compensation of multi-channel mismatches in high-speed

Compensation of multi-channel mismatches in high-speed high-resolution photonic analog-to-digital converter GUANG YANG, WEIWEN ZOU,* LEI YU, KAN WU, AND JIANPING CHEN State Key Laboratory of Advanced Optical Communication Systems and Networks, Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China *[email protected]

Abstract: 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

OCIS codes: (060.5625) Radio frequency photonics; (230.0250) Optoelectronics; (250.4745) Optical processing devices; (000.4430) Numerical approximation and analysis.

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1. Introduction 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 [6], and telecommunications [7]. 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 [8], 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 [9] 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. [18] 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 [25], 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. [3] 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 [17], 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 [23]. 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.

Fig. 1. Experimental setup of the TWI-PADC system. (a) Generation of time-wavelength interleaved sampling source, (b) Schematic of spectral slicing of mode locked laser and time-wavelength mapping of pulse trains, (c) Photonic sampling via a Mach-Zehnder modulator, (d) Schematic of photonic sampling in time domain, (e) Photodetection and electronic digitization, (f) Schematic of electronic digitization. WDM: wavelength-division multiplexer, TDL: tunable delay line, VOA: variable optical attenuator, FRM: Faraday rotator mirror, EOM: electro-optical modulator, PD: photo-detector, ADC: analog-to-digital converter.

The optical intensity of the sampling source (i.e. optical pulse train) of the TWI-PADC system can be expressed by

( ) ( )0

4 1 , 1, 2,3, 4,n n S S nk

p t A t kT n T t nδ∞

=

= − − − − Δ = (1)

where pn(t) corresponds to the pulse train in the nth channel. An and Δtn denote its amplitude and time skew, respectively. δ(·) represents the Dirac function, which is used as an approximation of the sampling source with a temporal interval of TS = 1/fS = 25 ps.


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, ( )INv t , as follows [9]

( ) ( )1 1cos ,

2 2M IN BT t v t VVπ

π= + + (2)

where Vπ and VB are the half-wave voltage and the DC bias of the EOM, respectively. In mathematics, the electronic sampled signal in the nth channel can be described by

( ) ( ) ( ), , , 1, 2,3,4,S n n PD n M nv t G R T t p t n= = (3)

where RPD,n and Gn are the responsivity of the PD and the transmittance gain (or loss) of the converted electronic signal in the nth channel, respectively. Consequently, the peak value of the sampled signal is captured and digitized by the multiple channels, which is schematically illustrated in Fig. 1(f).

The digitized data after channel mapping can be written by

[ ] ( )4

1

4 1 ,Q n M S S nn

v k a T kT n T t=

= + − + Δ (4)

where the integer k is the number of digitized samples. an = GnRPD,nAn and Δtn denote the digitized amplitude and the time skew in the nth channel, respectively.

3. Mathematical derivation

3.1 Spectral analysis

Consider the RF signal to be sampled is sinusoidal, which can be expressed by

( ) 0 cos( )IN INv t V t= Ω 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 follows

[ ] ( ) ( ) ( ) ( )( )4

2 +11 0

1 cos 2 1 1 ,2

mnQ n m IN n S

n m

av k a J M m k n t Tω

∞

= =

= + − − + − + Δ (5)

where ωIN = ΩINTS is the normalized angular frequency of the digitized data, M = πV0/Vπ is defined as the modulation index, and J2m+1 is the (2m + 1)th order Bessel function. In Eq. (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 by

[ ] ( ) ( ) ( )

( ) ( ) ( )

31

2 10 1

31

2 10 1

2= 1 2 1

4,

21 2 1

4

m

Q l m INl m

m

l m INl m

kV J M m

kJ M m

πω π β δ ω ω

πβ δ ω ω

∞++

−= =

∞+−

−= =

− − − − + − + − −

(6)

where ω = 2πfTS is the normalized angular frequency and

( ) ( )2 1 2 14 44 4

1 1

1 1, , 0,1,2,3.

4 4IN n IN n

l n l nj jj t j t

l n l nn n

a e e a e e lπ π

β β− −

− −Ω Δ − Ω Δ+ −

= =

= = = (7)


Note that the spectrum [ ]QV ω in Eq. (6) and lβ + ( lβ − ) in Eq. (7) are complex, which can be

derived from the digitized data. Furthermore, the spectrum [ ]QV ω 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 [26]. 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.

Fig. 2. Schematic of a typical DFT spectrum with channel mismatch effects exiting in a 4-channel TWI-PADC.

Based on the algorithm of dual channel mismatch compensation demonstrated in Appendix, the digitized data [ ]Qv k of a single-frequency RF signal at fIN can be reconstructed

by

[ ][ ]

[ ][ ]

( ) ( )( ) ( )

2 21 2 1 2-1

2 21 2 1 2

2 2, = ,

2 2

IN IN

IN IN

j f t j f tj k

Q Q

j f t j f tj kQ Q

a a e e a a ev k v k

v k v k e a a e a a e

π δ π δπ

π δ π δπ

−+ +

− − −−

+ − = − +

H H

(8)

where [ ] 1 [ 0]Q Qv k F V ω+ −= > and [ ] 1 [ 0]Q Qv k F V ω− −= < are data series in time domain

derived by an inverse DFT from the positive and negative band of the DFT spectrum of

[ ]QV ω , respectively. The reconstructed data can be expressed by [ ] [ ] [ ]= +Q Q Qv k v k v k+ − . a1 and

a2 are the amplitudes of the dual channels. δt = Δt2-Δt1 is the relative time skew between the dual channels, which can be calculated from the spectrum in Eq. (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 [28], the ENOB is defined by

1.76

,6.02PADCSINAD

ENOB−

= (9)

where SINADPADC is the signal to noise and distortion ratio (SINAD) of the PADC and given by [28]

signal

noise distortion

10 log ,+PADC

PSINAD

P P= (10)


where Psignal, Pdistortion, and Pnoise are the powers of the RF signal, distortion, and noise. They can be numerically derived from the DFT of the quantized data after channel mapping [see 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 follows

2 2 2 2

20 20 20 2020 log 10 10 10 10 ,Modulation Mismatch Noise JitterSDR SDR SN

PAD

R S R

C

N

SINAD− − − −

= − + + +

(11)

where SDRModulation, SDRMismatch, SNRNoise, or SNRJitter corresponds to the modulation nonlinearity, channel mismatch effects, amplitude noise, and timing jitter, respectively. They can be derived from the spectrum in Eq. (6). Since the modulation nonlinearity is dominantly by the 3rd harmonics, SDRModulation can be approximately expressed by

( ) ( ) ( )3 120 log 20 log ,ModulationSDR J M J M Mγ= − = − (12)

where γ(M) represents the effect of the modulation nonlinearity. SDRMismatch is determined by the ratio between the powers of the RF signal and

mismatches as follows

( )

32 -2

2 2 2 212 -2

0 0

20 log 20log 4 ,Mi

k kk

a Ism N tatchS fDRβ β

σ π σβ β

+

=+

+= − = − +

+

(13)

where σa is the normalized deviation of the amplitude an, defined by ( )4

2

1

14a n

n

a aa

σ=

= −

, and 4

1

4nn

a a=

= . σt is the root-mean-square (RMS) of time skew Δtn, i.e. 4

2

1

4t nn

tσ=

= Δ .

Similar to EADCs [29], SNRNoise can be described by

( ) ( )120 log 20log ,Nois Ne NS MN JR Mσ ρ= − = − (14)

where σN is the noise level including the intensity noise of the optical source, the shot noise of the photodetector, and the thermal noise. ρN(M) represents the ratio of σN to the RF amplitude.

Besides, SNRJitter is determined by the timing jitter as follows

20log 2 ,IN JJitterSNR fπ σ= − (15)

where σJ is the RMS timing jitter of the TWI-PADC sampling source. The entire SINAD of the PADC system can be derived from Eqs. (11)-(15), which is

written by

( ) ( ) ( )2 2 2 2 2 2 220log 4 + .PADC IN t J a NSINAD f M Mπ σ σ σ ρ γ= − + + + (16)


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 2 2,MLL ,RF=J J Jσ σ σ+ = 36 fs.

Fig. 3. SSB phase noise spectra of AMLL and electronic oscillator and their integral RMS timing jitters.


Fig. 4. Spectra of digitized data at the frequency of fIN = 3.1 GHz for different modulation index (M): (a) M = 0.05, (b) M = 0.15, (c) M = 0.25, (d) M = 0.50. (e) SINAD versus modulation index (M). The blue diamond (left vertical axis) represents the experimental results. The green curve (left vertical axis) denotes the fitting curve according to the theoretical model of Eq. (16) and the red dashed curve (right vertical axis) is its derivative. σN = 1.4 × 10−4, σJ = 36 fs, σa = 1.0 × 10−2, and σt = 95 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.

Fig. 5. Spectra of digitized data for different multi-channel mismatches: (a) σa = 8.2 × 10−2 and σt = 378 fs, (b) σa = 4.4 × 10−2 and σt = 197 fs, (c) σa = 2.1 × 10−2 and σt = 128 fs, (d) σa = 1.0 × 10−2 and σt = 95 fs. (e) SINAD for different multi-channel mismatches. The blue triangles represent the experimental results which are indicated in the parentheses. The contours denote the theoretical estimation based on Eq. (16).

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).


Fig. 6. Comparison between spectra of digitized data without (solid curves) and with (dotted curves) mismatch compensation. M = 0.17, (a) fIN = 1.1 GHz, (b) fIN = 3.1 GHz, (c) fIN = 6.1 GHz, (d) fIN = 12.1 GHz.

Fig. 7. ENOB of the TWI-PADC as a function of the RF input frequency. Diamonds and triangles represent the experimental values without and with channel mismatch compensation, respectively. The theoretical limitation corresponding to the ambiguity (i), timing jitter (ii), noise level (iii), multi-channel timing mismatch (iv), or multi-channel amplitude mismatch (v) is depicted for comparison, respectively. Numerical estimation with or without channel mismatch compensation is plotted by (vi) or (vii), respectively.

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 [30], 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.


Fig. 8. ENOB of PADCs as a function of bandwidth (a) and sampling rate (b). The blue circles indicate the relevant works published in the literatures (marked in brackets). The diamonds and triangles indicate our system performance before channel mismatch compensation and after compensation, respectively. The dashed lines represent the theoretical limitations determined by the timing jitter of 1 ps, 100 fs, and 10 fs, respectively.

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) [9] whereas the sampling rate is 505 MS/s. Similarly, 200-MS/s down-sampling with 7 bits (ENOB) and 40 GHz (bandwidth) was demonstrated [22]; a comparable performance of ENOB and bandwidth with 2 GS/s [23] or 10 GS/s [24] sampling rate was reported, respectively. In the work by W. Ng et al. [18], 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 [31]. 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.

5. Conclusion 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

lβ + or lβ − by an inverse DFT. It means that 2N number of amplitude and time skew can be

derived from both real and imaginary parts of lβ + or lβ − . Taking lβ + as example, it can be

expressed as

( ) ( )2 1 2 11 1

0 0

1, arg .

l n l nN Nj jN N

n l n ll lIN

a e t eπ π

β β− −− −

+ +

= =

= Δ = Ω (17)

Besides, lβ + represents the component of the digitization spectrum at f = lfS/N ± fIN as shown

in Eq. (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

[ ]QV ω and the mismatched spectrum [ ]QV ω can be derived from Eqs. (4) and (7) as follows

[ ] [ ] [ ]

[ ] [ ] [ ]

0 1

0 1

,2 2

,2 2

S SQ Q Q

S SQ Q Q

V H V H V

V H V H V

ω ωω ω ω ω ω

ω ωω ω ω ω ω

+ + −

− − +

= + − − − = − + + +

(18)

where [ ]QV ω+ , [ ]QV ω− and [ ]QV ω+ , [ ]QV ω− are the positive part and negative part of [ ]QV ω

and [ ]QV ω , respectively. And,

[ ] ( ) [ ] ( )0 1 2 1 1 2

1 1= , = ,

2 2S Sj T j TH a a e H a a eωδ ωδω ω+ − (19)

where a1 and a2 are the amplitudes of the two channels and δ = Δt2-Δt1 is the relative time skew between two channels which can be derived from Eq. (17). With an inverse DFT, Eq. (18) can be converted to

[ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ]

0 1

0 1

,

,

j kQ Q Q

j kQ Q Q

v k h k v k e h k v k

v k h k v k e h k v k

π

π

+ + −

− − − +

= ∗ + − ∗

= − ∗ + ∗

(20)

where h0[k] = F−1H0[ω], h1[k] = F−1H1[ω], and the compensated data

[ ] [ ] [ ]= +Q Q Qv k v k v k+ − . For a single-tone input at ωIN, Eq. (20) can be expressed by

[ ][ ]

( ) ( )( ) ( )

[ ][ ]

[ ][ ]

1 2 1 2

1 2 1 2

2 2.

2 2

IN IN

IN IN

j jj k

Q Q Q

j jj kQ QQ

a a e e a a ev k v k v k

v k v kv k e a a e a a e

δ δπ

δ δπ

Ω − Ω+ + +

− −− Ω − Ω−

+ − = = − +

H

(21)

When detH ≠ 0, i.e. cosΩINδ ≠ 0 (which is always tenable for δ much smaller than TS), the compensated data can be reconstructed according to Eq. (21), which is represented by Eq. (8).


Funding National Natural Science Foundation of China (NSFC) (61571292, 61535006, and 61505105); SRFDP of MOE (grant no. 20130073130005).

Acknowledgments We are grateful to all the referees for helpful criticisms of earlier versions of this paper.


compensation of multi-channel mismatches in high-speed

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