Multirate asynchronous sampling of sparse multiband signals

Abstract

Because optical systems have huge bandwidth and are capable of generating low noise short pulses they are ideal for undersampling multi-band signals that are located within a very broad frequency range. In this paper we propose a new scheme for reconstructing multi-band signals that occupy a small part of a given broad frequency range under the constraint of a small number of sampling channels. The scheme, which we call multi-rate sampling (MRS), entails gathering samples at several different rates whose sum is significantly lower than the Nyquist sampling rate. The number of channels does not depend on any characteristics of a signal. In order to be implemented with simplified hardware, the reconstruction method does not rely on the synchronization between different sampling channels. Also, because the method does not solve a system of linear equations, it avoids one source of lack of robustness of previously published undersampling schemes. Our simulations indicate that our MRS scheme is robust both to different signal types and to relatively high noise levels. The scheme can be implemented easily with optical sampling systems. I. INTRODUCTIONA multi-band signal is one whose energy in the frequency domain is contained in the finite union of closed intervals. A sparse signal is a signal that occupies only a small portion of a given frequency region. In many applications of radars and communications systems it is desirable to reconstruct a multi-band sparse signal from its samples. When the signal bands are centered at frequencies that are high compared to their widths, it is not cost effective and often it is not feasible to sample at the Nyquist rate F nyq ; the rate that for a real signal is equal to twice the maximum frequency of the given region in which the signal spectrum is located. It is therefore desirable to reconstruct the signal by undersampling; that is to say, from samples taken at rates significantly lower than the Nyquist rate. Sampling at any constant rate that is lower than the Nyquist rate results in down-conversion of all signal bands to a low frequency region called a baseband. This creates two problems in the reconstruction of the signal. The first is a loss of knowledge of the actual signal frequencies. The second is the possibility of aliasing; i.e. spectrum at different frequencies being down-converted to the same frequency in the baseband.Optical systems are capable of very high performance undersampling [1]. They can handle signals whose carrier frequency can be very high, on the order of 40 GHz, and signals with a dynamic range as high as 70 dB. The size, the weight, and the power consumption of optical systems make them ideal for undersampling. The simultaneous sampling of a signal at different time offsets or at different rates can be performed efficiently by using techniques based on wavelength-division multiplexing (WDM) that are used in optical communication systems.There is a vast literature on reconstructing signals from undersampled data. Landau proved that, regardles...

Multirate Synchronous Sampling of Sparse Multiband Signals

“…We describe this simple procedure for eliminating frequencies which, according to our assumption, cannot be part of the spectral support of the originating signal. The elimination is similar to one presented in asynchronous-MRS [9]. We denote the indicator function χ i [l] as follows:…”

“…The sampling frequencies are the same as are used in our experiments based on asynchronous MRS [9]. The number of bands was set to 8 (4 positive and 4 negative frequencies, assuming no carrier frequency so low as to have the 0 frequency in its spectrum).…”

“…A previous work has demonstrated a different scheme for reconstructing sparse multiband signals [9]. The scheme, called multirate sampling (MRS), entails gathering samples at P different rates.…”

“…The reconstruction method in [9] does not require synchronization of the different sampling channels. In addition to reducing the complexity of the sampling hardware, unsynchronized sampling relaxes the stringent requirement in multicoset sampling schemes of a very small timing jitter in the sampling time of the channels.…”

“…Accurate signal reconstruction using the unsynchronized MRS scheme of [9] requires that each frequency in the support of the signal be unaliased in at least one of the sampling channels. In this paper we describe a new reconstruction algorithm that overcomes this deficiency by using synchronized sampling channels.…”