Abstract

In this paper, we propose a simple frequency offset estimation (FOE) algorithm for high-speed coherent optical orthogonal frequency-division multiplexing (CO-OFDM) systems, which avoids the exhaustive search computations on the integer part of FOE by using a merit function. In time domain, the FOE algorithm can estimate all the frequency offset in the range of [-5GHz, +5GHz], for satisfying the practical application requirement, based on only one redesigned training symbol. In addition, we provide the theoretical analysis about the quantitative influences of the residual frequency offset (RFO) on CO-OFDM systems. In order to reduce the final estimation error, a new zero-overhead residual frequency offset estimation (RFOE) algorithm is also proposed in this paper. Finally, the feasibility and effectiveness of the proposed FOE and RFOE algorithms are demonstrated in a 464 Gbit/s polarization-division multiplexed (PDM) 16-ary quadrature amplitude modulation (QAM) CO-OFDM system by the simulation, and numerical results validate the proposed algorithms.

©2012 Optical Society of America

1. Introduction

As 100 Gb/s Ethernet (100 GbE) has become increasingly a commercial reality, the high-speed optical transmission beyond 100 Gb/s per-channel data rate is being actively researched for future optical transmission systems [13], likely employing 400 Gb/s and 1 Tb/s super channels. Coherent optical orthogonal frequency-division multiplexing (CO-OFDM) as one of the most promising technologies presents numerous flexibilities and advantages for the future high-speed transmission system. 1) High electrical and optical spectral-efficiency can be offered by the tight spectral structure of OFDM. 2) Only 10–20% oversampling is required by not filling the edge subcarriers [4,5], which alleviates the sampling rate need for ADC/DAC. 3) By taking advantage of fast Fourier transform (FFT) operation, OFDM signal can be easily processed in the large-scale to deal with larger channel dispersion and higher data rate [6]. 4) Each subcarrier can have different modulation format without any hardware change. Hence the channel data rate can be adjusted adaptively according to the optical channel condition [7].

However, the CO-OFDM signal is known to be high sensitive to the carrier frequency offset. The continuous phase change leads to serious inter-carrier interference (ICI) penalties. In order to compensate this impairment without using an optical phase-locked loop that has drawbacks in terms of higher cost and complexity, the digital frequency offset estimation (FOE) algorithm is essential to OFDM receivers [813]. The classic Schmidl algorithm [8,9] divides the frequency offset into the fraction and the integer part of the subcarrier spacing and estimates the two kinds of frequency offset respectively by using two training symbols (TSs). In theory, it has an infinite estimation range. However, the exhaustive search computations are required to estimation the integer part of frequency offset (IFO) by using a merit function after the FFT. Otherwise, since the IFO correction utilizes a cyclic sample shift in the frequency domain, the IFO corresponding to the cyclic prefix (CP) cannot be compensation, which results in attaching the different additional initial phase on the samples located in the different FFT window. For removing the residual phase effect on one-tap equalization, the extra computation has to be required. Another approach to estimation frequency offset is achieved by extracting the phase of a pilot subcarrier in the center of OFDM spectra [12,13], in which all the phase impairments induced by the laser linewidth and frequency offset can be estimated in time domain. However, the receiving filter cannot filter out the pilot subcarrier accurately without knowing the carrier frequency offset in practice. Therefore, additional coarse FOE should be worked before the pilot subcarrier method, as shown in [13]. Besides, in order to suppress the amplitude spontaneous emission (ASE) noise induced by the erbium doped fiber amplifiers (EDFA), the high pilot-to-signal power rate (PSR) must be required, which causes an obvious OSNR penalty.

In the actual CO-OFDM systems, the practical application requirement of the FOE algorithm will be satisfied as long as the estimation range is close to [-5GHz, +5GHz], since the commercially available tunable lasers may have a frequency accuracy within ±2.5GHz over the lifetime [14]. Based on the estimation requirement, we propose a simple and efficient FOE algorithm for the future high-speed CO-OFDM systems by using a redesigned training symbol, which is capable of estimating all the frequency offset in the time domain without the exhaustive search computation. In order to reduce the training overhead, only one training symbol is used for the OFDM symbol synchronization and the FOE. Subsequently, we quantitatively analyze the influences of the residual frequency offset (RFO) on CO-OFDM systems and propose a residual frequency offset estimation (RFOE) method, in which a pair of estimated channel transfer function vectors is employed, without other extra training overheads. By the simulation, the feasibility and effectiveness of the proposed FOE and RFOE algorithms are demonstrated with 464-Gbit/s PDM-16QAM-CO-OFDM signal.

2. Operation principle of frequency offset estimation

At the receiver, the output signal of the fiber is mapped from the optical field into the electrical domain by coherent detectors. The received electrical signal r(t) can be written as

r(t)=exp(j2πΔft+jϕ(t))(s(t)h(t)+w(t))
whereΔfandϕ(t)are the frequency and phase deviation between the transmitter laser and local oscillator respectively, s(t) is the transmitted OFDM baseband signal, h(t) is the channel impulse response function, w(t) is the ASE noise, represents the operator of convolution. After ADC, the nth sampler(n)is expressed as
r(n)=exp(j2πΔfnTs+jϕ(n))(s(n)h(n)+w(n))
where TS is the sampling period of ADC, Ts=1/fs. Here, the timing asynchronization is not considered. Then the sampling rate fS = NfSC, whereNis the size of FFT/IFFT and fSC is the subcarrier spacing.

Due to the basically time-invariant characteristic of the optical channel and the protection by CP, two identical neighboring samples will be suffered same damage after passing through the channel, except a phase difference between them which is caused by the carrier frequency offset. Based on above property, the Schmidl [9] use a training symbol with two identical halves in the time domain, as shown in Fig. 1(a) , to search the start of the OFDM frame as symbol synchronization. Here, the timing metric M(d) is calculated as

M(d)=|P(d)|2R(d)M(d)[0,1]
where P(d) is a correlate function,
P(d)=m=0N/21[r*(d+m)r(d+m+N/2)]
R(d) is a normalization factor of M(d) defined by
R(d)=(m=0N/21|r(d+m)|2)(m=0N/21|r(d+m+N/2)|2)
where d denotes a time index of the received symbols, and (.)* is the complex conjugate operation. The starting point d^is determined by the maximum point of M(d). Simultaneously, the △f can be estimated according to thep(d^). However, since the phase angle operation arg(.) returns the phase difference of two identical samples (πNf TS) in a value of [π,+π], only the fractional frequency offset ΔfFFO in the range of ±fSC can be estimated by

 figure: Fig. 1

Fig. 1 (a) The conventional and (b) the redesigned training symbol structures for OFDM symbol synchronization and FOE. CP: cyclic prefix.

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ΔfFFO=arg(p(d^))πfcs

Evidently, a shorter spacing distance of identical samples can increase the estimation range proportionately. To cope with the required estimation range of ±5GHz, the spacing distance k of a pair of identical samples must be satisfied

|2πk(±5GHz)Ts|π
where k should be an integer and smaller than fs /109. For further avoiding the extra training overhead, we plan to utilize a special training symbol to complete symbol synchronization and estimate FO in the full range. Here, Eq. (3) is also used for symbol synchronization, then the training symbol must satisfy the design requirement that the identical sample is repeated with the intervals of N/2 and k simultaneously. Therefore, the training symbol is divided into h identical blocks as shown in Fig. 1(b), and h can be calculated as
h=2ceil(log2Nmax(k))
where the ceil(.) represents the operator that rounds the input to the nearest integer greater than or equal to it, max(k) is the maximum of k calculated by Eq. (7). Then the estimated frequency offset△fest with a large theoretical estimation range (≥[-5GHz, +5GHz]) can be obtained as

Δfest=arg(n=0NNh1(r*(n)r(n+Nh)))hfs2πN

However, since the estimation accuracy is much more vulnerable to ASE noise with a smaller repeated space, the increase of estimation error is present by using Eq. (9). In order to improve the FOE accuracy, only the integer part of △fest is retained and combined with the △fFFO that is computed by Eq. (6), as

Δfest=integer(Δfestfsc)fcs+ΔfFFO
where integer(.) represents the operator that round the input to the closest integer. Considering that the miscalculate of Δfest will be prone to occur in the situation that the actual frequency offset is close to the integer multiple of fSC, the final estimation Δfestis given by

Δfest=integer(ΔfestΔfestfsc)fsc+Δfest

3. Residual frequency offset analysis and estimation

3.1 Effect of residual frequency offset

Due to in the presence of ASE noise, the FOE algorithm cannot estimate carrier frequency offset without any error. Therefore, the received digital signal after frequency offset compensation in time domain is expressed as

r(n)=r(n)exp(j2πΔfestnTs)=exp(j2πefscnTs+jϕ(n))(s(n)h(n)+w(n))
where efscdenotes the RFO (efsc = Δf-Δfestand the normalized RFO e is far less than 1). For the sake of simplicity, the phase effect induced by the laser linewidth-ϕ(n)is ignored in the following analyses. Subsequently, r'is converted from serial to parallel (S/P) for FFT operation after removing CPs. The nth sample in the ith OFDM symbol rin' is given by
rin=piexp(j2πen/N)(1Nk=0N1CikHikexp(j2πkn/N)+win)pi=exp(j2πe(i1)(N+Ncp)/N)
where pi is the initial phase of ith OFDM symbol induced by the RFO, Cik is the transmitted subcarrier, Hik is the channel transfer function in the frequency domain and Ncp is the number of CP. After FFT, the received sample for the kth subcarrier in the ith OFDM symbol can be calculated as
Cik=CikHikη0pi+Iik+WikIik=n=1,nkNCinHinηnkηm=1Nsin(π(m+e))sin(π(m+e)/N)exp(jπ(m+e)(11N))
where Iik is the ICI noise, ηm represents the ICI coefficient between two subcarriers with the space of m, Wis the additive white Gaussian noise (AWGN).

It can be seen that there are three kinds of effects related to RFO: η0, Iik, and pi. Since η0is just a fixed impairment, which can be easily compensated with Hik in the following channel estimation module, we mainly focus on the latter two impacts of Iik and pi. The first one Iik is a random varying interference, which can be assumed as AWGN [15]. The variance of the interference Iik can be calculated by

σICI2=σC2n=1,nkN|η|2nk
where σc2is the variances of the transmitted information signal. The effective signal-noise-ratio (SNR) is given by
SNR'=σc2σICI2+σw2=(n=1,nkN|η|2nk+SNR1)1
where SNR = σc2σw2is the original channel SNR without the effect of Iik. σw2is the variances of the ASE noise. For a 16QAM signal, the analytical form of the BER is given by [16]

BER=38erfc(SNR10)

In Fig. 2(a) , the BER performance of 128-subcarrier 16QAM modulated OFDM system in the presence of RFO is computed according to Eq. (16) and Eq. (17). It can be seen that the normalized RFO-e should be not more than 0.04 for a 1.5 dB SNR penalty at BER of 1E-3.

 figure: Fig. 2

Fig. 2 (a) The ICI influence induced by the RFO, (b) the variation of signal amplitude as a function of the RFO.

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The second one pi is an accumulated phase impairment, which varies on the per-symbol basis. After the channel estimation by using pilot carriers or pilot symbols, it will cause a random signal amplitude variation. As known to all, the channel transfer function Hik can be estimated in the frequency domain by

H^ik=CikCik=Hikη0pi+Iik+WikCik

Since the optical channel changes on the timescale of milliseconds [15], which is considered to be invariant (Hik = Hk) in an OFDM frame that contains a large number of OFDM symbols, an average operation is usually performed to suppress the AWGN effects and obtain a better channel estimate, as

H^k=1Li=1LH^ik=1Li=1LHkη0pi+Iik+WikCikHkη01Li=1Lpi
whereH^kis the estimated channel transfer function of the kth subcarrier of the symbol which is located in a same OFDM frame, and L is the number of training symbols for the channel estimate. Next, substituting pi of Eq. (14) into Eq. (19), we derive

H^k=Hkη01Li=1Lexp(j2πe(i1)(N+Ncp)/N)=Hkη0sin(πe(N+Ncp)L/N)Lsin(πe(N+Ncp)/N)exp(jπe(L1)(N+Ncp)/N)

Then the data subcarrier signal Cik (i>L) after equalization without considering the effect of Iik has

C^ik=H^*kCik=Hk*η0*Lsin(πe(N+Ncp)/N)sin(πe(N+Ncp)L/N)exp(jπe(L1)(N+Ncp)/N)×(CikHkη0pi+Wik)=αξiCik+Wik
where αand ξidenote the amplitude and phase impairments of C^ikrespectively, as

α=Lsin(πe(N+Ncp)/N)sin(πe(N+Ncp)L/N),ξi=exp(j2πe(N+Ncp)(iL)/N)

Obviously,ξi as a phase impairment that is not changed with subcarrier, can be easily compensated by the following carrier phase recovery module [17]. However, the uncertainty of αwill cause some difficulties in signal decision. If PDM is applied, theαwill be more sensitive to RFO, since the accumulated phase difference induced by RFO is added double based on the shifted orthogonal training symbols by using the channel estimation method as depicted in [18]. Then Eq. (22) can be expressed as

α=Lsin(dπe(N+Ncp)/N)sin(dπe(N+Ncp)L/N)ξi=exp(j2πde(N+Ncp)(iL)/N),d={1,withoutPDM2,withPDM

Figure 2(b) shows the variation of signal amplitude αas a function of the normalized RFO in the conditions of N = 128, Ncp = 8 and L = 20. It can be seen that the large amplitude variation will be caused by a small RFO. Due to the uncertainty of RFO, it can hardly find an optimal decision threshold to achieve a stable BER performance in all the situations. Based on the above detailed theoretical analyses, the impact of RFO should not be underestimated.

3.2 Residual frequency offset estimation

Depending on the slow-varying characteristic of optical channel and the phase induced by RFO that varies on the per-symbol basis, the RFO therefore can be estimated according to a pair of estimated channel transfer function vectors without any extra training overheads. The block diagram of the proposed RFOE method is illustrated in Fig. 3 .

 figure: Fig. 3

Fig. 3 Schematic illustration of RFO estimation algorithm.

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For the sake of simplicity, only the phase-related information is considered in the following calculation process. Firstly, the phase difference between two estimated channel transfer functions of the kth subcarrier, which are located in the ith and (i + m)th OFDM symbols respectively, is calculated by

H^(i+m)kH^ik*=exp((j2πde(m+i1)(N+Ncp)/N)(j2πde(i1)(N+Ncp)/N)+θn)=exp(j2πdem(N+Ncp)/N+θn)
where m is the symbol distance of two estimated channel transfer functions, which is an integer and bigger than 1, θndenotes all the AWGN phase. Obviously, the calculated phase difference in Eq. (24) is independent of the subcarriers and OFDM symbols. In order to wipe out the (zero mean) Gaussian noise and obtain a better estimate result, a summation operation that essentially performs an average operation in phase is used

PE=k=1NH^ikH^(i+m)k*exp(j2πdem(N+Ncp)/N)

Then, the estimated RFO can be easily calculated as

e^fsc=arg(PE)2πdm(N+Ncp)Ts

Here, since the phase of PE without ambiguity is chosen from-πtoπ, the estimation range of e^ is limited by

-N2dm(N+Ncp)e^N2dm(N+Ncp)
in which m not only relates to the estimation range, but also influences the estimation accuracy of the proposed RFOE algorithm. It as an important parameter of RFOE will be investigated in the next simulation section particularly. Finally, the estimated value e^fsc is fed back to before FFT to wipe out the impact of RFO.

4. Simulations and discussions

4.1 Simulation system model

To investigate the performance of the proposed FOE and RFOE algorithms, the extensive simulations have been carried out in a 464 Gb/s PDM-16QAM-CO-OFDM system which is built by the VPI transmission Maker 8.5 and MATLAB softwares, as shown in Fig. 4(a) .

 figure: Fig. 4

Fig. 4 (a) Simulation setup of the 464Gbit/s PDM-OFDM-16QAM system, the schematics of the DSP in the transmitter (b) and receiver (c), (d) OFDM frame structure.

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In the transmitter, the DSP is performed to generate the OFDM baseband signals. The different steps of DSP for one polarization state are shown in Fig. 4(b). The pseudo-random bit sequence (PRBS) with a length of 215-1 as the transmitted data stream is mapped into 16QAM, which adds 7 pilots information used for phase estimation to fill the middle 107 subcarriers. The time domain signal is generated by an IFFT operation of size 128, and after parallel-to-serial conversion (P/S) 8 CPs as guard interval are inserted for eliminate ISI due to channel dispersion, resulting in an OFDM symbol size of 136. Forty-one training symbols are subsequently inserted at the beginning of each OFDM frame, which consists of 1 training symbol for OFDM symbol synchronization and FOE, 20 training symbols for channel estimation and 20 null symbols, as shown in Fig. 4(d). Each OFDM frame includes 41 training and 3000 OFDM payload symbols. Afterwards, the real and imaginary parts of OFDM signal are uploaded into the DAC operated at 80GSam/s to generate IQ analog signals, and fed into two optical Mach-Zehnder (MZ) modulators respectively as the I/Q modulator to generate a 16QAM-OFDM signal of one polarization state. Here, a laser with 100 kHz linewidth is employed at 1550nm. After a polarization beam combiner (PBC), the PDM 16AM-OFDM signal is obtained with 464 Gb/s ( = 80GSam/s*8*100/136*3000/3041) data rate and 625 MHz ( = 80GHz/128) subcarrier spacing respectively.

Fiber link compose of single model fiber (SMF), erbium doped fiber amplifier (EDFA) and optical filter. Here, optical signal-to-noise ratio (OSNR) can be adjusted by changing the noise figure of EDFA. After fiber propagation, the received signals are divided into two arbitrary states of polarization via a polarization beam splitter (PBS). Then, both states of polarization are mapped from the optical field into four electrical signals by utilizing the passive quadrature hybrid with balanced detectors. Next, the electrical signals were digitized by ADCs at 80Gsam/s with 8-bit of resolution and stored for DSP using MATLAB, as shown in Fig. 4(c). Firstly, an overlapped frequency domain equalizer [19] is employed, which effectively compensates the most of CD and reduce the CP overheads simultaneously. Taking into account the compensation error in practices, the CD about 100ps/nm is always remained after CD compensation, which adds 10ps of PMD to be further compensated by CP and channel estimation module. The nonlinear influence is not considered in this paper. Subsequently, the OFDM symbol synchronization, FOE, channel estimation, RFOE and phase recovery are carried out before the data is finally recovered.

4.2 Simulation results and analysis

First of all, in order to investigate the performance of OFDM symbol synchronization by using the redesigned training symbol (which includes 16 identical blocks according to the computation of Eq. (7) and Eq. (8)), the timing metric M(d) of Schmidl algorithm under different OSNR conditions is illustrated in Fig. 5 , with a frequency offset of 5GHz being considered. In the simulation, the OFDM signal is transmitted starting from the 51st symbol. As can be seen in Fig. 5, the timing metrics keep a plateau whose level is reduced by the deterioration of OSNR, but in all the OSNR conditions the plateaus are kept between the 51st and the 60th symbols. It demonstrate that the start of OFDM symbol (corresponding to the maximum value of M(d)) can be taken within a correct window to guarantee the data integrity for the FFT operation. Next, the redesigned training symbol will continue to be used for FOE. Figure 6 shows the absolute values of the normalized FOE error (with respect to subcarrier spacing fsc) as a function of frequency offset by using the proposed FOE algorithm under a poor OSNR condition (OSNR = 24dB). To verify the estimation results statistically, more than 500 sets of data are run for each measurement point. The mean normalized estimation error can keep about 0.02 and the maximum is no more than 0.06 with the frequency offset changed from −4.9GHz to 4.9GHz, which basically achieves our anticipated design objective to guarantee the requirement of the practical application availably.

 figure: Fig. 5

Fig. 5 Timing metric of Schmidl algorithm by using the redesigned training symbol with a frequency offset of 5GHz.

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 figure: Fig. 6

Fig. 6 Absolute values of normalized FOE error under [-4.9GHz, +4.9GHz] frequency offset with OSNR of 24dB.

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Consequently, let us analyze the impact of RFO on the performance of PDM-16QAM- OFDM system. Figure 7 shows the constellation diagrams of the demodulated OFDM signals for X-polarization with different RFO. To clearly observe the changes induced by the different RFO, a better OSNR (OSNR = 28dB) is chosen in this case. The constellation diagram without RFO is shown in Fig. 7 (a) as a reference, in which the bit error rate (BER) is 7.73E-5. Comparing the three constellation diagrams, we can find an obvious change on the signal amplitudes in the presence of RFO. For the situation of e = 0.01 and e = 0.015, the signal amplitudes are respectively increased 2.9dB and 7.1dB compared to that of without RFO. The simulation results in agreement with the theoretical analyses (refer to Fig. 2(b)). Otherwise, the spread of the constellation points increases dramatically also in the cases of the presence of RFO impairment (see Figs. 7(b), 7(c)), due to the increased ICI impairment induced by RFO. It causes a further deterioration of the BER performance. In above simulations, the BERs are degraded to 9.11E-3 and 0.135 when e = 0.01 and e = 0.015. Figure 8 depicts more the simulation results of BER performance as a function of the normalized RFO for 24dB, 26dB and 28dB of OSNR. As shown in Fig. 8, since the higher OSNR can make the dominant impairment convert from the ASE noise to the RFO more rapidly, the BER is more sensitive to the RFO in the higher OSNR. In order to achieve the absences of the noticeable RFO penalty, the normalized RFO should be controlled below 0.003, which is obviously difficult to the proposed FOE algorithm (see Fig. 6). Therefore, an additional effective RFOE algorithm should be required to provide enough estimation accuracy.

 figure: Fig. 7

Fig. 7 The output constellations of one polarization after phase recovery at OSNR = 28dB, (a) without RFO, (b) e = 0.01, and (c) e = 0.015.

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 figure: Fig. 8

Fig. 8 BER performance of 464Gbps PDM-OFDM-16QAM signal as a function of the normalized RFO for OSNR = 24dB, 26dB and 28dB.

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Figure 9 depicts the influence of algorithm parameter m (described in Section 3.2) on the performance of the proposed RFOE algorithm, where e denotes a normalized RFO that is required to be estimated and its maximum value is set to be 0.06 according with the maximum estimation error in Fig. 6. It can be seen that the higher estimation accuracy is obtained by the increase of m. However, with the increase of m the RFO, the estimation range is reduced correspondingly. In the conditions of e = 0.02, 0.04 and 0.06, the m cannot be exceed 3, 5 and 11 respectively, which is consistent with the analysis results calculated by Eq. (27). Obviously, there is a trade-off between estimation accuracy and range. Nevertheless, the final estimation error can be decreased to less than 0.003 as long as m ≥ 2. Referring to Fig. 8, the influence of final RFO (e < 0.003) can be ignored. In the following simulations, m is set to be 3 for RFOE.

 figure: Fig. 9

Fig. 9 The absolute normalized estimation error (mean value) versus algorithm parameter m for various e at OSNR = 26dB.

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Lastly, the final estimation error and the BER performance of the joint FOE scheme that combines the proposed FOE algorithm with the RFOE algorithm is investigated and compared with the classic Schmidl FOE algorithm. Note that in order to remove the CP related phase impairment inducing by IFO, in the situation of Schmidl algorithm being used, additional computation of compensation is carried out before the channel estimation. In Fig. 10(a) , the joint FOE scheme shows a significant advantage in the estimation accuracy comparing the Schmidl FOE algorithm. Its normalized estimation errors are almost kept within 0.002 in the all OSNR scenarios. By taking advantage of the excellent estimation performance, the joint FOE scheme can guarantee basically same BERs as that measured in the ideal FOE (see Fig. 10(b)). The required OSNR at a BER of 1E-3 for the proposed FOE algorithm with RFOE is about 26dB, which is approximately 2dB better than that of the Schmidl FOE method.

 figure: Fig. 10

Fig. 10 (a)The absolute normalized estimation error (mean value), and (b) BER performance as a function of the OSNR.

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5. Conclusion

A simple FOE algorithm based on a redesigned training symbol for the high-speed CO-OFDM systems was proposed, and its performance was numerically investigated at a 464-Gbit/s PDM-16QAM-CO-OFDM system. This algorithm can successfully track [-4.9GHz, 4.9GHz] frequency offset at the poor OSNR condition by avoiding an exhaustive search calculation. In order to further improve the FOE precision, a new zero-overhead RFOE algorithm, which can effectively reduce the final normalized estimation error to less than 0.003, was also proposed. By taking advantage of the excellent estimation performance, the joint FOE scheme that combines the proposed FOE and RFOE algorithms can almost remove all the influences of frequency offset in terms of the BER performance of the system. The required OSNR at a BER of 1E-3 for the joint FOE scheme is about 26dB, which is approximately 2dB better than that of the Schmidl FOE method.

Acknowledgments

This study is supported by the National Basic Research Program of China (2012CB315905), the National Natural Science Foundation of China (61172050, 61172048, 61100184, and 61173149), China Postdoctoral Science Foundation funded project, National Key Projects, and Beijing Science and Technology Program.

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References

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  1. R.W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14(4), 3–9 (2010).
    [Crossref]
  2. X. Liu, S. Chandrasekhar, B. Zhu, P. J. Winzer, A. H. Gnauck, and D. W. Peckham, “448-Gb/s reduced-guard-interval CO-OFDM transmission over 2000km of ultra-large-area fiber and five 80 Ghz-grid ROADMs,” J. Lightwave Technol. 29(4), 483–490 (2011).
  3. Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s per channel coherent optical OFDM transmission with subwavelength bandwidth access,” in in National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper PDPC1.
  4. E. Ip, A. P. Lau, D. J. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express 16(2), 753–791 (2008).
    [Crossref] [PubMed]
  5. W. Shieh, X. W. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking 7(2), 34–55 (2008).
  6. Y. Tang and W. Shieh, “Coherent optical OFDM transmission up to 1 Tb/s per channel,” J. Lightwave Technol. 27(16), 3511–3517 (2009).
    [Crossref]
  7. X. W. Yi, W. Shieh, and Y. Ma, “Phase noise effects on high spectral efficiency coherent optical OFDM transmission,” J. Lightwave Technol. 26(10), 1309–1316 (2008).
    [Crossref]
  8. T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
    [Crossref]
  9. Z. Zhang, W. Jiang, H. Zhou, Y. Liu, and J. Gao, “High accuracy frequency offset correction with adjustable acquisition range in OFDM systems,” IEEE Trans. Wirel. Comm. 4(1), 228–237 (2005).
    [Crossref]
  10. S.-H. Fan, J. Yu, D. Qian, and G.-K. Chang, “A fast and efficient frequency offset correction technique for coherent optical orthogonal frequency division multiplexing,” J. Lightwave Technol. 29(13), 1997–2004 (2011).
    [Crossref]
  11. Z. Zhang, K. P. Long, M. Zhao, and Y. N. Liu, “Joint frame synchronization and frequency offset estimation in OFDM systems,” IEEE Trans. Broadcast 51(3), 389–394 (2005).
    [Crossref]
  12. S. L. Jansen, I. Morita, T. C. W. Schenk, N. Takeda, and H. Tanaka, “Coherent optical 25.8-Gb/s OFDM transmission over 4160-km SSMF,” J. Lightwave Technol. 26(1), 6–15 (2008).
    [Crossref]
  13. F. Buchali, R. Dischler, M. Mayrock, X. Xiao, and Y. Tang, “Improved frequency offset correction in coherent optical OFDM systems,” in 34th European Conference on Optical Communication, 2008. ECOC 2008 (2008), Paper Mo.4.D.4.
  14. Optical Internetworking Forum, “Integrable Tunable Laser Assembly MSA,” OIF-ITLA-MSA-01.1 (Nov. 22, 2005).
  15. W. Shieh and I. Djordjevic, OFDM for Optical Communications (Elsevier, 2011).
  16. B. Aklar, Digital Communications: Fundamentals & Applications (Prentice-Hall, 2001).
  17. X. W. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photon. Technol. Lett. 19(12), 919–921 (2007).
    [Crossref]
  18. S. L. Jansen, I. Morita, T. C. Schenk, and H. Tanaka, “Long-haul transmission of 16 ×52.5 Gbit/s polarization-division-multiplexed OFDM enabled by MIMO processing (Invited),” J. Opt. Networking 7(2), 173–182 (2008).
    [Crossref]
  19. K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
    [Crossref]

2011 (2)

2010 (1)

R.W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14(4), 3–9 (2010).
[Crossref]

2009 (1)

2008 (6)

X. W. Yi, W. Shieh, and Y. Ma, “Phase noise effects on high spectral efficiency coherent optical OFDM transmission,” J. Lightwave Technol. 26(10), 1309–1316 (2008).
[Crossref]

E. Ip, A. P. Lau, D. J. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express 16(2), 753–791 (2008).
[Crossref] [PubMed]

W. Shieh, X. W. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking 7(2), 34–55 (2008).

S. L. Jansen, I. Morita, T. C. W. Schenk, N. Takeda, and H. Tanaka, “Coherent optical 25.8-Gb/s OFDM transmission over 4160-km SSMF,” J. Lightwave Technol. 26(1), 6–15 (2008).
[Crossref]

S. L. Jansen, I. Morita, T. C. Schenk, and H. Tanaka, “Long-haul transmission of 16 ×52.5 Gbit/s polarization-division-multiplexed OFDM enabled by MIMO processing (Invited),” J. Opt. Networking 7(2), 173–182 (2008).
[Crossref]

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[Crossref]

2007 (1)

X. W. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photon. Technol. Lett. 19(12), 919–921 (2007).
[Crossref]

2005 (2)

Z. Zhang, W. Jiang, H. Zhou, Y. Liu, and J. Gao, “High accuracy frequency offset correction with adjustable acquisition range in OFDM systems,” IEEE Trans. Wirel. Comm. 4(1), 228–237 (2005).
[Crossref]

Z. Zhang, K. P. Long, M. Zhao, and Y. N. Liu, “Joint frame synchronization and frequency offset estimation in OFDM systems,” IEEE Trans. Broadcast 51(3), 389–394 (2005).
[Crossref]

1997 (1)

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
[Crossref]

Barros, D. J.

Chandrasekhar, S.

Chang, G.-K.

Cox, D. C.

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
[Crossref]

Fan, S.-H.

Gao, J.

Z. Zhang, W. Jiang, H. Zhou, Y. Liu, and J. Gao, “High accuracy frequency offset correction with adjustable acquisition range in OFDM systems,” IEEE Trans. Wirel. Comm. 4(1), 228–237 (2005).
[Crossref]

Gnauck, A. H.

Ip, E.

Ishihara, K.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[Crossref]

Jansen, S. L.

S. L. Jansen, I. Morita, T. C. Schenk, and H. Tanaka, “Long-haul transmission of 16 ×52.5 Gbit/s polarization-division-multiplexed OFDM enabled by MIMO processing (Invited),” J. Opt. Networking 7(2), 173–182 (2008).
[Crossref]

S. L. Jansen, I. Morita, T. C. W. Schenk, N. Takeda, and H. Tanaka, “Coherent optical 25.8-Gb/s OFDM transmission over 4160-km SSMF,” J. Lightwave Technol. 26(1), 6–15 (2008).
[Crossref]

Jiang, W.

Z. Zhang, W. Jiang, H. Zhou, Y. Liu, and J. Gao, “High accuracy frequency offset correction with adjustable acquisition range in OFDM systems,” IEEE Trans. Wirel. Comm. 4(1), 228–237 (2005).
[Crossref]

Kahn, J. M.

Kobayashi, T.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[Crossref]

Kudo, R.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[Crossref]

Lau, A. P.

Liu, X.

Liu, Y.

Z. Zhang, W. Jiang, H. Zhou, Y. Liu, and J. Gao, “High accuracy frequency offset correction with adjustable acquisition range in OFDM systems,” IEEE Trans. Wirel. Comm. 4(1), 228–237 (2005).
[Crossref]

Liu, Y. N.

Z. Zhang, K. P. Long, M. Zhao, and Y. N. Liu, “Joint frame synchronization and frequency offset estimation in OFDM systems,” IEEE Trans. Broadcast 51(3), 389–394 (2005).
[Crossref]

Long, K. P.

Z. Zhang, K. P. Long, M. Zhao, and Y. N. Liu, “Joint frame synchronization and frequency offset estimation in OFDM systems,” IEEE Trans. Broadcast 51(3), 389–394 (2005).
[Crossref]

Ma, Y.

X. W. Yi, W. Shieh, and Y. Ma, “Phase noise effects on high spectral efficiency coherent optical OFDM transmission,” J. Lightwave Technol. 26(10), 1309–1316 (2008).
[Crossref]

W. Shieh, X. W. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking 7(2), 34–55 (2008).

Masuda, H.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[Crossref]

Matsui, M.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[Crossref]

Miyamoto, Y.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[Crossref]

Mizoguchi, M.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[Crossref]

Morita, I.

S. L. Jansen, I. Morita, T. C. W. Schenk, N. Takeda, and H. Tanaka, “Coherent optical 25.8-Gb/s OFDM transmission over 4160-km SSMF,” J. Lightwave Technol. 26(1), 6–15 (2008).
[Crossref]

S. L. Jansen, I. Morita, T. C. Schenk, and H. Tanaka, “Long-haul transmission of 16 ×52.5 Gbit/s polarization-division-multiplexed OFDM enabled by MIMO processing (Invited),” J. Opt. Networking 7(2), 173–182 (2008).
[Crossref]

Peckham, D. W.

Qian, D.

Sano, A.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[Crossref]

Schenk, T. C.

S. L. Jansen, I. Morita, T. C. Schenk, and H. Tanaka, “Long-haul transmission of 16 ×52.5 Gbit/s polarization-division-multiplexed OFDM enabled by MIMO processing (Invited),” J. Opt. Networking 7(2), 173–182 (2008).
[Crossref]

Schenk, T. C. W.

Schmidl, T. M.

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
[Crossref]

Shieh, W.

Y. Tang and W. Shieh, “Coherent optical OFDM transmission up to 1 Tb/s per channel,” J. Lightwave Technol. 27(16), 3511–3517 (2009).
[Crossref]

X. W. Yi, W. Shieh, and Y. Ma, “Phase noise effects on high spectral efficiency coherent optical OFDM transmission,” J. Lightwave Technol. 26(10), 1309–1316 (2008).
[Crossref]

W. Shieh, X. W. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking 7(2), 34–55 (2008).

X. W. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photon. Technol. Lett. 19(12), 919–921 (2007).
[Crossref]

Takatori, Y.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[Crossref]

Takeda, N.

Tanaka, H.

S. L. Jansen, I. Morita, T. C. W. Schenk, N. Takeda, and H. Tanaka, “Coherent optical 25.8-Gb/s OFDM transmission over 4160-km SSMF,” J. Lightwave Technol. 26(1), 6–15 (2008).
[Crossref]

S. L. Jansen, I. Morita, T. C. Schenk, and H. Tanaka, “Long-haul transmission of 16 ×52.5 Gbit/s polarization-division-multiplexed OFDM enabled by MIMO processing (Invited),” J. Opt. Networking 7(2), 173–182 (2008).
[Crossref]

Tang, Y.

Y. Tang and W. Shieh, “Coherent optical OFDM transmission up to 1 Tb/s per channel,” J. Lightwave Technol. 27(16), 3511–3517 (2009).
[Crossref]

X. W. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photon. Technol. Lett. 19(12), 919–921 (2007).
[Crossref]

Tkach, R.W.

R.W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14(4), 3–9 (2010).
[Crossref]

Winzer, P. J.

Yamada, E.

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[Crossref]

Yang, Q.

W. Shieh, X. W. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking 7(2), 34–55 (2008).

Yi, X. W.

W. Shieh, X. W. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking 7(2), 34–55 (2008).

X. W. Yi, W. Shieh, and Y. Ma, “Phase noise effects on high spectral efficiency coherent optical OFDM transmission,” J. Lightwave Technol. 26(10), 1309–1316 (2008).
[Crossref]

X. W. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photon. Technol. Lett. 19(12), 919–921 (2007).
[Crossref]

Yu, J.

Zhang, Z.

Z. Zhang, K. P. Long, M. Zhao, and Y. N. Liu, “Joint frame synchronization and frequency offset estimation in OFDM systems,” IEEE Trans. Broadcast 51(3), 389–394 (2005).
[Crossref]

Z. Zhang, W. Jiang, H. Zhou, Y. Liu, and J. Gao, “High accuracy frequency offset correction with adjustable acquisition range in OFDM systems,” IEEE Trans. Wirel. Comm. 4(1), 228–237 (2005).
[Crossref]

Zhao, M.

Z. Zhang, K. P. Long, M. Zhao, and Y. N. Liu, “Joint frame synchronization and frequency offset estimation in OFDM systems,” IEEE Trans. Broadcast 51(3), 389–394 (2005).
[Crossref]

Zhou, H.

Z. Zhang, W. Jiang, H. Zhou, Y. Liu, and J. Gao, “High accuracy frequency offset correction with adjustable acquisition range in OFDM systems,” IEEE Trans. Wirel. Comm. 4(1), 228–237 (2005).
[Crossref]

Zhu, B.

Bell Labs Tech. J. (1)

R.W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14(4), 3–9 (2010).
[Crossref]

Electron. Lett. (1)

K. Ishihara, T. Kobayashi, R. Kudo, Y. Takatori, A. Sano, E. Yamada, H. Masuda, M. Matsui, M. Mizoguchi, and Y. Miyamoto, “Frequency-domain equalization without guard interval for optical transmission systems,” Electron. Lett. 44(25), 1480–1481 (2008).
[Crossref]

IEEE Photon. Technol. Lett. (1)

X. W. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photon. Technol. Lett. 19(12), 919–921 (2007).
[Crossref]

IEEE Trans. Broadcast (1)

Z. Zhang, K. P. Long, M. Zhao, and Y. N. Liu, “Joint frame synchronization and frequency offset estimation in OFDM systems,” IEEE Trans. Broadcast 51(3), 389–394 (2005).
[Crossref]

IEEE Trans. Commun. (1)

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
[Crossref]

IEEE Trans. Wirel. Comm. (1)

Z. Zhang, W. Jiang, H. Zhou, Y. Liu, and J. Gao, “High accuracy frequency offset correction with adjustable acquisition range in OFDM systems,” IEEE Trans. Wirel. Comm. 4(1), 228–237 (2005).
[Crossref]

J. Lightwave Technol. (5)

J. Opt. Networking (2)

S. L. Jansen, I. Morita, T. C. Schenk, and H. Tanaka, “Long-haul transmission of 16 ×52.5 Gbit/s polarization-division-multiplexed OFDM enabled by MIMO processing (Invited),” J. Opt. Networking 7(2), 173–182 (2008).
[Crossref]

W. Shieh, X. W. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Networking 7(2), 34–55 (2008).

Opt. Express (1)

Other (5)

Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s per channel coherent optical OFDM transmission with subwavelength bandwidth access,” in in National Fiber Optic Engineers Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper PDPC1.

F. Buchali, R. Dischler, M. Mayrock, X. Xiao, and Y. Tang, “Improved frequency offset correction in coherent optical OFDM systems,” in 34th European Conference on Optical Communication, 2008. ECOC 2008 (2008), Paper Mo.4.D.4.

Optical Internetworking Forum, “Integrable Tunable Laser Assembly MSA,” OIF-ITLA-MSA-01.1 (Nov. 22, 2005).

W. Shieh and I. Djordjevic, OFDM for Optical Communications (Elsevier, 2011).

B. Aklar, Digital Communications: Fundamentals & Applications (Prentice-Hall, 2001).

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Figures (10)

Fig. 1
Fig. 1 (a) The conventional and (b) the redesigned training symbol structures for OFDM symbol synchronization and FOE. CP: cyclic prefix.
Fig. 2
Fig. 2 (a) The ICI influence induced by the RFO, (b) the variation of signal amplitude as a function of the RFO.
Fig. 3
Fig. 3 Schematic illustration of RFO estimation algorithm.
Fig. 4
Fig. 4 (a) Simulation setup of the 464Gbit/s PDM-OFDM-16QAM system, the schematics of the DSP in the transmitter (b) and receiver (c), (d) OFDM frame structure.
Fig. 5
Fig. 5 Timing metric of Schmidl algorithm by using the redesigned training symbol with a frequency offset of 5GHz.
Fig. 6
Fig. 6 Absolute values of normalized FOE error under [-4.9GHz, +4.9GHz] frequency offset with OSNR of 24dB.
Fig. 7
Fig. 7 The output constellations of one polarization after phase recovery at OSNR = 28dB, (a) without RFO, (b) e = 0.01, and (c) e = 0.015.
Fig. 8
Fig. 8 BER performance of 464Gbps PDM-OFDM-16QAM signal as a function of the normalized RFO for OSNR = 24dB, 26dB and 28dB.
Fig. 9
Fig. 9 The absolute normalized estimation error (mean value) versus algorithm parameter m for various e at OSNR = 26dB.
Fig. 10
Fig. 10 (a)The absolute normalized estimation error (mean value), and (b) BER performance as a function of the OSNR.

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

r ( t ) = exp ( j 2 π Δ f t + j ϕ ( t ) ) ( s ( t ) h ( t ) + w ( t ) )
r ( n ) = exp ( j 2 π Δ f n T s + j ϕ ( n ) ) ( s ( n ) h ( n ) + w ( n ) )
M ( d ) = | P ( d ) | 2 R ( d ) M ( d ) [ 0 , 1 ]
P ( d ) = m = 0 N / 2 1 [ r * ( d + m ) r ( d + m + N / 2 ) ]
R ( d ) = ( m = 0 N / 2 1 | r ( d + m ) | 2 ) ( m = 0 N / 2 1 | r ( d + m + N / 2 ) | 2 )
Δ f F F O = arg ( p ( d ^ ) ) π f c s
| 2 π k ( ± 5 G H z ) T s | π
h = 2 c e i l ( log 2 N max ( k ) )
Δ f e s t = arg ( n = 0 N N h 1 ( r * ( n ) r ( n + N h ) ) ) h f s 2 π N
Δ f e s t = integer ( Δ f e s t f s c ) f c s + Δ f F F O
Δ f e s t = integer ( Δ f e s t Δ f e s t f s c ) f s c + Δ f e s t
r ( n ) = r ( n ) exp ( j 2 π Δ f e s t n T s ) = exp ( j 2 π e f s c n T s + j ϕ ( n ) ) ( s ( n ) h ( n ) + w ( n ) )
r i n = p i exp ( j 2 π e n / N ) ( 1 N k = 0 N 1 C i k H i k exp ( j 2 π k n / N ) + w i n ) p i = exp ( j 2 π e ( i 1 ) ( N + N c p ) / N )
C i k = C i k H i k η 0 p i + I i k + W i k I i k = n = 1 , n k N C i n H i n η n k η m = 1 N sin ( π ( m + e ) ) sin ( π ( m + e ) / N ) exp ( j π ( m + e ) ( 1 1 N ) )
σ I C I 2 = σ C 2 n = 1 , n k N | η | 2 n k
S N R ' = σ c 2 σ I C I 2 + σ w 2 = ( n = 1 , n k N | η | 2 n k + S N R 1 ) 1
B E R = 3 8 e r f c ( S N R 10 )
H ^ i k = C i k C i k = H i k η 0 p i + I i k + W i k C i k
H ^ k = 1 L i = 1 L H ^ i k = 1 L i = 1 L H k η 0 p i + I i k + W i k C i k H k η 0 1 L i = 1 L p i
H ^ k = H k η 0 1 L i = 1 L exp ( j 2 π e ( i 1 ) ( N + N c p ) / N ) = H k η 0 sin ( π e ( N + N c p ) L / N ) L sin ( π e ( N + N c p ) / N ) exp ( j π e ( L 1 ) ( N + N c p ) / N )
C ^ i k = H ^ * k C i k = H k * η 0 * L sin ( π e ( N + N c p ) / N ) sin ( π e ( N + N c p ) L / N ) exp ( j π e ( L 1 ) ( N + N c p ) / N ) × ( C i k H k η 0 p i + W i k ) = α ξ i C i k + W i k
α = L sin ( π e ( N + N c p ) / N ) sin ( π e ( N + N c p ) L / N ) , ξ i = exp ( j 2 π e ( N + N c p ) ( i L ) / N )
α = L sin ( d π e ( N + N c p ) / N ) sin ( d π e ( N + N c p ) L / N ) ξ i = exp ( j 2 π d e ( N + N c p ) ( i L ) / N ) , d = { 1 , without PDM 2 , with PDM
H ^ ( i + m ) k H ^ i k * = exp ( ( j 2 π d e ( m + i 1 ) ( N + N c p ) / N ) ( j 2 π d e ( i 1 ) ( N + N c p ) / N ) + θ n ) = exp ( j 2 π d e m ( N + N c p ) / N + θ n )
P E = k = 1 N H ^ i k H ^ ( i + m ) k * exp ( j 2 π d e m ( N + N c p ) / N )
e ^ f s c = arg ( P E ) 2 π d m ( N + N c p ) T s
- N 2 d m ( N + N c p ) e ^ N 2 d m ( N + N c p )

Metrics