 π f t_{0}^{(γ)}] P_{(I, Q)} T^{(γ)}

P _(I,Q) and T ^(γ) have been already defined in Appendix B.

D. Variance at the equalizer output

The signal at equalizer output can be written as s ^(γ) e[k]=s ^(γ) _FFT [k]H ^(γ) _e[k]. Thus, the I and Q components of the each subcarrier at the equalizer output are given by:

s_{e, I}^{(γ)} [k] = A_{e}^{(γ)} [k] \cos (Φ_{e}^{(γ)} [k]) s_{FFT, I}^{(γ)} [k] + A_{e}^{(γ)} [k] \sin (Φ_{e}^{(γ)} [k]) s_{FFT, Q}^{(γ)} [k]

s_{e, Q}^{(γ)} [k] = A_{e}^{(γ)} [k] \cos (Φ_{e}^{(γ)} [k]) s_{FFT, Q}^{(γ)} [k] - A_{e}^{(γ)} [k] \sin (Φ_{e}^{(γ)} [k]) s_{FFT, I}^{(γ)} [k]

The variance of the I and Q components of each subcarrier can then be evaluated following a similar procedure to the one of Eq. 19. Assuming that the frequency dependency of the ASE noise PSD along the OFDM signal bandwidth is small and using the relation between the elements of the P_I and P _Q matrixes defined in Appendix B, it can be shown that the covariance between the I and Q components of each subcarrier at FFT output is null and thus, E[s ^(γ) _FFT,(I,Q)[k]s ^(γ) _FFT,(Q,I)[k]]=E[s ^(γ) _FFT,(I,Q)[k]]E[s ^(γ) _FFT,(Q,I)[k]]. Hence, the variance of the I and Q components of the k-th subcarrier of the γ-th OFDM symbol at the equalizer output can be written (α ^(γ) _e,(I,Q)[k])²=(A ^(γ) _e[k])² (σ ^(γ) _FFT,(I,Q)[k])² where we have used that (σ ^(γ) _FFT,I[k])²=(σ ^(γ) _FFT,Q[k])².

References and links

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Semi-analytical approach for performance evaluation of direct-detection OFDM optical communication systems

Abstract

1. Introduction

2. Theory

2.1. System modeling

2.2. Gaussian distribution characterization

2.3. Performance evaluation

3. Results and analysis for OFDM-UWB radio signals

3.1. Probability density function of the signal at equalizer output

3.2. Mean and variance of the signal at equalizer output

3.3. BER

4. Results and analysis of OFDM signals proposed for long-haul networks

5. Conclusion

Acknowledgments

A. Mean and variance of the signal at the equalizer output

B. Signal-noise beat contribution

C. Noise-noise beat contribution

D. Variance at the equalizer output

References and links

Cited By

Figures (6)

Equations (67)

Optics Express