## Optical Analog-to-Digital Conversion System based on Compressive Sampling

1  Introduction

With the development of radio frequency (RF) technology and digital signal processing, there has been an ever-growing demand for high-rate and high-precision analog-to-digital converters (ADCs). Due to the wide spectral that the multiband radio signals might lie in, their Nyquist rates could be far beyond the specifications of the state-of-the-art commercial ADCs. An innovative system called Modulated Wideband Converter (MWC) is recently proposed by Yonina Eldar et al. based on Compressive Sampling (CS) techniques, which makes it possible to acquire adequate information of a sparse signal with limited sampling rate.

Considering the severe demands imposed on the accuracy of amplitude and phase, as well as the stability of sampling pulses, it is difficult to realize a practical and realtime MWC system in electrical circuit. Synthesizing the advantages of optical sampling pulses and electrical devices, we proposed a novel scheme of optical analog-to-digital conversion system in the paper Optical Analog-to-Digital Conversion System based on Compressive Sampling, IEEE Photonics Technology Letters, 23(2), 2011 by Hao Nan, Yuantao Gu, and Hongming Zhang.

2  System Description

The scheme of Optical MWC (OMWC) is shown in Figure 1. Multiple channels of radio-frequency optical pulses are generated simultaneously by optical pulse sources with different wavelengths. Then these sampling pulses with different wavelengths in separate channels, are modulated respectively by predefined $T_p$-periodic sequences that contain $M$ sampling pulses in each period. After that, all sampling pulses are multiplexed to one channel by an optical multiplexer (MUX) and then enter an Electro-Optic Modulator (EOM).

After passing through a high-pass (HP) filter, the analog signal modulates the sampling pulses in the EOM. The EOM can modulate the amplitude of the optical signal with the electrical signal using a Mach-Zehnder interferometer. Then the modulated sampling pulses are separated by a wavelength demultiplexer (DEMUX) which can split the optical signal into several channels by their wavelengths. Photo Diode (PD) of each channel detects optical signals and then transforms them to electrical analog signals. After passing through the low-pass (LP) filters, electrical analog signals are sampled by parallel electrical ADCs whose sampling rate is the same as the band width of the LP filters. Finally, all these digital signals are fed into a sparse signal reconstruction algorithm to recover a Nyquist rate sampling sequence.

Figure 1: The scheme of Optical Modulated Wideband Converter (OMWC).

Compared to Electrical MWC (EMWC), there are several advantages in the proposed optics based implementation.

1.     Because the optical pulses width is less than one picosecond (ps), their periods can be rather short. Thus we can realize $100$GS/s (samples per second) and even faster effective sampling. Besides, the modulation bandwidth can extend into an ultra wide frequency range by using EOM.

2.     Utilizing the Wavelength Division Multiplexing (WDM) technique, the proposed scheme requires only one modulator to sample the signal. This simple and compact setup could reduce the system complexity and, very importantly, avoid the synchronization of multiple EOMs.

3.     The jitter of optical pulses produced by mode-locked lasers is less than 1 ps. It is one order of magnitude smaller than electrical sampling pulses and as a result high quantization precision can be achieved using such ultra-stable sampling pulses.

4.     Considering the narrow width of optical pulse, its spectrum is more flat than that of the electrical square sampling pulse (as depicted in Fig.2). Therefore in OMWC, every bin of the signal spectrum will be regarded with the same weight level when being modulated to the baseband. Thus the desired recovery probability and recovered SNR at all frequency bins are approximately the same.

Figure 2: Comparison of optical Gaussian sampling pulse and electrical square sampling pulse in waveform (top) and frequency spectrum (bottom). The spectrum amplitude is normalized.

4  Numerical Results

In the following two simulations, 40 channels with 10MHz ADCs are used to sample a sparse signal at Nyquist rate 10GHz. Each sampling sequence contains 999 pulses in one period. Consequently the spectrum range of -5GHz to 5GHz is divided into 999 bins. As a result, the width of each frequency bin is about 10MHz. Optical sampling pulses are set with width 2 ps and period 100 ps.

In the first simulation, the sparse signal with Gaussian white noise contains 3 narrow band components (5MHz of each and their locations are unknown) and sampled at SNR 2.21dB on the whole and 20dB in their specific bins. The power spectrum density (PSD) of original and recovered signals are depicted in Figure 3. It can be readily accepted that all support bins are correctly identified. Though the background noise outsides the supports are removed, the recovered noise in support bins increases. The details are showed in Figure 3(b) and (d).

Figure 3: Power spectrum density of (a) original signal and (c) recovered signal, where (b) and (d) are the detail of (a) and (c), respectively. The x-axis denotes frequency in GHz.

In the second simulation, the behaviors of OMWC and EMWC are compared in various scenarios. In the following text, we count bins from the baseband. For example, bin 20 denotes frequency range 200MHz$\pm$5MHz. Suppose there is a unique narrow band signal of 5MHz, which locates at bin 20, 100, 200, 300, 400, or 490, respectively. The input SNR in its specific bin ranges from 6dB to 18dB. In each condition, the simulations are conducted 1000 times and the recovery probability results are plotted in Figure 4. It is readily read that as the SNR increases the recovery probability approaches to 1 and OMWC performs similarly regardless of which bin the signal locates in. Although recovery probability of EMWC is better than OMWC in low frequency, it decreases sharply in high frequency. Furthermore, the improvement of EMWC in low frequency bins is far less than the lost in high ones.

Figure 4: Recovery probabilities of OMWC (blue solid line with circle marker) and EMWC (red dashed line with star marker) in different frequency bins, where x-axis denotes input SNR in the selected band and y-axis denotes recovery probability.