We now begin applying the machinery developed in previous chapters to generalized sampling. In this chapter we focus on subspace priors and linear sampling. Our goal is to recover a signal x from its samples, when we know a priori that x lies in some subspace A of a Hilbert space H. For example, x may be a bandlimited signal, a piecewise polynomial signal, or a pulse amplitude modulation (PAM) signal with known pulse shape. We focus on SI subspaces, which were treated in the previous chapter, and uniform sampling grids. In this setting, we will show that sampling and reconstruction can be performed by filtering operations. However, all the results herein hold in more abstract Hilbert space settings including finite-dimensional spaces and spaces that are not SI, and for sampling grids that are not necessarily uniform.

Sampling and reconstruction processes

Sampling setups

Although our focus in this chapter is on subspace priors, many of the essential ideas will also be used in Chapter 7 when treating smoothness and stochastic priors. Therefore, in the next section we provide an overview of the different setups and criteria that will be used in both chapters. We then focus on the subspace setting and defer the discussion on other signal classes to the next chapter. In Chapter 8 we revisit subspace priors and generalize the sampling mechanism to include nonlinear sampling. Subsequent chapters are devoted to nonlinear signal priors taking the form of unions of subspaces.