Effects of the photonic sampling pulse width and the photodetection bandwidth on the channel response of photonic ADCs

Feiran Su; Guiling Wu; Lei Ye; Rui Liu; Xiang Xue; Jianping Chen

doi:10.1364/OE.24.000924

1. Introduction

Photonic analog-to-digital converter (PADC) is a promising alternative to its electronic counterpart for measuring ultra-wide bandwidth signals [1–3]. Up to now, several PADC schemes have been proposed [4–7]. Among them the time-interleaved PADC (TIPADC) is one of the most practically feasible schemes for its potential to convert multi-ten-GHz signals with high effective number of bits (ENOB) by combining advantages of photonics in ultra-low jitter and wideband processing and electronics in high precision quantization [7–10]. The photonic sampling pulse’s temporal width (pulse width for short hereafter) and the photodetection bandwidth are two key parameters for TIPADCs. The pulse width is related to the performance of adopted laser sources, the bandwidth of optical devices and the distortion in lightwave propagation. The photodetection bandwidth is determined by all devices from photodiode to electronic ADC (EADC) in a sampling channel. They dominate the fundamental performance, especially the channel frequency response, of a TIPADC. Juodawlkis et al. [11] investigated the impact of the pulse width on the frequency response of photonic sampling systems using integrate-and-reset detection, which converts the pulse energy to a voltage through photodetection and integration processes, and quantizes the voltage with EADCs. Valley et al. [12] analyzed the effect of finite pulse width on TIPADCs where the difference between the ideal signal value at the sampling time and the average value over the corresponding Gaussian photonic pulse is considered as the error. Khilo et al. [10] have suggested that photodetection bandwidth should be several times larger than the single channel sampling rate to avoid inter-symbol interference (ISI) between pulses. However, there is still no quantitative analysis about the effect of photodetection bandwidth on TIPADCs.

In this paper, we investigate the effects of the photonic sampling pulse width and the photodetection bandwidth on TIPADC performance in terms of channel response. By modeling the sampling process, each channel is equivalent to a linear analog channel with an ideal sampler, and the equivalent response of channel is derived accordingly. Then the impact of the photonic sampling pulse width and the photodetection bandwidth on channel response is analyzed theoretically. The results indicate that finite temporal width of the photonic sampling pulse will limit channel analog bandwidth and distort the input signal. The minimum feasible photodetection bandwidth is the half of single channel sampling rate. The ISI induced by finite photodetection bandwidth will cause periodic ripples on channel frequency response. The channel frequency responses of a TIPADC under different pulse widths and photodetection bandwidths are measured experimentally. The results are consistent with the theoretical ones.

2. Modeling and analysis

Figure 1(a) illustrates the diagram of typical N-channel TIPADCs. RF input is modulated onto a high-speed time-wavelength interleaved photonic sampling pulse train via an electro-optic modulator. The modulated photonic sampling pulse train is then demultiplexed into N parallel channels by optical demultiplexing and converted to electrical signals by photodiodes, which are then sampled and quantized by EADCs. Finally, the quantized data in all channels are combined to reconstruct the sampled RF input by digital processing.

Fig. 1 (a) TIPADC scheme and (b) the equivalent sampling procedure.

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The photonic sampling pulse train, which is typically generated based on a mode-locked laser via spectrum slicing and time-wavelength mapping [7–10], is composed of N sub-trains. Each sub-train has the same wavelength and can be considered uniform since it is from the same mode-locked laser and goes through the same path. Therefore, the temporal shape of the n-th sub-train can be expressed as

p_{n} (t) = P_{A, n} \sum_{m = - \infty}^{\infty} p_{S, n} (t - m T_{S} - d_{P, n}),

where

P_{A, n}

is the average power of the n-th sub-train,

p_{S, n} (t)

is the photonic sampling pulse power normalized by

P_{A, n}

,

T_{S}

is the single channel sampling period,

d_{P, n}

is the delay of the n-th sub-train.

The lithium niobate Mach-Zehnder modulator (MZM) is usually employed as electro-optic modulator, whose response to the voltage of RF input $v_{I} (t)$ is a cosine function. In the small-signal input condition, the response of MZM biased at quadrature can be approximated linearly [13]. Hence the n-th modulated sub-train can be expressed as

p_{M, n} (t) \approx 0.5 α_{n} [1 - h_{M, n} (t) * v_{I} (t)] p_{n} (t),

where

α_{n}

is the attenuation factor of MZM to the n-th sub-train,

h_{M, n} (t)

is the small-signal impulse response of MZM.

The temporal shape of pulse after optical demultiplexing can be considered unchanged as long as the bandwidth of optical demultiplexing is larger enough. The effect of nonlinearity, mainly the self-phase modulation (SPM), in the fiber after modulator can be neglected provided that the fiber length is less than $L_{N L, n} / 10$ , where $L_{N L, n}$ is the nonlinear length threshold of SPM [14]:

L_{N L, n} = \frac{1}{α_{n} γ_{n} P_{A, n} p_{S, n} (0)},

where

γ_{n}

is the nonlinear coefficient of fiber. For a 100 MHz sub-train with an average power of 0 dBm, if sampling pulses are Gaussian with full-width at half-maximum (FWHM) of 3 ps,

L_{N L, n}

of a standard single-mode optical fiber with

γ_{n} \approx

2 W⁻¹km⁻¹ at 1550 nm is ~160 m in the worst case where

α_{n}

= 1 and the pulses have the maximum peak power after MZM. Hence the SPM effect can be ignored as long as the fiber length is less than 16 m in this case. For a practical implementation, it is easy to control the fiber length within this limit.

The responsivity of photodiode can also be considered linear when the pulse energy is far less than the saturation energy [15]. Therefore, the processing after modulator can be approximately linear, and the samples extracted by the EADC in the n-th channel can be expressed as

v_{Q, n} [k] = {h_{E, n} (t - d_{E, n}) * p_{M, n} (t) |}_{t = k T_{S}},

where

h_{E, n} (t)

is the impulse response including all the devices from the photodiode to the EADC and

d_{E, n}

is the delay used to adjust the timing of pulses for the sampling of EADC in the n-th channel.

After a simple derivation (detailed in Appendix 1), we have

{\begin{cases} v_{Q, n} [k] = V_{Q 0, n} + v_{Q 1, n} [k] \\ {V_{Q 0, n} = 0.5 α_{n} h_{E, n} (t - d_{E, n}) * p_{n} (t) |}_{t = 0} \\ {v_{Q 1, n} [k] = h_{A, n} (t) * v_{I} (t) |}_{t = k T_{S}} \end{cases},

where

h_{A, n} (t)

is the equivalent channel impulse response

h_{A, n} (t) = - 0.5 α_{n} [h_{E, n} (t - d_{E, n}) p_{n} (- t)] * h_{M, n} (t) .

In Eq. (5),

V_{Q 0, n}

is the DC component irrelevant to RF input. We can get the RF input from the component

v_{Q 1, n} [k]

. From the equation we can see that each channel can be considered as a sampling channel with an equivalent channel impulse response

h_{A, n} (t)

, and an N-channel TIPADC can be accordingly equivalent to a generalized sampling system [16]. Figure 1(b) shows the equivalent sampling procedure.

When channels are matched both in gain and delay ( $d_{P, n} = (n - 1) T_{S} / N$ and $d_{E, n} = (N - n + 1) T_{S} / N$ ), we have

h_{A, n} (t) = h_{A, 1} (t + \frac{n - 1}{N T_{S}}) .

In such case, one can interleave

v_{Q 1, n} [k]

of all channels directly to reconstruct the RF input since the difference of response among channels is only the ideal delay. It is easy to prove that the system response

h_{A} (t)

is the same as that of channel 1:

h_{A} (t) = h_{A, 1} (t) .

Generally, there are mismatches between the channels, which will induce undesired spurs in the spectrum of the interleaved output and degrade the performance of the system. Reconstruction method similar to those for traditional time-interleaved EADCs is needed to eliminate the mismatch effect [17].

For the equivalent sampling procedure, the total sampling rate should be sufficient at first to avoid aliasing in the final output according to sampling theorem. At the same time, for a time-interleaving ADC, since the RF input passes through each channel and is then sampled by a sampler at each channel, the analog bandwidth of each channel should be larger than that of input signal to avoid signal component losing before sampling operation.

Channel analog bandwidth can be investigated through its frequency response. From Eq. (6), this response can be derived as

H_{A, n} (Ω) = - \frac{α_{n} P_{A, n}}{2 T_{S}} H_{M, n} (Ω) \sum_{m = - \infty}^{\infty} c_{m, n} {H^{'}}_{E, n} (Ω + m Ω_{S}),

where

{\begin{cases} c_{m, n} = P_{S, n} (m Ω_{S}) \exp (- j m Ω_{S} d_{P, n}) \\ {H^{'}}_{E, n} (Ω) = H_{E, n} (Ω) \exp (- j Ω d_{E, n}) \end{cases} .

In Eq. (9) and Eq. (10), the functions denoted by a upper case letter are the Fourier transform of the functions expressed by the corresponding lower case letter, and

Ω_{S} = 2 π / T_{S}

.

H_{A, n} (Ω)

is the weighted summing of the shifted replicas of

{H^{'}}_{E, n} (Ω)

, where the weights,

c_{m, n}

, are determined by the photonic sampling pulse train. Figure 2 illustrates the formed channel frequency responses in two bandwidth cases of

H_{E, n} (Ω)

. One can see that since the interval is

Ω_{S}

, the adjacent replicas can overlap each other and form a continuous passband only when the ratio of the bandwidth of

H_{E, n} (Ω)

to

Ω_{S}

, called normalized bandwidth

β_{E, n}

, is not less than 0.5. This case implies that channel analog bandwidth can be much larger than photodetection bandwidth. Of course, the bandwidth of the formed continuous passband is limited by the bandwidth of

P_{S, n} (Ω)

since the weights for the replicas out of the bandwidth of

P_{S, n} (Ω)

will descend at least 3dB compared to

c_{0, n}

. When

β_{E, n} < 0.5

, no overlap occurs between the shifted replicas and there are multiple discrete narrow passbands in channel frequency response. Therefore, for a channel frequency response with a continuous wide passband, we have the following constrains:

{\begin{matrix} β_{C, n} \leq β_{M, n} \\ β_{C, n} \leq β_{P, n} \\ β_{E, n} \geq 0.5 \end{matrix},

where

β_{C, n}

,

β_{M, n}

and

β_{P, n}

are the normalized bandwidth of

H_{A, n} (Ω)

,

H_{M, n} (Ω)

and

P_{S, n} (Ω)

to

Ω_{S}

, respectively.

Fig. 2 Channel frequency responses in two bandwidth cases of $H_{E, n} (Ω)$ .

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Since $β_{P, n}$ is reversely proportional to pulse width, the finite temporal width of the photonic sampling pulse will limit the bandwidth of equivalent channel according to Eq. (11) and hence distort the input signal. For a Gaussian pulse with a temporal FWHM of $σ$ , $β_{P, n} = \sqrt{2} \ln 2 / π σ$ , and the channel analog bandwidth is limited within 100 GHz for a 3 ps Gaussian pulse. Nevertheless, finite pulse width will not result in excess noise. This is because the channel response for each given pulse width is deterministic, which will not induce random fluctuations on output signals.

From Eq. (11), the minimum feasible photodetection bandwidth is half of the single channel sampling rate in order to form a continuous wide passband. The pulses after a photodetection with low $β_{E, n}$ will be broadened, which may cause overlap between pulses and induce ISI in the sampling of EADC. For typical TIPADC configurations, where the temporal width of modulated photonic pulses is sufficient short compared to $h_{E, n} (t)$ , we have the following expression (detailed in Appendix 2) when there is ISI:

\frac{1}{T_{S}} \sum_{m = - \infty}^{\infty} {H^{'}}_{E, n} (Ω + m Ω_{S}) \exp [- j (Ω + m Ω_{S}) d_{P, n}] = R_{n} (Ω),

where

R_{n} (Ω)

is a periodic function with a period of

Ω_{S}

.

Considering $P_{S, n} (Ω) \approx 1$ in the passband of equivalent channel, the channel frequency response can be derived from Eq. (9) and Eq. (12) as (detailed in Appendix 2)

H_{A, n} (Ω) = - \frac{α_{n} P_{A, n}}{2} H_{M, n} (Ω) R_{n} (Ω) \exp (j Ω d_{P, n}) .

Due to the periodicity of

R_{n} (Ω)

, Eq. (13) implies that the ISI induced by photodetection will cause ripples with a period of

Ω_{S}

in channel frequency response. Similar to the effect of finite pulse width on channel response, this ISI will also not result in excess noise since channel response is deterministic for a given

h_{E, n} (t)

.

3. Experiment verification

In order to verify the above theoretical analyses, we measure the frequency responses of a sampling channel in TIPADC. Figure 3 shows the schematic setup of experiment. The passive mode-locked fiber laser (MLL) (Precision Photonics, FFL1560) produces 70 fs pulses at a repetition rate of 36.456 MHz with an average power of 40 mW. An optical tunable filter (OTF) (Alnair Labs, CVF-300CL) is used to generate sampling pulses with controllable bandwidth $β_{P, n}$ by adjusting the linewidth of passing-by pulses. An Erbium-doped optical fiber amplifier (EDFA) along with an attenuator is adopted to maintain the average power of the generated pulses to be −6 dBm, since the average output power of OTF is changed as its bandwidth is tuned during the experiment. At this power level, the nonlinear length threshold of SPM is ~1.4 km at least (the FWHM of the generated shortest pulse is measured as ~18 ps). The channel length from modulator to photodiode is kept within 10 m and hence SPM can be ignored. A 70 GHz sampling oscilloscope (Keysight, DCA-X 86100D with 86118A module) along with a 95 GHz photodetector (Finisar, XPDV4120R) is used to measure the sampling pulse shapes.

Fig. 3 Experiment setup. MLL: Mode locked laser; OTF: Optical Tunable Filter; EDFA: Erbium-doped optical fiber amplifier; OSC: oscilloscope; PC: Polarization Controller; MZM: Mach-Zehnder modulator; AWG: Arrayed waveguide grating; ODL: Optical Delay line; PD: Photodiode; LPF: Lowpass filter; PLL: phase locked loop.

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The single tone from a 40 GHz microwave signal generator (Rohde & Schwarz, SMF 100A) is fed to the RF input of a 20 Gb/s MZM which has a half wave voltage of ~4.5 V and is biased at quadrature. We set the power level of microwave generator to be 0 dBm (0.32 V at 50 Ω load) to ensure the amplitude of single tone input is far less than the half wave voltage of modulator, so that small signal approximation in the proposed model is valid [13] and the interference of nonlinear distortion is avoided in the measurement. An arrayed waveguide grating with channel spacing of 0.8 nm is used as optical demultiplexer. Although there is only one channel and no pulses are demultiplxed, we still use an AWG to make the channel be consistent with the sampling channel in multi-channel setup. An optical delay line (ODL) is used to adjust the timing of pulses.

The adopted photodetector is composed of a photodiode (Thorlabs, DET08CFC/M) and a transimpedence amplifier, which has a bandwidth of 100 MHz. The input average power to photodiode is ~-18 dBm and the photodetected pulses are monitored to avoid nonlinearity. 2nd-order Butterworth lowpass filters (LPF) with different bandwidths are employed to change overall photodetection bandwidth $β_{E, n}$ . Signals after LPFs are sampled by a digitizer (Keysight, M9703A) with an analog bandwidth of 650 MHz and an ENOB of 9.0. The digitizer is synchronized with the MLL via a dual-loop charge-pump phase-locked loop (PLL) circuit (Texas Instruments, LMK04828). The PLL circuit has a reference clock from the synchronizing output of MLL and provides a locked-in clock required by the digitizer. In the measurement LPFs with bandwidths of ~9.1 MHz, ~18.3 MHz, ~36.5 MHz and ~73.0 MHz are used. Comparing to all the other cascaded devices from photodiode to EADC, the bandwidth of LPF is the lowest, so photodetection bandwidth is only limited to be LPF bandwidth. According to the principle of TIPADCs, digitizer is required to operate with a sampling rate of $f_{S}$ , where $f_{S} = 1 / T_{S}$ . However, the adopted digitizer can only operate at a sampling rate from 1.4 GS/s to 1.6 GS/s, which is larger than $f_{S}$ . Hence, in order to sample at $f_{S}$ equivalently, the sampling rate of digitizer is set to 42 times of $f_{S}$ (~1.53 GS/s) then the sampling result of the digitizer is directly down-sampled by 42 as the final output. We employ an ODL to adjust the timing, so that the final output is the peak values of each photodetected pulses when RF input is off.

The channel frequency response at a given frequency is measured by stimulating the channel with a single tone at this given frequency from the microwave signal generator and getting the output from the digitizer via LabVIEW. The amplitude of the output tone can characterize the magnitude of channel response at the frequency. By stimulating the channel in different frequency points of a certain range, channel frequency response in the corresponding range can be obtained. It is worth noting that the tone in the output is aliased from the input when the measured frequency is beyond the first Nyquist zone of sampling channel. The measurement, however, can still obtain channel frequency response via receiving the aliased tone since the aliasing does not change the amplitude of the tone.

Figure 4(a) shows the measured photonic sampling pulse shapes in different OTF linewidths. All the shapes are normalized by their own peak value. One can see that pulses with different temporal widths, i.e., different bandwidth $β_{P, n}$ , are obtained by adjusting the filter linewidth. In order to calibrate the effect of the devices (including the 95GHz PD, cables and the 70 GHz sampling oscilloscope) on the measurement of pulse shape, the pulse shape without OTF is also measured. Since laser pulse in this case is sufficient short, the measured shape can be taken as the impulse response of the devices for calibration. The channel frequency responses in different OTF linewidths are measured and shown in Fig. 4(b). The adopted 2nd-order Butterworth LPF in the measurement has a bandwidth of $2 f_{S}$ (~73.0 MHz). All the responses are normalized by their own value at the lowest measured frequency. One can observe that although photodetection bandwidth is only ~73.0 MHz, the channel analog bandwidth can be much larger than the photodetection bandwidth since $β_{E, n} \geq 0.5$ , and increases with the decrease of pulse width. The response of MZM with the cables and connectors is measured via a network analyzer (Keysight, PNA-X N5247A) and also shown in Fig. 4(b). The result shows that the bandwidth of MZM along with its accessories is no more than ~10 GHz and its response is not flat even within 3dB bandwidth, which limits the increase of channel analog bandwidth with the pulse bandwidth as Eq. (11) indicates.

Fig. 4 (a) The measured photonic sampling pulse shapes in different OTF linewidths and (b) the measured channel frequency responses in different OTF linewidths.

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Figure 5(a) shows the experimental and theoretical channel frequency responses without the effect of MZM in different OTF linewidths. The bandwidth of the adopted LPF is fixed to $2 f_{S}$ , i.e. $β_{E, n} = 2$ . The experimental responses are obtained by removing the frequency response of MZM from the measured channel frequency response in Fig. 4(b). The theoretical responses are calculated according to Eq. (6) by using the measured sampling pulse shapes and the theoretical response of adopted LPF while taking $h_{M, n} (t)$ as Dirac function. We can see that the experimental and theoretical channel frequency responses are consistent. Figure 5(b) summarizes the measured sampling pulse bandwidth $β_{P, n}$ and channel analog bandwidth $β_{C, n}$ without the effect of MZM as a function of OTF linewidth. The sampling pulse bandwidth $β_{P, n}$ in each OTF linewidth is calculated from the corresponding measured pulse shape while $β_{C, n}$ is obtained from the measured channel response. We can see that $β_{C, n}$ is corresponding to the $β_{P, n}$ just as Eq. (11) indicates when the effect of MZM is removed.

Fig. 5 (a) The theoretical and experimental channel frequency responses in different OTF linewidths when the effect of MZM is removed and (b) the channel and sampling pulse bandwidth versus OTF linewidth.

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Figure 6(a) shows the measured channel frequency responses without the effect of MZM in different $β_{E, n}$ . Figure 6(b) shows the detailed experimental and theoretical channel frequency response in different $β_{E, n}$ observed in a small frequency range from 9.5 GHz to 10.5 GHz. The experimental response is extracted from the results in Fig. 6(a). The theoretical response is calculated from the measured sampling pulse and the theoretical response of LPF while taking $h_{M, n} (t)$ as Dirac function. The OTF linewidth is fixed to 0.4 nm during the measurement. The LPFs with bandwidth of $0.25 f_{S}$ , $0.5 f_{S}$ , $f_{S}$ and $2 f_{S}$ are used respectively to change $β_{E, n}$ from 0.25 to 2. All the responses are normalized by the value at the lowest measured frequency when $β_{E, n} = 2$ . One can see that the passband of channel responses can be up to ~15 GHz as long as $β_{E, n}$ is not less than $0.5$ . The ripples become severer as the bandwidth of LPF decreases. This is because the lower the bandwidth of LPF is, the more broadly the pulse is broadened, and the severer the ISI becomes. When the LPF bandwidth is decreased to $0.25 f_{S}$ (~9.1 MHz), ripples with a fluctuation of larger than 3 dB and a period of $f_{S}$ (~36.5 MHz) as Eq. (13) indicates can be observed in Fig. 6(b), which results in multiple discrete narrow passbands. These results are coincident with the theoretical analyses in section 2.

Fig. 6 (a) The measured channel frequency response in different $β_{E, n}$ and (b) the detailed experimental and theoretical channel frequency response in different $β_{E, n}$

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4. Conclusion

The effects of the photonic sampling pulse width and the photodetection bandwidth on TIPADCs were investigated. The expression of the channel frequency response was presented. The theoretical analysis showed that the bandwidth of channel response is limited by the finite temporal width of the photonic sampling pulse. The minimum feasible bandwidth of photodetection is half of the single channel sampling rate in order to form a continuous wide passband. The photodetection induced ISI will cause periodic ripples in the channel response. The theoretical results were verified by measuring the channel frequency responses of a TIPADC experimentally under different pulse widths and photodetection bandwidths.

Appendix 1

This appendix presents the derivation of Eq. (5) and (6), which gives the photodetected signal and channel impulse response for TIPADC.

From Eq. (1), (2) and (4), the signal sampled by EADC can be derived as

v_{Q, n} [k] = {0.5 α_{n} p_{n} (t) * h_{E, n} (t - d_{E, n}) - 0.5 α_{n} [h_{M, n} (t) * v_{I} (t)] p_{n} (t) * h_{E, n} (t - d_{E, n}) |}_{t = k T_{S}} .

The first term is a constant since

p_{n} (t)

has a period of

T_{S}

, and can be denoted as

V_{Q 0, n}

:

V_{Q 0, n} = 0.5 α_{n} \int p_{n} (k T_{S} - τ) h_{E, n} (τ - d_{E, n}) d τ = 0.5 α_{n} \int p_{n} (- τ) h_{E, n} (τ - d_{E, n}) d τ .

The second term is related to RF input and can be denoted as

v_{Q 1, n} [k]

. Let

v_{M, n} (t) = h_{M, n} (t) * v_{I} (t)

, then this term can be derived as

\begin{matrix} v_{Q 1, n} [k] = - 0.5 α_{n} \int v_{M, n} (k T_{S} - τ) p_{n} (k T_{S} - τ) h_{E, n} (τ - d_{E, n}) d τ \\ = - 0.5 α_{n} \int v_{M, n} (k T_{S} - τ) p_{n} (- τ) h_{E, n} (τ - d_{E, n}) d τ \\ = {- 0.5 α_{n} v_{M, n} (t) * [p_{n} (- t) h_{E, n} (t - d_{E, n})] |}_{t = k T_{S}} \end{matrix} .

Hence

v_{Q, n} [k]

can be expressed as Eq. (5) and the equivalent channel impulse response

h_{A, n} (t)

can be introduced as Eq. (6).

Appendix 2

This appendix presents the derivation of Eq. (12) and Eq. (13), which gives the characteristic of sampling channel with the ISI induced by photodetection.

For typical TIPADC configurations, the width of modulated photonic pulses is sufficient short compared to $h_{E, n} (t)$ such that the photonic pulses can be regarded as an impulse input to $h_{E, n} (t)$ . In this case, the shape of each pulse after photodetection is proportional to $h_{E, n} (t - d_{P, n} - d_{E, n})$ . Hence when ISI occurs in the sampling of the pulse after photodetection, ISI will also occur in the sampling of the pulse $h_{E, n} (t - d_{P, n} - d_{E, n})$ . According to Nyquist ISI criterion, when there is ISI in the sampling of the pulse $h_{E, n} (t - d_{P, n} - d_{E, n})$ with a period of $T_{S}$ , the Fourier transform of the sampled pulse $R_{n} (Ω)$ cannot be a constant, where

R_{n} (Ω) = \frac{1}{T_{S}} \sum_{m = - \infty}^{\infty} {H^{'}}_{E, n} (Ω + m Ω_{S}) \exp [- j (Ω + m Ω_{S}) d_{P, n}] .

Then from Eq. (17) and the fact

R_{n} (Ω)

cannot be a constant when ISI occurs, we can obtain that

R_{n} (Ω)

is a periodic function with a period of

Ω_{S}

, as Eq. (12) indicates.

Considering $P_{S, n} (Ω) \approx 1$ in the passband of equivalent channel, we observe that the channel frequency response is related to $R_{n} (Ω)$ in the passband:

\begin{matrix} H_{A, n} (Ω) = - \frac{α_{n} P_{A, n}}{2 T_{S}} H_{M, n} (Ω) \sum_{m = - \infty}^{\infty} c_{m, n} {H^{'}}_{E, n} (Ω + m Ω_{S}) \\ = - \frac{α_{n} P_{A, n}}{2 T_{S}} H_{M, n} (Ω) \sum_{m = - \infty}^{\infty} {H^{'}}_{E, n} (Ω + m Ω_{S}) \exp (- j m Ω_{S} d_{P, n}) \\ = - \frac{α_{n} P_{A, n}}{2} H_{M, n} (Ω) R_{n} (Ω) \exp (j Ω d_{P, n}) \end{matrix} .

Then Eq. (13) is obtained.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (NSFC) (61127016, 61535006).

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Effects of the photonic sampling pulse width and the photodetection bandwidth on the channel response of photonic ADCs

Abstract

1. Introduction

2. Modeling and analysis

3. Experiment verification

4. Conclusion

Appendix 1

Appendix 2

Acknowledgments

References and links

Cited By

Figures (6)

Equations (18)

Optics Express