## Abstract

We compare formats for optical intensity modulation limited by thermal noise with the assumption of having ideal devices. At the same bitrate and bandwidth, a hitherto unknown format turns out to be more power efficient than known formats. This new modulation, which is a hybrid between on-off keying and phase-shift keying, belongs to the subcarrier modulation family. At asymptotically high signal-to-noise ratios, this hybrid scheme has a 1.2 dB average electrical power gain and 0.6 dB average optical power gain compared to OOK, while it has a 3.0 dB average electrical power gain and 2.1 dB average optical power gain compared to subcarrier QPSK.

© 2010 OSA

## 1. Introduction

Multilevel modulation, with information encoded onto the amplitude and phase of the optical carrier, has attracted significant research interest in fiber-optical communication to boost the transmission rates and spectral efficiencies [1]. However, the enabling technology consisting of coherent transmitters and receivers is not feasible for short-haul applications such as local area networks, data centers, and computer interconnects [2], where the overall cost and complexity has to be kept down. Therefore, intensity modulation and direct detection (IM/DD), where the information is modulated onto the intensity of the optical carrier using a laser diode and the receiver detects the instantaneous power of the received signal using a simple photodiode, is more suitable for such types of low cost and complexity applications. For such links, there arises a challenge in selecting a modulation format which offers a good trade-off between spectral and power efficiency as well as having low peak amplitudes in the generated electrical waveform; in the case of high peaks, the laser diode will operate in a highly nonlinear fashion [3, 4].

Since the optical phase in an IM/DD system cannot be used to carry information, multilevel pulse amplitude modulation (*M*-PAM), which directly modulates the optical carrier, is a simpler way to achieve higher spectral efficiency than the widely spread on-off keying (OOK) modulation format while keeping the hardware complexity low. A link analysis was performed in [5] for a vertical-cavity surface-emitting laser (VCSEL) operating with 4-PAM signaling, and the first sensitivity measurements were presented. Upper and lower bounds on the capacity of 2-, 4-, 8-, and 16-PAM were derived in [6], and the power efficiency of *M*-PAM was shown to be low in [7] due to the increased number of levels. Further, *M*-ary pulse-position modulation (*M*-PPM) has been considered in free-space optics due to its high power efficiency, but it has a poor spectral efficiency [8, 9], making it less suitable for fiber-optical communication.

On the other hand, the subcarrier modulation (SCM) family described in the wireless infrared communications context [10, Ch. 5] allows the adoption of any bandwidth-efficient multilevel modulation for IM/DD systems. The concept is to first modulate the data onto an electrical subcarrier, whose frequency is of the same order as the symbol rate, using the standard I/Qmodulation formats such as multilevel phase-shift keying (*M*-PSK) or multilevel quadrature amplitude modulation (*M*-QAM). A direct current (DC) bias is added in order to make the modulated signal nonnegative, before it directly modulates the light source. The DC bias is crucial to avoid signal clipping by the laser diode which deteriorates the system performance. Compared with coherent transmission, the complexity in the transmitter and receiver of SCM is shifted to the electrical domain. An experimental demonstration of the SCM concept can be found in [11], and a novel transmitter design for the subcarrier quadrature phase-shift keying (QPSK) and 16-QAM using fast digital exclusive-OR (XOR) gates is presented in [12] and [13]. However, the drawback in using SCM at the same baud rate as the modulation formats that directly modulate the optical carrier, e.g., the aforementioned OOK and *M*-PAM, is that it occupies twice the bandwidth of such modulations due to the intermediate step of modulating the information onto an electrical subcarrier before modulating the optical carrier [10, p. 125]. In addition, the DC bias, which carries no information, consumes transmission power without adding any noise resilience since it is independent of the transmitted information. To increase the power efficiency of SCM, [14] minimized the bias for the transmitted signal on a symbol-by-symbol basis, and [15] allowed the bias to vary even within the symbol interval.

An extension of the above subcarrier modulation family is to superimpose several carriers resulting in a multiple-subcarrier modulation (MSM) [10, p. 122], or orthogonal frequency-division multiplexing (OFDM) if the carriers are orthogonal. Further, a subclass of OFDM known as discrete multitone (DMT), where the output of the inverse fast Fourier transform modulator is real instead of complex, is investigated in [16]. A demonstration of a dual carrier system with 16-QAM can be found in [17], and transmission using DMT over a multimode fiber in [18]. The main drawback of the MSM family is the poor optical average power efficiency due to the high peak-to-average power ratio (PAPR) in the MSM electrical signal, which requires an increased DC bias to make the signal nonnegative. [15, 19] investigated power reduction techniques for MSM, [20] investigated PAPR reduction for OFDM, and [4,21] studied the light emitting diode (LED) nonlinearity effects on OFDM and DMT, respectively.

In this work, our contribution is to present a novel quaternary subcarrier modulation scheme called on-off phase-shift keying (OOPSK_{SCM}), and compare it with OOK and the subcarrier QPSK (QPSK_{SCM}), since they have the same spectral efficiency. According to [10, Ch. 5], [9], OOK is the most power efficient among the known modulation formats having the same spectral efficiency. However, as we will see, OOPSK_{SCM} turns out to be the most power efficient among known formats.

## 2. System model

The system under investigation is depicted in Fig. 1(a). The transmitter consists of a modulator which maps the symbol *u*(*k*) ∈ {0,1, …,*M* − 1} at instant *k* to a waveform belonging to the modulation signal set *S* = {*s*
_{0}(*t*), *s*
_{1}(*t*), …, *s*
_{M−1}(*t*)}. The resulting electrical waveform *x*(*t*) is positive and directly modulates the laser diode. Its output is the optical field

which propagates in the optical link, where *c* is the slope efficiency of the laser [3], *ν*
_{0} is the optical carrier frequency, and *θ* is a random phase, uniformly distributed in [0,2*π*). The photodiode does the optical-to-electrical conversion, which is then followed by a demodulator whose output is *û*(*k*), an estimate of *u*(*k*). The demodulator is a correlator or matched filter receiver, which minimizes the symbol error rate at a given signal-to-noise ratio (SNR) [29, Sect. 4.1]. In the special case of SCM, this demodulator involves mixing the signal with the subcarrier and a 90° phase-shifted subcarrier, and using those and the detected total power as detection variables [22].

Due to the nature of the applications involving short-haul links, the dominant channel impairment is thermal noise in the photodetector [23], [24, p. 155]. Therefore, the system under investigation can be well modeled as a modulator and demodulator with additive white Gaussian noise (AWGN), where the propagating signal satisfies the nonnegativity constraint as shown in Fig. 1(b) [9], [24, p. 153]. The AWGN has a double-sided power spectral density *N*
_{0}/2 and models the thermal noise. The same baseband model can be used to represent the wireless infrared channel [6, 9, 22], [10, Ch. 5].

## 3. Investigated modulation schemes

We propose the so called OOPSK_{SCM} subcarrier modulation, which is a hybrid between OOK and phase-shift keying. Therefore, the DC bias required to make the signals positive is also utilized to carry information. It consists of a signal set with size *M* = 4 defined for 0 ≤ *t* < *T _{s}* as

$${s}_{1}\left(t\right)=A[1+\mathrm{sin}\left(2\pi {f}_{c}t\right)]$$

$${s}_{2}\left(t\right)=A[1+\mathrm{sin}(2\pi {f}_{c}t+\frac{2\pi}{3})]$$

$${s}_{3}\left(t\right)=A[1+\mathrm{sin}(2\pi {f}_{c}t-\frac{2\pi}{3})],$$

where
$A=\left(\frac{2}{3}\right)\sqrt{\frac{2{E}_{s}}{{T}_{s}}}$
, *E _{s}* =

*E*log

_{b}_{2}

*M*is the average symbol energy,

*E*is the average bit energy,

_{b}*T*= 1/

_{s}*R*is the symbol period, and

_{s}*R*is the symbol rate. In this case, we consider

_{s}*f*= 1/

_{c}*T*as the subcarrier frequency. Other hybrids between amplitude-shift keying and phase-shift keying have been studied in [25], where their performances were evaluated for the AWGN channel using coherent detection, and in [26], where analytical expressions for the error probabilities over fading channels have been obtained. However, such modulation formats do not satisfy the nonnegativity constraint; therefore, they do not belong to the SCM family and cannot be used over intensity modulated channels.

_{s}Our investigation encompasses modulation formats with spectral efficiency 1 (bits/s)/Hz, where the OOK modulation defined for 0 ≤ *t* < *T _{s}* as

will be used as a benchmark as in [9], [10, Ch. 5], and compared with QPSK_{SCM}, defined as

for 0 ≤ *t* < *T _{s}* [10, Ch. 5] and

*i*∈ {0, …,3} with Gray mapping, and with the proposed OOPSK

_{SCM}. The above definitions ensure that all three modulation formats have the same average symbol energy.

## 4. Performance investigation

Figure 2 shows the electrical spectrum
${S}_{x}\left(f\right)=F\left[\overline{{R}_{x}\left(\tau \right)}\right]$
of the transmitted signal *x*(*t*) [27, p. 70], where *F* is the Fourier transform and
$\overline{{R}_{x}\left(\tau \right)}$
is the average autocorrelation function of the cyclostationary process *x*(*t*), at the same bit rate *R _{b}* =

*R*log

_{s}_{2}

*M*using OOK, QPSK

_{SCM}, and OOPSK

_{SCM}. For the subcarrier baseband waveforms

*x*(

*t*), the normalized carrier frequency is

*f*/

_{c}*R*= 1/2. With the definition of the absolute bandwidth as the width of the spectral main lobe [28, Ch. 12], Fig. 2 shows that all the considered schemes occupy the same absolute electrical bandwidth although OOK has twice the symbol rate. This is because OOK has half the number of levels compared with the other considered modulation formats. An interesting observation is the spectral dependence of OOPSK

_{b}_{SCM}, making it a blend between the spectrum of OOK and QPSK

_{SCM}.

Figure 3 shows the optical spectrum
${S}_{z}\left(f\right)=F\left[\overline{{R}_{z}\left(\tau \right)}\right]$
centered at the optical carrier frequency *ν*
_{0} of the signal *z*(*t*) propagating in the fiber using the different modulation schemes. The modulations still have the same spectral efficiency, although OOK has higher side lobes than the subcarrier modulations. A clear difference compared to the electrical spectra is the existence of spectral spikes for the QPSK_{SCM} and OOPSK_{SCM} modulation formats. These spikes are the result of the nonlinear relationship between *x*(*t*) and *z*(*t*) due to the laser diode, as shown in Eq. (1). This effect is not present in the OOK spectrum due to the fact that rectangular pulse shaping was used in all the simulations, and that the baseband waveform of OOK does not contain a carrier as in the case of the SCM family. The square root of a rectangular pulse is still rectangular, while the square root of a sinusoid is not sinusoidal.

Figure 4 shows the simulated bit error rate (BER) performances of the three modulation schemes using the baseband model depicted in Fig. 1(b), where ideal hardware is assumed, together with their theoretical results. The used model is accurate for links not close to the bandwidth limits of the channel or the electronic bandwidth limits of the transmitter and receiver. From the above definition of OOK, the BER derivation [27, p. 175] yields

whereas the theoretical BERs of QPSK_{SCM} and OOPSK_{SCM} were approximated using the standard union bound found in [29, Eq. (4.88)]. This bound can be simplified to

where *d* is the minimum signal distance defined by

*K* is the number of distinct signal pairs (*s _{i}*(

*t*),

*s*(

_{j}*t*)) with

*i*<

*j*for which ∫(

*s*(

_{i}*t*) −

*s*(

_{j}*t*))

^{2}

*dt*=

*d*

^{2}, and

*B*is the average number of bits that differ between these

*K*pairs. Evaluating (7) for all pairs of signals is straightforward for the signal sets in (2)–(4). The results for Gray-mapped QPSK

_{SCM}are

*M*= 4,

*K*= 4,

*B*= 1, and

*d*

^{2}= 4

*E*/3, which yields

_{b}and for OOPSK_{SCM}
*M* = 4, *K* = 6, *B* = 8/6, and *d*
^{2} = 8*E _{b}*/3, which yields

For OOK, finally, the parameters are *M* = 2, *K* = 1, *B* = 1, and *d*
^{2} = 2*E _{b}*, in which case the union bound (6) actually yields the exact expression (5).

It should be noted that simulations are accurate at low SNR, and union bounds at high SNR. Furthermore, there will be performance degradation when the physical realization and its influence on the signals are considered. Since different modulation formats are affected differently by a specific implementation choice, making fair performance comparisons becomes challenging. Therefore, it is beyond the scope of this work to study limitations from the system hardware viewpoint.

To achieve a BER of 10^{−6}, OOPSK_{SCM} offers a 1 dB average electrical power gain compared to OOK, the widely used modulation in this field, and 2.8 dB gain compared to QPSK_{SCM}. These performance gains do not include the power consumption of the drive and decision circuits.

Table 1 presents the average electrical power gain, the average optical power gain, and the electrical peak power gain, all with respect to OOK at the same BER. The comparison is valid asymptotically at high SNR, where the error performance is determined by *d*, see (6). Thus the signal amplitudes were normalized to give the same *d* for all modulation formats.

The average electrical power gain is defined as

at asymptotically high SNR *E _{b}*/

*N*

_{0}, where

*P̄*

_{e(OOK)}is the average electrical power of OOK, and

*P̄*=

_{e}*E*/

_{s}*T*is the average electrical power of the modulation format under study. As a result, OOPSK

_{s}_{SCM}is shown to have a 1.2 dB average electrical power gain compared to OOK, while it has a 3.0 dB gain compared to QPSK

_{SCM}.

Table 1 also presents the average optical power gain of the different modulation formats relative to OOK

to achieve the same BER at asymptotically high SNR. The average optical power of a modulation format *P̄ _{o}* depends solely on the DC bias required to make

*x*(

*t*) ≥ 0 [6]. OOPSK

_{SCM}offers a 0.6 dB average optical power gain relative to OOK and 2.1 dB relative to QPSK

_{SCM}.

From the expressions of the signal waveforms, we derived the electrical peak power gain of the different considered modulation schemes with respect to OOK. This entity is the ratio of the electrical peak power of OOK to the peak power of the other modulations under study. The electrical peak power of a modulation is

The electrical peak power gain gives a measure of tolerance against the nonlinearities in the laser diode which limit the link performance [4]. In the SCM family, QPSK_{SCM} is 3.0 dB worse than OOK, whereas OOPSK_{SCM} is 1.2 dB worse than OOK, making it 1.8 dB better than QPSK_{SCM}, and thus more robust to the nonlinearity impact by the laser diode and this leads to a better link performance.

## 5. Conclusion

Among the studied modulation schemes, OOK, QPSK_{SCM}, and OOPSK_{SCM} occupy the same electrical and optical bandwidth. If power consumption is the main concern for environmental and cost reasons, OOPSK_{SCM} is the best choice, whereas if reducing the impact of laser nonlinearity on the link performance is the main concern, OOK is the best choice.

A promising track for future work is to implement this new modulation scheme experimentally, where the effect of nonlinearities and the other impairments present can be quantified, and theoretical models describing the overall system can be derived. This will be the basis for investigations of transceiver complexity and exploration of even more power-efficient higher level modulation formats.

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