**4.5 Dependence on dimensions of the mixing matrix**

We have investigated the requirements on *M*, the small dimension of the measurement matrix, to recover a signal composed of a small number of sinusoids using the OMP-NLS algorithm. Fig. 4 shows the fraction of failed recoveries as a function of M for a problem in which the signal is composed of 1,3,5, or 7 sinusoids and *N* = 128. For each value of *K* we performed 1000 trials so a failure fraction of 0.1 corresponds to 100 failures. The conventional relation between *K*, *M*, and *N* for recovery is given by *M* **=** *C K* **log(***N/K***)** (Baraniuk, 2007; Candes and Wakin, 2008). From Fig. 4 we see that the curves for *K* = 3,5 and 7 are equispaced and correspond to *C* ~ 1.5.

We have also investigated several different types of the measurement matrix as displayed in Fig. 5. The three curves correspond to three different measurement matrices. For the blue curve the mixing matrix is generated from the sum of random integers drawn from {-1,0,1} plus i times different random integers drawn from {-1,0,1}; for the red curve, complex numbers with the real and imaginary parts given by reals uniformly distributed between -1 and 1 and i times uniformly distributed reals; for the magenta curve, the mixing matrix is generated from randomly chosen -1's and 1's. The magenta curve for a real mixing matrix made from 1's and -1's is inferior to the blue and red curves for the two complex mixing matrices. The differences between the red and blue curves in Fig. 5 appear to be random fluctuations and are in agreement with other CS results that Gaussian and Bernoulli measurement matrices perform equally well (Baraniuk, 2007; Candes and Wakin, 2008). Fig. 6 compares calculations with the weighting matrix given by eq. (7) to calculations with the weighting matrix set to the identity matrix. One can see that the green curve with the weighting matrix set to the identity matrix is significantly worse in the important region of less than 1% failure.

Applications of the Orthogonal Matching Pursuit/ Nonlinear

weighting matrix set to the identity matrix.

magnitude larger than Vf.

**5. Results for sparse sinusoids with noise** 

**5.1 Signal composed of a single sinusoid with noise** 

Least Squares Algorithm to Compressive Sensing Recovery 181

Fig. 6. Fraction of failed recoveries as a function of the small dimension of the mixing matrix M. The red curve is with the weighting matrix defined by eq. (7). The green curve has the

Figs. 7 (a) and (b) show the error in the recovery of a single-frequency, unity amplitude signal as a function of the small dimension *M* of an *M*x1024 mixing matrix **Ʒ** with V = 10-2 for 100 realizations of the noise. As *M* increases the standard deviations of the errors in both frequency and amplitude, Vf and Va, decrease as expected since more measurements are made to average a given noise level. The decrease of about a factor of 3 in Vf and Va for a factor of 10 increase in *M* is consistent with estimates based on SNR (Shaw and Valley, 2010; Davenport et al., 2006). Fig. 8 shows Vf and Va as a function of s averaged over 20 different 4x1024 mixing matrices. Both Vf and Va are proportional to V with Va about 2 to 3 orders of

**(a)**

Fig. 4. Fraction of failed recoveries as a function of the small dimension of the mixing matrix *M* for signals consisting of 1 (magenta), 3 (red), 5 (blue) and 7 (green) sinusoids. The large dimension of the mixing matrix is *N* = 128 and 1000 trials were performed for each value of *M*.

Fig. 5. Fraction of failed recoveries as a function of the small dimension of the mixing matrix M. For the blue curve the mixing matrix is generated from the sum of random integers drawn from {-1,0,1} plus i times different random integers drawn from {-1,0,1}; for the red curve, complex numbers with the real and imaginary parts given by reals uniformly distributed between -1 and 1; for the magenta curve, the entries of the mixing matrix are randomly chosen from -1 and 1.

180 Applications of Digital Signal Processing

Fig. 4. Fraction of failed recoveries as a function of the small dimension of the mixing matrix *M* for signals consisting of 1 (magenta), 3 (red), 5 (blue) and 7 (green) sinusoids. The large dimension of the mixing matrix is *N* = 128 and 1000 trials were performed for each value of *M*.

Fig. 5. Fraction of failed recoveries as a function of the small dimension of the mixing matrix M. For the blue curve the mixing matrix is generated from the sum of random integers drawn from {-1,0,1} plus i times different random integers drawn from {-1,0,1}; for the red curve, complex numbers with the real and imaginary parts given by reals uniformly distributed between -1 and 1; for the magenta curve, the entries of the mixing matrix are

randomly chosen from -1 and 1.

Fig. 6. Fraction of failed recoveries as a function of the small dimension of the mixing matrix M. The red curve is with the weighting matrix defined by eq. (7). The green curve has the weighting matrix set to the identity matrix.
