Distortion-Rate Function of Sub-Nyquist Sampled Gaussian Sources

The amount of information lost in sub-Nyquist sampling of a continuous-time Gaussian stationary process is quantified. We consider a combined source coding and sub-Nyquist reconstruction problem in which the input to the encoder is a noisy sub-Nyquist sampled version of the analog source. We first derive an expression for the mean squared error in the reconstruction of the process from a noisy and information rate-limited version of its samples. This expression is a function of the sampling frequency and the average number of bits describing each sample. It is given as the sum of two terms: Minimum mean square error in estimating the source from its noisy but otherwise fully observed sub-Nyquist samples, and a second term obtained by reverse waterfilling over an average of spectral densities associated with the polyphase components of the source. We extend this result to multi-branch uniform sampling, where the samples are available through a set of parallel channels with a uniform sampler and a pre-sampling filter in each branch. Further optimization to reduce distortion is then performed over the pre-sampling filters, and an optimal set of pre-sampling filters associated with the statistics of the input signal and the sampling frequency is found. This results in an expression for the minimal possible distortion achievable under any analog to digital conversion scheme involving uniform sampling and linear filtering. These results thus unify the Shannon-Whittaker-Kotelnikov sampling theorem and Shannon rate-distortion theory for Gaussian sources.Comment: Accepted for publication at the IEEE transactions on information theor

On the Minimax Capacity Loss under Sub-Nyquist Universal Sampling

This paper investigates the information rate loss in analog channels when the sampler is designed to operate independent of the instantaneous channel occupancy. Specifically, a multiband linear time-invariant Gaussian channel under universal sub-Nyquist sampling is considered. The entire channel bandwidth is divided into $n$ subbands of equal bandwidth. At each time only $k$ constant-gain subbands are active, where the instantaneous subband occupancy is not known at the receiver and the sampler. We study the information loss through a capacity loss metric, that is, the capacity gap caused by the lack of instantaneous subband occupancy information. We characterize the minimax capacity loss for the entire sub-Nyquist rate regime, provided that the number $n$ of subbands and the SNR are both large. The minimax limits depend almost solely on the band sparsity factor and the undersampling factor, modulo some residual terms that vanish as $n$ and SNR grow. Our results highlight the power of randomized sampling methods (i.e. the samplers that consist of random periodic modulation and low-pass filters), which are able to approach the minimax capacity loss with exponentially high probability.Comment: accepted to IEEE Transactions on Information Theory. It has been presented in part at the IEEE International Symposium on Information Theory (ISIT) 201

The Panchromatic Hubble Andromeda Treasury

The Panchromatic Hubble Andromeda Treasury (PHAT) is an on-going HST Multicycle Treasury program to image ~1/3 of M31's star forming disk in 6 filters, from the UV to the NIR. The full survey will resolve the galaxy into more than 100 million stars with projected radii from 0-20 kpc over a contiguous 0.5 square degree area in 828 orbits, producing imaging in the F275W and F336W filters with WFC3/UVIS, F475W and F814W with ACS/WFC, and F110W and F160W with WFC3/IR. The resulting wavelength coverage gives excellent constraints on stellar temperature, bolometric luminosity, and extinction for most spectral types. The photometry reaches SNR=4 at F275W=25.1, F336W=24.9, F475W=27.9, F814W=27.1, F110W=25.5, and F160W=24.6 for single pointings in the uncrowded outer disk; however, the optical and NIR data are crowding limited, and the deepest reliable magnitudes are up to 5 magnitudes brighter in the inner bulge. All pointings are dithered and produce Nyquist-sampled images in F475W, F814W, and F160W. We describe the observing strategy, photometry, astrometry, and data products, along with extensive tests of photometric stability, crowding errors, spatially-dependent photometric biases, and telescope pointing control. We report on initial fits to the structure of M31's disk, derived from the density of RGB stars, in a way that is independent of the assumed M/L and is robust to variations in dust extinction. These fits also show that the 10 kpc ring is not just a region of enhanced recent star formation, but is instead a dynamical structure containing a significant overdensity of stars with ages >1 Gyr. (Abridged)Comment: 48 pages including 22 pages of figures. Accepted to the Astrophysical Journal Supplements. Some figures slightly degraded to reduce submission siz

Sampling versus Random Binning for Multiple Descriptions of a Bandlimited Source

Random binning is an efficient, yet complex, coding technique for the symmetric L-description source coding problem. We propose an alternative approach, that uses the quantized samples of a bandlimited source as "descriptions". By the Nyquist condition, the source can be reconstructed if enough samples are received. We examine a coding scheme that combines sampling and noise-shaped quantization for a scenario in which only K < L descriptions or all L descriptions are received. Some of the received K-sets of descriptions correspond to uniform sampling while others to non-uniform sampling. This scheme achieves the optimum rate-distortion performance for uniform-sampling K-sets, but suffers noise amplification for nonuniform-sampling K-sets. We then show that by increasing the sampling rate and adding a random-binning stage, the optimal operation point is achieved for any K-set.Comment: Presented at the ITW'13. 5 pages, two-column mode, 3 figure