H Taki

S Azou

A Hamie

A Al Housseini

A Alaeddine

A Sharaiha

Improving the Power Eff ciency of SOA-based UWB over Fiber Systems via Pulse Shape Randomization

A simple pulse shape r andomization scheme is considered in this paper for improving the per for mance of Ultr a Wide band (UWB) communication systems using On Off Keying (OOK) or Pulse Position Modulation (PPM) for mats. The advantage of the proposed scheme, which can be either employed for Impulse Radio (IR) or for car r ier-based systems, is f r st theoretically studied based on closed-for m der ivations of power spectr al densities. Then, we investigate an application to an IR-UWB over optical f ber system, by utilizing the 4-th and 5-th or der s of Gaussian der ivatives. Our approach proves to be effective for 1 Gbps-PPM and 2 Gbps-OOK tr ansmissions, with an advantage in ter ms of power eff ciency for shor t distances. We also examine the per for mance for a system employing an in-line Semiconductor Optical Amplif er (SOA) with the view to achieve a reach extension, while limiting the cost and system complexity.

I. INTRODUCTION

As there is a growing demand for high speed and low power transmission systems nowadays, ultra wide band (UWB) radio technology becomes an important technique to be used in the new generation of short-range broadband wireless communications [START_REF] Ghavami | Ultra wideband signals and systems in communication engineering[END_REF] [START_REF] Aiello | Ultra-wideband wireless systems[END_REF]. The main advantages offered by UWB are the high data rate, immunity to multipath propagation, f exibility in reconf guring data rate and power, very good time domain resolution, accurate mobile user location, and easy data protection. UWB applications include local and wide area networks, sensor networks, emergency communications, radar, remote sensing, and military applications [START_REF] Oppermann | UWB: theory and applications[END_REF]. Widening UWB system coverage via optical access networks has become an important research topic over the past few years [START_REF] Lim | Fiber-wireless networks and subsystem technologies[END_REF]- [START_REF] Gomes | Radio-over-fiber transport for the support of wireless broadband services[END_REF]. Such UWB-over-f ber infrastructures require direct optical pulse generation [START_REF] Zeng | All-optical UWB impulse generation based on cross-phase modulation and frequency discrimination[END_REF], [START_REF] Bolea | Optical UWB pulse generator using an N tap microwave photonic filter and phase inversion adaptable to different pulse modulation formats[END_REF] or electro-optical conversion utilizing intensity modulators [START_REF] Yu | Ultra-wideband generation based on cascaded Mach-Zehnder modulators[END_REF]. IR-UWB technique has shown a great interest to be used due to its low complexity and cost transceiver architectures [START_REF] Win | Impulse radio: How it works[END_REF], [START_REF] Fernandes | Recent advances in IR-UWB transceivers: An overview[END_REF]. In 2002, the Federal Communication Commission (FCC) allowed UWB devices to operate within a particular spectral mask in order to avoid dangerous interference with other narrow band communication systems [START_REF] Sheng | On the spectral and power requirements for ultra-wideband transmission[END_REF]. According to this regulation the transmitted power spectral density of UWB has to stay below the limit of -41.3dBm/MHz in the frequency range of [3.1,10.6] GHz [15]. This power limitation has raised the need for highly eff cient IR-UWB signals [START_REF] Yajnanarayana | Spectral efficient IR-UWB communication design for low complexity transceivers[END_REF], hence pulse shaping and spectrum smoothing are mandatory to increase the transmitted power without violation of FCC regular limit [START_REF] Nakache | Spectral shaping of UWB signals for time-hopping impulse radio[END_REF]. A special effort has recently been made by Abraha et al. [START_REF] Abraha | Routing of power efficient IR-UWB wireless and wired services for in-building network applications[END_REF]- [START_REF] Abraha | Power-efficient impulse radio ultrawideband pulse generator based on the linear sum of modified doublet pulses[END_REF] for improving the power eff ciency of Gaussian pulses with a lower order of complexity, by using a combination of monocycles and delayed doublets. Pulse shaping can improve the envelope of the overall spectrum, but has no inf uence on the discrete spectral peaks resulting from modulation patterns.Hence smoothing the spectrum of modulated signal requires a change in the statistical properties of the pulse train, time randomization being important so as to reduce the intensity of frequency comb lines. Direct sequence (DS) and Time Hopping (TH) are the mostly utilized randomization techniques, based on varying the pulse amplitude or time position respectively [START_REF] Maggio | Pseudo-chaotic time hopping for UWB impulse radio[END_REF]- [START_REF] Romme | On the power spectral density of time-hopping impulse radio[END_REF]. The main drawback for DS is the complexity associated with multi-level generation, and randomizing in terms of polarity [START_REF] Pearce | Spectral spike reduction for ultrawideband impulse radio system[END_REF] is not suitable for unipolar encoding like OOK or PPM. Hopping the pulse position between different chips in case of TH requires a larger symbol duration, which consequently lowers the data rate and may not be pertinent for high speed applications. In this work, a simple randomization technique is adopted for improving the system performance; it consists in changing the pulse waveform over time while keeping the same time location and scaling factor. The energy per bit is conserved for all transmitted frames/symbols, in order to f t the requirement of non-coherent energy-detection based UWB systems [START_REF] Cui | UWB system based on energy detection of derivatives of the Gaussian pulse[END_REF]. The proposed Multi-Waveform OOK (MWOOK) and Multi-Waveform PPM (MWPPM) modulation schemes have been applied to a Radio over Fiber (RoF) system. Our scheme has proved a better power eff ciency at short distances of f ber, which can be extended into several kilometers using a Semiconductor Optical Amplif er (SOA) for in-line amplif cation purpose.

II. PROPOSED MULTI-WAVEFORM OOK/PPM MODULATION FORMATS

Conventional OOK and PPM impulse radio signals correspond to a stream of modulated pulses via amplitude or position, respectively. The same pulse is systematically repeated along different transmitted frames, leading to a periodicity in time domain and hence to high spectral spikes. Our approach aims to alleviate this undesirable properties by changing the pulse shape over time while conserving the digital modulation format principle. Adjusting the pulse waveform is not new in UWB communications [START_REF] Silva | Spectrally efficient UWB pulse shaping with application in orthogonal PSM[END_REF], but here it is for randomization purpose and not for modulation. In case of an OOK-modulated signal, the proposed Multi-Waveform OOK scheme consists in transmitting different pulse shapes for the bit '1', each pulse being selected randomly from a particular set P containing N candidate waveforms, and no transmission is done for bit '0' as in the conventional scheme. The transmitted signal is then expressed as

w M W OOK (t) = k N j=1 b j,k d k p j (t -kT ) (1)
where d k ∈ {0,1} denotes the binary data to transmit, T is the frame duration, and b j,k ∈ {0, 1} stands for the j th bit of the N -bits coded number B k = (2 R(k) ) 2 , with R(k) being a random integer from the uniform distribution on the interval

[0, N -1]. Hence, N j=1
b j,k = 1 which means that only one pulse p j is triggered in the set P . Similarly, as illustrated in Fig. 1, a Multi-Waveform PPM can be expressed as

w M W P P M (t) = k N j=1 b j,k p j (t -kT -d k Γ) (2)
where Γ denotes the modulation index. As different pulses are involved in the randomization process, no correlation must appear between any 2 waveforms from the set P over the pulse interval T, (notice that frame duration is considered to be equal to the pulse period in this work) :

T 0 p m (t)p n (t)dt = 0 ∀m = n (3)
Following a similar approach than that used in [START_REF] Padgett | Physical-layer modeling of UWB interference effects[END_REF], closedform expressions of the power spectral densities (PSDs) can be easily derived for the proposed MWOOK/MWPPM formats (see Appendix A for an outline of the proof). For MWOOK, we get the following PSD expression:

S M W OOK (f ) = N j=1 P j (f ) 2 µ 2 d T 2 N - 1 N 2 + µ 2 d N 2 T 2 k δ f - k T (4)
where µ d stands for the expectation of the binary data d k and P j (f ) is the spectrum of pulse p j (t). For the MWPPM format, the PSD takes the form

S M W P P M (f ) = N j=1 P j (f ) 2 1 T 1 N - |µ m | 2 N 2 + |µ m | 2 N 2 T 2 k δ f - k T (5)
where µ m = cos(πf Γ) is the expectation of the random process m k = e -j2πf ε k in the spectral domain, with

ε k = d k Γ.
With the view to characterize the spectrum smoothness, we define the Discrete to Continuous Spectral Ratio (DCSR) as the factor multiplying the series of frequency comb lines divided by the continuous part of the spectrum. For the MWOOK scheme we obtain the following expression

DCSR M W OOK = 1 T 2N -1) (6)
whereas for MWPPM the ratio takes the form

DCSR M W P P M = |µ m | 2 T N -|µ m | 2 ) ( 7
)
The DCSR criterion has been established based on theoretical PSD expressions; so, this criterion will no longer be suited to spectral estimates. In the sequel, we will adopt a complementary principle for assessing the benefits of randomization, the Peak-to-Average Spectral Ratio (P ASR), corresponding to be the mean of discrete lines intensity divided by the average of the overall power spectral density. We will also consider the important criterion of power efficiency, defined as

η = P F max(P F ) (8)
where P F stands for the power collected over a frequency band of interest F (typically the [3.1-10.6] GHz band) for the electrical signal at antenna input, and max(P F ) denoting the total power evaluated over the same band for an OOK or PPM modulated signal based on the sinc pulse, which is optimal in the sense that it corresponds to a 100% spectrum use (full coverage of the spectral mask). From Eqs. 6 and 7, it can be clearly seen that increasing the number N of pulses tends to decrease the DCSR, which can result in a better power efficiency as will be illustrated in the next section. Note also that for a unit processing gain transmission, the data rate is R b = 1/T , so the DCSRs are linear with R b . A critical step for taking advantage of the randomized signals is the design of the set of candidate pulses P . One obvious constraint is to conserve energy per bit for all transmitted frames. In addition, we must ensure that the pulses are all mutually uncorrelated (Eq. 3). There are mainly two approaches to achieve this goal, either the transmitter relies on sine and cosine waves with use of local oscillators, or the transmitter directly operates in baseband (Impulse Radio system). The two solutions are investigated in the next section.

III. CARRIER-BASED VERSUS CARRIERLESS RANDOMIZATION A. Carrier-Based Randomization

To satisfy the decorrelation constraint for a system using local oscillators with inphase and quadrature components, we can design a set P of rectangular pulses modulated by sine waves at frequencies f n , n = 1, 2, ..., N/2; in this case, the orthogonality is ensured provided that

+ T 2 -T 2 sin( 2πf i t T )sin( 2πf j t T )dt = 0, ∀i = j (9)
This is also true in case of sine and cosine waves operating at same frequency. For the purpose of illustration, if we consider a system operating with a pulse period T of 0.5 ns, we can identify 4 frequencies satisfying the orthogonality condition over the [3.1,10.6] GHz band:

f 1 = 4 GHz , f 2 = 6 GHz , f 3 = 8 GHz , f 4 = 10
GHz; thus, with 4 local oscillators and using inphase and quadrature components we can get N = 8 pulses. With this approach, the pulse shape randomization results from a random selection of the sinusoid. As observed in fig. 2 and fig. 3, for both modulation types (MWOOK and MWPPM, respectively), the PASR decreases when increasing the number N of pulses, which means a lower intensity of discrete spectral lines and hence a much more effective randomization. As already pointed out, the frequency comb lines grow linearly with R b , so the benefits of using a large set P is more pronounced at high bit rate but at the price of an increased complexity. one based on 4 pulses. In fig. 5, the spectrum corresponds to 1 Gbps classical and multi-waveform PPM with same N as in MWOOK (4 pulses). The absence of discrete lines at even multiples of data rate is due to the zero-crossings of sinc spectrum resulting from the 0.5 ns rectangular pulse. PPM time shift is fixed at Γ = 0.5 ns, which is required for non-coherent energy detector receiver. The discrete spectral term is scaled by cos 2 (.) as described in ( 5) associated with spectral peaks located at even multiples of data rate, this is a bad spectral phenomenon as comb lines have a higher intensity for PPM than OOK. On the other hand, thanks to the zero crossings of oscillating term which eliminate spectral lines at odd multiples of bit rate, specially at 1 GHz inside GPS band ([0.96-1.61] GHz). As smoothing the power spectral density permits to step-up the spectrum level without violating FCC mask, we can see in fig. 6 a raise in the power efficiency as we increase the number N of randomizing pulses. Simulations were done for a 1 Gbps-MWPPM and 2Gbps-MWOOK. The lower power efficiency associated with PPM is due to the strong spectral peaks described above. In terms of average power, our randomization scheme shows a consequent improvement with respect to OOK and PPM as observed in fig. 7. That promises with a SNR enhancement when applied in real transmission systems.

B. Carrierless Randomization

The interest of the previous approach is that it gives us some performance bounds for MWOOK and MWPPM schemes, but evidently the use of local oscillator(s) translates into an increase of complexity with respect to a carrierless scheme, which may not be suitable for practical applications. In this section, we investigate the alternate solution of a baseband system (IR-UWB). With the aim of designing a simple low cost system, we focus on the case where only two pulse waveforms are used for achieving the randomization. There is not a unique choice for the two elementary pulses, but we typically have to respect key constraints such as zero correlation (3), spectral mask compatibility, constant energy, and low complexity of generation. In this study, we considered the 4-th and 5-th order Gaussian monocycles {g 4 (t), g 5 (t)}, with same pulse shaping factor σ, as they can easily be generated via CMOS circuits while offering favorable spectral properties [START_REF] Phan | 4.7 pJ/pulse 7th derivative Gaussian pulse generator for impulse radio UWB[END_REF]:

g 4 (t) = ( t 4 σ 8 - 6t 2 σ 6 + 3 σ 4 )e -t 2 2σ 2 (10)
g 5 (t) = ( -t 5 σ 4 + 10t 3 σ 2 -15t)e -t 2 2σ 2 (11)
For the conventional modulation schemes (OOK, PPM), the 5th Gaussian pulse with σ = 51 ps is known to be the optimal choice as it provides the maximum power efficiency [START_REF] Abraha | Routing of power efficient IR-UWB wireless and wired services for in-building network applications[END_REF]. If MWOOK or MWPPM is adopted, a slightly different shaping factor σ = 49 ps has to be chosen so as keep a constant energy. Thus, each pulse is no longer optimal with respect to the spectral mask but, as will be shown in the following, favorable power efficiencies will be achieved thanks to the proposed randomizing techniques. Time waveforms and corresponding spectra of utilized pulses are plotted in fig. 8 and fig. 9, respectively. The PASR versus data rate is studied in fig. 10 and fig. 11 for conventional and multi-waveform schemes. Due to the modulation index used for PPM (Γ = 0.5 ns), the maximum rate that can be achieved is 1 Gbps. Since the data rate R b is multiplied by the series of frequency combs (described in Eq. 4 and Eq. 5), we can observe a linear response for both modulation types.

IV. AN APPLICATION TO AN IR-UWB OVER FIBER SYSTEM

Until now, the proposed modulation schemes have been analysed only from a theoretical point of view and their benefits in the context of a realistic channel remains to be investigated. In this section, we consider an application to the Radio-over-Fiber (RoF) depicted in fig. 12. The transmitter relies on a Mach-Zehnder Modulator (MZM) with half-wave voltage V π of 6 V, biased at V DC = 1.5V π by the electrical IR-UWB signal (peak-to-peak voltage V pp = 0.25V π ) with a continuous wave light applied at its input (laser diode signal). The channel is made of a Single Mode Fiber (SMF) with an eventual use of an in-line amplifier (SOA) for the reach extension purpose. At the receiver side (just before wireless transmission), a photo-detector converts the optical power into an electrical voltage signal to be attenuated before entering the antenna (so as to meet FCC mask). The same ADS softwarebased model as in [START_REF] Khaleghi | Experimental validation of numerical simulations and performance analysis of a coherent optical-OFDM transmission system employing a semiconductor optical amplifier[END_REF] is adopted for the SOA, as it has proved to be highly accurate and in very good agreement with experimental data. For comparison purpose, we have examined first the performance gain of multi-waveform transmission in the absence of optical amplification. In fig. 13 and fig. 14 we can observe that the power efficiency initially exhibits an almost flat shape over several kilometers whatever the source power P s is. This is due to FCC spectrum violation before attenuator, hence the signal is systematically scaled so as to meet FCC limit. The slight decrease of the power efficiency in this portion of the characteristics, whose width depends upon the source power, is due to the signal attenuation in optical fiber (0.2 dB/km). It can be clearly seen that MWOOK and MWPPM schemes offer a significant improvement at short reach, as a result of randomization (lower PASRs). After a certain distance related to P s , there is a quick drop-off of the efficiency gain (fig. 14) which then tends towards zero because no spectral lines reduction can be achieved via randomization if the PSD at the attenuator input is already below FCC mask. The efficiency improvement at short reach is larger for MWOOK (close to 9%) than for MWPPM (slightly above 4%). For P s = 12 dBm, a performance increase with respect to standard modulation schemes is obtained up to 20 km and 25 km for OOK and PPM, respectively. The maximum distance over which the proposed schemes brings an improvement is around 40 km for a 15 dBm laser power (at this distance the spectrum at the attenuator input falls below the FCC limit). By considering the Fiber-to-the-Home (FTTH) network coverage zones mentioned in the literature [START_REF] Llorente | Ultra-wideband radio signals distribution in FTTH networks[END_REF], 40 km is not sufficiently large for evaluating the benefits of randomized transmission; thus a SOA was placed as in-line amplifier at this distance to get a reach extension at limited cost. A comparison of the PSDs corresponding to standard modulation schemes and MWOOK/MWPPM, for the whole stream of symbols at SOA output, is illustrated by fig. 15 and fig. 16. Lower PASRs can be clearly observed. For the PPM modulations, the spectrum envelope is influenced by the oscillating term with a period of 2GHz resulting from the 0.5ns shifting in time domain.

Power efficiency and corresponding performance gain are plotted in fig. 17 and fig. 18 versus fiber length, considering different values of the biasing current (I bias ), for a laser power of 15 dBm. An evident performance improvement can again be observed for proposed schemes until the distance matches a certain limit related to I bias . Injecting high values of biasing current extends the target distance nonlinearly, a better efficiency being obtained for MWOOK and MWPPM up to 95 km and 105 km respectively, when the SOA is biased at 200 mA. The received optical power at SOA input is -1 dBm, boosted by an optical gain of 4.47 dB, 10.58 dB, or 11.75 dB ; for I bias = 100 mA, 150 mA, or 200 mA, respectively. Another point to mention is that high biasing current tends to slightly lower the performance gain at short distances due to the nonlinear effects inherent to SOA amplification. So when the spectrum at the photo-detector output is above the FCC limit, linearity of the optical amplifier has a priority over output power response, that is why the best performance gain (close to 10%) is obtained when applying I bias =100 mA, provided that the receiver is located at a distance below 85 km (above this limit, MWOOK/MWPPM do not offer any advantage). Operating at larger biasing currents means slightly less power efficiency gain; we then have a trade off between reach extension and power efficiency.

V. CONCLUSION

In this paper, a new spectrum smoothing technique is developed on the basis of OOK or PPM modulation schemes, by applying a pulse shape randomization. The benefits of the approach is first demonstrated by deriving closed-form expressions of the power spectral densities. It is shown that using a greater number of pulses leads to a better spectral behavior, and consequently higher power efficiency. The technique is compatible both with carrier-based transmitters and with impulse radio (IR) systems. With the view to propose a very simple implementation, we investigated the particular case of an IR using two pulses belonging to the gaussian family (4-th and 5-th derivative). Further, we investigated an application to an IR-UWB over optical fiber transmission system operating at 1 Gbps for PPM format and at 2 Gbps for OOK. A significant improvement at short reach of the optical link has been demonstrated if a randomized transmission is adopted. An eventual in-line optical amplification via SOA is examined for reach extension purpose at low-cost, a realistic model of the component being used. The active distance over which the multi-waveform modulation schemes perform better is controlled by the biasing current injected into the SOA in a flexible way. Considering the antenna and wireless channel is among our future objectives, so as to quantify the benefits of the proposed solutions in terms of Bit-Error-Rate.

VI. APPENDIX

We give here an outline of the proof of equations ( 4) and (5), by following similar steps to those used in [START_REF] Padgett | Physical-layer modeling of UWB interference effects[END_REF]. Let's first adopt a general expression for the multi-waveform modulation schemes:

w(t) = k N j=1 b j,k a k p j (t -kT -ε k ) (12)
where the amplitude coefficient a k equals the binary data d k in the MWOOK case and equals zero for MWPPM; regarding the time shift, we have ε k = 0 for MWOOK and ε k = d k Γ for MWPPM. The transmitted signal can be alternatively expressed as a convolution product:

w(t) = N j=1 d j (t) * p j (t) (13)
where S D (f ) stands for the PSD of the pulse train and P j (f ) being the Fourier transform of p j (t). The discrete spectral lines resulting from the modulation pattern correspond to the term S D (f ) only; with the view to derive its expression, it can be written as

S D (f ) =< E d|D(f, t)| 2 dt > (15)
where < . > corresponds to time averaging and the accumulated energy spectral density at time t taking the form

|D(f, t)| 2 = | t -∞ d(τ )e -j2πf τ dτ | 2 (16)
After some algebra, it can be seen that the following equations holds:

|D(f, t)| 2 = k l
b k b k+l a k a k+l e 2πf T lj e 2πf j(ε k+l -ε k ) × u(t -kT ) u(t -kT -lT )

where u(t) is the unit step function. Then, it is straightforward to show that Finally, after time averaging and applying the Poisson formula l e j2πf T l = 1 T k δ(f -k T ), we get the expression

S D (f ) = 1 T σ 2 b σ 2 â + σ 2 b µ 2 a + µ 2 b σ 2 â + µ 2 b µ 2 a -µ 2 b µ 2 a |µ m | 2 + µ 2 b µ 2 a |µ m | 2 T k δ(f - k T ) (19)
where µ × corresponds to the mean of the process × and σ denoting the standard deviation of the corresponding zero mean process (e.g. b k = µ b + bk with E bk bk+l = σ 2 b δ[l]).

Fig. 1 :

 1 Fig. 1: A simple description for Multi-Waveform PPM transmission, where each frame consists of 2 time slots.

Fig. 2 :Fig. 3 :

 23 Fig.2: The Peak-to-Average Spectral Ratio versus number of pulses used for OOK randomization, at different data rates.The power spectral density for 2 Gbps classical OOK is shown in fig.4, besides to the spectrum of multi-waveform

Fig. 4 :

 4 Fig. 4: Power Spectral Densities of 2Gbps standard OOK (left) and multi-waveform OOK based on 4 pulses (right).

Fig. 5 :

 5 Fig. 5: Power Spectral Densities of 1 Gbps classical PPM (left) and MWPPM based on 4 pulses (right).

Fig. 6 :

 6 Fig. 6: Power Efficiency versus the number of pulses used for MWOOK and MWPPM.

Fig. 7 :Fig. 8 :

 78 Fig. 7: Average Power versus the number of pulses used for MWOOK and MWPPM.

Fig. 9 : 2 -

 92 Fig.9: The power spectral densities of 5-th optimal Gaussian used for standard OOK-PPM, besides to that of 4-th and 5-th pulses utilized with MWOOK-MWPPM.

Fig. 10 :

 10 Fig. 10: Peak-to-Average Spectral Ratio versus data rate for OOK-MWOOK based on 4 th and 5 th derivatives of Gaussian.

3 PPM

 3 using optimal 5-th gaussian 2-MWPPM using 4-th & 5-th gaussians

Fig. 11 :

 11 Fig. 11: Peak-to-Average Spectral Ratio versus data rate for PPM-MWPPM based on 4 th and 5 th derivatives of Gaussian.

Fig. 12 :

 12 Fig. 12: Block diagram of the Impulse Radio over Fiber system with Matlab/ADS co-simulation.

Fig. 13 :Fig. 14 :

 1314 Fig. 13: Power Efficiency versus Fiber Length, where MZM is injected by 15 dBm Laser Power.

Fig. 15 :

 15 Fig. 15: Power Spectral Densities for 2 Gbps OOK-MWOOK based on 4 th -5 th derivatives of Gaussian Pulses at SOA output, where I bias =100 mA.

Fig. 16 :

 16 Fig. 16: Power Spectral Densities for 1 Gbps PPM-MWPPM based on 4 th -5 th derivatives of Gaussian Pulses at SOA output, where I bias =100 mA.

Fig. 17 :Fig. 18 :

 1718 Fig. 17: Power Efficiency versus Fiber Length, using SOA placed at 40 km, and biased with 150 mA at 15 dBm.

R

  b (0)R a (0) E [δ(t -kT )] + l<0 k R b (l)R a (l)R m (l) e j2πf T l E [δ(t -KT )] + l>0 k R b (l)R a (l)R m (l) e j2πf T l E [ δ(t -KT -lT )] (18)where R b (l), R a (l) and R m (l) denote the auto-correlation functions of the independent processes b k , a k and m k = e -j2πf ε k , respectively.


H Taki

S Azou

A Hamie

A Al Housseini

A Alaeddine

A Sharaiha

Simple Pre-distortion Schemes for Improving the Power Efficiency of SOA-Based IR-UWB over Fiber Systems

Keywords:

In this paper, we investigate the usage of SOA for reach extension of an impulse radio over fiber system. Operating in the saturated regime translates into strong nonlinearities and spectral distortions, which drops the power efficiency of the propagated pulses. After studying the SOA response versus operating conditions, we have enhanced the system performance by applying simple analog pre-distortion schemes for various derivatives of the Gaussian pulse and their combination. A novel pulse shape has also been designed by linearly combining three basic Gaussian pulses, offering a very good spectral efficiency (>55%) for a high power (0 dBm) at the amplifier input. Furthermore, the potential of our technique has been examined considering a 1.5 Gbps-OOK and 0.75 Gbps-PPM modulation schemes. Pre-distortion proved an advantage for a large extension of optical link (150 Km), with an inline amplification via SOA at 40 Km. MZM Electrical IR-UWB signal SMF

I. INTRODUCTION

UWB is known to be a promising technology for shortrange broadband wireless communications, thanks to its many interesting features including high data rate, immunity to multipath propagation, flexibility in reconfiguring data rate and power, very good time domain resolution, accurate mobile user location, and easy data protection [START_REF] Ghavami | Ultra wideband signals and systems in communication engineering[END_REF] [START_REF] Aiello | Ultra-wideband wireless systems[END_REF]. Hence, UWB is a good candidate for various applications like local and wide area networks, sensor networks, emergency communications, radar, remote sensing, and military applications [START_REF] Oppermann | UWB: theory and applications[END_REF]. In 2002, the Federal Communication Commission (FCC) endorsed UWB devices to use the unlicensed frequency band [3.1,10.6] GHz, where the maximum emitted power must not exceed -41.3 dBm/MHz, in order to avoid interference with existing communication systems [4]. Impulse Radio (IR), consisting in transmitting ultra short duration pulses in baseband, is an attractive UWB technology due to its low cost and complexity [START_REF] Win | Impulse radio: How it works[END_REF], [START_REF] Fernandes | Recent advances in IR-UWB transceivers: An overview[END_REF]. Extending the coverage areas of UWB systems using optical fiber technology, is a pertinent approach that has been demonstrated in recent studies [START_REF] Lim | Fiber-wireless networks and subsystem technologies[END_REF]- [START_REF] Gomes | Radio-over-fiber transport for the support of wireless broadband services[END_REF]. UWB transmission over fiber requires direct optical pulse generation [START_REF] Zeng | All-optical UWB impulse generation based on cross-phase modulation and frequency discrimination[END_REF], [START_REF] Bolea | Optical UWB pulse generator using an N tap microwave photonic filter and phase inversion adaptable to different pulse modulation formats[END_REF] or electro-optical conversion via intensity modulators [START_REF] Yu | Ultra-wideband generation based on cascaded Mach-Zehnder modulators[END_REF]. Inline amplification is a key function for achieving the reach extension of any optical communication network. As low cost devices, Semiconductor Optical Amplifiers (SOAs) [START_REF] Connelly | Semiconductor Optical Amplifiers[END_REF] may be good candidates in the context of IR over fiber transmissions, the essence of IR being the simplicity with the objective of a wide use. Unfortunately, a main drawback for this approach is the non-linear effects inherent in the amplification, causing strong distortions in the propagating signals and a degradation in the power efficiency. Due to the regulation constraints associated to UWB free space propagation, highly efficient transmission is required which means that the pulse shapes must be preserved along the transmission in order to collect the maximum possible power under the FCC spectral mask. Hence, overcoming SOA non-linear distortions becomes a key objective. In the literature, numerous linearization techniques dealing with over fiber transmission have been investigated [START_REF] Roberts | Electronic precompensation of optical nonlinearity[END_REF]- [START_REF] Lowery | Fiber nonlinearity pre-and post-compensation for longhaul optical links using OFDM[END_REF]. Here, we examine the potential of analog predistortion against SOA non-linearity; by tuning some key parameters of pulses belonging to the Gaussian family or of their combinations [START_REF] Abraha | Routing of power efficient IR-UWB wireless and wired services for in-building network applications[END_REF], we have succeeded in designing highly efficient pulse shapes at the receiver side (photodetector output). The best parameters of pre-distorted input pulses were extracted via global optimization based on the interior point algorithm. The resultant waveforms have been examined considering modulated IR signals; it is shown that a significant enhancement in the efficiency can be achieved for a large optical link extension, while operating at limited cost and complexity.

II. IR-UWB OVER FIBER SYSTEM ARCHITECTURE AND SOA IMPACT ON OUTPUT PULSES RESPONSE

The Radio-over-Fiber (RoF) utilized is depicted in fig. 1. The transmitter relies on a Mach-Zehnder Modulator (MZM) with half-wave voltage V π of 6 V, biased at V DC = 1.5V π by the electrical IR-UWB signal (with a peak-to-peak voltage V pp ), a continuous wave light being applied at its input (laser diode signal). The channel is made of a Single Mode Fiber (SMF) and an in-line amplifier (SOA) for the reach extension purpose, with an injected bias current I bias . At the receiver side (just before wireless transmission), a photo-detector converts the optical power into an electrical voltage signal to be eventually attenuated before entering the antenna (so as to meet FCC mask). The same ADS software-based model as in [START_REF] Khaleghi | Experimental validation of numerical simulations and performance analysis of a coherent optical-OFDM transmission system employing a semiconductor optical amplifier[END_REF] is adopted for the SOA (INPHENIX-IPSAD1501), as it has proved to be highly accurate and in very good agreement with experimental data. Increasing the transmission distance requires operating in the saturated region of the SOA, which yields various nonlinear impairments affecting the transmitted signal. On the non-coherent receiver side, the resulting distortion will be characterized in terms of power efficiency with respect to the FCC mask, defined as:

η = 10.6 GHz 3.1 GHz S p (f ) df 10.6 GHz 3.1 GHz S F CC df × 100% (1)
where S p (f ) denotes the power spectral density (PSD) for a pulse p(t) [START_REF] Abraha | Routing of power efficient IR-UWB wireless and wired services for in-building network applications[END_REF]. η is computed after photo-detector while considering antenna effects are ignored. To have a look at SOA impact, fig. 2 shows the spectral response of the 5-th order derivative Gaussian pulse in electrical domain (solidblue) and after being passed through SOA (dashed-green), without propagation in fiber. The PSD of the optical pulse violates FCC in the most restricted region corresponding to GPS band [0.96 -1.61] GHz, where the low power level (-75.3 dBm/MHz) is greatly challenging. So the average signal power has to be reduced so as to meet this limit, leading to a significant impact on the power efficiency (degradation > 30%). Hence, spectrum enhancement requires eliminating the low frequency components induced by SOA. A simple criterion for evaluating this spectral impact corresponds to the ratio between the power collected in the [0,2] GHz band with respect to the overall PSD:

P SR 2GHz = 2GHz 0 S p (f )df ∞ 0 S p (f )df × 100% (2)
This criterion is analyzed in fig. 3 for the case of the 5 th derivative Gaussian pulse, with different peak-to-peak voltage and laser power, where I bias = 250 mA. A quick rise of the PSR can be clearly observed, especially for V pp > 0.5 V π where strong nonlinearities act on the pulse waveform. In the following, we investigate the benefits of electrical pulse precompensation with the view to avoid any power efficiency degradation. The power efficiency of the 5 th order Gaussian pulse is plotted in fig. 4 versus SOA input power, for I bias = 250 mA. When the Amplified Spontaneous Emission (ASE) noise is ignored, a degradation in the power efficiency is observed as we move towards higher input powers. The bad impact of SOA particularly occurs at large V pp (e.g. V pp = 0.75V π), where the drop in the efficiency results from the large spectral distortion. On the opposite, for V pp = 0.25V π, a semi-linear gain behavior is ensured, leading to a very good spectral efficiency (approximately 50%) over a broad range of input power, reaching the saturation region (0 dBm). In the noisy case, the power efficiency is evaluated via Monte-Carlo simulations with the computation of the average overs 100 runs; as can be clearly seen, the pseudo-linear gain characteristics is no longer preserved due to the larger noise influence at lower input power. Thus, there is a compromise between linearity and optical signal to noise ratio (OSNR), which depends upon the peak-to-peak voltage. As can be seen, the ASE impact on pulse dynamics tends to be reduced for large V pp ; it is a positive aspect but at the cost of nonlinearities. For V pp = 0.25V π, the rise in efficiency is due to the increasing OSNR, combined with limited nonlinear effects. Prior to strong saturation, a slight increase in the efficiency (< 5%) is noticed for the noisy signals over normal ones, for any V pp , that is due to the power added by ASE noise within the UWB mask. We can systematically verify the convergence between solid and dashed lines at very high input power, resulting from a negligible noise power. For the remaining part of this paper, we will assume a medium value V pp = 0.5V π, which is interesting in the sense that it both represents the noisy and nonlinear characteristics typically associated with SOA.

III. SOA NONLINEARITIES MITIGATION VIA ELECTRICAL PULSES PRE-DISTORTION

A broad variety of linearization techniques are discussed in the literature [START_REF] Roberts | Electronic precompensation of optical nonlinearity[END_REF]- [START_REF] Lowery | Fiber nonlinearity pre-and post-compensation for longhaul optical links using OFDM[END_REF], [START_REF] Zhang | Linearization Technologies for Broadband Radio-Over-Fiber Transmission Systems[END_REF] for reducing the nonlinear effects associated to optical communication systems. The different approaches can be classified into two main groups, electrical linearization and optical linearization. Optical schemes can operate over a wide bandwidth and some techniques are even specifically designed for SOA [START_REF] Tai | Reduction of nonlinear distortion in mqw semiconductor optical amplifier using light injection and its application in multichannel m-qam signal transmission systems[END_REF], [START_REF] Tabatabai | Feed-Forward Linearization Technique for Reducing Non-Linearity in Semiconductor Optical Amplifier[END_REF], but due to either high cost or bandwidth limitations associated to radio frequency control circuits, these schemes are not considered in our study. Regarding the methods operating in the electrical domain, digital linearization (pre-distortion, post-distortion) is interesting for its flexibility but we see the bandwidth/sampling requirements as a drawback for a low cost IR-UWB system; so, we preferred to investigate the alternative of analog predistortion, consisting in designing circuits at the transmitter side so as to suppress the nonlinear distortion components resulting from SOA amplification. As reported in various studies, tunable UWB pulse generators can be designed at low cost in CMOS technology [START_REF] Han | Ultra-wideband electronically tunable pulse generators[END_REF]- [START_REF] Thai-Singama | Demonstration of a Low-Cost Ultrawideband Transmitter in the 3.1-10.6-GHz Band[END_REF]. Considering such techniques, we made the assumption of a tunable Gaussian pulse generator, so that the generated pulse x(t, p) can be shaped with arbitrary parameters p: shaping factor for the case of a single Gaussian pulse, or pulse shaping factors, weighting coefficients, besides to the delay for a linear combination of Gaussian pulses. Then our aim is to identify the best tuning parameters p* in the sense of a maximized power efficiency at the SOA output. In this study, we solve this global optimization problem via an interior point algorithm [START_REF] Mitchell | Interior point methods for combinatorial optimization[END_REF], using a numerical model of the component, as illustrated in fig. 5. All distinct parameters are initialized at their optimum in electrical domain, so the algorithm can start from these values to re-optimize the pulse shape obtained at SOA output. Note that our approach presents some similarities with a work of Mirshafiei et al. [START_REF] Mirshafiei | Ultra-wideband pulse shaping: bypassing the inherent limitations of the Gaussian monocycle[END_REF], who addressed the pre-compensation against antenna effects, based on combinations of Gaussian pulses generated in the optical domain. Our contribution, whose a first investigation has been reported in [START_REF] Taki | Pulse Shape Pre-distortion for Improving the Power Efficiency of SOA-based IR-UWB over Fiber Systems[END_REF], is different in the sense that we aim at mitigating SOA nonlinearities, which are very specific, and we consider that the pulses are generated/pre-distorted in the electrical domain. Regarding the practical aspect, this process could be automatically implemented in a closed loop transceiver system, utilizing software-to-hardware interfaces.

In our setup, SOA was placed at 40 km, the operating point corresponding to an input power P SOA = 0 dBm, which is a strong gain saturation point, while MZM is driven by V pp = 0.5V π. These operating conditions are fixed for the remaining part of the study.

A. Case of 5-th derivative Gaussian pulse

The high power efficiency offered by the 5 th derivative Gaussian pulse, besides to the acceptable order of generation complexity, both make it an attractive target for impulse radio system designers [START_REF] Jin | A CMOS fifth-derivative Gaussian pulse generator for UWB applications[END_REF]. As shown in (3), only one variable could be calibrated in this waveform, which is the pulse shaping factor (σ). Such a parameter has a simultaneous influence on the 10 dB bandwidth (B 10 ) and on the central frequency. Figure 6 describes the original and pre-distorted (PD) pulses in electrical domain and at SOA output, in the case of placing the photo-detector directly after SOA, where the spectra are plotted in the noise-free case (ASE noise not considered for spectrum readability). We can clearly see that the pre-distorted pulse exhibits only a small improvement in power efficiency (< 5%), this is due to the single degree of freedom. Therefore it is better to search for more appropriate pulse shapes to be adopted with pre-distortion. x

5 (t) = A √ 2π -t 5 σ 11 + 10t 3 σ 9 - 15t σ 7 exp - t 2 2σ 2 (3)

B. Higher order of Gaussian derivative

In reality, to avoid hitting the GPS band, the spectrum must be shifted towards higher frequencies. The spectral peak for any n th order Gaussian pulse derivative

x n (t) is f peak = √ n 2πσ
, where the spectrum is expressed as

X n (f ) = A j2πf n exp - 2πf σ 2 2 (4)
To step up the power spectral density under FCC, we have to reduce B 10 by utilizing higher values of σ, as illustrated in (4). Increasing σ can be compensated by augmenting n, hence making a separate control for B 10 and f peak . Adopting a higher order of derivation gains more flexibility in shaping the spectrum, as the latter being far from the most severely powerrestricted band (corresponding to GPS), and consequently the design of a pre-distorted waveform becomes much easier. The benefit of the 6-th derivative Gaussian appears in fig. 7, with a time domain expression given in [START_REF] Win | Impulse radio: How it works[END_REF]. By lowering the 10 dB bandwidth of the pre-distorted pulse spectrum, we have enabled an increase in the efficiency (> 20%) without violating the regular limit. In this case, the pre-distortion block could be implemented as a simple differentiator, which can enhance the performance at limited complexity.

x 6 (t) = A √ 2π - 15 σ 7 + 45t 2 σ 9 - 15t 4 σ 11 + t 6 σ 13 exp - t 2 2σ 2 (5)

C. Abraha's Combination of Gaussian Doublets

Good results have been achieved with single pulse predistortion; unfortunately, the number of adjustable parameters limits the improvement in the spectral efficiency. A more effective technique is to apply a combination of waveforms, in order to get a superior control of the resultant pulse dynamics, as will be presented in the sequel. Some combinations have been proposed in the literature [START_REF] Wang | A GA-based UWB pulse waveform design method[END_REF] [START_REF] Matsuo | Design of UWB pulses based on B-splines[END_REF], but taking into account practical constraints, it is better to conserve simple combinations within the Gaussian family. We have considered the work of Abraha et al., who have recently developed highly effective pulse shapes based on either monocycles or doublets [START_REF] Abraha | Routing of power efficient IR-UWB wireless and wired services for in-building network applications[END_REF]. As the latter approach relies on higher order of Gaussian derivative, with a better spectral efficiency, it will be the candidate in our work. Hence, two basic Doublets x 21 (t), x 22 (t) will be considered, having different weighting coefficients and shaping factors:

C D (t) = a 1 x 21 (t) + a 2 x 22 (t -τ ) (6)
x 2i (t) = A √ 2πσ 3 i k t 2 σ 2 i -1 exp - t 2 2σ 2 i (7)
a 1 and a 2 are the weighting factors, τ is a time delay between pulses, fixed at τ =25 ps, and k = 1.16 is an arbitrary scaling of the modified Doublet. Table I displays all the parameters of utilized pulses, besides to the power efficiencies in electrical domain and at SOA output (the spectral characteristics being evaluated at this point). BL 34 is defined as the left -34 dB bandwidth (frequency space between f L and f peak ), this criteria is useful to evaluate the spectrum in correspondence with GPS band. The term A is always set at 1, as it has no influence on time dynamics for single waveforms; and in case of combinations, only the relative amplitude between basic pulses is important, since the real intensity for signals at photodetector output is controlled by the laser power and peak- to-peak voltage. In Abraha's work, the doublets are similar with opposite polarities, as a high pass filtering is made via spectral oscillating term resulting from time delay. We don't preserve these conditions with pre-distortion, as the target is to mitigate SOA non-linearity by exploiting all the degrees of freedom associated to the linear combination of pulses. Hence, different shaping factors and weighting coefficients have been considered in our work. A significant jump in the power efficiency (> 45%) is illustrated in fig. 8 for the predistorted combination of Doublets. The increase in the 10 dB bandwidth from 6.84 GHz to 7.34 GHz has enlarged the spectral coverage area under FCC mask, besides to the 1.5 GHz reduction in BL 34 , which permits a step up in the PSD without exceeding GPS limit. The peak of the spectrum exactly matches FCC limit (-41.3 dBm/MHz) at f peak = 6.2 GHz, which is not so far from the center of the UWB mask, thanks to pre-distortion.

D. A new waveform based on a combination of fundamental Gaussians

Increasing the number of tuning parameters in a waveform has shown to be more pertinent for pre-distortion. A new pulse shape is now proposed, based on a linear combination of three fundamental Gaussian pulses:

C G (t) = a 1 g 1 (t) + a 2 g 2 (t) + a 3 g 3 (t) (8)
where

g i (t) = exp - t 2 2σ 2 i ( 9
)
Such a combination enables a pre-distortion with 5 variables (σ 1 , σ 2 , σ 3 , a 2 /a 1 , a 3 /a 1 ), which promises for a better power efficiency obtained at SOA output. The weighting coefficients have been optimized as described in table I. The resultant waveform is depicted in fig. 9 (left); it has a negligible DC component due to the opposite polarities of Gaussian amplitudes, with an energy being concentrated at a higher frequency range due to zero crossings. The peak frequency is 6.14 GHz, where the 3 dB and 10 dB bandwidths are B 3 = 4.1 GHz and B 10 =7.5 GHz respectively, which is in very good agreement with the UWB spectral mask. As illustrated in fig. 9 (right), a highly efficient pulse waveform is obtained, offering a power efficiency of η=56.47%, which outperforms the 5-th derivative Gaussian pulse (50.7%) and is close to Abraha's combination of Doublets (57.2%) in electrical domain. An important feature associated with our approach is the absence of any time differentiator nor delay line, which reduces the system cost and complexity [START_REF] Abraha | Routing of power efficient IR-UWB wireless and wired services for in-building network applications[END_REF], [START_REF] Jin | A CMOS fifth-derivative Gaussian pulse generator for UWB applications[END_REF], [START_REF] Sheng | On the spectral and power requirements for ultra-wideband transmission[END_REF]. As can be seen in fig. 10, our combination has carried out the best power efficiency at SOA output (η = 56.1%), which performs better than Abraha's combination of Doublets (54.1%), thanks to the larger number of tuning parameters. After SOA, the pre-distorted spectrum has a close peak frequency to the original one (6.1 GHz), but covering a larger 10 dB bandwidth (7.45 GHz). Here there is a special case, the original spectrum at SOA output hits FCC mask at low frequencies due to the presence of a significant left side lobe, the main lobe staying far from the FCC limit even at the 34 dB band. This spectral characteristic results from the combining of time spread and windowing effect, which can be efficiently overcome by designing a predistorted pulse (better distribution of the pulse energy inside the time window). Unfortunately, the pulse undergoes a time spread after amplification, thus the energy outside the 0.5 ns slot is no more negligible, which translates into significant sidelobes in frequency domain (sinc influence). The predistortion processing has enabled a significant change in the pulse dynamics so as to render this time windowing effect negligible.

E. Bias current influence

Efficient pulses in electrical domain are usually more affected by the optical system non-linearity. We can see in fig. 11 that for low biasing currents (100 mA), the 5 th and 6 th Gaussian pulses have a small degradation in the power efficiency; whereas a significant decrease (>30%) is noticed for the combination of Doublets. Applying a larger biasing current increases the order of nonlinearity, which leads to lower spectral efficiencies for all pulses. Nevertheless, we observe that no great change occurs after 200 mA, due to the optical gain saturation at 0 dBm input power. Pre-distortion could be applied with all biasing currents, but with the view to achieve a reach extension the value of 250 mA is adopted, as associated with the highest optical power at SOA output. For such conditions, our pre-distorted pulse offers a neat advantage over the 5-th or 6-th derivative Gaussian waveforms and is slightly more effective than Abraha's pre-distorted pulse. Until now only theoretical models have been considered for the different pulse shapes. But from a practical point of view, some uncertainties are expected to occur in the CMOS circuits that could be designed to generate the various pulses. So, we investigate in this sub-section the impact of any parametric deviation with respect to the optimum solution discussed previously. Figure 12 illustrates how the power efficiency is affected by a parametric uncertainty for the 5-th and 6-th Gaussian pulses, together with the Doublets combination. G 5 and G 6 show a kind of immunity against systematical errors, as σ is the unique variable which controls the dynamics of a single pulse. The problem appears with Abraha's combination, which exhibits a relative robustness against σ 1 and a 2 deviations; unfortunately, any slight change in σ 2 is enough to drop off the spectral efficiency, specially for negative deviation errors, knowing that it is not a simultaneous change in the combination variables. We can figure out that the newly designed pulse is the most sensitive against system uncertainties, as illustrated in fig. 13. In fact, it is much more influenced than the other waveforms, as it relies on a larger number of pulses, and all parameters have been optimized simultaneously, so any change in one of the variables makes all the set no more optimal. Besides, pure Gaussian pulses have strong DC component, which can re-appear in our combination due to any lose in the basic pulses symmetry (positive/negative symmetry which cancels the DC), and consequently lead to a fast drop in the power efficiency ( η < 5% for a deviation error of 10%).

F. Robustness against uncertainty for single waveforms and combinations

IV. CASE OF A MODULATED IMPULSE RADIO SIGNAL PROPAGATING OVER OPTICAL FIBER LINK

In our simulations, we take into account the effects of the optical link (attenuation and chromatic dispersion) before SOA, as they have an influence on the pulse dynamics entering the amplifier, depending on the covered distance, fiber input power, and peak-to-peak power swing. Therefore, the optimum parameters of pre-distorted waveforms have been extracted considering SOA located at 40 km, where the operating conditions are the same as mentioned in table I. In order to evaluate our pre-distortion scheme considering a stream of modulated pulses, a new criterion of power efficiency has been defined as η M = P F max(P F )

where P F stands for the power collected over a frequency band of interest F (typically the [3.1-10.6] GHz band) for the electrical signal at antenna input, and max(P F ) denoting the total power evaluated over the same band for an OOK or PPM modulated signal based on the sinc pulse, which is optimal in the sense that it corresponds to a 100% spectrum use (full coverage of the spectral mask). The advantage of pre-distortion is depicted in fig. 14 for a 1.5 Gbps OOK, so we can gain an improvement over a large enough traveling distance (40 km to 150 km), knowing that no pre-amplifier has been placed for modulated signals before the antenna. The degradation in the curves is due to frequency shaping and deviation caused in fiber [START_REF] Pan | Performance evaluation of UWB signal transmission over optical fiber[END_REF] (with a chromatic dispersion of 17 ps/nm/km), so the central frequency and B 10 are no more preserved specially at large extension link. This improvement is justified by the fact that, the envelope of spectral spikes for OOK or PPM takes the shape of basic transmitted waveform, so perfect pulse shapes lead to an efficient spectrum associated with the modulated impulse radio signal. Between 40 km and 100 km, we can see a flat response in the curves corresponding to combinations of pulses, which results from the fact that all spectra are already above FCC limit, thus the signal has to be systematically attenuated so as to respect the mask. Our waveform exhibits a faster degradation than Doublets at large distances, as it has less immunity against fiber effects. Regarding practical aspects, we can say that power efficiency enhancement comes at the price of a larger complexity; depending on application constraints, adopting higher order of derivation with a single pulse shape could be more suitable than applying a combination of pulses, and utilizing a combination of two pulses may be much simpler than using a set of three pulses. The PSD of the C G -based OOK signal at SOA output is plotted in fig. 15, with no attenuation. Apparently, the spectral shaping at low frequency band is the reason beyond efficiency enhancement, so a higher power level can be achieved by the spectra when adapted to FCC mask, as shown in fig. 16 for a 0.75 Gbps PPM. Figure 17 proves that the same conclusion could be drawn for PPM regardless the covered distance in the optical link, where PPM index is 0.5 ns. Therefore, the proposed pre-distortion technique has always validated a good potential, while reaching a coverage of 150 km, fitting the request of Fiber-to-the-Home (FTTH) networks reported in the literature [START_REF] Llorente | Ultra-wideband radio signals distribution in FTTH networks[END_REF]. The improvement in the power efficiency translates into a higher signal-to-noise ratio obtained at the antenna output, which promises with a better bit error rate performance in energy detection receivers.

V. CONCLUSION

A low cost SOA in-line amplifier has been considered in this study for the purpose of reach extension of IR-UWB over fiber systems. Based on a realistic SOA physical model, analog pre-distortion schemes have been investigated for coping with SOA nonlinearities and with the view to improve power efficiency. Two linear combinations of Gaussian pulses were utilized, the first is based on Abraha's combination of Doublets, and the second has been newly designed based on pure Gaussians. Both have proved to be highly effective for correcting SOA impact. Using higher orders of Gaussian derivatives is an alternative solution offering a lower potential but with less complexity. Over a large extension of the optical link, our pre-distortion approach has exhibited an enhancement in the power efficiency of 1.5 Gbps-OOK and 0.75 Gbps-PPM. Experiments will be conducted in our future work so as to validate the proposed schemes, while considering bit error rate analysis and including antenna effects.

Fig. 1 :

 1 Fig. 1: Block diagram of the Impulse Radio over fiber system with Matlab/ADS co-simulation.

Fig. 2 :Fig. 3 :

 23 Fig. 2: Power Spectral Density of 5 th Gaussian in electrical domain and after SOA (regular and irregular with FCC).

Fig. 4 :

 4 Fig. 4: Power Efficiency of 5 th Gaussian versus SOA input powerfor several peak-to-peak voltages, where no fiber is considered.

Fig. 5 :

 5 Fig. 5: Block diagram for the optimization process used for designing pre-distorted electrical waveforms.

Fig. 6 :

 6 Fig. 6: Original (Org) and pre-distorted (PD) 5 th Gaussian, in electrical domain (Elec) and after SOA (Opt).

Fig. 7 :

 7 Fig. 7: Original (Org) and pre-distorted (PD) 6 th Gaussian, in electrical domain (Elec) and after SOA (Opt).

Fig. 8 :

 8 Fig. 8: Original (Org) and pre-distorted (PD) C Doub , in electrical domain (Elec) and after SOA (Opt).

Fig. 9 :

 9 Fig. 9: Our proposed combination CG based on the pulses gi, in time (left) and frequency (right) domain.

Fig. 10 :

 10 Fig. 10: Original (Org) and pre-distorted (PD) CG, in electrical domain (Elec) and after SOA (Opt).

Fig. 11 :

 11 Fig. 11: Power Efficiency of original and pre-distorted pulses directly after SOA (at 40 km) for several biasing currents.

6 Fig. 12 :

 612 Fig. 12: Robustness against uncertainty for the 5 th and 6 th Gaussian pulses, besides to Abraha's combination of Doublets.

3 Fig. 13 :

 313 Fig. 13: Robustness against uncertainty for our proposed pulse based on fundamental Gaussians.

Fig. 14 :

 14 Fig. 14: Power Efficiency versus fiber length for 1.5 Gbps OOK based on the utilized pulses, P-SOA=0 dBm, I bias = 250 mA, Vpp = 0.5V π, where SOA is placed at 40 km.

Fig. 15 :

 15 Fig. 15: Irregular PSD of 1.5 Gbps OOK based on original (left) and pre-distorted (right) CG(t) after SOA (at 40km), where P-SOA=0 dBm, I bias = 250 mA, Vpp = 0.5V π.

Fig. 16 :Fig. 17 :

 1617 Fig. 16: Regular PSD of 0.75 Gbps PPM based on original (left) and pre-distorted (right) CG(t) after SOA (at 40km), where P-SOA=0 dBm, I bias = 250 mA, Vpp = 0.5V π.

TABLE I :

 I Parameters (σ1, σ2, σ3, a2/a1, a3/a1) of electrical generated pulses, and the spectral characteristics (f peak , B10, BL34 ) of corresponding waveforms at SOA output (placed at 40 km). Besides to the power efficiencies (%) in electrical domain and after SOA, with PSOA = 0 dBm, I bias = 250 mA, and Vpp = 0.5 Vπ.