Due to the capacity in simultaneously down-converting and receiving ultra-wideband, multi-carrier radio frequency (RF) or microwave signals, the photonic bandpass sampling has found more and more applications in multi-carrier communication, frequency-agile coherent radar, compressive sensing, etc. The nonlinear transfer during the electronics-to-optics conversion results in distortions, which are bandpass sampled and frequency-folded within the first Nyquist zone, together with the target signals. Because of the multi-octave-span operation, all nonlinearities must be considered besides the usually-concerned third-order inter-modulation distortion (IMD3). We show theoretically that a photonic bandpass sampling link is equivalent to a baseband digital nonlinear link, and then propose a corresponding linearization scheme for the output signal. Such digital linearization is capable of suppressing all types of distortions. Both numerical and experimental examples are demonstrated, where all of the 3rd-order nonlinearities, including the internal and external IMD3, the cross modulation, and 3rd-order harmonics, are well eliminated.
© 2015 Optical Society of America
In commercial or military software defined radios (SDRs), it is urgent to communicate with single radio with rapidly-changed carrier frequencies, or to communicate simultaneously with many radios under different radio frequency (RF) bands. In order to digitalize the multi-carrier RF signal, a wideband analog-to-digital converter (ADC) can be used, with usually too high sampling rate if the Nyquist sampling theorem is satisfied. The bandpass sampling is currently considered as a promising alternative technique, instead of multiple traditional down-conversions, which supports the direct down-conversion of multi-carrier signals using a single low-speed ADC . Recently, the photonic technology shows another attractive trend for processing high frequency and multi-carrier RF signals, with the advantages of low insertion loss, broad bandwidth, immunity to electromagnetic interference, etc. For example, the photonic channelization has shown the ultra-wideband capacity over the traditional electronic technologies [2, 3 ]. Naturally, photonic bandpass sampling, the combination of the above two techniques, can also push the SDR applications with much higher carrier frequencies (tens of GHz). In , directly down-converting and sampling of microwave signal with 40-GHz carrier and 200-MHz bandwidth has been experimentally demonstrated, with an effective number of bits (ENOB) larger than 7. The photonic bandpass sampling is now benefiting from rapidly improved performance of femtosecond lasers. Some key characteristics, such as repetition rate larger than 1 GHz, pulse width around 100 fs, as well as sampling jitter less than 10 fs, have been experimentally demonstrated [5–8 ]. As a result, photonic bandpass sampling is widely studied recently and is proposed for multi-carrier communication [9, 10 ], frequency-agile coherent radar , compressive sensing [11, 12 ], and so on.
Due to its simple structure, capacity of high speed, and commercial availability, the external intensity modulation based on Mach-Zehnder modulator (MZM) is mostly adopted in photonic bandpass sampling. Therefore the photonic bandpass sampling is essentially a special analog photonic link, of which the fidelity has to be carefully considered because of the intrinsical nonlinear transfer function of MZM. In a general analog photonic link, nonlinear distortions are considered as the major contributor for the degradation of spurious free dynamic range (SFDR). Besides the well-known third-order inter-modulation distortion (IMD3), multiple types of distortions, including harmonic crosstalk and cross modulation distortion (XMD), must be considered simultaneously due to the multi-octave-span and multi-carrier operation of the photonic bandpass sampling link. Moreover, unlike traditional photonic link, all of the above distortions are bandpass sampled into the first Nyquist zone and overlaps with the target signal, which may not be eliminated by filtering.
Over the past years, various approaches have been reported to improve the link linearity, and schemes include both the analog and the digital signal processing (DSP). Many are actually band-limited within only one octave span, though the processing RF bandwidth is indeed enlarged compared with the traditional electronic-only technology. For example, the IMD3 and XMD information was found to be contained by the baseband of the photo current in an intensity-modulation direct-detection (IMDD) link if the MZM is low-biased, which was used, both in analog processing  and in DSP [14, 15 ], to eliminate one or two types of distortions. However, since the MZM is low-biased, such processing cannot cover a continuous multi-octave span as well as all types of distortions, though a few multi-carrier experiments have been reported . The same limitation can also be found in optical processing schemes [16, 17 ]. Some techniques support ultra-broadband linearization in theory, but are practically limited. For example, the bandwidth of MZMs combination in parallel  may be limited by the assisted RF splitter, since keeping a precise and constant power split ratio under widely changed RF carrier frequency range is a challenge for current RF devices. DSP algorithm can also support multi-octave-span operation, both for IMDD link [19, 20 ] and for phase-modulated link [21, 22 ], but the required Nyquist bandwidth of the ADC is equal or even larger than that of the input RF signal.
In this paper we propose and demonstrate a digital linearization scheme for ultra-broadband, multi-carrier photonic bandpass sampling link, by which all types of the nonlinear distortions could be eliminated. Both in a numerical and in an experimental example, we show that all 3rd-order nonlinearities, including the internal and external IMD3, XMD, and harmonic spurs, are folded and overlap the down-converted signals within the first Nyquist zone after bandpass sampling, which can be greatly suppressed by the proposed method. Under a typical input RF power, the 3rd-order nonlinearities are suppressed 30 dB around, and the two-tone SFDR is improved to 117 dB within 1 Hz bandwidth. The nonlinearity generation during the photonic bandpass sampling is theoretically modeled, as well as its digital compensation.
2. Principle and simulation
Figure 1(a) shows the schematic diagram of the proposed digitally linearized photonic bandpass sampling link. The input x(t), which contains multiple bandpass signals with different RF frequencies, is firstly carried by a pulsed optical source with a repetition rate of f, then recovered to y(t) by a photo receiver, and finally digitalized by an ADC which is synchronized by the above pulsed source. Assume that the voltage-to-voltage link transfer function under a continuous-wave (CW) light is T(x), and each pulse of the sampling source is described by its normalized intensity profile, s(t), where ∫s(t)dt = 1, then the transfer function of the photonic bandpass sampling link before the ADC is
Equation (2), which is actually the transfer function of the photonic bandpass sampling link with ADC, shows that when the pulse duration is short enough, “nonlinear transmission before bandpass sampling” is equivalent to “bandpass sampling before nonlinear transmission,” i.e. the photonic bandpass sampling link in Fig. 1(a) is equivalent to a digital baseband link in Fig. 1(b). Such equivalence simplifies the linearization greatly: after bandpass sampling, the bandwidth of the input signal, xk, is limited within [-f/2, f/2], so that the original ultra-wideband linearization can be obtained in a narrow digital baseband. Until now, several DSP algorithms have been demonstrated to linearize a baseband analog photonic link. For example, the iteration of distortion regenerating and subtracting can linearize the output of a general IMDD link [19, 20 ]; the in-phase/quadrature (I/Q) demodulation can recover the input both in a phase-modulated link  and in a polarization-modulated one . The above algorithms designed for analog link are also able to linearize the digital baseband link shown in Eq. (2). Here we employ the single distortion regenerating and subtracting algorithm. In the digital domain, we build a transfer function, TE(x), to emulate the physical one T(x). The bandpass sampling output yk is divided by the link amplitude gain G 1/2, and then passes through the emulator. The regenerated output is expected to contain approximately twice of the distortions inside yk. As a result, a linear combination of yk and its regeneration can achieve the linearized output y ’ k as (a detailed explanation can be found in )
We should note that though the above baseband linearization algorithm is similar to that for analog link, they have different physical view as well as the mathematical implementation, resulting in very different distortion elimination capacity. The key for distortion elimination is to build a digital transfer function which is exactly the same as the physical process or is its inverse, so that the newly-generated distortion can matches the original one precisely. Under a given sampling rate, the digital transfer emulator has also a fixed bandwidth. After the corresponding transfer, the newly-generated harmonics may overrange such bandwidth and then are relocated. For example, we assume that the sampling rate is 200 MHz and the baseband signal is a sinusoidal wave at 41 MHz, then its 3rd-order harmonics, which is at 123 MHz, will however be folded digitally at 77 MHz. This will never happen physically in a usual analog link, so that the regenerated 77-MHz distortion is fake and does not match any real one, which fails the distortion elimination. As a result, up-resampling before passing through the digital emulator is usually required in order to avoid the above frequency folding. Otherwise the ADC bandwidth should be enlarged enough to effectively contain all of the concerned nonlinear harmonics. However, our theory shows that the photonic bandpass sampling link is equivalent to a digital baseband link, and all the distortions are physically folded into its first Nyquist zone governed by Eq. (2), which matches the frequency folding during the digital regeneration in Eq. (3). We can then conclude that there is no need to employ a bandwidth-enlarged ADC nor to up-resample the raw data, and that all types of distortions can be compensated under a well-designed emulator. Note that in order to achieve the above equivalence and good nonlinearity elimination, the optical sampling pulse should approach impulse function not only for x(t) but also for high-order harmonics under consideration. For example, if a 10-GHz tone is input and at least the 3rd order nonlinearity should be concerned, then the optical pulse should be short enough to bandpass sample the 30-GHz harmonics with little power fading. Limited sampling bandwidth results in imperfect cancellation as well as linearization. However, such requirement can be easily achieved by current femtosecond fiber laser: the pulse duration is much less than 1 ps which can support bandpass sampling of harmonics of microwaves.
The multiple types of distortions as well as their elimination are illustrated numerically. Assume that the optical pulse train has a repetition rate of 80 MHz and pulse duration of 1 ps, the half-wave voltage of the quadrature-biased MZM is 6 V, the photo receiver has a responsivity of 50 V/W, and the average received optical power is 5 dBm. The input SA and SB are two dual-tone signals, which are around 1.81 GHz and 15.53 GHz, respectively. Their dual-tone frequency intervals are 3.2 MHz and 1.9 MHz, respectively. When the input RF power is 0 dBm per tone, Fig. 2(a) shows the received intermediate frequency (IF) signals that are simultaneously down-converted by the photonic bandpass sampling (the green lines). Obviously considerable nonlinear distortions are also found, which have been classified into five types in Fig. 2. The internal IMD3 (the red lines) around SA/SB is induced by the modulation from SA/SB itself. The XMD (the blue lines) around SA/SB is induced by the modulation from the other SB/SA. The 3rd-order harmonics (the black lines) is the sum-frequency spurs of any three of the four tones, while the external IMD3 (the magenta lines) can be seen as an internal one if one treats SA and SB as a single dual-tone signal. The internal IMD3 and XMD are usually considered in single- or multi-carrier link [15, 23, 24 ], since they cover the same band as the received signals. The 3rd-order harmonics and external IMD3 are usually out of the signal band, but have to be included in multi-octave-span scenario. All the above four are 3rd-order nonlinearities, and share almost the same major significance (note that the 2nd-order nonlinearities are ignored since the MZM is quadrature-biased). After bandpass sampling, they are distributed inside the first Nyquist zone, so that the elimination of all is a must. The residual minor distortions (the cyan lines) have much lower power, and are higher-order ones.
The linearization is achieved according to Eq. (3), and the final output is shown in Fig. 2(b). All of the 3rd-order distortions are suppressed with as large as 30 dB, which are as the same magnitude as the 5th-order ones. Note that the amplitude of the higher-order distortions are clearly enlarged, which is because of the nonlinear mixing between the fundamental and the numerous original 3rd-order spurs during the digital emulating. Higher-order nonlinearities can be further suppressed by iterations of Eq. (3) .
3. Experiment results
The above theory and numerical example are experimentally demonstrated. The setup is shown in Fig. 1. The home-made femtosecond laser has a repetition rate of 79.82 MHz, and the pulse duration is measured to be 401 fs around. The bias angle of the MZM (EOSpace, 20 GHz) is fixed at its quadrature by a home-made dither-free bias controller. The half voltage (Vπ) is 5 V. The average power of the lightwave hitting on the PD (EM4, 20 GHz bandwidth with responsivity of 0.92 A/W) is 6 dBm. All loads are 50 Ω. A second low-speed PD is used to receive directly the femtosecond pulse train and to synchronize the ADC (ADLink with 14-bit level and 200 MS/s maximum). The output discrete data undergo an offline DSP processing shown in Eq. (3). Two dual-tone signals, SA and SB, are combined and input to the MZM. SA contains two sinusoidal waves at 15.571 GHz and 15.573 GHz, respectively, and SB contains 1.8041 GHz and 1.8027 GHz, respectively. When the powers of the four sinusoidal waves are all 0 dBm at the MZM, the received IF signal by the ADC is shown in Fig. 3(a) .
As can be seen, SA is down-converted around 6 MHz, while the IF signal corresponding to SB is around 32 MHz. The loss of down-conversion link is about 18 dB. The short optical pulse duration does not show additional loss. Since the repetition rate is low, the photo-generated current can still be fully collected in the PD, so that the average optical power dominates the link loss, just like the ordinary analog photonic link.
A lot of distortions are found inside the first Nyquist zone, which is similar to the numerical example. Here we emphasize the internal IMD3 and the XMD in each signal, as shown in Fig. 3(a). The average power ratios between fundamental and IMD3 and between fundamental and XMD are 48.0 dB and 35.4 dB, respectively. For linearization, the digital emulator is a simple IMDD link (T in Fig. 1(a)) with the above parameters. Here we assume a perfect quadrature-biased MZM and a linear PD. After the linearization shown in Fig. 3(b), the average ratios are 65.1 dB and 70.6 dB, respectively. Other 3rd-order nonlinear spurs are also well suppressed (e.g. those between 15 MHz and 25 MHz). We note that there are still significant nonlinear spurs left after the linearization, which are the residual 2nd-order distortions and will influence the application of multicarrier communication. The 2nd-order may result from the imbalance inside the MZM or from the nonlinearity of the PD, which has not been considered by our current digital emulator. Such distortions could also be eliminated as long as the corresponding nonlinear transfer function is precisely estimated.
The powers of the both signals are scanned respectively while the down-converted SB (i.e. the target) as well as the XMD around SB is monitored. When SA (i.e. the interferer) with a fixed power of 0 dBm and SB with a varying power from −5 dBm to 4 dBm are applied on the MZM, the received down-converted SB fundamental and XMD sidebands are measured, and the suppression ratio is plotted in Fig. 4(a) . With a fixed interferer power, the power ratios are kept unchanged when the target changes. The XDM sidebands are suppressed about 33.6 dB by the linearization algorithm. Similarly, when the input target is fixed at 0 dBm and the interferer signal power varies from −5 to 4 dBm, the XMD suppression is shown in Fig. 4(b). As the theory predicts, the XMD power changes along the interferer with a slope of two. 30.7-dB XMD suppression is experimentally observed.
By varying the power of SA, both the down-converted target and the IMD3 sidebands are monitored. Figure 5 shows the measured fundamental and IMD3 powers as a function of the input RF power for both cases with and without linearization. The measured noise floor after the PD is −158 dBm/Hz. So the SFDR before and after the linearization is 109 dBm·Hz2/3 and 117 dBm·Hz2/3, respectively. We note that the current design is still limited by the quantization noise of the ADC. Such limitation is expected to be removed by a low-noise electronic amplifier before the ADC .
As a summary, we in this paper proposed and demonstrated a digital linearization scheme for the typical photonic bandpass sampling link. Different from a usual analog photonic link, all of the nonlinear distortions should be considered here due to the multi-octave-span operation. Meanwhile, the nonlinear distortions are also bandpass sampled and then frequency-relocated within the first Nyquist zone, together with the target signals. The much more complicated nonlinearities than usual analog photonic link were demonstrated by a numerical example. Correspondingly, we showed theoretically that a typical photonic bandpass sampling link could be equivalent to a digital baseband link, and then proposed a digital linearization scheme where all of the frequency-folded distortions could be well suppressed. Experimentally, we showed that all four types of 3rd-order distortions were suppressed with 30 dB, and a two-tone SFDR of 117 dB within 1-Hz bandwidth was achieved.
This work was supported in part by NSFC Program (61471065), National 973 Program (2012CB315705), NSFC Program (61401411 and 61302016), and NCET-13-0682.
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