## Abstract

We present an equalization based inter symbol interference (ISI) mitigation equalization method for time-interleaved photonic analog-to-digital converters (TIPADCs) by modelling the photodetection as a pulse shaper. A zero-forcing equalizer is designed and applied in a single-channel TIPADC system. The periodic ripples in equivalent channel frequency responses under three different photodetection bandwidths are suppressed from ∼2 dB, ∼5 dB, ∼10 dB to ∼0.5 dB, ∼0.5 dB, ∼1 dB, respectively, by the designed equalizer. The TIPADC after equalization has a bandwidth of 15 GHz, mainly limited by MZM. The ability for wideband signal distortion suppression of the proposed method is demonstrated by sampling an equal-power two-tone signal. The output power difference induced by ISI between two tones can be suppressed from ∼10.9 dB to ∼1.3 dB after equalization.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Photonic analog-to-digital converters (PADCs) have attracted a plenty of interests due to its potential ability to break the bottlenecks of traditional electronic analog-to-digital converters (EADCs) [1–9]. Among various kinds of PADCs, time-interleaved PADCs (TIPADCs) combine advantages of photonics (like ultralow jitter and wideband processing) and electronics (like high-precision quantization and mature development), which makes TIPADCs one of the most practically feasible schemes [5–8,10,11]. Photodetection, accounting from the photodiode (PD) to the EADC, is used to realize the conversion from optical signals to electrical signals and quantization, which is an important part in TIPADCs. Feiran Su *et al.* have suggested that the minimum feasible photodetection bandwidth is a half of the single channel sampling rate [12], in order to avoid the distortion induced by the overlap of the adjacent electrical pulses, i.e., inter symbol interference (ISI). However, the suggestion does not take the digital signal processing (DSP) technique into consideration.

In this paper, we model the photodetection in TIPADCs as a pulse shaper and propose an equalization based method to mitigate the ISI in TIPADCs caused by a limited photodetection bandwidth. The proposed ISI mitigation principle is theoretically analyzed based on the equivalent channel frequency response of TIPADCs. The proposed method is experimentally verified in a single-channel TIPADC system by adopting a zero forcing equalization method. An approach to high-precisely acquire zero forcing equalizer tap coefficients is given. The equivalent channel frequency responses with/without the designed equalizers under three different photodetection bandwidths are measured. The results show that the periodic ripples can be suppressed from ∼2 dB, ∼5 dB, ∼10 dB to ∼0.5 dB, ∼0.5 dB, ∼1 dB, respectively, by adopting zero forcing equalizers. An equal-power two-tone signal is sampled by the TIPADC to verify the ability for wideband signal distortion suppression of the proposed method. The results show that the power difference induced by ISI between two tones can be suppressed from ∼2.1 dB, ∼5.3 dB and ∼10.9 dB to ∼0.4 dB, ∼0.4 dB, ∼1.3dB, respectively, after equalization, under three different photodetection bandwidths.

## 2. System modeling and ISI mitigation principle for TIPADCs

The structure of a general TIPADC is illustrated in Fig. 1 [7, 8, 10, 11]. A time-wavelength interleaved optical sampling pulse train is generated from a mode locked laser (MLL) and wavelength division multiplexers (WDMs). Radio frequency (RF) signals are modulated onto the optical sampling pulse train via a Mach-Zehnder modulator (MZM). A subsequent WDM demultiplexes the modulated optical sampling pulse train into parallel channels. Each optical pulse is detected and converted into an electrical pulse by a photodiode, and then sampled and quantized to digital data by a cascaded EADC which is synchronized with the MLL. The sampled RF signals are reconstructed by combining the digital data from all channels with DSP.

Considering the effect of the temporal width of an optical pulse on the equivalent channel frequency responses [12–14], for a TIPADC, the bandwidth of optical pulse should be much larger than that of MZM in order to take full advantage of MZM bandwidth. Thus, the temporal shape of an optical pulse can be approximate to Dirac delta function and the temporal shape of the optical sampling train in the *n*-th channel can be expressed as:

*P*is the average power of the

_{A}*n*-th channel,

*T*is the sampling interval,

_{s}*d*

_{p,n}= (

*n*− 1)

*T*/

_{s}*N*is the delay of the

*n*-th sub-train, and

*δ*(

*t*) is the Dirac delta function.

The modulation via the MZM can be approximated to linear sampling to signals when the input signal power satisfies the small-signal condition and the MZM is biased at quadrature [15]. The responsivity of photodiode can be considered linear under the condition that the pulse energy is far less than the saturation energy [16]. Consequently, the optical pulse is converted and quantized into digital data by photodetection. The modulated components in sampling results can be expressed as [12–14]:

*h*(

_{M}*t*) is the small-signal impulse response of MZM,

*v*(

_{I}*t*) is the sampled signal,

*h*(

_{E}*t*) is the photodetection impulse response,

*d*

_{E,n}represents the delay from the MZM to the EADC in the

*n*-th channel, and * denotes the convolution operation. The unmodulated component in the sampling results are neglected, since it is irrelative to the sampled signals and easy to remove [12,14].

From Eq. (2), besides the conversion from the optical signal to electrical signal, the photodetection will also broaden optical pulse for its limited bandwidths. The optical pulse broadening effect of photodetection can be regarded as pulse shaping to optical pulses. As the optical pulses are narrow enough, the shape of electrical pulses after pulse shaping is approximate to that of the photodetection impose response, *h _{E}* (

*t*). The shaped electrical pulses are then sampled and quantized by EADCs. Consequently, the digital copy of the sampled signal is output. Figure 2 shows the equivalent sampling procedure of a single channel.

The pulse shaping of photodetection with different bandwidths can result in different sampling results, as illustrated in Fig. 3. When the photodetection bandwidth is wide enough, as shown in Fig. 3(a), the temporal width of shaped pulses is sufficient narrow and shaped pulses will not interfere with adjacent ones. EADC sampling values on the shaped pulses will only depend on the sampled signal of *v _{I}*(

*t*) at the corresponding optical sampling time. However, when the photodetection bandwidth is not wide enough, as shown in Fig. 3(b), shaped pulses will overlap with the subsequent ones. Even worse, the former shaped pulses can be non-zero at the EADC sampling time of the current shaped pulse. As a result, a sampling value is not only dependent on the sampled signal at the corresponding optical sampling time, but also previous shaped signals, i.e. ISI in TIPADCs.

The ISI can affect the equivalent channel frequency response of TIPADCs which is expressed as [12,13]:

In Eqs. (3) and (4) the functions denoted by a upper case letter are the Fourier transform of the functions expressed by the corresponding lower case letter.

When there is no ISI in the EADC sampling of the shaped pulses, the magnitude of *R* (Ω) is a constant number and the equivalent channel frequency response is flat. Nevertheless, the magnitude of *R*(Ω) is a periodic function with a period of Ω* _{s}* = 2

*π*/

*T*when there is ISI, which induces ripples in the equivalent channel frequency response. In the cases of sampling wide bandwidth signals, the frequency selection of the channel frequency response with ripples will distort the sampling results.

_{s}In order to form a continuous wide passband and avoid sampling distortion, Feiran Su *et al.* have suggested that the minimum feasible bandwidth of photodetection is a half of the single channel sampling rate [12]. In the case, a wideband photodetection is needed for TIPADCs with higher single channel sampling rate. An analog equalizer can be used to increase photodetection bandwidth and hence mitigate ISI. However, an analog equalizer with wide bandwidth is hard to realize and will increase system complexity [17–19]. For TIPADCs, the sampling data only relay on the samples at intervals *T _{s}* at the EADC. Therefore, digital equalizers can be used to mitigate ISI in digital domain. The equivalent sampling procedure with an equalizer is shown in Fig. 4.

Suppose the frequency response of the digital equalizer is *H _{EQ}*(Ω). In order to mitigate ISI, the equalized photodetection bandwidth,

*H′*(Ω), should satisfy the Nyquist ISI criterion:

_{E}Using the Fourier series expansion and the periodicity of digital filter, *H _{EQ}*(Ω) can be expressed as:

The impulse response of the equalizer can be obtained with Fourier inverse transform and expressed in digital domain:

After equalization, *H _{E}* (Ω) is replaced with

*H′*(Ω). From Eqs. (3)–(5), the magnitude of equivalent channel frequency response can be expressed as:

_{E}From Eq. (9), one can see that there is no periodic ripples in the channel frequency response after digital equalization. A photodetection with a narrow bandwidth will not lead to ISI induced sampling distortion. In this case, the minimum feasible bandwidth of photodetection can be less than a half of the single channel sampling rate by introducing digital equalization without increasing the complexity of the system hardware. Therefore, TIPADCs can reach a higher single sampling rate, hence a higher system sampling rate with a limited spectra width of optical pulses, without the demand of wideband photodetection. The bandwidth of the equivalent channel frequency response is only limited by the frequency response characteristic of the MZM.

## 3. Zero-forcing equalization implement

Equations (5) and (8) indicate that an ideal digital equalizer which can eliminate ISI completely should be an infinite impulse response (IIR) filter. However, an IIR filter cannot be realized in practice. It is necessary to approximate the ideal IIR filter with a finite impulse response (FIR) filter. Considering the stable and linear time-invariant characteristics of the frequency response of the photodetection, an effective digital zero-forcing equalizer, as one kind of FIR filter, can be employed to mitigate the ISI in TIPADCs [20,21].

In the zero-forcing equalization, the continuous-time system model is translated to an equivalent discrete-time model, where *h _{E}*(

*t*) is expressed as a discrete copy of it,

*h*[

_{E}*k*].

*h*[

_{E}*k*] is determined by the temporal shape of

*h*(

_{E}*t*), the EADC sampling point on

*h*(

_{E}*t*), and the channel sampling rate. The taps coefficients of the zero-forcing equalizer need to satisfy 2

*N*+ 1 linear equations when the equalizer has 2

*N*+ 1 taps:

Equation (10) indicates that the photodetection impulse response after zero-forcing equalization has only one non-zero value, and the rest of the other values are all forced to zeros, which ensues there is no ISI at these sampling points.

According to Eqs. (8) and (10), *h _{EQ}*[

*k*] can be determined by

*h*[

_{E}*k*]. For a TIPADC, the EADC sampling point on

*h*(

_{E}*t*), and the channel sampling rate are determinate.

*h*[

_{E}*k*] can be obtained as long as

*h*(

_{E}*t*) is known. A precise measuring method for

*h*(

_{E}*t*) based on equivalent time sampling is presented and shown in Fig. 5(a), where all the devices in photodetection are taken into consideration. A low repetition rate optical pulse train with a narrow enough pulse width is employed to avoid the effect of ISI. After passing through an MZM without applying microwave signal and a WDM, the optical pulse train is detected by a photodiode. The detected electrical pulses are sampled by an EADC with a sampling clock having a small frequency deviation from the repetition rate of the optical pulse train. Because of the frequency deviation, a little incremental delay will be added before the next sampling to the electrical pulse by the EADC. The entire electrical pulse shape can be sampled at a high equivalent sampling rate as long as the sampling time is long enough, since the sampled electrical pulses is a periodic signal. Finally, the electrical pulse shape can be reconstructed according to the discrete sampling values, as shown in Fig. 5(b).

The measured result of *h _{E}*(

*t*) can be expressed as

*τ*is the equivalent sampling time interval, i.e. incremental delay, and determined by the repetition rate of the optical pulses,

*f*, and the EADC clock frequency,

_{m}*f*:

_{c}From Eq. (11), when Δ*f* = *f _{c}* −

*f*is a positive number, the measurement is equivalent to sample

_{m}*h*(

_{E}*t*) with a sampling rate of 1/

*τ*. However, when Δ

*f*is a negative number, the measurement is equivalent to sample

*h*(−

_{E}*t*) with a sampling rate of 1/

*τ*. Under this circumstance, the sampling result should be inversed in time-domain to acquire the correct measurement result. When Δ

*f*is zero, the measurement is equivalent to sample

*h*(

_{E}*t*) at a fixed point, which induces the sampling result is a constant number.

After the measurement, *h _{E}*[

*k*] can be obtained from

*h*

_{E,m}[

*l*] according to the sampling point on the electrical pulse and the single channel sampling interval,

*T*, in TIPADCs:

_{s}*L*is a offset corresponding to the sampling point on the electrical pulse.

## 4. Experimental results

The proposed equalization based method is verified on a single-channel TIPADC shown in Fig. 6. A passive MLL (OneFive, Origami) generates the optical pulses at a repetition rate of 250 MHz. RF signals are generated from a signal generator (Rohde & Schwarz, SMF 100A) and modulated onto the optical pulses with a 40 Gbps MZM (Fujitsu, FTM7939EK) biased at quadrature. The WDM with 200 GHz bandwidth in C-band is used to be consistent with the sampling channel in multi-channel setup. A variable delay line (VDL) with a delay resolution of 1 fs and a span of 560 ps is used to adjust the EADC sampling point on the electrical pulses. A variable optical attenuator (VOA) attenuates the average power of optical pulses to ∼−20 dBm to ensure the pulse energy is in the linear region of the responsivity of photodiode. The bandwidth of the PD with trans-impedance amplifier (TIA) is 300 MHz and the average optical power at 1 dB compression point of responsivity is −14 dBm. The minimum detectable average optical power is −40 dBm. Electrical pulses are quantized by a EADC (Keysight, M9703A) with an analog bandwidth of 1.2 GHz and an ENOB of 9.0. The EADC is clocked by the MLL via a phase locked loop (PLL). 2nd-order Butterworth lowpass filters (LPFs) with different bandwidths are employed between the PD and the EADC to change the photodetection bandwidth and the electrical pulse shapes, which simulates a PADC system with different photodetection bandwidth. The bandwidth of the cascaded LPFs are 90 MHz, 70 MHz, and 40 MHz, respectively. Comparing to the PD and the EADC, the LPFs have the lowest bandwidth, so the photodetection bandwidth can be approximately considered as the LPF bandwidth.

The photodetection impulse response is measured via the setup shown in Fig. 7. All the optical and electrical devises are the same as the ones in Fig. 6, except the optical source and the EADC clock source. Optical pulses with a low repetition rate of 36.456 MHz are generated from another MLL (Precision Photonics, FFL1560). A microwave signal is generated from a signal generator (SG) and clock the EADC. The frequency deviation between the SG and the optical pulse repetition rate is 2 KHz, which is equivalent to sample the electrical pulses at a sampling rate (1/*τ*) of ∼664.5 GS/s. Considering the duration of a shaped electrical pulse is tens of nanoseconds under the photodetection bandwidths in the experiment, the equivalent sampling rate is enough to extract the detail of shaped electrical pulses.

The measurement results of the photodetection impulse response with a bandwidth of 90 MHz, 70 MHz, and 40 MHz are illustrated in Fig. 8(a), where all the impulse responses are normalized by the corresponding maximum value. The sampling results with frequency deviation of −2 KHz and zero are illustrated in Figs. 8(b) and 8(c). When the frequency deviation is −2 KHz, the sampling results is the time reversal of the sampling results with a frequency deviation of 2 KHz. When the frequency deviation is zero, the sampling results are constant. The results are consistent with the theoretical ones. From the figure, the duration of shaped pulses increases with the decease of LPF bandwidths. Considering the channel sampling interval of 4 ns in the experimental TIPADC, a shaped pulse will interfere with subsequent shaped pulses, and the interference becomes worse with the decease of LPF bandwidths.

Figure 9 shows the channel equivalent frequency responses with different photodetection bandwidth with and without equalization. The channel frequency response is measured by stimulating the channel with a single tone at different frequencies and getting the output from the EADC. The input RF power is 0 dBm to satisfy the small signal condition [15]. In the results, the photodetection with a bandwidth of 90 MHz, 70 MHz, and 40 MHz induce different level power ripples of ∼2 dB, ∼5 dB, ∼10 dB, correspondingly. The ripples indicate that these photodetection bandwidths are too narrow comparing with the sampling rate to form a continuous passband. Moreover, the lower the bandwidth of LPF is, the more broadly the pulse is broadened, and the severer the ISI becomes. After equalized by the proposed method, the power ripples are suppressed to ∼0.5 dB, ∼0.5 dB, ∼1dB, respectively. And the bandwidth of TIPADC is widened to 15 GHz. The decrease of the equalized channel frequency response with the frequency is due to the frequency response of MZM, as shown in Fig. 9(d), which is consistent with Eq. (9).

To demonstrate the ability for wideband signal distortion suppression of the proposed method, an equal-power two-tone signal (2001 MHz and 2124 MHz) is fed into the TIPADC with a photodetection bandwidth of 90 MHz, 70 MHz, and 40 MHz, respectively. The power of equal-power two-tone signal is 0 dBm. The normalized power spectrum of the sampling results before and after equalization is shown in Fig. 10. The power is normalized by the max power in the power spectrum, and the frequency is normalized by a half of the sampling rate. In the figure, the effect of the MZM bandwidth to the power spectrum has been compensated. Before equalization, the two tones in the output power spectrum of the TIPADC are different under the input of the equal-power two-tone signal for ISI. With the decrease of photodetection bandwidth from 90 MHz to 40 MHz, the power differences increase from ∼2.1 dB to ∼10.9 dB. After equalization, the power difference is mitigated to ∼0.4 dB, ∼0.4 dB, ∼1.3 dB, and the distortions are corrected by ∼1.8 dB, ∼4.9 dB, ∼9.6 dB, respectively.

In above experiments, the channel sampling rate of the established TIPADC is 250 MHz. According to the theory proposed by Feiran Su *et al.*, minimum feasible photodetection bandwidth to form a continuous passband should be half of the channel sampling rate (125 MHz). The experimental results show that when the photodetection bandwidth (90 MHz, 70 MHz and 40 MHz in experiment) is less than a half of the channel sampling rate, the ripple in the equivalent channel frequency response can be compensated by equalization, and a continuous pass band can be formed. Hence, the signal distortion induced by ISI can be suppressed. Correspondingly, under the existing photodetection bandwidth, the introduction of equalization makes it possible to further increase the single channel sampling rate, thereby increasing the system sampling rate of TIPADCs.

## 5. Conclusion

The photodetection in TIPADCs is modeled as a pulse shaper to optical pulses. According to the model, the equalization is introduced for mitigating ISI caused by a limited photodetection bandwidth and correcting sampling distortion. A theoretical analysis based on the equivalent channel frequency response is given. An equalization method based on zero-forcing equalization is designed in detail, including the high-precise obtainment of tap coefficients method. The zero-forcing equalization based method is applied to a single-channel TIPADC to experimentally verify the proposed equalization method. In the experiment, the channel equivalent frequency responses with/without equalizers under three different photodetection bandwidths are measured. The results show that the designed equalizer suppresses the periodic ripples from ∼2 dB, ∼5 dB, ∼10 dB to ∼0.5 dB, ∼0.5 dB, ∼1 dB, respectively, and widen the bandwidth of TIPADC to 15 GHz. To demonstrate the ability of the proposed method for wideband signal distortion suppression, an equal-power two-tone signal is sampled. The power difference induced by ISI between two tones is suppressed from ∼2.1 dB, ∼5.3 dB and ∼10.9 dB to ∼0.4 dB, ∼0.4 dB, ∼1.3dB by the proposed equalization.

## Funding

National Natural Science Foundation of China (NSFC) (61535006, 61627817).

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