Frames as codes

Abstract

This chapter reviews the development of finite frames as codes for erasures and additive noise. These types of errors typically occur when analog signals are transmitted in an unreliable environment. The use of frames allows one to recover the signal with controllable accuracy from part of the encoded, noisy data. While linear binary codes have a long history in information theory, frames as codes over the real or complex numbers have only been examined since the 1980s. In the encoding process, a vector in a finite-dimensional real or complex Hilbert space is mapped to the sequence of its inner products with frame vectors. An erasure occurs when part of these frame coefficients is no longer accessible after the transmission. Additive noise can arise from the encoding process, such as when coefficients are rounded, or from the transmission. This chapter covers two of the most popular recovery algorithms: blind reconstruction, where missing coefficients are set to zero, and active error correction, which aims to recover the signal perfectly based on the known coefficients. The erasures can be modeled as either having a deterministic or a random occurrence pattern. In the deterministic regime it has been customary to optimize the frame performance in the worst-case scenario. Optimality for a small number of erasures then leads to geometric conditions such as the class of equiangular tight frames. Random erasure models are often used in conjunction with performance measures based on averaged reconstruction errors, such as the mean-squared error. Frames as codes for erasures are also closely related to recent results on sparse recovery. Finally, fusion frames and packet erasures introduce an additional structure which imposes constraints on the construction of optimal frames.