**6. Comparison with other recovery methods**

In this section we compare our version of OMP with an NLS optimization step for the sinusoid frequency and amplitude at each iteration to two common methods for CS recovery: OMP with a linear least squares amplitude estimator at each iteration and convex optimization based on the ell-1 norm of the sparse target vector plus the ell-2 norm of the measurement constraint given by eq. (2). It should be noted that most of the cases presented in the previous sections cannot be solved with OMP/LS or penalized ell-1 norm methods so it is necessary to pick a special case to even perform the comparison. Consider a noise-free signal that consists of 5 unity amplitude sinusoids at 5 different frequencies. We assume N=1024 time samples and an M=30 x N=1024 complex measurement matrix made up of the sum of random reals plus i times different random reals, both sets of reals uniformly distributed between -1 and 1.

#### **6.1 Baseline case OMP-NLS**

We performed 100 different calculations with the frequencies chosen by a pseudo-random number generator. In order to control the number of significant figures, we took the frequencies from rational numbers uniformly distributed between 0 and 1 in steps of 10-6. Table 2 shows the fraction of failed recoveries and the average standard deviation in the value of the recovered frequency as a function of the oversampling ratio.

#### **6.2 OMP with Linear Least Squares**

We performed the same 100 calcuations using conventional OMP in which the NLS step is replaced by LS as in Tropp and Gilbert (2007). Note that the number of failed recoveries is 184 Applications of Digital Signal Processing

Fig. 11. Standard deviation in frequency Vf (red-lower curve) and amplitude Va (green upper curve) for the case with input frequencies {0.3389, 0.3390}, unity amplitude and a 16x1024

In this section we compare our version of OMP with an NLS optimization step for the sinusoid frequency and amplitude at each iteration to two common methods for CS recovery: OMP with a linear least squares amplitude estimator at each iteration and convex optimization based on the ell-1 norm of the sparse target vector plus the ell-2 norm of the measurement constraint given by eq. (2). It should be noted that most of the cases presented in the previous sections cannot be solved with OMP/LS or penalized ell-1 norm methods so it is necessary to pick a special case to even perform the comparison. Consider a noise-free signal that consists of 5 unity amplitude sinusoids at 5 different frequencies. We assume N=1024 time samples and an M=30 x N=1024 complex measurement matrix made up of the sum of random reals plus i times different random reals, both sets of reals uniformly

We performed 100 different calculations with the frequencies chosen by a pseudo-random number generator. In order to control the number of significant figures, we took the frequencies from rational numbers uniformly distributed between 0 and 1 in steps of 10-6. Table 2 shows the fraction of failed recoveries and the average standard deviation in the

We performed the same 100 calcuations using conventional OMP in which the NLS step is replaced by LS as in Tropp and Gilbert (2007). Note that the number of failed recoveries is

value of the recovered frequency as a function of the oversampling ratio.

mixing matrix.

distributed between -1 and 1.

**6.1 Baseline case OMP-NLS** 

**6.2 OMP with Linear Least Squares** 

**6. Comparison with other recovery methods** 

about the same as the baseline OMP-NLS but the frequency error is huge by comparison. This is the natural result of the frequency grid, which is the limit on the OMP resolution. Timing comparisons with our software show that OMP-NLS takes about 50% longer than conventional OMP. We have also windowed the OMP calculations in order to reduce "spectral leakage" and hopefully achieve better performance. Aside from the lowered failure fraction for *Nf* = 2, windowing OMP appears to have no statistically significant effect.


Table 2. Comparing OMP with NLS to OMP and OMP with windowing for 4 values of the overcomplete dictionary Nf = 1,2,4,8. (a) failure fraction, %. (b) rms error in recovered frequencies.

We have also compared windowed OMP to OMP/NLS in the presence of noise. Fig. 12 shows the frequency and amplitude errors, Vf and Va, as a function of the noise standard deviation V for OMP (blue) and OMP-NLS (red) for a signal composed of two sinusoids with *N* = 128, *M* = 20 and *N*f = 4 averaged over 100 trials with randomly chosen input frequencies. Note that the OMP frequency error drops to an asymptote of about 6 x 10-4 and the OMP amplitude error to about 0.23 for V < 0.1 while the OMP-NLS errors continue to drop linearly proportional to V for V < 0.1.
