## Abstract

The electronic aperture jitter effect originating from the electronic digitization is investigated in the channel-interleaved photonic analog-to-digital converter (PADC) system. The influence of the electronic aperture jitter to the effective number of bits (ENOB) of the PADC system is extracted and evaluated. According to the theoretical analysis, the electronic aperture jitter can be significantly suppressed by the channel-interleaving scheme. In the experiment, the effect of electronic aperture jitter is measured under different optical-electronic conversion (OEC) bandwidths and channel numbers. It is eventually found that the condition of the OEC bandwidth equaling to the Nyquist frequency of one single channel is critical to optimize the electronic aperture jitter and ENOB.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

As a microwave photonic solution towards the signal reception and processing in next-generation radar and communication systems [1–5], photonic analog-to-digital converter (PADC) technology achieves ultra-broad input bandwidth as well as ultra-high sampling rate. According to the working mechanism, PADC schemes can be basically divided into several typical classes [6], which are the photonic assisted one [7,8], the photonic sampled and digitized one [9,10], and the photonic sampled and electronic digitized one [11,12]. Compared with the others, the photonic sampled and electronic digitized PADC shows its advantage in the fidelity of high-resolution, broadband, and ultra-high-rate digitization and has been becoming the mainstream scheme. Generally speaking, this PADC scheme is basically constructed of a photonic frontend and an electronic backend and the channel-interleaving architecture is always adopted to implement an ultra-high sampling rate [13,14]. In our previous works, the distortions induced by both channel mismatch and nonlinearity in the photonic frontend of the channel-interleaved PADC scheme have been studied and effectively eliminated [15,16]. Consequently, the performance of the channel-interleaved PADC system would get close to the limitation determined by the noise and/or timing jitter. Hence, the investigation of the timing jitter in the channel-interleaved PADC is of great significance for the applications in modern radar system, which requires high-frequency broadband digitization with high resolution.

The timing jitter in the channel-interleaved PADC system is contributed by both the photonic frontend and electronic backend. In the photonic frontend, the timing jitter is mainly induced by both the timing jitter of the sampling clock and the relative drift between the analog input signal and the sampling clock. However, in the electronic backend, the timing jitter is intrinsically induced by the time aperture of the digitization performed by the electronic analog-to-digital converter (EADC). Thus, the electronic aperture jitter would become an inherent limitation to the channel-interleaved PADC system. Provided that the digitization by the EADC of the channel-interleaved PADC system follows the photonic sampling and channel interleaving process by the photonic frontend, the effect of electronic aperture jitter on the channel-interleaved PADC system would be different from the direct sampling and digitization by the traditional EADC. Moreover, note that the electronic aperture jitter is always superposed with the other noises in the PADC system, its precise extraction and measurement would become another crucial issue.

In this work, we theoretically and experimentally investigate the effect of the electronic aperture jitter in a channel-interleaved PADC system. The relationship between the optical-electronic conversion (OEC) bandwidth and the effect of electronic aperture jitter is studied. A novel method to measure the intrinsic influence of the electronic aperture jitter to the channel-interleaved PADC system is proposed and experimentally verified. It is found that the channel-interleaved PADC system is more advantageous than the traditional EADC since the OEC bandwidth, equaling to the Nyquist frequency of a single channel, effectively suppresses the electronic aperture jitter effect.

## 2. Principles

The schematic of the channel-interleaved PADC system is illustrated in Fig. 1. There are two different noises induced by timing jitters of the photonic frontend and electronic backend, respectively. In the photonic frontend, the analog input signal is sampled at the sampling rate of *f*_{s} via an electronic-optical modulation (EOM) and then the pulse train is demultiplexed into *N* parallel channels. The timing jitter is mainly induced by the electro-optical sampling and called the sampling jitter. In the electronic backend, the demultiplexed pulse trains are firstly optical-electronic converted (OEC) by a photodetector (PD) array and then digitized by an EADC array. The timing jitter is determined by the time aperture of the digitizing circuit of the EADC array and nominated to be the electronic aperture jitter. Figure 2(a) illustrates a schematic of the spectrum of the digitized output from the PADC system, where the distortions and noises induced by both photonic frontend and electronic backend are intrinsically superposed. It is worth noting that the relative amplitude level between the timing jitters are uncertain as marked in Fig. 2(a). Hence, the electronic aperture jitter should be precisely extracted from the other of the photonic frontend.

In mathematics, the working mechanism of the channel-interleaved PADC can be expressed as follows [16]

*T*

_{s}= 1/

*f*

_{s}is the temporal interval of the sampling clock,

*v*(

_{in}*t*) is the analog input, and

*v*[

_{out,n}*k*] is the digitized output in the

*n*th channel.

*h*(

_{EO}*t*) and

*h*(

_{OE}*t*) represent the impulse responses of the EOM in the photonic frontend and the OEC in the electronic backend, respectively. Taking the timing jitters into account, Eq. (1) can be further modified by

*δ*

_{EO}(

*t*) denotes the sampling jitter and

*δ*

_{OE}(

*t*) corresponds to the electronic aperture jitter.

*α*

_{OE}(

*t*) represents the amplitude noise induced by the OEC and digitization.

In this study, two methods are adopted to extract the electronic aperture jitter induced noise from the others. Firstly, a phase-locking loop that synchronizes the analog input and the sampling clock is adopted to suppress the sampling jitter as shown in Fig. 2(b). After synchronization, the sampling jitter induced noise would be cancelled as *δ*_{EO}(*t*)≈0 and might be neglected in Eq. (2). Secondly, a cross-correlation method is developed as shown in Fig. 2(c). Based on two arms divided from one single channel, it is possible to eliminate the amplitude noise of *α*_{OE}(*t*) in this channel. In mathematics, the cross-correlation method can be described by

*δ*

_{OE,1}(

*t*) =

*δ*

_{OE,2}(

*t*) =

*δ*

_{OE}(

*t*), which makes <

*δ*

_{OE,1}(

*t*),

*δ*

_{OE,2}(

*t*)> = ||

*δ*

_{OE}(

*t*)||

^{2}. Since the amplitude noises in both two arms are uncorrelated, it turns out that <

*α*

_{OE,1}(

*t*),

*α*

_{OE,2}(

*t*)> = 0 [17] and eventually arrives at Eq. (3).

With two methods explained above, the terms *δ*_{EO}(*t*) and *α*_{OE}(*t*) are neglected in Eq. (2), *δ*_{OE}(*t*) could be extracted from the noises. Considering a single tone input *v _{in}*(

*t*) =

*V*

_{0}exp(

*j*2

*πf*

_{0}

*t*) and using the analyses from [16], Eq. (2) could be further derived as

*T*

_{M}_{, 0}=

*π*/2

*V*and

_{π, RF}*m*

_{0}satisfies |

*f*

_{0}-

*m*

_{0}

*f*/

_{s}*N*|≤

*f*/2

_{s}*N*.

*H*(

_{EO}*f*),

*H*(

_{OE}*f*), and

*P*(

*f*) are the Fourier transforms of

*h*(

_{EO}*t*),

*h*(

_{OE}*t*), and

*p*(

*t*), respectively.

*p*(

*t*) is the instantaneous power of the sampling clock. In Eq. (6),

*ε*

_{OE}(

*t*) is the electronic aperture jitter which induces the noise

*δ*

_{OE}(

*t*) and its root mean square (RMS) integral is defined by ||

*ε*

_{OE}(

*t*)|| =

*σ*.

From Eqs. (4)–(6), after both the EOM and OEC, the input frequency *f*_{0} is down-converted by each harmonics *mf _{s}*/

*N*into

*f*(

_{d}*m*) as described by Eq. (5). After digitization by the EADC array, the input signal of

*f*

_{0}is eventually aliased into

*f*(

_{d}*m*

_{0}), which is defined as the digitized frequency within the so-called single-channel Nyquist bandwidth (i.e.

*f*/2

_{s}*N*) [18]. Both temporal and spectral behaviors of the down-conversion in one interleaved channel are illustrated in Figs. 3(a) and 3(b), respectively.

To further analyze the effect of electronic aperture jitter on the PADC system, we derivate the noise power induced by the electronic aperture jitter (*P _{N}*) and the signal power (

*P*) from Eqs. (4)–(6), which are described by

_{S}*f*is called an effective frequency, which takes the formula:

_{eff}Equations (8) and (9) show that the SNR induced by the electronic aperture jitter is determined by the effective frequency *f _{eff}*, which is dependent on the frequency response of

*H*(

_{EO}*f*),

*H*(

_{OE}*f*), and

*P*(

*f*) in Eq. (5), respectively. Thus, the effective number of bits (ENOB) can be evaluated by the definition of

*ENOB*= (

*SNR*-1.76)/6.02 [19]. In comparison, Eq. (8) represents the SNR and can be used to calculate the ENOB for the situation where the input signal is directly detected by the EADC as long as

*f*is simply replaced by

_{eff}*f*

_{0}.

In the channel-interleaved PADC system, *H _{EO}*(

*f*) and

*P*(

*f*) could be approximated as constants due to the ultra-broad input bandwidth of the EOM and ultra-short duration of the sampling clock. Hence, the OEC frequency response

*H*(

_{OE}*f*) is the main factor which denotes the noise induced by the electronic aperture jitter. For simplicity, the OEC frequency response can be approximately described as a low-pass filter, such as, a rectangular model of

*H*(

_{OE}*f*) = 1 when |

*f*|≤

*B*and

_{OEC}*H*(

_{OE}*f*) = 0 when |

*f*|>

*B*, where

_{OEC}*B*represents the OEC bandwidth. Using this approximation, Eq. (9) could be further derived as

_{OEC}*f*(

_{d}*m*) within the OEC bandwidth.

Figure 3(c) compares the electronic aperture jitter induced noise in the EADC direct detection and the channel-interleaved PADC with two different OEC bandwidths of *B _{OEC}* =

*f*/2

_{s}*N*and

*B*≫

_{OEC}*f*/2

_{s}*N*, respectively. When

*B*=

_{OEC}*f*/2

_{s}*N*, the effective frequency becomes

*f*=

_{eff}*f*

_{0}-

*m*

_{0}

*f*/

_{s}*N*. Thus,

*f*≤

_{eff}*f*

_{0}and |

*f*|≤

_{eff}*f*/2

_{s}*N*, which means that the electronic aperture jitter induced noise can be effectively suppressed when compared with the EADC direct detection. When

*B*≫

_{OEC}*f*/2

_{s}*N*, for an example of

*f*(

_{d}*m*)≈

*mf*/

_{s}*N*, the effective frequency can be calculated as

*f*= [

_{eff}*K*(

*K*+ 1)(2

*K*+ 1)/6]

^{1/2}×

*f*/

_{s}*N*where the integer

*K*= ⌊

*B*/(

_{OEC}*f*/

_{s}*N*)⌋. Note that

*f*increases with

_{eff}*B*and thus the electronic aperture jitter induced noise is larger than the EADC direct detection for an input frequency of

_{OEC}*f*

_{0}>

*f*. Therefore, it is worth noting that the condition of

_{eff}*B*=

_{OEC}*f*/2

_{s}*N*, the OEC bandwidth of an interleaved channel being a half of its sampling rate, not only guarantees a continuous frequency response of the channel-interleaved PADC [20], but also optimizes the electronic aperture jitter noise as well as its determined ENOB limitation.

## 3. Experimental results

#### 3.1. Cancellation of the sampling jitter based on synchronization method

In order to verify the feasibility of the synchronization method to the suppression of the sampling jitter noise, a dual-output Mach-Zehnder modulator (DO-MZM) based phase-locking loop (PLL) is developed as depicted in Fig. 4(a). In the PLL, the analog input signal generated from a RF source (Keysight N5183B) is sampled by the optical sampling clock via a DO-MZM (EOSpace, AZ-1X2-AV5-40). The sampled signal from dual outputs are converted into the electrical signal. Later, it is low-pass filtered and heterodyned for the phase detection. The detected phase error is feed back to the microwave synthesizer so that the synchronization between the optical sampling clock and the analog input signal is achieved. To evaluate the residual error after synchronization, a FFT analyzer (Keysight MSOS804A) is used to analyze the output from the phase detection.

In the experiment, two kinds of the optical sampling clocks are used. One is an actively mode locked laser (AMLL, Calmar PSL-40-TT) seeded by a synthesizer (Keysight E8257D) at 20 GHz and the other is a passively mode locked laser (PMLL) with 100 MHz repetition rate (MenloSystem C-Fiber Sync). The analog input signal of the RF source at ~20 GHz is sampled and digitized. To investigate the sampling jitter, the phase noises of the AMLL, PMLL, and the RF source are measured via a phase noise analyzer (Keysight E5052B) and depicted in Fig. 4(b). The phase noise spectra is sketched from 1 kHz to 100 MHz and the timing jitters are calculated by integral of the phase noise spectra over this frequency range. The corresponding integrated timing jitters are also illustrated in Fig. 4(c). The RMS timing jitter is 19 fs for the RF source, 17 fs for the AMLL, and 11 fs for the PMLL, respectively. Furthermore, the sampling jitter can be derived as the root of the quadratic sum of both timing jitters of sampling clock and RF source, which is 25 fs between the AMLL and the RF source or 22 fs between the PMLL and the RF source, respectively.

The spectra of the residual phase error between the sampling clocks and the RF source after the synchronization are also characterized and plotted in Fig. 4(b). Compared with the phase noise spectra of the AMLL, PMLL and RF source, the residual phase noise within the locking bandwidth (~1 MHz) is significantly suppressed whereas the phase noise beyond the locking bandwidth follows its original conditions. Figure 4(c) depicts the integral timing jitter that corresponds to the residual phase error after synchronization for both cases, which reach ~1.5 fs and is much lower than the original values. It confirms that the sampling jitter could be effectively suppressed when the sampling clock and the analog input signal are synchronized. Since the relative timing jitter after synchronization is far lower than the original values, the approximation of *δ*_{EO}(*t*)≈0 applied to Eq. (2) is verified. Note that the PLL scheme was also implemented for a parametric sampling gate, which provides a synchronization between different pulse laser sources [21].

#### 3.2. Dependence of ENOB on electronic aperture jitter

In order to experimentally investigate the effect of the electronic aperture jitter to the ENOB of the PADC under different OEC bandwidths, a single-channel quantization with 100 MS/s sampling rate is developed based on Fig. 1(a). In this scheme, the sampling gate is a MZM with 40 GHz input bandwidth (Photoline MXIQ-LN-40), the sampling clock is generated from the 100 MHz PMLL (MenloSystem C-Fiber Sync) and the digitization is performed by a 100-MS/s EADC (Texas Instruments ADS42JB69). In the experiment, the analog input signal is synchronized with the sampling clock using the method depicted in Fig. 4(a) to eliminate the sampling jitter. The cross-correlation method is used to extract the electronic aperture jitter induced noise. PDs with 50 MHz and 2 GHz bandwidth are used to implement the OEC bandwidth of *B _{OEC}* =

*f*/2

_{s}*N*and

*B*≫

_{OEC}*f*/2

_{s}*N*, respectively.

The spectra of the digitized data of ~10 GHz and ~30 GHz analog input signals are illustrated in Figs. 5(a) and 5(b), respectively. The measured OEC frequency responses for 50 MHz and 2 GHz OEC bandwidths are illustrated in Fig. 5(c). Figure 5(a) and Fig. 5(b) illustrate a comparison of the spectra with and without the cross correlation for the 50 MHz OEC bandwidth and show an effective elimination of *α*_{OE}(*t*) that is the amplitude noise induced by the OEC and digitization. It experimentally verifies the theoretical estimation of Eq. (3). The ENOB is calculated from the spectra of Fig. 5(a) and 5(b) and plotted as a function of the analog input frequency in Fig. 5(d) for different OEC bandwidths. Compared with the EADC direct detection, the PADC with 50 MHz OEC bandwidth optimizes the electronic aperture jitter determined ENOB limitation whereas the 2 GHz OEC bandwidth degenerates it. The numerical simulations based on the theoretical analyses of Eqs. (8)–(10) are also depicted in Fig. 5(d), providing a consistence with the experimental observations.

To investigate the effect of electronic aperture jitter in a high-speed channel-interleaved PADC system, an experimental configuration with 40 GS/s total sampling rate is developed. The 40-GHz MZM (Photoline MXIQ-LN-40) is still used as the sampling gate and the AMLL (Calmar PSL-40-TT) is seeded at 40 GHz to generate the optical sampling clock. Note that the channel number determines the sampling rate of a single channel and its Nyquist bandwidth, which influences the optimal OEC bandwidth and the electronic aperture jitter determined ENOB limitation as depicted in Fig. 5(c). Two experimental schemes with 4 and 2 channels are developed based on Fig. 1(a), respectively. The single-channel sampling rate in 4-channel and 2-channel schemes are 10 GS/s and 20 GS/s, respectively. The digitization in both experiments are performed by the EADCs of a real-time oscilloscope (Keysight MSOS804A). Due to the limited input bandwidths of the EADCs, the OEC bandwidth can reach 4.2 GHz in 4-channel scheme and 8.4 GHz in 2-channel scheme, which gets close to the condition of *B _{OEC}* = ~

*f*/2

_{s}*N*.

Figure 6 shows the spectra of the digitized outputs from both experiments, where the sampling jitter is canceled via the synchronization method and the electronic aperture jitter induced noise is extracted via the cross-correlation method. The spectra of 12.5 GHz and 32.5 GHz analog input measured by the 4-channel scheme are shown in Figs. 6(a) and 6(b), respectively. The spectra of 15 GHz and 35 GHz analog input digitized by the 2-channel scheme are illustrated in Figs. 6(c) and 6(d), respectively. It can be found that the noise floor is significantly reduced by the cross-correlation method so as to extract the electronic aperture jitter induced noise. The measured OEC frequency responses of 4.2 GHz and 8.4 GHz in both 4-channel and 2-channel schemes are depicted in Fig. 7(a), which indeed satisfies *B _{OEC}* = ~

*f*/2

_{s}*N*. The ENOB limitation in both schemes are derived from the spectra and depicted in Fig. 7(b). It is found that the electronic aperture jitter determined ENOB limitation is optimized in both schemes, which matches the solid curves in Fig. 7(b) representing the theoretical analyses of Eqs. (8)–(10). Moreover, the optimization in the 4-channel scheme is more obvious than that in the 2-channel one. Consequently, for a given high-speed sampling rate, more demultiplexed channels provide better optimization of the electronic aperture jitter determined ENOB limitation if the OEC bandwidth is optimally satisfied by

*B*= ~

_{OEC}*f*/2

_{s}*N*.

## 4. Conclusion

We have investigated the electronic aperture jitter effect and its determined ENOB limitation in the channel-interleaved PADC system. Based on the theoretical analyses, the electronic aperture jitter induced noise can be inherently suppressed owing to the channel interleaving configuration and the OEC bandwidth approximately equaling to the single-channel Nyquist frequency leads to an optimal ENOB. In the experiment, the noise induced by the electronic aperture jitter in the PADC channel-interleaved system has been extracted and measured under different OEC bandwidths and channel numbers, which verifies the theoretical analyses. It is found that for a given high-speed sampling rate, more demultiplexed channels can enhance the ENOB limitation determined by the electronic aperture jitter if the condition of the OEC bandwidth approximately equaling to the single-channel Nyquist frequency is preset.

## Funding

National Natural Science Foundation of China (61822508, 61571292, and 61535006).

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