## Abstract

A semi-analytical simulation method (SASM) is proposed to evaluate the signal-to-noise ratio (SNR) of time stretched signals at the output of photonic analogue-to-digital converter (Ph-ADC) system. Analytical expressions of the signal at Ph-ADC output considering generic electrical signals applied to the electro-optic modulators of the Ph-ADC are derived. The contribution to the total variance of the received signal from the noise introduced by the electrical transmitter and receiver, and by the optical amplifier are derived analytically taking into account the pulsed nature of the optical signal. The proposed SASM shows excellent agreement of SNR estimates with the estimates provided by Monte Carlo simulation. This result is confirmed for variance dominantly imposed by the noise introduced by the electrical transmitter, by the optical amplifier and by the electrical receiver. A simplified approach is also proposed and compared with previous work. It is shown that mean power estimates obtained from this simplified approach are valid while the modulator is operating in the linear region and the signal is not affected by the frequency response of the electrical receiver filter. Additionally, it is concluded that the estimates of the noise variance due to the electrical transmitter are acceptable when a small signal analysis of noise along the Ph-ADC is valid.

© 2011 Optical Society of America

## 1. Introduction

Currently, state-of-the-art arbitrary waveform generators and real-time scopes are used in high bit-rate optical fiber communications experiments as commercially digital-to-analogue (DAC) and analogue-to-digital converters (ADC) with adequate specifications are not still available [1]. In order to increase the effective bandwidth of electronic ADCs, time stretching (TS) photonic (Ph) ADC systems have been proposed [2–6]. These systems are based on a photonic structure where the electrical signal modulates a tight optical pulsed signal previously chirped due to propagation over a spool of dispersive optical fiber. The optical signal is then propagated over a second spool of fibre in order to achieve the desired TS [2].

Recently, a different application for the TS Ph-ADC has been proposed [7–9]. In this novel application, the TS Ph-ADC system is used to compress the spectrum of ultra-wideband (UWB) radio signals captured from sensor antennas that are strategically located inside home premises. Digital signal processing (DSP) is then applied to the compressed signal, and localization, fingerprinting and spectral management of UWB transceivers is performed. With this approach, standard electronic ADCs can be used to monitoring the whole UWB band (from 3.1 until 10.6 GHz) and management protocols can be used to increase the low equivalent isotropic radiated power (EIRP) allowed for UWB picocells. Some of the main constraints of this TS Ph-ADC system are the very low signal power captured by the sensor antenna due to the low EIRP of UWB transceivers, additional losses due to signal propagation from the transceiver up to the sensor and the fiber losses of the photonic structure of the Ph-ADC. Thus, a strong electrical and optical amplification needs to be performed in the sensor and in the optical part, respectively, resulting in signal-to-noise ratio (SNR) degradation due to the noise introduced.

Monitoring of the SNR of the TS received signal is of special interest as it will be conditioning the performance of DSP algorithms. The SNR analysis of conventional TS Ph-ADC systems was already performed in [2]. In that work, the impact of the TS effect on the SNR dominantly impaired by different noise types was discussed and it was concluded that the SNR is usually dominantly impaired by amplified spontaneous emission (ASE) noise. However, there are two important issues in the novel TS Ph-ADC system that have not been analyzed in detail yet: i) although the SNR of the TS signal presents a significant fluctuation along time due to the pulsed nature of the signal generated by the optical source, as experimentally shown in [8], the SNR expressions presented in [2] do not depend on time as, to simplify the analysis, the pulsed nature of the signal was neglected and ii) the SNR may be dominantly impaired by the noise introduced at the sensor rather than by ASE noise due to the high amplification of the captured UWB signal that may be required.

In this work, analytical expressions for the current at the Ph-ADC output and for the variance contributions of the TS received signal due to the electrical noise added in the sensor, the ASE noise introduced by the optical amplifier and the noise introduced in the electrical receiver are derived. With these expressions, a semi-analytical method is proposed to evaluate the SNR of conventional TS Ph-ADC systems and of the novel TS Ph-ADC system as well, being a powerful tool for the TS Ph-ADC system optimization. In addition, a simplified approach with low analytical and computational complexity that allows evaluating the mean power of the received signal and the variance due to the electrical transmitter noise is also proposed.

## 2. TS Ph-ADC system description

Figure 1 shows the diagram of the TS Ph-ADC system considered in this work [9]. The optical source generates narrow pulses with a repetition rate, *f _{rep}*, that are launched into a dispersive spool of fiber with a length,

*L*

_{1}. Before electro-optic modulation, the optical signal is amplified, filtered and divided by

*N*branches. The optical chirped pulses of each one of the

*N*branches are used to sample (the term “sample” here means that, in a given time instant, the electrical signal is converted to the optical domain only when there is an optical pulse feeding the modulator) and map into the optical domain snapshots of the UWB radio signals captured by the sensors. The

*N*sensors are spread along one UWB picocell [7] and each sensor consists in one antenna, one electrical amplifier and one band-pass filter (BPF) used to reduce the noise power. After electro-optic modulation, the signal in each branch is delayed, combined and launched to a second spool of dispersive fiber with length,

*L*

_{2}. The time delays used in each branch are designed in order to ensure that, after combination, the pulses snapshots coming from each branch are not overlapped in time. This condition restricts the range of the pulse repetition rate. After combination, the time period of the multiplexed signal is given by

*T*≥ (

_{p}*Nf*)

_{rep}^{−1}. The second spool of fiber allows for the TS of the different optical pulses and, consequently, of the radio signal snapshots. The TS factor,

*M*, depends on the ratio between the accumulated dispersion of the first and second spools of fibre [2], $M=1+\frac{{L}_{2}{D}_{2}}{{L}_{1}{D}_{1}}$, where

*D*

_{1}and

*D*

_{2}are the dispersion parameters of first and second spools of fiber, respectively. After the second spool of fiber, a polarizer can be used to reduce the influence of the noise power on the system performance. When used, the polarizer is aligned with the optical field polarization as this is the desired situation in actual systems. After photodetection, the (compressed) spectrum of the received signal is composed by two main components: the compressed radio-frequency (RF) spectrum of the UWB signals captured by each sensor and the low frequency spectrum that arises due to the optical pulses. The BPF located after the PIN is used to reduce the power of the low frequency spectrum, as will be seen afterwards.

## 3. Signal at TS Ph-ADC output

Lets define the optical signal at the input of the electro-optic modulator (EOM) of each branch by
${e}_{EOM,i}^{\left(k\right)}\left(t\right)$ (the noise added by the optical amplifier is included in this signal). The index *k* identifies the sensor number, 1 ≤ *k* ≤ *N*, with *N* the number of sensors. In this way, the theoretical derivations presented in this work remain valid for optical sources generating chirped or chirpless pulses, with arbitrary shapes and optical power levels, and for any number of sensors.

Considering that the power of the UWB radio signal is sufficiently low to keep the EOMs operating in the linear regime (as it may be desired in order to avoid harmonics and intermodulation products occurring with high power levels), the nonlinear EOMs characteristic [10] can be linearized around the bias point. Hence, the optical signal at the output of each EOM can be represented as
${e}_{EOM,o}^{\left(k\right)}\left(t\right)={e}_{EOM,i}^{\left(k\right)}\left(t\right){v}^{\left(k\right)}\left(t\right)$, where the signal *v*^{(}^{k}^{)}(*t*) is given by:

*m*

^{(k)}is a parameter proportional to the EOM modulation index,

*b*

^{(k)}is the EOM bias point and ${n}_{e}^{\left(k\right)}\left(t\right)$ represents the electrical noise arriving at the EOM. Before propagation over the 2

*spool of fiber, the signal of each sensor is delayed and combined in time domain with signals coming from the other sensors. Hence, the resulting multiplexed signal can be described as ${e}_{mux}\left(t\right)={\sum}_{k=1}^{N}{e}_{EOM,o}^{\left(k\right)}\left(t-{T}^{\left(k\right)}\right)$, where*

^{nd}*T*

^{(k)}is the delay applied to the signal coming from the

*k*-th sensor. Describing the 2

*spool of fiber by its impulse response,*

^{nd}*h*

_{2}(

*t*), the complex amplitude of the optical field arriving at the PIN input can be represented as:

*R*, the photocurrent can be written as:

_{λ}*z*} is the real part of

*z*. Equation (5) considers that the frequency rate of the optical source and the time delays of each sensor have been carefully designed to avoid overlapping between consecutive pulses of the TS multiplexed signal at PIN input. This means that, the beats between the pulses carrying UWB signal snapshots from different sensors performed by the PIN are null as they are separated in time. In addition, Eq. (5) shows that the photocurrent may be split in three contributions: i) the first term represents the static photocurrent and the shape of this term is strongly related with the shape of the optical pulsed source; ii) the second term gives the electrical time stretched signal and iii) the third term represents distortion components. These results are in agreement with the particular case studied in [2], where a sinusoidal wave was considered as the electrical signal applied to the EOM.

The current at the output of the BPF of the electrical receiver can then be written as:

*h*(

_{r}*t*) is the impulse response of the BPF.

## 4. Characterization of the noise at the Ph-ADC output

In this section, the variance of the signal at the TS Ph-ADC output due to the noise introduced by the electrical sensor, the optical amplifier and the electrical receiver is analytically derived.

Due to the losses introduced by the two spool of fibres used in the photonic structure of the Ph-ADC system to provide the adequate TS of the RF electrical signal, an optical amplifier must be included along the system. The ASE noise introduced by this amplifier will contribute for the SNR degradation of the TS signal at Ph-ADC output being necessary, for SNR purposes, to evaluate its power contribution.

The main sources of electrical noise may be located in two different points of the Ph-ADC structure: at each sensor and at the receiver side. Though the signal variance due to the noise of the electrical receiver is of straightforward evaluation, the influence of the sensor noise on the received TS signal must be carefully evaluated as it will be converted into the optical domain being its variance at the Ph-ADC output affected by the high peak power of the optical pulses.

#### 4.1. Noise introduced by the sensors

In order to derive analytically the contribution to the variance of the received TS signal due to the noise introduced by each sensor, noiseless optical amplifiers and electrical receiver are considered in this section.

The variance due to the sensors noise is given by:

*i*(

_{o}*t*) is given by Eq. (6) and

*R*is the load resistance. The derivation of a closed form expression that allows evaluating the variance from Eq. (7) is performed in Appendix A. This variance can be approximated by:

_{L}*w*

^{(k)}(

*t*,

*f*) is presented in Appendix A and it represents the pulsed nature (due to the optical pulsed signal) of the variance contribution due to the sensor noise. The voltage noise arriving at the EOM is modeled as zero mean additive Gaussian noise with two-sided power spectral density (PSD) ${S}_{e}^{\left(k\right)}\left(f\right)=2{k}_{B}T{R}_{L}{f}_{e}^{\left(k\right)}{\left({g}_{e}^{\left(k\right)}\right)}^{2}{\left|{H}_{BPF,i}^{\left(k\right)}\left(f\right)\right|}^{2}$, where

*k*is the Boltzmann constant and

_{B}*T*is the absolute temperature in Kelvin. ${f}_{e}^{\left(k\right)}$ and ${g}_{e}^{\left(k\right)}$ are the noise figure and the voltage gain of the electrical amplifier of the

*k*-th sensor. It should be stressed that, along this work, letter

*g*is used to identify the gain in linear units while

*G*is used to represent the gain in decibel.

#### 4.2. Noise introduced by the optical amplifier

The derivation of the expressions that allow evaluating the contribution to the variance due to the ASE noise introduced by the optical amplifier considers noiseless sensors and receiver. The optical amplifier is an erbium doped fiber amplifier (EDFA) modeled by a zero mean additive white Gaussian noise with ASE PSD along one (⊥ or || to the signal) polarization direction given by *S _{o}* =

*hνn*(

_{sp}*g*– 1), where

_{o}*h*is the Planck’s constant,

*ν*is the central frequency of the optical signal spectrum,

*n*is the spontaneous emission factor and

_{sp}*g*is the EDFA gain.

_{o}Although conventional intensity modulation/direct detection optical communication systems have been considered in [11] to evaluate the variance contribution due to the ASE noise, a similar procedure can be used in TS Ph-ADC systems by taking into account that a pulsed optical signal arrives at the PIN input. From [11], the total variance due to ASE noise after a direct-detection optical receiver and electrical filtering is given by:

*p*describes the influence of the polarizer on the ASE-ASE noise beat variance and

*p*= 1 or

*p*= 2 in the presence or absence of a polarizer, respectively.

*S*is the PSD level of the ASE noise along one polarization direction at the optical filter input,

_{ASE}*h*(

_{o}*t*) is the inverse Fourier transform of the low-pass equivalent of the optical filter transfer function,

*H*(

_{o}*f*). For an optical amplifier located at a point of the system from which the optical power conversion up to the PIN input is Γ,

*S*is obtained from the PSD of the ASE noise added by the EDFA through

_{ASE}*S*= Γ

_{ASE}*S*.

_{o}In the case of the TS Ph-ADC system, some simplifications to Eq. (9) can be accomplished taking into account the nature of the signals involved in the system. For low modulation indexes, we can assume for the noise variance analysis that the instantaneous optical power is approximately the power of the optical pulses. Additionally, as the bandwidth of the electrical part of the receiver should be higher than the received signal bandwidth, we can approximate also *e _{PIN}* (

*χ*)

*h*(

_{r}*t*–

*χ*) ≈

*e*(

_{PIN}*t*)

*h*(

_{r}*t*–

*χ*). The bandwidth of the optical devices should be very large to avoid the broadening of the time aperture of each pulse along the optical structure. As a consequence, the bandwidth of

*H*(

_{o}*f*) is much larger than the bandwidth of

*H*(

_{r}*f*) and therefore,

*h*(

_{o}*χ*) can be approximated by

*h*(

_{o}*χ*)≈

*H*(0)

_{o}*δ*(

*χ*), leading to

*h*(

_{r}*t*–

*χ*) *

*h*(

_{o}*χ*) ≈

*h*(

_{r}*t*–

*χ*)

*H*(0). Substituting these simplifications in Eq. (9), the first term, corresponding to the s-ASE term, can be rewritten using the Parseval’s relation as:

_{o}From the comparison between the s-ASE noise beat term given by Eq. (10) and the ASE-ASE noise beat term presented in Eq. (9), it can be concluded that the s-ASE noise beat term dominates at the peak time instant when |*e _{PIN}*(

*t*)|

^{2}≫

*S*

_{ASE}*B*

_{n,o}, where it has been considered that the frequency-dependence of the PSD of the ASE noise can be approximated by a rectangular shape with width

*B*

_{n,o}(equivalent filtered ASE bandwidth).

#### 4.3. Noise due to the electrical receiver

The equivalent transfer function, *H _{r}*(

*f*), of the receiver BPF represented in Fig. 1 accounts for the frequency limitations imposed by the electrical part of the Ph-ADC receiver on the TS received signal and also the gain of the electrical receiver amplifier. Thus, this equivalent filter is composed by an electrical amplifier characterized by its current gain,

*g*, and the noise figure,

_{r}*f*, and the filter used to eliminate the low-frequency high power spectrum due the optical pulsed signal and to reduce the noise power:

_{r}*h*(

_{r}*t*) =

*g*

_{r}

*h*

_{BPF,r}(

*t*). The electrical current noise due to the receiver referred to the amplifier output is modeled as additive Gaussian electrical noise with two-sided PSD, ${S}_{r}=2{k}_{B}T{f}_{r}{g}_{r}^{2}/{R}_{L}$. Thus, the variance due to the electrical receiver noise at the system output is given by:

## 5. Evaluation of the SNR at the Ph-ADC output

Due to the pulsed nature of the signal and of the noise variance at the output of the TS Ph-ADC system, the evaluation of the SNR is performed along one period of the received TS multiplexed signal. The period of the multiplexed signal is given by the ratio between the period of the signal generated by the optical source and the number of sensors of the system. The following procedure is used to estimate the SNR of the received TS multiplexed signal:

- Evaluation of the instantaneous power of the TS received signal, from Eq. (6) or by numerical simulation.
- Evaluation of the mean power of the TS received signal. The mean power is evaluated over the set of periods of the received multiplexed signal and thus, the variation of the mean power along the period is obtained. In order to get stabilized mean power estimates, several signal periods must be considered. The number of periods needed to get stabilized estimates will be analyzed later in this work.
- Evaluation of the total variance along one period by performing the mean of the variances over the set of periods of the received multiplexed signal. Notice that, as the different variance contributions are mainly imposed by the power of the optical pulses, the variance of the different periods should be similar.
- Evaluation of the SNR along one period of the received multiplexed signal given by the ratio between the mean power estimated in step 2 and the variance estimated in step 4.

The procedure mentioned above can be represented analytically by:

*SNR*(

*t*),

*p*(

_{m}*t*) and ${\sigma}_{t,m}^{2}\left(t\right)$ are the SNR, the mean power and the variance of the received TS signal along the period of the multiplexed signal, respectively, and

*N*is the number of periods.

_{p}*p*(

_{k}*t*) and ${\sigma}_{k}^{2}\left(t\right)$ are the instantaneous power and the total variance of the received signal along the

*k*-th period, given by:

*i*(

_{o}*t*), can be evaluated using the expression given by Eq. (6). That expression considers only the propagation of one optical pulse and that linearized modulators (assumption valid when the electrical power of the signal applied to the modulator is low) are used, neglecting the distortion induced by the modulator nonlinearity. Alternatively, these model limitations can be overcome by evaluating the current at the output of the Ph-ADC through numerical simulation.

When compared with the SNR expressions presented in [2], Eq. (13) shows two main upgrades: it considers the variance contribution due to the noise introduced in each sensor of the Ph-ADC system and it allows obtaining the SNR variation with time imposed by the signal and by the noise. Henceforth, the evaluation of the SNR through the expressions presented in this section is referred to as the semi-analytical simulation method (SASM).

## 6. Results and discussion

This section is divided in three sub-sections. In the first one, the application scenario, the main parameters of the TS Ph-ADC system introduced in section 2 and the signal at different points of the system setup are presented. In the second one, the variance derived in section 4 is validated by comparison with Monte Carlo (MC) results. In the third one, the SNR at the Ph-ADC output derived in section 5 is validated also by comparison with MC results for different gain levels of the amplifiers used in the TS Ph-ADC system.

#### 6.1. Application scenario and UCELLS TS Ph-ADC system setup

The main target of the European Union “Ultra-wide band real-time interference monitoring and cellular management strategies” (UCELLS) project is the demonstration of cellular UWB capabilities employing a spectrum monitoring system based on a high performance Ph-ADC system in order to efficiently enable coexistence and compatibility of UWB-based wireless communications with existing and future mobile and fixed wireless systems [12]. In that project, the TS Ph-ADC system shown in Fig. 1 is implemented inside an office/home room and the different sensors are spread along the whole room area to sense online the UWB signals that are in use at a given time instant. The captured signals are then passed through the Ph-ADC system in order to relax the bandwidth requirements of the ADC used to digitize the received signal prior localization, fingerprinting and spectrum management algorithms are applied to. Due to DSP requirements, the aperture time of the optical pulses at the input of each EOM must be of the order of a few ns and the pulse repetition rate of a few MHz. In order to satisfy these requirements, the analysis presented in this work considers an optical source generating Gaussian pulses with −3 dB bandwidth of 1.7 nm and an accumulated dispersion in the first spool of fiber of −1513 ps/nm (dispersion parameter of −183 ps/nm/km and attenuation coefficient of 0.81 dB/km). The aperture time at the input of each EOM is approximately 1.7 nm×1513 ps/nm=2.55 ns. The noise figure of the EDFA is 7 dB and the optical filter used to reduce the ASE noise power is modeled by a 2* ^{nd}* order super-Gaussian filter with −3 dB bandwidth of 1.5 THz. Five sensors (with identical characteristics) and a pulse repetition rate of 3.3 MHz are considered. Orthogonal frequency division multiplexing (OFDM) UWB radio signals are used as the electrical signals captured by each sensor antenna [13] and the analysis performed along this work consider identical power levels for the UWB signals captured by the different sensor antennas. The electrical amplifier of each sensor has a noise figure of 6 dB and the BPF following the amplifiers is a 6

*order filter with −3 dB passband between 2.0 GHz and 14.7 GHz. The electro-optic conversion is performed by Mach Zehnder modulators biased at the quadrature point. The length of the second spool of fiber is 22.2 km, leading to*

^{th}*M*=3.4, and a polarizer (

*p*= 1) before the photo-detector is considered. At the receiver side, the electrical amplifier has a noise figure of 6 dB and a 6

*order BPF with −3 dB passband between 600 MHz and 2.0 GHz is used. In addition, it is considered that*

^{th}*R*= 50Ω.

_{L}Figure 2 shows the time waveform at different points of the TS Ph-ADC system in order to clarify the system operation. A unitary antenna gain and a free space distance of 3 m between the wireless terminals and each sensor are considered for power levels purposes. Figure 2(a) depicts the pulsed signal waveform generated by the optical source, and Fig. 2(b) shows a zoom of Fig. 2(a). From Fig. 2(b), it can be seen that very tight optical pulses are generated by the optical source according to the time-wavelength mapping validity [2]. Figure 2(c) shows one pulse of the optical chirped signal arriving at the input of the EOM of each sensor. The chirped signal is then used to sample the OFDM-UWB radio signal captured by each sensor as indicated by Fig. 2(d) and 2(e) for sensor 1. Finally, Fig. 2(f) shows the TS UWB snapshots at the Ph-ADC output after removing the high power/low frequency spectrum.

Figure 3 shows the spectrum of the signal at different points of the TS Ph-ADC system. Figure 3(a) depicts the spectrum of the OFDM-UWB radio signal applied to the EOM arms. In this example, only UWB sub-band 1 is being used by the wireless terminals. Figure 3(b) shows the spectrum of the signal at the PIN output. The spectrum is composed by two main components: the compressed high power/low frequency spectrum due to the optical pulsed signal and a compressed version of the OFDM-UWB signal spectrum located at an intermediate frequency.

Figure 3(c) shows the compressed spectrum after removing the high power/low frequency spectrum using the BPF. This spectrum corresponds to the time waveform shown in Fig. 2(f), where the OFDM-UWB signal is already clearly identified due to the low frequency spectrum removal.

#### 6.2. Variance contributions validation

The variance results presented in this section consider the TS Ph-ADC system parameters described in section 6.1, the generation of optical pulses with a peak power of 40 dBm and the RF power at the input of the sensor amplifier, *P _{in}*, set to −40 dBm. In addition, it is considered that the spectrum of the radio signal captured by each sensor lies into the first three UWB sub-bands, centered at 3.43 GHz, 3.96 GHz and 4.49 GHz. The variance estimates obtained from MC simulation are evaluated over 100 runs corrupted by uncorrelated noise samples. For MC and SASM estimates, a multiplexed signal with 2000 periods is considered. Due to memory limitations, the 2000 periods have not been transmitted together over the Ph-ADC system. Instead, a signal at the source output comprising four optical pulses has been considered. After propagation along the first spool of fiber, this signal is applied to the five sensors of the UCELLS architecture and, after time multiplexing, the signal consists in 20 time periods (or 20 consecutive optical pulses). The 2000 time periods have been achieved by repeating the simulation and saving the TS signal at the output of the Ph-ADC 100 times. It should be stressed that, in each one of these 100 runs, different binary sequences for the OFDM-UWB radio signal generation, random time delays for the signals arriving at each sensor antenna and different random time delays between numerical and MC simulation cases have been considered.

The validation study is performed for three different cases: the total variance is dominated by i) the noise of each sensor, ii) the ASE noise and iii) the noise of the electrical receiver. The gain of the different amplifiers used to obtain these three cases correspond to practical gain levels that may be used in the TS Ph-ADC to: i) increase the power level of the captured OFDM-UWB radio signal due to the low EIRP of these signals and also the transmission losses introduced between the terminals and the sensor antenna, ii) compensate for the losses introduced by the optical fibre and the optical components and iii) provide adequate signal levels to the ADC card used after the Ph-ADC.

Figure 4 shows the different variance contributions and the total variance evaluated analytically and estimated from MC simulation. In the case of Fig. 4(a) and 4(c), the −3 dB bandwidth of the filter used in the simulation is 1.5 THz while, in the case of Fig. 4(b), it is 750 GHz. The optical filter bandwidth reduction has been performed in order to avoid an increase of the computation time requirements to correctly characterize the ASE-ASE noise beat term with an optical filter bandwidth of 1.5 THz. This issue only appears for the simulation of the results of Fig. 4(b) as only in this case the ASE-ASE noise beat term cannot be neglected. The comparison between the total variance evaluated analytically with the variance obtained from MC simulation shows excellent agreement, confirming that the total noise variance at the output of the TS Ph-ADC system can be evaluated from the analytical expressions, independently of the noise source dominance. Figure 4(a) and 4(c) show also that the total variance obtained along the time instants where the level of the optical pulsed signal is low, is well described by the variance contribution due to the receiver noise as this term does not depend on the power of the optical pulsed signal. In addition, Fig. 4(b) shows that, for those time instants, the total variance is higher than the one predicted by the variance contribution due to the receiver noise. As shown in Fig. 4(b), the ASE-ASE variance contribution cannot be neglected and, therefore, the total variance at the time instants where the level of the optical pulsed signal is low is given by the sum of two contributions: the ASE-ASE variance and the variance due to the receiver noise.

#### 6.3. SNR validation

Equation (13) shows that the evaluation of the SNR depends on the number of periods considered to get stabilized mean power and noise variance estimates. Although the variance has a low dependence on the electrical signal level applied to the EOM, the number of periods used to evaluate the mean power is of special concern and, thus, it should be carefully analyzed.

Figure 5 shows the mean power of the signal at the output of the TS Ph-ADC system as a function of the number of periods considering the mean power evaluated from MC simulation (noise is added to the signal where each amplifier is located) and noiseless simulation. Although the results are presented only for one of the three amplification cases under study, similar conclusions have been obtained for the other cases. The different lines shown in Fig. 5 represent the mean power obtained along different time instants of the period of the pulsed signal. The same instants have been considered for the results obtained by MC simulation and by noiseless simulation. The main target of this study is to analyze how fast the power along the pulse stabilizes with MC and noiseless simulation in order to identify how many periods are required to evaluate correctly Eq. (13). The comparison between the results obtained by both approaches show that stabilized mean power estimates can be obtained with 1500 periods. Nevertheless, all the results presented in this work are evaluated with 2000 periods in order to have a good confidence interval on the mean power estimates.

Figure 6 depicts the mean power and the SNR obtained along the pulsed signal period for the three different cases under study. Results obtained from MC and from the analytical expressions derived in section 4 and 5 are presented. Figure 6 shows that the mean power and the SNR provided by the analytical expressions are in excellent agreement with the estimates provided by MC simulation, independently of the dominant noise type. The results depicted in Fig. 6 show that different gains of the electrical and optical amplifiers used along the Ph-ADC system may lead to significantly different SNR levels, suggesting that a trade-off among all the gains of the system may lead to an optimized system operation.

## 7. Simplified analysis

In this section, a simplified approach is presented, compared with the work of [2] and its validity range is discussed. The main purpose of this simplified approach is to provide an alternative low (analytical and computational) complexity method that allows obtaining acceptable SNR estimates under a given set of system parameters conditions avoiding the higher complexity of the SASM.

#### 7.1. Mean power

Lets focus the analysis only in one optical pulse, and consider that the receiver BPF eliminates the distortion components of the received TS signal without reducing the power of the desired signal component. In this case, the mean power along the period of the optical pulse can be obtained from Eq. (6):

*P*(

_{PIN}*t*) is the instantaneous power of the optical pulse at the PIN input and

*p*takes into account the influence of the fiber dispersion of the 2

_{PF}*spool of fiber on the RF signal power. This effect can be evaluated by considering the fiber dispersion-induced power fading on each OFDM-UWB signal [2]. As several OFDM-UWB sub-bands can be transmitted simultaneously, the power fading can be estimated using the power reduction at the center of each sub-band. Therefore,*

^{nd}*P*

_{RF,i}is the power of the signal transmitted in the

*i*-th UWB sub-band at the EOM input,

*f*

_{RF,i}is the central frequency of the

*i*-th UWB sub-band (1 ≤

*i*≤ 14),

*λ*is the operating optical wavelength and

*N*is the number of UWB sub-bands transmitted in the snapshot.

_{UWB}The mean power evaluated from Eq. (15) is similar to the one presented in [2] with two main differences: i) Eq. (15) depends on the instantaneous power of the optical pulse at the PIN input while the mean power used in [2] depends on the mean optical power as the time dependence was neglected and ii) the fiber dispersion-induced power fading on the RF signal is clearly identified in Eq. (15) while it is not referred to in [2].

The mean power evaluation from Eq. (15) allows overcoming the computational complexity required by the SASM as, instead of evaluating the electrical power by averaging over a set of several snapshots, the power of the RF signals applied to the EOM is characterized analytically. However, the mean power estimates provided by Eq. (15) are valid only for RF signals that operate in the linear region of the EOM. Some inaccuracy is also expected when tight receiver BPFs are used as its influence on the signal spectrum is neglected in Eq. (15).

Figure 7(a) depicts the mean power obtained through numerical simulation and from the simplified expression given by Eq. (15) for the system parameters described in section 6 and *G _{e}*=70 dB,

*G*=20 dB and

_{o}*G*=30 dB. Figure 7(a) shows a significant mismatch (∼5dB) between the peak of the mean power obtained by both methods and, consequently, this mismatch will lead to significant inaccuracy on the SNR estimates obtained using the simplified expression derived for the mean power. This is due to the high amplification level at the sensor that leads to the operation of the EOM in the non-linear regime and, consequently, to inaccuracy of Eq. (15). This high amplification level may be required by two different reasons: i) the power of the UWB radio signals captured by the sensor antenna may be very low due to the low EIRP of the transmitted UWB signals and the additional losses introduced in the UWB signals by the radio propagation path from the terminals until the sensor antenna and ii) the balance between the different noise variance contributions in order to reach an optimized SNR operation. Further investigation has shown that excellent mean power estimates accuracy are obtained from Eq. (15) when

_{r}*G*<60 dB.

_{e}Figure 7(b) shows results similar to the ones of Fig. 7(a) but with *G _{e}*=40 dB,

*G*=30 dB,

_{o}*G*=50 dB and the receiver BPF designed to ensure that its −3dB bandwidth is coincident with the bandwidth occupied, at the PIN output, by the three UWB sub-bands. This tight BPF may be required for purposes of noise power reduction. Figure 7(b) shows that discrepancies exceeding 1 dB occur between the estimates obtained through numerical simulation and from Eq. (15).

_{r}#### 7.2. Noise variance contributions

The derivation of a simplified version of Eq. (8) is performed considering also one optical pulse. It is also considered that the dispersion of the 2* ^{nd}* spool of fiber affects the PSD of the sensor noise as it affects the UWB signals, i. e., the noise PSD generated at the electrical transmitter is converted to the PIN output by considering the spectrum compression (due to TS), an increase of the PSD level (in order to keep the same signal energy) and the noise PSD is still decreased due to the power fading effect. Hence, Eq. (8) can be simplified to:

Regarding the ASE noise contribution to the total variance, Eq. (10) represents already a simplified version of the ASE noise variance. That equation requires low analytical and computational complexity. Equation (10) can be viewed as a generalization of the ASE noise expression presented in [2], as it depends on the instantaneous power of the optical signal at the PIN input (taking into account the pulsed nature of the optical signal) rather than on the mean optical power.

The variance contribution due to the electrical receiver noise given by Eq. (11) is similar to the one presented in [2] and does not require any simplification as it is already of easy evaluation.

## 8. Conclusion

A SASM that allows evaluating the SNR of the signal at the output of TS Ph-ADC systems has been proposed. The method takes into account the noise introduced by the electrical part of the transmitter and receiver of the Ph-ADC, and by the optical amplifier. The SASM considers also the pulsed nature of the optical source and the TS of generic electrical signals.

The comparison with Monte Carlo simulation results has shown that the proposed method provides reliable SNR estimates under different noise (amplification) conditions and that the SNR of the TS Ph-ADC system used to implement the new application proposed in [7, 8] can be limited by any of the different noise types introduced along the Ph-ADC system, rather than be imposed by the ASE noise as indicated in [2]. It has been also shown that the variance and SNR dependence along time due to the optical pulsed signal are properly described by the SASM and, thus, the SASM presented in this work can be viewed as a generalization of the study presented in [2], providing an accurate approach to evaluate rigorously the SNR of the Ph-ADC system. However, when compared with [2], the accuracy improvement is achieved at the expense of analytical and computational complexity increase. As an alternative, a simplified approach that allows overcoming that complexity has been presented and shown that it leads to good SNR performance estimates while the EOMs are operating in its linear region and the receiver bandwidth is larger than the TS signal bandwidth.

## A. Contribution to the variance due to noise introduced in the sensors

In order to obtain simple closed form expressions, the derivation of the variance contribution due to the electrical noise considers that the Ph-ADC system is composed only by one sensor.

Taking into account the simplification mentioned above, the index *k* used in section 3 to identify the signals and parameters of the different sensors will be dropped. Hence, from Eq. (6), the mean of the current at the output of the TS Ph-ADC system is given by:

*g*(

_{c}*τ*) is deterministic as the noise introduced by the optical amplifier has been neglected.

By evaluating the square of the mean and the mean square of *i _{o}*(

*t*) from Eq. (18) and Eq. (6), respectively, considering that the mean of the electrical noise is null, that odd order moments of Gaussian processes with zero mean are also null and by replacing

*g*

_{RF,c}(

*t*) by Eq. (4), the contribution to the variance due to the electrical noise given by Eq. (7) can be written as:

*s*(

*t*) = (

*e*

_{EOM,i}(

*t*)

*v*(

_{RF}*t*)) *

*h*

_{2}(

*t*) and

*n*(

*t*) = (

*e*

_{EOM,i}(

*t*)

*n*(

_{e}*t*))*

*h*

_{2}(

*t*). Equation (19) shows that the variance due to the electrical noise is composed by three main contributions: the optical pulsed signal-noise beat term (PSNBT), the electrical signal-noise beat term (ESNBT) and the noise-noise beat term (NNBT). In the TS Ph-ADC system, the dominant contribution to the variance due to the noise introduced in the sensor is given by the PSNBT as, though the average power of the pulsed signal is usually low due to the low pulse repetition rate of the optical source, in the time interval of interest (when an optical pulse is present) there is a strong power peak leading to a strong PSNBT power level when compared with the power of the ESNBT and the NNBT. Therefore, the electrical noise variance of Eq. (19) can still be approximated, in the time interval where the optical pulses occur, by:

*n*(

*t*) mentioned above, Eq. (21) can be written as:

*R*(

_{e}*τ*

_{3}–

*τ*

_{4}) is the auto-correlation function of the electrical noise arriving at the EOM arms given by the Wiener-Kintchine theorem, ${R}_{e}\left(\tau \right)={\int}_{-\infty}^{+\infty}{S}_{e}\left(f\right)\text{exp}\left[j2\pi f\tau \right]df$. Thus, using this theorem in Eq. (22) and after performing some algebra, we obtain:

*w*(

*t*,

*f*) = (

*g*(

_{c}*t*)

*r*

^{*}(

*t*,

*f*)) *

*h*(

_{r}*t*) and ${r}^{*}\left(t,f\right)=\left[{e}_{EOM,i}^{*}\left(t\right)\text{exp}\left(j2\pi ft\right)\right]*{h}_{2}^{*}\left(t\right)$.

Following a similar procedure, the second term of Eq. (20) can be easily written as:

Equation (25) allows evaluating the electrical variance contribution of the received TS signal when one sensor is considered. When more than one sensor and, consequently, more than one amplifier is used in the system, the variance contribution due to the noise added in other sensors can be also evaluated from Eq. (25) by replacing each one of the signals/parameters used in Eq. (25) by the ones corresponding to the sensor under analysis:

As the variance contribution generated by the noise added in independent sensors is separated in time due to the optical delays represented in Fig. 1, the total variance is obtained by summing the variance contributions from the different sensors, ${\sigma}_{{n}_{e}}^{2}\left(t\right)={\sum}_{k=1}^{N}{\left({\sigma}_{{n}_{e}}^{\left(k\right)}\right)}^{2}\left(t\right)$.

## Acknowledgments

The work of Tiago Alves was supported by Fundação para a Ciência e a Tecnologia from Portugal under contract SFRH/BD/29871/2006 and the project TURBO-PTDC/EEA-TEL/104358/2008. This work was also supported in part by the EU project UCELLS-FP7-IST-1-216785. The authors would like to thank also to UCELLS’ partners by the fruitful discussions about the structure and the parameters of the Ph-ADC system.

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