**2. Background**

170 Applications of Digital Signal Processing

to work; scanning the DFT off integer values requires performing the CS recovery algorithm over and over again with an unknown sparse transform and becomes prohibitively

Here we present a new approach to recovering sparse signals to arbitrary accuracy when the parameters of the signal do not lie on a grid and the sparsifying transform is unknown. Our approach is based on orthogonal matching pursuit (OMP), which has been applied to recovering CS signals by many authors (Donoho et al., 2006; Tropp and Gilbert, 2007; Liu and Temlyakov, 2010; Huang and Zhu, 2011). The major difference between our work and previous work is that we add a nonlinear least squares (NLS) step to each OMP iteration. In the first iteration of conventional OMP applied to finding sinusoids, one finds the frequency that maximizes the correlation between the measurement matrix evaluated on an overcomplete dictionary and the CS measurement, solves a linear least squares problem to find the best estimate of the amplitude of the sinusoid at this frequency, and subtracts this sinusoid multiplied by the measurement matrix from the CS measurement. In the second iteration, one finds the frequency that maximizes the correlation between the measurement matrix and the residual measurement, solves a least squares problem for both frequencies to get new estimates of both amplitudes, and subtracts the sum of the two sinusoids multiplied by the measurement matrix from the previous residual. This process is described in detail in "Algorithm 3 (OMP for Signal Recovery)" in the paper by Tropp and Gilbert (2007) and in our Table 1 in Section 3. Our approach proceeds in the same way as conventional OMP but we substitute a **Nonlinear Least Squares** step for the linear least squares step. In the NLS step, we use a minimizer to find better values for the frequencies without reference to a discrete grid. Because the amplitudes are extremely sensitive to the accuracy of the frequencies, this leads to a much better value for the amplitudes and thus to a much more accurate expansion of the input signal. Just as in conventional OMP, we continue our algorithm until a system level threshold in the residual is reached or until a known number of sinusoids is extracted. A related procedure for matching pursuit but not yet applied to compressive sensing or orthogonal matching pursuit is described by Jacques & De Vleeschouwer (2008). What we refer to as the NLS step appears in their Section V, eq. (P.2). Our approach to CS recovery differs from most methods presented to date in that we assume our signal (or image) is sparse in some model as opposed to sparse under some transform. Of course, for every sparse model there is some sparsifying transform, but it may be easier in some problems to find the model as opposed to the transform. Models inevitably involve parameters, and in most cases of practical interest, these parameters do not lie on a discrete grid or lie on a grid that is too large for efficient discrete processing techniques (see the discussion in Section 1 of Jacques & De Vleeschouwer, 2008). For instance, to recover the frequency of a sinusoid between 0 and 1 to precision of 10-16 would require 1016 grid points. While we first developed our method to find the frequency and amplitude of sinusoids, like OMP it is readily adaptable to signals that are the superposition of a wide range of other models. In Section 2, we present background material on the OMP, NLS and CS methods on which our method is based. In Section 3, we develop the modelbased OMP/NLS formulation. Sections 4 and 5 contains the application to signals that consist of a sum of a small number of sinusoids. Section 6 compares performance of our algorithm to conventional OMP using a linear least square step and to penalized ell-1 norm

expensive when the number of sinusoids in the signal exceeds 1.

methods.

Our method and results rely heavily on work in three well-known areas: orthogonal matching pursuit, nonlinear least squares and compressive sensing.
