## Abstract

In this paper, we theoretically and experimentally prove that sub-carriers in double-side band fast orthogonal frequency division multiplexing (DSB-FOFDM) are orthogonal over a symbol interval independent of the signal phase and amplitude. Therefore, the commonly utilized DSB-FOFDM is quadrature amplitude modulation (QAM) accommodated; while previously DSB-FOFDM was usually modulated by amplitude shift keying (ASK) or binary phase shift keying (BPSK). In our proof-of-concept experiments, bit error ratio (BER) performance of 10 Gb/s quadrature phase shift keying (QPSK) modulated DSB-FOFDM was equivalent to that of 10 Gb/s QPSK modulated OFDM after 500 km standard single mode fiber (SSMF) transmission. 10 Gb/s QPSK modulated DSB-FOFDM largely outperformed the commonly utilized 4-ASK modulated DSB-FOFDM in BER performance. Additionally, BER performance of 10 Gb/s 16-QAM modulated DSB-FOFDM was equivalent to that of 10 Gb/s 16-QAM modulated OFDM after 500 km SSMF transmission.

© 2013 Optical Society of America

## 1. Introduction

Multiple sub-carriers technology has gained much attention in recent years as it provides a relatively straightforward way to accommodate high data rate link. Orthogonal frequency division multiplexing (OFDM) is a typical multiple sub-carriers modulation format [1–3]. It enjoys a much higher spectrum efficiency (SE) than many other multiple sub-carriers technologies as a result of overlapped orthogonal sub-carriers. Additionally, it can be easily implemented by inverse fast Fourier transform (IFFT) and FFT. These merits make OFDM widely used and investigated in optical fiber communications [4–6].

In 2010, A. D. Ellis and Lei proposed a fast OFDM (FOFDM) scheme in optical fiber communications to halve the sub-carrier spacing of OFDM based on discrete cosine transform (DCT) [7, 8] and interleaved discrete-Fourier transform (DFT) [9, 10], respectively. In this scheme, Nyquist rate is achieved [7–11]. Moreover, FOFDM largely outperforms OFDM in frequency offset compensation [12] and enjoys an improved performance in channel estimation [13]. Therefore, FOFDM has been a promising scheme especially in long-haul high-speed optical fiber communication systems [14–16]. Additionally, many practical techniques for FOFDM to improve its performance have been widely investigated [17–22].

However, there is a tradeoff between sub-carrier spacing and modulation formats in FOFDM. [7–9,11–13,17] pointed out that this scheme can only be modulated by 1-dimension modulation formats, such as amplitude shift keying (ASK) and binary phase shift keying (BPSK). The commonly used 2-dimension modulation formats with higher SE, such as quadrature phase shift keying (QPSK) or quadrature amplitude modulation (QAM), can not be utilized in FOFDM [11–13, 17–20]; while OFDM is commonly QAM modulated [1–6].

To address this problem, in 2013, A. D. Ellis presented a 16-QAM FOFDM scheme implemented by an optical I/Q modulator [22]. In each branch of the optical I/Q modulator, 4-ASK modulated FOFDM was utilized. After the optical I/Q modulator, 16-QAM modulated FOFDM was generated. However, in this scheme, each branch was still 4-ASK modulated, two inverse DCT (IDCT) and two DCT modules were required, and this scheme did not improve SE of each branch in essence. Thus, this scheme did not well address the problem [7–9,11–13,17] pointed out that FOFDM can only be ASK or BPSK modulated actually.

In this paper, we theoretically and experimentally prove that orthogonality of sub-carriers in the widely utilized double-side band FOFDM (DSB-FOFDM) is independent of the signal amplitude and phase during a symbol interval in fact. Therefore, DSB-FOFDM is QAM accommodated, and thus its SE can be further improved. Moreover, sub-carrier spacing of QAM modulated DSB-FOFDM is just half of that of OFDM, identical to that of the ASK or BPSK modulated DSB-FOFDM.

In our proof-of-concept experiment, 10 Gb/s QPSK modulated DSB-FOFDM and 16-QAM modulated DSB-FOFDM were successfully transported over 500 km standard single mode fiber (SSMF). Bit error ratio (BER) performance of the QPSK modulated DSB-FOFDM and OFDM was equivalent. QPSK modulated DSB-FOFDM largely outperformed 4-ASK modulated DSB-FOFDM, the commonly utilized modulation format in DSB-FOFDM, in BER performance after 500 km SSMF transmission. In addition, BER performance of 16-QAM modulated DSB-FOFDM was equivalent to that of 16-QAM modulated OFDM.

## 2. Principle

To investigate the minimum sub-carrier spacing for crosstalk-free operation, sub-carrier orthogonality should be considered firstly. Generally, a sub-carrier can be written as,

where*ω*denotes the sub-carrier frequency,

_{c}*a*(

*t*) and

*b*(

*t*) are the input complex signal data. Then,

*t*

_{0},

*t*

_{0}+

*T*] is a symbol interval,

*s*(

_{k}*t*) represents the

*k*-th sub-carrier, ${s}_{l}^{*}(t)$ denotes conjugation of the

*l*-th sub-carrier. In crosstalk-free operation, we have

*ξ*(

*t*) = 0. For QAM modulated

*a*,

_{k}*a*,

_{l}*b*and

_{k}*b*, they are constant complex during [

_{l}*t*

_{0},

*t*

_{0}+

*T*]. Therefore, to make

*ξ*(

*t*) = 0 for arbitrary

*a*,

_{k}*a*,

_{l}*b*and

_{k}*b*, we have

_{l}In Eq. (3), if ${a}_{k}{a}_{l}^{*}-{b}_{k}{b}_{l}^{*}=0$ and ${a}_{k}{a}_{l}^{*}+{b}_{k}{b}_{l}^{*}=0$, *ω _{c}* is
determined only by equation ${\int}_{{t}_{0}}^{{t}_{0}+T}\text{sin}\left\{(k\pm l){\omega}_{c}t\right\}dt=0$. One solution of equation ${\int}_{{t}_{0}}^{{t}_{0}+T}\text{sin}\left\{(k\pm l){\omega}_{c}t\right\}dt=0$ is

*ω*=

_{c}*mπ*/{(

*k*±

*l*)

*T*} with

*t*

_{0}= (2

*n*+ 1)

*T*/(2

*m*), where

*m*and

*n*are positive integers and

*t*

_{0}determines the initial phase of sub-carriers. Furthermore, to make this solution correct for arbitrary

*k*and

*l*, we have

*ω*=

_{c}*mπ/T*with

*t*

_{0}= (2

*n*+ 1)

*T*/(2

*m*). Then, we have the minimum sub-carrier spacing Δ

*ω*=

*π/T*. Similarly, if ${b}_{k}{a}_{l}^{*}+{a}_{k}{b}_{l}^{*}=0$ and ${b}_{k}{a}_{l}^{*}-{a}_{k}{b}_{l}^{*}=0$, we have Δ

*ω*=

*π/T*with

*ω*=

_{c}*mπ*/{(

*k*±

*l*)

*T*} and

*t*

_{0}=

*nT/m*.

In other conditions excluding ${a}_{k}{a}_{l}^{*}\pm {b}_{k}{b}_{l}^{*}=0$ and ${b}_{k}{a}_{l}^{*}\pm {a}_{k}{b}_{l}^{*}=0$, *ω _{c}* =
2

*mπ*/{(

*k*±

*l*)

*T*} is the only solution of Eq. (3) and

*t*

_{0}is arbitrary in this case. Therefore, we have a minimum sub-carrier spacing Δ

*ω*= 2

*π/T*for arbitrary integers

*k*and

*l*. This is the standard OFDM with Δ

*ω*= 2

*π/T*.

Note that in Eq. (3),
*t*_{0} can only determine whether equation ${\int}_{{t}_{0}}^{{t}_{0}+T}\text{sin}\left\{(k\pm l){\omega}_{c}t\right\}dt=0$ or ${\int}_{{t}_{0}}^{{t}_{0}+T}\text{cos}\left\{(k\pm l){\omega}_{c}t\right\}dt=0$ which includes the sub-carrier information is true;
whether equation ${a}_{k}{a}_{l}^{*}\pm {b}_{k}{b}_{l}^{*}=0$ or ${b}_{k}{a}_{l}^{*}\pm {a}_{k}{b}_{l}^{*}=0$ which contains the signal data is true is not
determined by *t*_{0}. Therefore,
*t*_{0} can only determine the initial sub-carrier phase,
not the signal data phase. Moreover, this indicates that the orthogonality of these
sub-carriers is independent of the signal phase and amplitude.

Therefore, according to the solutions to Eq.
(3), to achieve DSB-FOFDM with Δ*ω*
= *π/T*, identical to half of that of standard OFDM,
we must have

In addition, sub-carriers *a _{k}*
cos(

*kω*) +

_{c}t*b*sin(

_{k}*kω*) and

_{c}t*a*cos(

_{l}*lω*) +

_{c}t*b*sin(

_{l}*lω*) should have the same form. Therefore, combined with Eq. (4), we have

_{c}t*a*=0 and

_{k}*a*= 0 or

_{l}*b*= 0 and

_{k}*b*= 0 to achieve the DSB-FOFDM scheme with a halved minimum sub-carrier spacing Δ

_{l}*ω*=

*π/T*.

This indicates that the sub-carrier spacing can be reduced if only cosine or sine
sub-carriers are utilized instead of the exponential sub-carriers in the standard
OFDM. Furthermore, in our derivations, *a _{k}*,

*a*,

_{l}*b*, and

_{k}*b*are constant complex signal during a symbol interval. Thus, DSB-FOFDM is QAM accommodated actually while previously it was only ASK or BPSK modulated [7–22].

_{l}Moreover, the QAM modulated DSB-FOFDM can be implemented by IDCT with
*s*(*t*) =
*a*(*t*)cos(*ω _{c}t*)
when

*b*(

*t*) = 0 or inverse discrete sine transform (IDST) with

*s*(

*t*) =

*b*(

*t*) sin(

*ω*) when

_{c}t*a*(

*t*) = 0. We can denote a DSB-FOFDM symbol as follows,

*a*and

_{k}*b*are constant complex during a DSB-FOFDM interval.

_{k}Figure 1 depicts the spectrum of DSB-FOFDM and the standard OFDM. It demonstrates that the minimum sub-carrier spacing of DSB-FOFDM is just half of that of the standard OFDM. Fig. 1(b) indicates that DSB-FOFDM includes a positive band and its negative image. DSB-FOFDM has many advantages in residual frequency offset compensation, channel estimation and chromatic dispersion compensation, as reported in [12, 13, 21, 22].

## 3. Experiment

#### 3.1. Experiment setup

Proof-of-concept experiments on QPSK modulated DSB-FOFDM were carried out. Figure 2 shows the experiment setup for 10 Gb/s QPSK modulated DSB-FOFDM coherent optical communication system. At transmitter, the initial binary sequence was firstly distributed to 256 sub-carriers by a serial to parallel (S/P) module. Then time-domain QPSK modulated DSB-FOFDM signal was generated through QPSK modulation and IDCT modules. 8 sub-carriers were utilized as pilot sub-carriers. 7 sub-carriers around DC were left empty. 1/16 cyclic prefix samples were employed. For every 64 DSB-FOFDM symbols, 2 training symbols were transmitted.

The generated time-domain QPSK modulated DSB-FOFDM signal was then uploaded into an arbitrary waveform generator (Tektronix AWG7122C) operating at 5 GSa/s to generate analog signals. As SE of QPSK is 2 bit/s/Hz, the overall link rate was about 10 Gb/s and the data rate in our experiment was approximately 8.5 Gb/s determined by the ratio of the data sub-carriers to the total sub-carriers and the overall link rate((256 – 7 – 8)/256 × 15/16 × 62/64 × 10 Gb/s ≈ 8.5 Gb/s).

The optical source in our experiment was a commercially available external cavity
laser (ECL) operating at a wavelength of 1550.05 nm with a line-width of about
100 kHz. To fix polarization state of signal light, a polarization controller
(PC) was utilized following ECL but before optical I/Q modulator. Subsequently,
the generated analog waveforms were amplified by SHF 100 AP with a maximum
response bandwidth of 25 GHz. Then, the amplified analog waveforms were fed into
an integrated dual-parallel dual-electrodes MZM (FUJITSU FTM7920EX) worked as an
optical I/Q modulator to up-convert the baseband signal. 3-dB bandwidth of this
MZM is about 40 GHz. The typical *V _{π}* of our
integrated dual-parallel dual-electrodes MZM (FUJITSU FTM7920EX) is 3.5 V.
Optical insertion loss is about 9.0 dB. In our experiment, the two sub-MZMs
worked at the null-bias point; the main-MZM worked at the quadrature-bias
point.

Then, the up-converted signal was launched into 5 × 100 km G. 652 SSMF
with 5 EDFAs. The typical noise figure of our utilized Opeak EDFAs is about 5.5
dB. The input optical power into the SSMF fiber is about −2.0 dBm in our
experiment. Fiber chromatic dispersion is about 17 ps/nm/km, attenuation loss is
about 0.2 dB/km. Before the coherent receiver, a variable optical attenuator
(VOA) was utilized to regulate the received optical power. Single-polarization
coherent receiver was utilized in our experiment and its 3-dB bandwidth is about
43 GHz. The input local oscillator (LO) power is about 5 dBm. LO utilized in the
coherent receiver is the output of a 1 × 2 optical power splitter
following PC at the transmitter. Our balanced coherent receiver is based on the
*u*^{2}*t* photodiodes BPDV2150R. The
maximum photodiodes bias voltage of PD1 and PD2 are +3.5 V and
−3.5 V, respectively. Their typical 3 dB cut-off frequency is about 42
GHz.

The coherent received signal was subsequently fed into a digital phosphor oscilloscope (Tektronix DPO72004C) operating at 50 GSa/s after amplification to implement A/D conversion. Then we downloaded the digital signal into Matlab, separated the training symbols and data symbols and removed CP. Subsequently, this digital signal was fed into a DCT module. Channel estimation was implemented by the simple least square (LS) algorithm. Phase estimation was implemented by the commonly utilized pilot-aided algorithm. At last, the equalized signal was then fed into a QPSK demodulation module.

In our experiments, the data processing algorithms were all the same as that utilized in the standard OFDM optical communication systems to fairly compare their performance. Performance of DSB-FOFDM and its advantages have been widely investigated in the previous works [7–22], so this paper mainly focuses on whether DSB-FOFDM is QAM accommodated to further improve its SE.

#### 3.2. Experiment results

Figure 3 shows the experiment constellations of the proposed 10 Gb/s QPSK modulated DSB-FOFDM after 500 km SSMF transmission at different received optical power. These constellations experimentally prove that DSB-FOFDM is QPSK accommodated indeed.

Figure 4 shows BERs of 10 Gb/s QPSK
modulated OFDM and DSB-FOFDM versus the average received optical power after
back-to-back (B2B) and 500 km SSMF transmission, respectively. After B2B
transmission, BER of 10^{−3} can be achieved when the received
optical power was about −40 dBm. After 500 km transmission, BER of
10^{−3} can be achieved when the received optical power was
about −38 dBm. Also, as Fig. 4
demonstrated, BER performance of QPSK modulated DSB-FOFDM and OFDM was
equivalent after B2B transmission and 500 km SSMF transmission,
respectively.

4-ASK modulated DSB-FOFDM is commonly utilized in the previous investigations [7–22]. As SE of QPSK is equal to that of 4-ASK, we also compared the performance of 4-ASK modulated DSB-FOFDM with that of QPSK modulated DSB-FOFDM in our proof-of-concept experiments.

Figure 5 demonstrates that BERs of 10 Gb/s 4-ASK modulated DSB-FOFDM and 10 Gb/s QPSK modulated OFDM after 500 km SSMF transmission versus the average received optical power. Constellations of the 4-ASK modulated DSB-FOFDM and the QPSK modulated DSB-FOFDM are also inserted in Fig. 5.

Figure 5 clearly shows that 10 Gb/s QPSK
modulated DSB-FOFDM largely outperforms 10 Gb/s 4-ASK modulated DSB-FOFDM after
500 km SSMF transmission in BER performance. 10 Gb/s QPSK modulated DSB-FOFDM
can achieve a BER of 10^{−3} when the average received optical
power was about −38 dBm while 10 Gb/s 4-ASK modulated DSB-FOFDM can
achieve a BER of 10^{−3} when the average received power was
about −36 dBm after 500 km SSMF transmission. And also, the difference
in BER performance of 10 Gb/s QPSK modulated DSB-FOFDM and 10 Gb/s 4-ASK
modulated DSB-FOFDM after 500 km transmission is in proportion to the average
received optical power shown in Fig.
5.

To investigate performance of the high-order QAM modulated DSB-FOFDM and verify that DSB-FOFDM is also the higher-order QAM accommodated, 16-QAM modulated DSB-FOFDM was also carried out in our proof-of-concept experiments. Its experiment setup is the same as that shown in Fig. 2. The AWG operated at 2.5 GSa/s to generate the 10 Gb/s 16-QAM DSB-FOFDM signal. Parameters including the total sub-carriers, data sub-carriers, CP length were also set the same as that in the 10 Gb/s QPSK DSB-FOFDM experiments.

Figure 6 demonstrates BER performance of
10 Gb/s 16-QAM modulated DSB-FOFDM and 10 Gb/s 16-QAM modulated OFDM after 500
km SSMF transmission versus the average received optical power. Constellation of
the 16-QAM modulated DSB-FOFDM is also inserted in Fig. 6. It clearly shows that DSB-FOFDM is high-order
QAM accommodated as OFDM does. And also, BER performance of 10 Gb/s 16-QAM
modulated DSB-FOFDM and 10 Gb/s 16-QAM modulated OFDM was equivalent. They can
achieve a BER of 10^{−3} when the average received optical power
was about −28 dBm as demonstrated in Fig. 6.

## 4. Conclusion

In this paper, we theoretically and experimentally prove that DSB-FOFDM is QAM accommodated while previously DSB-FOFDM was usually modulated by ASK or BPSK. Therefore, SE of DSB-FOFDM can be further improved with the introduction of high-order QAM. In addition, the QAM modulated DSB-FOFDM halves the sub-carrier spacing of the standard OFDM as the conventional ASK or BPSK modulated DSB-FOFDM does. In our proof-of-concept experiment, 10 Gb/s QPSK modulated DSB-FOFDM and 10 Gb/s 16-QAM modulated DSB-FOFDM were successfully transported over 500 km SSMF, respectively. Experiment results show that BER performance of 10 Gb/s QPSK modulated DSB-FOFDM was equivalent to that of 10 Gb/s QPSK modulated OFDM after 500 km SSMF transmission. 10 Gb/s QPSK modulated DSB-FOFDM largely outperformed 10 Gb/s 4-ASK modulated DSB-FOFDM, the commonly utilized modulation format in DSB-FOFDM, in BER performance after 500 km SSMF transmission. Also, BER performance of 10 Gb/s 16-QAM modulated DSB-FOFDM was equivalent to that of 10 Gb/s 16-QAM modulated OFDM.

## Acknowledgments

This work was supported in part by the National Basic Research Program of China (973 Program) under Grant 2012CB315605 and 2014CB340102, in part by the National High-Tech Research and Development Program of China (863 Program) under Grant 2013AA013401, in part by the National Natural Science Foundation under Grant 61271191, 61271193, and 61072054, in part by the Fund of State Key Laboratory of Information Photonics and Optical Communications, and in part by the Fundamental Research Funds for the Central Universities.

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