Sub-Nyquist SAR via Fourier Domain Range Doppler Processing
Kfir Aberman, Yonina C. Eldar
SYNTHETIC aperture radar (SAR) is a well proven radar imaging technology that enables the production of high- resolution images of targets and terrain. SAR can be oper- ated at night and in adverse weather conditions, overcoming limitations of optical and infrared systems. The basic idea of SAR is that a single monostatic radar transmits pulses at microwave frequencies at a uniform pulse repetition interval (PRI) as it moves along a path. The echoes coming from ground scatterers are then collected and processed in order to generate a focused image. The coherent information recorded at the different positions is used to synthesize a long antenna in order to improve resolution.
Processing of SAR data requires two-dimensional space- variant correlation of the raw data with the point scatter response of the SAR data acquisition system . A full two- dimensional time domain correlation can handle the space- variance, but is computationally inefficient. In order to acceler- ate computation time, various algorithms have been developed that impose different approximations on the correlation kernel
The Range-Doppler Algorithm (RDA) is the most widely used approach for high resolution processing of SAR data. It is conceptually the simplest, can accommodate range varying parameters and is independent of the transmitted pulse structure. An important part of RDA is the Range Cell Migration Correction (RCMC) operation, which is aimed at decoupling the dependency between the two dimensions of the system, range and azimuth, which are also known as fast-time and slow-time, respectively. This step requires fine delay resolution in the Range-Doppler domain, which is typically obtained by digital interpolation . Interpolation allows to reduce the sampling rate at the cost of additional digital computations which effectively increase the rate in the digital domain. In practice, oversampling is often employed to eliminate artifacts caused by digital implementation of standard RDA processing.
According to the Shannon-Nyquist theorem, the minimal sampling rate at the SAR receiver should be at least twice the bandwidth of the detected signal in order to avoid aliasing . In addition, the need to avoid azimuth ambiguities in the resulting image is translated into a minimal pulse repetition frequency (PRF) requirement. The PRF has to be greater than the Doppler bandwidth of the received signals which is dictated by several system parameters, i.e, platform velocity, carrier frequency and the real antenna aperture. This, in fact, limits the maximal swath of the system . Consequently, this two-dimensional dense sampling results in large data rates, requiring large on board memory which may be restricted by downlink throughput requirements, especially for orbital missions.
The emerging theory of compressive sensing (CS) states that a signal which is sparse in some basis, can be reconstructed from highly incomplete samples or measurements , . Since a SAR image is a map of a spatial distribution of the reflectivity function of stationary targets and terrain, many SAR images are sparse or compressible under an appropriate basis such as wavelet, curvelet or total variation . In this paper we show that CS can be applied on both dimensions of SAR. Rate reduction in range is realized by low rate analog-to-digital conversion (ADC) at the receiver and azimuth subsampling is expressed by the transmission of a smaller number of pulses during a coherent processing interval (CPI).
Conventional Synthetic Aperture Radar (SAR) systems are limited in their ability to satisfy the increasing requirement for improved spatial resolution and wider coverage. The demand for high resolution requires high sampling rates, while coverage is limited by the pulse repetition frequency. Consequently, sampling rate reduction is of high practical value in SAR imaging. In this paper, we introduce a new algorithm, equivalent to the well-known Range-Doppler method, to process SAR data using the Fourier series coefficients of the raw signals. We then demonstrate how to exploit the algorithm features to reduce sampling rate in both range and azimuth axes and process the signals at sub-Nyquist rates, by using compressed sensing (CS) tools. In particular, we demonstrate recovery of an image using only a portion of the received signal’s bandwidth and also while dropping a large percentage of the transmitted pulses. The complementary pulses may be used to capture other scenes within the same coherent processing interval. In addition, we propose exploiting the ability to reconstruct the image from narrow bands in order to dynamically adapt the transmitted waveform energy to vacant spectral bands, paving the way to cognitive SAR. The proposed recovery algorithms form a new CS-SAR imaging method that can be applied to high-resolution SAR data acquired at sub-Nyquist rates in range and azimuth. The performance of our method is assessed using simulated and real data sets. Finally, our approach is implemented in hardware using a previously suggested Xampling radar prototype.