We demonstrate a method to compensate multi-channel mismatches that intrinsically exist in a photonic analog-to-digital converter (ADC) system. This system, nominated time-wavelength interleaved photonic ADC (TWI-PADC), is time-interleaved via wavelength demultiplexing/multiplexing before photonic sampling, wavelength demultiplexing channelization, and electronic quantization. Mismatches among multiple channels are estimated in frequency domain and hardware adjustment are used to approach the device-limited accuracy. A multi-channel mismatch compensation algorithm, inspired from the time-interleaved electronic ADC, is developed to effectively improve the performance of TWI-PADC. In the experiment, we configure out a 4-channel TWI-PADC system with 40 GS/s sampling rate based on a 10-GHz actively mode-locked fiber laser. After multi-channel mismatch compensation, the effective number of bit (ENOB) of the 40-GS/s TWI-PADC system is enhanced from ~6 bits to >8.5 bits when the RF frequency is within 0.1-3.1 GHz and from ~6 bits to >7.5 bits within 3.1-12.1 GHz. The enhanced performance of the TWI-PADC system approaches the limitation determined by the timing jitter and noise.
© 2016 Optical Society of America
Photonic analog-to-digital converter (PADC) technology has been developing rapidly in recent decades because it benefits from the low timing jitter of optical pulse trains [1–3]. The PADCs provide alternative solutions to electronic analog-to-digital converters (EADCs) for diverse applications of radar [4,5], surveillance , and telecommunications . As comprehensively summarized in [2,3], one of the typical PADC with high speed and high resolution is called the photonic sampled and electronic quantized PADC. In this PADC [2,3], a stable pulsed laser works as a sampling source , an optical modulator serves as a sampling gate for RF signal, and an array of photo-detectors (PDs) is employed to convert the sampled optical signal to the sampled electronic signal that is electronically quantized by an array of EADCs. Each PD [9,10] and EADC  is reduced in bandwidth and sampling rate when compared to the PADC’s overall bandwidth and sampling rate because multiple channelization is carried out by multiplexing and demultiplexing processes. There are two schemes to reduce bandwidth and sampling rate: the segment-interleaved PADC [11–15] and time-interleaved (or sample-interleaved) PADC [9,16,17]. The segment-interleaved PADC is channelized by wavelength demultiplexing [14,15] and each channel is stretched in time domain or compressed in frequency domain via the dual-stage dispersive time-wavelength mapping before and after the photonic sampling [11–13]. In contrast, the time-interleaved PADC, also nominated time-wavelength interleaved PADC (TWI-PADC), is channelized by two steps. First, the sampling rate, i.e. the repetition rate of the pulsed laser, is multiplied to generate a time-wavelength interleaved high-speed sampling source by either the wavelength multiplexing [18,19] or time-wavelength mapping [3,20]. Second, the high-speed sampling source after the modulator samples optically the RF signal and the sampled optical signal is time-interleaved by wavelength demultiplexing so that each channel is correspondingly reduced in bandwidth and sampling rate. In comparison, the TWI-PADC is more advantageous than the segment-interleaved PADC because the TWI-PADC provides a higher resolution.
Up to date, various works [3,4,9,17–24] have been demonstrated to improve the performance of the TWI-PADC. For instance, more than 40-GHz sampling bandwidth with the effective number of bits (ENOB) of about 7 bits were demonstrated [3,4,22–24]. However, all the sampling rates were lower than the sampling bandwidth and didn’t satisfy the Nyquist theorem since one channel [22,24] or dual channels [3,4,23] was carried out. Ng et al.  four-fold multiplied the 10-GHz repetition rate of an actively mode-locked laser (AMLL) to achieve very fast sampling rate of 40 GS/s and a high ENOB of 8.3 bits whereas the sampling bandwidth was limited to be only 1.6 GHz. This is possibly because there are large mismatches among the multiple channels during the generation of the ultrahigh-speed PADC sampling source, the optical sampling, and multi-channel electronic quantization. Most recently , we proposed a spectral analysis and compensation method to partially overcome the mismatches during the generation of the time-wavelength interleaved PADC sampling source. In order to realize a good-performance TWI-PADC with high speed, high resolution, and wide bandwidth, there is still a well-known issue, that is, how to estimate and compensate the multi-channel mismatches. There is successful evidence to fully solve this issue in the segment-interleaved PADCs [14,15]. Besides, Khilo et al.  has addressed that the architecture of TWI-PADC is in principle similar with that of time-interleaved EADC and this issue in TWI-PADC might be proceeded if referred to the calibration and compensation algorithms successfully applied in modern EADCs. In , the effects of the multiple-channel mismatches in TWI-PADC were analyzed and estimated for hardware adjustments. However, to the best of our knowledge, there is no successful demonstration to compensate the multi-channel mismatches beyond the hardware limitation in TWI-PADC.
In this paper, we present a spectral analysis and compensation of the multi-channel mismatches in TWI-PADC. The effects of multi-channel mismatches, electro-optical modulation nonlinearity, and timing jitter on the ENOB of TWI-PADC are investigated. Inspired from modern time-interleaved EADCs [26,27], the compensation algorithm of multi-channel mismatches is developed to effectively enhance the ENOB of TWI-PADC from the hardware limitation to the timing jitter and noise determined limitation. In the experiment, high-speed and high-resolution TWI-PADC with 40-GS/s sampling rate and >7.5 ENOB covering 12.1 GHz bandwidth is successfully demonstrated.
2. Experimental setup
Figure 1 depicts the experimental setup of the TWI-PADC system, which is a typical architecture of the photonic sampled and electronic quantized PADC [2,18]. The laser source is an AMLL (Calmar PSL-10-TT), which is seeded by an electronic oscillator (Keysight E8257D) at 10 GHz. The pulse width of the AMLL is temporally compressed into 1.2 ps by a pulse compressor (Calmar PCS-2). After wavelength demultiplexing, multi-channel time/amplitude adjustment, and wavelength multiplexing processes, shown in Figs. 1(a) and 1(b), the repetition rate of the AMLL is four-fold multiplied to fS = 40 GHz. Note that the generation of the sampling source was previously presented in . The photonic sampling via an electro-optical modulator (EOM) is schematically depicted in Figs. 1(c) and 1(d). The RF signal to be sampled is provided by another microwave synthesizer (Rhode & Schwarz SMA 100A) and modulated to the sampling source via a Mach-Zehnder intensity modulator (EO Space AZ-1X2-AV5-40) with the bandwidth of 40 GHz and the half-wave voltage of Vπ = 3 V. As illustrated in Figs. 1(e) and 1(f), the photonic sampled signal is demultiplexed into 4 parallel channels by a wavelength division multiplexer (WDM). The WDM with 1.6-nm bandwidth in each channel is identical to the one used in Fig. 1(a). A tunable delay line (TDL) with an accuracy of ~100 fs is inserted in each channel for multi-channel time adjustment. In each channel, the average optical power into a PD with 20 GHz bandwidth (Discovery DSC-R401HG-59) is ~0 dBm. It is converted into the electronic sampled signal and digitized by a 10 GS/s real-time oscilloscope with 4 channels (Keysight MSOS804A). The oscilloscope is clocked by the electronic oscillator (Keysight E8257D) also seeding the AMLL.
The optical intensity of the sampling source (i.e. optical pulse train) of the TWI-PADC system can be expressed by
The normalized transmittance of the EOM (i.e. the relation between the output and input optical intensity), TM, is determined by the RF signal to be sampled,, as follows 
In mathematics, the electronic sampled signal in the nth channel can be described byFig. 1(f).
The digitized data after channel mapping can be written by
3. Mathematical derivation
3.1 Spectral analysis
Consider the RF signal to be sampled is sinusoidal, which can be expressed by with a frequency of fIN = ΩIN/2π and an amplitude of V0. When a quadrature bias (i.e. VB = -Vπ/2) is applied to the EOM, the digitized data after channel mapping is derived from Eq. (4) as followsEq. (5), the item of an/2 is the offset in each channel, which can be digitally eliminated by a DC-block after digitization.
The discrete Fourier transform (DFT) of the digitized data [see Eq. (5)] after the elimination of the offset (an/2) can be expressed byEq. (6) and () in Eq. (7) are complex, which can be derived from the digitized data. Furthermore, the spectrum shows that the nonlinearity of EOM leads to the higher odd-order harmonics of the input frequency at f = (2m + 1)fIN (m is an integer). The channel mismatch effect in frequency domain is characterized by the distortion spurs at f = lfS/4 ± (2m + 1)fIN (l = 0,1,2,3). It is similar to the case in EADCs . Figure 2 illustrates a typical DFT spectrum of a 4-channel TWI-PADC according to the spectral analysis in Eqs. (6) and (7). Note that only the interval of [0, fS/2] is plotted due to the symmetry of the spectrum.
Based on the algorithm of dual channel mismatch compensation demonstrated in Appendix, the digitized data of a single-frequency RF signal at fIN can be reconstructed byEq. (6) according to Appendix.
In consequence, for a 4-channel TWI-PADC, the compensation algorithm in Eq. (8) can be applied to the dual channels of the first and third or the second and fourth channel, respectively. Later, the algorithm is performed to the two data series that are previously reconstructed and thus all four channels are reconstructed eventually. Note that this algorithm can be effectively applied to a TWI-PADC with an even number of multiple channels.
3.2 SINAD and ENOB estimation
According to IEEE standard for terminology and test methods for ADCs , the ENOB is defined by28]Eq. (6)].
In the TWI-PADC system, the factors of distortions and noise are the higher odd-order harmonics induced by the modulator nonlinearity, the distortion spurs induced by channel mismatch, the amplitude noise, and the timing jitter. Hence, the SINADPADC is a summation of the signal-to-noise ratio (SNR) and signal-to-distortion ratio (SDR) for different factors. Note that the SNR is defined by the ratio of the signal power to the power of stochastic noises whereas the SDR represents the ratio of the signal power to the power of the mismatch and nonlinearity induced distortions. The distortions are related with the RF signal to be sampled.
The SINADPADC can be expressed as followsEq. (6). Since the modulation nonlinearity is dominantly by the 3rd harmonics, SDRModulation can be approximately expressed by
SDRMismatch is determined by the ratio between the powers of the RF signal and mismatches as follows
Similar to EADCs , SNRNoise can be described by
Besides, SNRJitter is determined by the timing jitter as follows
The entire SINAD of the PADC system can be derived from Eqs. (11)-(15), which is written by
4. Experimental details
4.1 Estimation of the multi-channel mismatches
From Eqs. (12) and (14), it is found that γ(M) increases whereas ρN(M) decreases with the modulation index of M. Hence, there is a value of Moptimal corresponding to the maximum SINADPADC in Eq. (16) for specific parameters of σN, σa, σt, σJ, and fIN. To experimentally determine Moptimal, we manipulate the output power of the microwave synthesizer (RF signal) to change M. The parameters of σa and σt are calculated from an and Δtn, which can be derived from the spectra of digitized data according to Eq. (A.1) in Appendix.
Due to the device-limited accuracy, the parameters of multi-channel mismatches are σa = 1.0 × 10−2 and σt = 95 fs, respectively. Figure 3 illustrates the single sideband (SSB) phase noise spectra of the AMLL and electronic oscillator, which are both measured by a signal analyzer (Rhode & Schwarz FSUP 50). The integral RMS timing jitters of the SSBM phase noise spectra are calculated to be σJ,MLL = 20.9 fs for the AMLL and σJ,RF = 29.3 fs for the electronic oscillator, respectively. Assuming the timing jitters of the AMLL and RF source are uncorrelated, the total RMS timing jitter of the TWI-PADC sampling source is determined by = 36 fs.
Figures 4(a)-(d) illustrate the spectra of digitized data at the RF frequency of fIN = 3.1 GHz for M = 0.05, 0.15, 0.25, and 0.50, respectively. The number of samples is 1 × 106 in each channel, corresponding to 10 kHz spectral resolution. Both the signal and the nonlinear distortion (i.e. the 3rd harmonic) increase with M. The SINAD is calculated according to Eq. (10) and depicted as a function of M in Fig. 4(e). According to Eqs. (14) and (15), the amplitude noise becomes dominant in the lower frequency range (i.e. SNRJitter→ + ∞ when fIN/fS → 0). The noise level is calculated to be ~-56 dBc, which means that σN = 1.4 × 10−4 according to Eq. (14).
With all the derived parameters σN, σa, σt, and σJ, the theoretical model [see Eq. (16)] is utilized to the least-squares fitting of experimental results. The fitting curve and its derivative are depicted in Fig. 4(e). It indicates that the fitting curve has a good consistence with the experimental results. The SINAD first increases and then decreases with M for a certain input frequency. The maximum SINAD appears at M = ~0.17, corresponding to the zero derivative of the fitting curve.
Since the amplitude and time skew of each channel can be extracted from the spectral analysis of the digitized data through Eq. (7), hardware adjustments of multi-channel mismatches are performed by manipulation of the VOAs and TDLs [see Fig. 1]. Figures 5(a)-5(d) present the spectra of digitized data after four-step hardware adjustments and Fig. 5(e) summarizes the SINAD for different multi-channel mismatches. Note that M = 0.17. It shows that the distortion spurs are gradually suppressed. In Fig. 5(e), the theoretical estimation based on Eq. (16) is depicted by contours and the measured SINAD values are marked in the parentheses. The SINAD increases from 26.2 dB to 41.5 dB after the hardware adjustments. The consistence of the experimental results and the theoretical estimation verifies the feasibility of the spectral analysis demonstrated above.
4.2 Compensation of the multi-channel mismatches
Figure 6 shows the spectra of the digitized data at different RF frequencies of fIN = 1.1 GHz, 3.1 GHz, 6.1 GHz, and 12.1 GHz, respectively. The modulation index is set to M = 0.17. It is shown that the mismatch spurs can be effectively eliminated by the mismatch compensation algorithm [see Eq. (8)]. As an example, for fIN = 3.1 GHz, the SINAD is enhanced from ~39 dB to ~54 dB and the ENOB is correspondingly improved from ~6.2 bits to ~8.7 bits. Note that the compensation algorithm is carried out as long as data acquisition is finished. It takes ~100 MB of memory and ~1 s (Intel Pentium P6100 CPU).
Figure 7 represents the ENOB as a function of the RF frequency. The discrete points indicate the experimental results of TWI-PADC system, which are calculated by Eqs. (9) and (10). The ENOB without the mismatch compensation is ~6 bits within 0.1~12.1 GHz whereas the ENOB with the mismatch compensation reaches ~7.5 bits within 3.1~12.1 GHz and even approaches ~8.5 bits within 0.1-3.1 GHz. According to the theoretical analyses in Eqs. (11)-(16) and the parameters of σN = 1.4 × 10−4, σa = 1.0 × 10−2, σt = 95 fs, and σJ = 36 fs, the limitations of our TWI-PADC system determined by noise, timing jitter, and channel mismatch are estimated, respectively. Referred to , the ambiguity limitation can be expressed by: ENOB = log2[2ln2/π√6(fINτD)2], where τD = 1.2 ps is the pulse duration of the sampling source in our system. The above limitations are all compared in Fig. 7. It is found that the original performance of the TWI-PADC is mainly limited by multi-channel mismatches. After the mismatch compensation algorithm, it is essentially enhanced to the noise and timing jitter determined limitations.
In Fig. 8, the performances of our TWI-PADC system before and after mismatch compensation are compared with those of the published relevant works. Note that the reference numbers of relevant works are also marked within brackets in Fig. 8. The limitations determined by the timing jitter σJ of 1 ps, 100 fs and 10 fs are estimated by Eq. (15) and depicted as dashed lines in Fig. 8. Juodawlkis et al. achieved 9.8 bits (ENOB) at 3 GHz (bandwidth)  whereas the sampling rate is 505 MS/s. Similarly, 200-MS/s down-sampling with 7 bits (ENOB) and 40 GHz (bandwidth) was demonstrated ; a comparable performance of ENOB and bandwidth with 2 GS/s  or 10 GS/s  sampling rate was reported, respectively. In the work by W. Ng et al. , a 10-GHz AMLL was adopted for a 4-channel TWI-PADC system with 40 GS/s sampling rate. However, the sampling range covers only 1.6 GHz and a narrow-bandwidth filter used in the EADC reduces the sampling bandwidth. Based on the time-stretched scheme, J. Chou et al. reported 4.5 bits (ENOB) at 95 GHz (bandwidth) with an effective sampling rate of 10 TS/s . Note that the original sampling rate 40 GS/s at 95 GHz is illustrated in Fig. 8 to take full advantage of the horizontal axis. It is reasonable because the sampling rate after time-stretching is multiplied whereas the bandwidth after time-stretching is compressed. Compared to the works with high resolutions [4,9,22–24,32,33], our work achieves a high resolution of >7.5 bits and higher sampling rate of 40 GS/s. In comparison to [18,34,35], the sampling rate of our TWI-PADC is comparable but the bandwidth of 12.1 GHz is more dominant. Moreover, the time-stretched scheme shows significant advantages in both the bandwidth and sampling rate whereas the time aperture is intrinsically limited.
We have demonstrated a multi-channel mismatch compensation to improve the performance of the TWI-PADC system with 40 GS/s sampling rate. First, the dependence of multi-channel mismatches, modulation nonlinearity, noise, and timing jitter on the ENOB is analyzed in frequency domain. Second, the compensation of multi-channel mismatches is effectively applied to enhance the ENOB. In consequence, the TWI-PADC is experimentally increased from ~6 bits to >8.5 bits within the bandwidth of 0.1~3.1 GHz and from ~6 bits to >7.5 bits within the bandwidth of 3.1~12.1 GHz. The experimental results are in good agreement with the theoretical analysis and the enhanced performance of the TWI-PADC approaches the limitation determined by the noise level of −56 dBc and timing jitter of 36 fs. In terms of the sampling rate, ENOB, and bandwidth, the performance of the TWI-PADC is comparable and even superior to the published relevant works [9,18,22–24,31–35].
Appendix Dual channel mismatch compensation algorithm
Eq. (7) presents a formula of N-length DFT. an and Δtn in each channel can be derived from or by an inverse DFT. It means that 2N number of amplitude and time skew can be derived from both real and imaginary parts of or . Taking as example, it can be expressed asEq. (15). Hence, Eq. (A. 1) indicates that the amplitude and time skew in each channel can be calculated from the distortion spurs on the digitization spectrum.
For a dual channel TWI-PADC system, the relation between the mismatch-free spectrum and the mismatched spectrum can be derived from Eqs. (4) and (7) as followsEq. (17). With an inverse DFT, Eq. (18) can be converted toEq. (20) can be expressed byEq. (21), which is represented by Eq. (8).
National Natural Science Foundation of China (NSFC) (61571292, 61535006, and 61505105); SRFDP of MOE (grant no. 20130073130005).
We are grateful to all the referees for helpful criticisms of earlier versions of this paper.
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