Wideband photonic blind source separation with optical pulse sampling

We propose and experimentally demonstrate an optical pulse sampling method for photonic blind source separation. The photonic system processes and separates wideband signals based on the statistical information of the mixed signals and thus the sampling frequency can be orders of magnitude lower than the bandwidth of the signals. The ultra-fast optical pulse functions as a tweezer that collects samples of the signals at very low sampling rates, and each sample is short enough to maintain the statistical properties of the signals. The low sampling frequency reduces the workloads of the analog to digital conversion and digital signal processing systems. In the meantime, the short pulse sampling maintains the accuracy of the sampled signals, so the statistical properties of the undersampling signals are the same as the statistical properties of the original signals. With the optical pulses generated from a mode-locked laser, the optical pulse sampling system is able to process and separate mixed signals with bandwidth over 100GHz and achieves a dynamic range of 30dB.

mixing matrix. The sampling frequencies can be orders of magnitude smaller than the bandwidths of the mixed signals, which greatly reduces the workloads of ADC and DSP. To sample the ultra-wideband signals (100GHz) at sub-Nyquist sampling rate, the major challenge is the sampling time, and it is impractical for the sampling circuit to achieve pico-second or femto-second sampling time with MHz or kHz sampling rate.
In this paper, we take advantage of the undersampling properties of photonic BSS for wideband RF signal processing and modulate the RF signals on optical pulses that pre-sample the signals before ADC. By using a mode-locked laser, the optical pulse width is in the order of 100fs. The ultra-narrow optical pulse function as a tweezer that collects samples of the mixed signals. Only a tiny potion of the mixed signals (1/10 2 to 1/10 5 ) is collected, which greatly reduces the workload of ADC and DSP, while each collected sample is short and thus accurate enough to represent the statistical information of the mixed signals. The DSP system calculates the de-mixing matrix based on the statistical information and controls the photonic circuit to separate the SOI from interferences with the de-mixing matrix.
The relationship between the optical pulse sampling and photonic BSS is similar to the relationship between the biopsy sample preparation and curative surgery. The biopsy (pulse sampling) collects samples for analysis (DSP) and curative surgery (BSS) is performed based on the analysis. Traditional digital BSS method digitizes the mixed signals at Nyquist sampling rate and works in a similar way as cutting the whole tissue off for sample analysis, which is not efficient and introduce extra workloads to both sample preparation and analysis. The ultra-fast optical pulse acts as a tweezer for sample preparation. With a small portion of the mixed signals sampled for statistical analysis, the workloads of both sample preparation (ADC) and analysis (DSP) are reduced for real-time and ultra-wideband signal processing.

Principle and experimental setup
This section discusses the principle and the experimental setup of photonic BSS system with optical pulse sampling. Section 2.1 discusses the principle of photonic BSS, and shows that the system can separate the mixed signals with sampling frequency lower than the Nyquist sampling rate. Section 2.2 discusses the principle of optical pulse sampling. Fig. 1 shows the schematic diagram of the photonic circuit that implement BSS algorithm. The photonic BSS system processes mixed signals from a MIMO receiver. The mixed signals are represented by: is the mixing matrix. and are received mixed signals from MIMO antennas. To demonstrate the principle of photonic BSS, a 16QAM is used as the SOI , and a Gaussian noise is used as the interference . Both SOI and interference are baseband signals with the same bandwidth. Fig. 2 (a) is the scatter plot that shows the real parts of the mixed signals and . The four straight lines match with the pattern of 16QAM, which corresponds to the projection of 16QAM on the real axis. Within each line, the density of the dots is Gaussian distributed, which represents the pattern of the interference. Separating SOI and interference is to find the de-mixing matrix , which includes two steps, principal component analysis (PCA) and independent component analysis (ICA) [7]:

Principle of blind source separation with undersampling
Where represents the PCA, and represents the ICA. To perform PCA, the 2 nd order moments of the mixed signals are measured with different weights applied to the signals. By using the tunable attenuations in the photonic circuit ( Fig. 1), the following weights are added to the mixed signals: Where is defined in Fig. 2 (a). Based on Equation 3, the 2 nd order moment of is a function of (red curve in Fig. 2 (a)): The PCA normalizes the 2 nd order moments and whitens the mixed signals. To perform ICA, the 4 th order moments (kurtosis) of the whitened signals are measured (red curve in Fig.  2 (b)) with new weights applied: Where is defined in Fig. 2 (b). With four random values, the unknown parameters , , , and are determined, and ICA can be performed with = , where: The de-mixing matrix is solved as = .
With the de-mixing matrix, the SOI is separated from the interference. The tunable delays and attenuators in Fig. 1 control the complex weights for and and the circuits within the dashed lines represent half of the demixing matrix , which recovers the SOI. Based on Equation 4 and Equation 8, only statistical properties of the mixed signals are needed to find the de-mixing matrix . To obtain the statistical properties, which are 2 nd and 4 th order moments, a sampling rate orders of magnitude lower than the Nyquist sampling rate can be applied. The undersampling rate reduces the workload of DSP, and the proposed system is able to process mixed signals with bandwidth that is beyond ADC speed limit.

Optical pulse sampling
When the sampling rate is orders of magnitude smaller than the Nyquist frequency, the duty cycle / also needs to be small enough, so the sampled signals can accurately represent the statistical information of the original signals. If the time duration  of each sample is longer than the bit length of SOI, the sampled signals do not accurately represent the statistical information of the mixed signals. This can be explained by ( ) ≠ ( ) , or in another word, when measuring the 2 nd order moments, the time average of the square does not equal to the square of the time average. The sine function in Fig. 3 (a) shows the maximum frequency of the mixed signal . In a photonic BSS system, the sampling frequency (1/ ) can be less than the frequency of the sine function, while the duration of each sample has to be small enough (τ 1 in Fig. 3 (a)). If the duration of each sample (τ 2 in Fig. 3 (a)) is comparable to , each sample is an average of the changing signals, and the sampled signals cannot represent the distribution of the original signals. To obtain extremely small duty cycle (10 -4 to 10 -8 ) for undersampling, the RF signals are modulated on a pulse laser that pre-samples the signals before ADC. The output of the optical modulator is the product of the periodic laser pulses and the RF signals ( Fig. 3 (b)). The sampling time is determined by the laser pulse width, so even ADC with long sampling time τ2 is used afterwards; the sampled signals still represent accurate statistical information from the short sample duration τ1.

Experimental results and analysis
In this section, optical pulse sampling for BSS is demonstrated with experimental results. The optical pulses are generated with both optical intensity modulator (section 3.1) and modelocked laser (section 3.2). With optical intensity modulator, optical pulses sample RF signals with sub-6GHz frequencies. Such system can be used to separate multiple wireless systems that coexist at the same RF frequencies, including 4G LTE, weather radar, radio telescope, etc. With the mode-locked laser, the optical pulses sample signals with 100GHz bandwidth and beyond. The pulse width of the mode-locked laser is in the order of 100fs, which functions as a tweezer that collect samples from ultra-wideband signals and obtain accurate statistical information. Mixed signals with mmWave frequencies or even THz frequency can be sampled with mode-locked laser pulses, which meets with the growing needs for 5G communication network [11,12], mmWave radar [13,14], and THz communication [15,16].

Demonstration of BSS with optical pulse sampling
In this section, we experimentally demonstrated that signals sampled with optical pulses accurately represent the statical properties. With 2 nd and 4 th order moments measured from the sampled signals, de-mixing matrix is solved. Fig. 4 shows the basic concept of optical sampling with a binary signal. The sampled signals (Fig. 4 (d)) is a product between the laser pulse sequence (Figs. 4 (a) and (b)) and the unsampled signals (Fig. 4 (c)). In this experiment, the laser pulse width is 5ns, and the signal bandwidth is 200MHz. Interference is introduced to demonstrate the effectiveness of the optical pulse sampling for BSS. Fig. 5 shows that the 2 nd order moments measured from pulse-sampled signals match with the theoretical results. The theoretical results calculate the 2 nd order moments with signals sampled at Nyquist frequency. In this experiment, the SOI is random binary signal with data rate of 200Mbps, and the interference is Gaussian noise with bandwidth overlapped with the bandwidth of SOI and clock unsynchronized with SOI. The mixing matrix (Equation 1) used in the experiment is = = 1, and = = 0.5. An arbitrary waveform generator is used to generate the mixed signals and . Fig. 5 (a) shows the and used in the experiment (black dots), and the theoretically calculated 2 nd order moments (red curve). Each black dot is a sampled data point. The dots form the shape of two straight lines because the SOI is binary signal. Within each line, the dots are Gaussian distributed, which reflect the distribution of the interference. The six large black dots in Fig. 5 (b) show the experimentally measured 2 nd order moments in six different angles ( in Equation 3 and Fig. 2 (a)). The black curve is the fitting result based on the experimental measurement. The fitting curve matches with the theoretical result (red curve). Figs. 5 (c) and (d) show the time domain signals measured with an oscilloscope and are used to calculate 2 nd order moments in Fig. 5 (b). Two points out of the six points are shown, and correspond to , where = 0º (Fig. 5 (c)), and , where = 90º (Fig. 5 (d)), respectively. In Fig. 5 (b), the 2 nd order moment of is significantly smaller than the 2 nd order moment of , which is also clearly shown in Figs. 5 (c) and (d), where the variance of the signals in Fig. 5 (c) is significantly smaller than the variance of the signals in Fig. 5 (d). The de-mixing matrix is calculated with the pulse-sampled signals, and is applied to the mixed signals and . Fig. 6 (a) shows that the recovered SOI matches with the original SOI, and a clear eye pattern is obtained Fig. 6 (b). The pulse width in Figs. 6 (a) and (b), or in another word, the duration of each sample is the same as the bit length of the original SOI. The bit error rate of the recovered signals is less than 1×10 -5 . Figs. 6 (c) and (d) show that if the short optical pulse is not used, and the duration of each sample is five times the bit length of the SOI, the SOI is not separated with the Gaussian interference. The recovered SOI does not match with the original SOI (Fig. 6 (c)) and a noisy eye pattern is obtained (Fig. 6 (d)). The bit error rate is 9×10 -2 , which is beyond the forward error correct limit [17].  To explain the results in Fig. 6, we test the system with 15 groups of signals, and each group includes a random binary SOI and a random Gaussian interference (Fig. 7). ′ and ′ in Fig.  7 (a) are the mixed signals after PCA. Based on Equations 7 and 8, 4 th order moments of ′ and ′ are calculated to perform ICA. The key parameter for ICA is in Equation 8. This is because in a scatter plot, matrix (Equation 9) corresponds to rotate the scatter plot by an angle . If is applied in a right way, the projections of the two lines on horizontal axis of the scatter plot does not have any overlaps (Fig. 7 (b)), which means the SOI is completely separated from the interference. If is not applied in a right way, the projections of the two lines on horizontal axis of the scatter plot have overlaps (Fig. 7 (d)), which means the SOI is not completely separated from the interference. Figs. 7 (a) and (b) use short pulse sampling, where the duration of each sample is the same as the bit length of SOI. The , which is marked by red arrows in Fig. 7 (a), from all the 15 groups of tests varies in a very small range (less than 10º). The projections of the rotated signals on the horizontal axis of the scatter plot does not have any overlaps (Fig. 7 (b)). Fig. 7 (b) plots the rotated signals for 15 tests together, and each parallel line corresponds to one test. All the lines in Fig. 7 (b) are close to each other with a variance of angle less than 10º. Figs. 7 (c) and (d) use long pulse sampling, where the duration of each sample is five times of the bit length of SOI. The (red arrow in Fig. 7 (c)) from 15 groups of tests varies in a large range (larger than 45º). The projections of the rotated signals on horizontal axis of the scatter plot have overlaps (Fig. 7 (d)). Such overlaps result in bit errors and a noisy eye pattern (Fig. 6 (d)).

Linearity analysis of mode-locked laser
Mode-locked laser generates femto-second laser pulses, which can be used to sample signals with 100GHz to THz bandwidth. Although femto-second pulses achieve the ultimate goal of ultra-fast sampling, the peak pulse power is orders of magnitude larger than its average power, and thus can saturate the photodetector of the BSS system (Fig. 1). By intensity modulating the RF signals on the laser pulses, the pulse power has to be within the linear range of the photodetector to perform effective signal processing. Both the measurement of 2 nd and 4 th order moments calculation and interference cancellation require the system to operate in the linear range. More importantly, the linear range determines the capability of the system to separate strong interference from weak SOI. The upper bound of the linear range is defined by the saturation power of the photodetector, and the lower bound of the linear range is defined by the noise-equivalent power. This section studies the saturation effect of the photodetector and the linear range of the sampling system with a mode-locked laser. Experimental results show that the linear range is at least 30dB, which means that the system is able to separate interference that is 30dB stronger than the SOI. The mode-locked laser used in this experiment has a pulse width of 70fs ( Fig. 8 (b)), center wavelength of 1560nm, and a repetition rate of 37MHz (Precision Photonics FFL1560-MP [18]). The laser pulses are attenuated and then received with a photodetector with 50GHz bandwidth (Finisar XPDV21). The RF output of the photodetector is amplified by an RF amplifier with 9GHz bandwidth and then measured with an oscilloscope with 6GHz bandwidth (Keysight 6000X). Fig. 8 (a) shows the measured peak pulse voltage from the oscilloscope with the average power of laser scans from -45dBm to 0dBm. The average power is controlled by a digital attenuator. The measured peak pulse voltage increases linearly with the average power and saturates at -10dBm. The saturation power of a photon detection system to the pulse laser can be calculated as [19]: Where the is the average power of the laser pulses that saturate the photodetector, is the power the continuous wave laser that saturates the photodetector, is the pulse period, and the is the full width at half maximum of the system impulse response. The photodetector used in this experiment has a of 16dBm, the is measured to be 90ps (Fig. 8 (c)). To measure the of the system, a 70fs pulse from the mode-locked laser can be treated as an impulse (70fs << 90ps). With , , and , is calculated as -9dBm, which is close the measured saturation power -10dBm.
The SNR at -10dBm average power is 39dBm (Fig. 8 (c)). The signal is defined as the peak voltage and the noise is defined as the variance of the signals at the peak. The SNR drops with the average power. The SNR is 7.8dB when the average power is -41dBm, and the received histogram have observable pulse peak (Fig. 8 (d)). The SNR is close to 1dB when the average power is -44dBm. The pulse sampling system has a linear power range of over 30dB (-10dBm to -44dBm). The linear power range for the sampling system is important for BSS, since the separation of strong interference from weak SOI is a widely existing application scenario for the system. Such application requires the system to be able to address both the high and low power levels at the same time. In addition to the 30dB range, we find that beyond the saturation power (-10dBm), the peak pulse voltage response of the photodetector still increases monotonously and significantly (Fig.  9). The results are repeatable, which means the photodetector does not break down. The significant increase of pulse voltage beyond -10dBm is not observed in Fig. 8 (a), because it reaches maximum output power of the RF amplifier. To remove the saturation effect of RF amplifier, Fig. 9 is measured without RF amplifier and shows that although the output response of the photodetector is no longer linear beyond the saturation power of -10dBm, the peak voltage still increases significantly with the input average power and the SNR is over 40dB. By calibrating the peak pulse voltage of the system, the nonlinear response can be corrected with the calibrated scale, and the -10dBm to 0dBm region can also be used to carry RF signals for BSS, which increase the dynamic range of the BSS system with another 10dB.

Conclusion
We proposed and experimentally demonstrated an optical pulse sampling system for photonic BSS. The photonic BSS separates the mixed signals in an analog way, and only statistical information of the mixed signals is needed to find the de-mixing matrix. The system is able to separate the SOI from interference at sampling rate orders of magnitude smaller than the data rate. The optical pulses sample the fast-changing signals for low-speed ADC without interference from adjacent bits. Experimental results show that with the pulse sampled signals, the statistical information can be accurately measured for BSS. By using femto-second pulses generated from mode-locked lasers, signals with bandwidth of 100GHz or more and power range of 30dB can be effectively sampled.

Funding
This research is supported by Rowan University startup grant and New Jersey Health Foundation (Grant # PC 77-21).