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

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

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 magnitude larger than Vf.

Applications of the Orthogonal Matching Pursuit/ Nonlinear

noise.

Least Squares Algorithm to Compressive Sensing Recovery 183

Fig. 9. Standard deviation of the error in the recovered frequency Vf as a function of noise standard deviation V for an input signal that consists of two complex sinusoids with amplitudes 1 and 0.01. The green, short dashed curve corresponds to the strong signal; the red, long dashed, to the weak signal. Each curve is averaged over 20 realizations of the

Fig. 10. Standard deviation of the amplitude error VD as a function of noise standard

**5.3 Signal composed of 2 sinusoids with closely spaced frequencies in noise** 

the weak signal. Each curve is averaged over 20 realizations of the noise.

frequencies are roughly correct but are not separated for V > 10-2.

deviation V for an input signal that consists of two complex sinusoids with amplitudes 1 and 0.01. The green, short dashed curve corresponds to the strong signal; red, long dashed, to

We have also investigated the ability of our algorithm to separate two closely spaced frequencies in the presence of noise. Fig. 11 shows Vf and Va for the case with input frequencies {0.3389, 0.3390}, unity amplitude and a 16x1024 mixing matrix. Note that significant amplitude error occurs at V > 10-4 compared to the single frequency results. The

Fig. 7. Standard deviation of the errors in frequency and amplitude of sinusoids mixed by a mixing matrix ) with dimensions *M* x 1024 recovered using OMP/NLS as a function of the small dimension *M* of the mixing matrix ) for V = 10-2. The results are obtained from the average of 100 independent calculations. (a) Frequency, (b) amplitude error.

Fig. 8. Standard deviation of the frequency and amplitude errors, Vf (lower red curve) and V<sup>a</sup> (upper green curve), as a function of V averaged over 20 different 4x1024 mixing matrices.

#### **5.2 Signal composed of 2 sinusoids with 100:1 dynamic range**

Noise also affects the ability of our algorithm to recover a small signal in the presence of a large signal. Figs. 9 and 10 showVf and Va for a test case in which the amplitudes are given by {1.0, 0.01}, *M* = 10, *N* =1024 and the frequencies are well separated. These results are for a single realization of the mixing matrix and averaged over 20 realizations of the noise. Note that as expected, the frequency and amplitude of the large-amplitude component are much better recovered than those of the small-amplitude component. Knowledge of the parameters of the small component essentially disappears for V greater than about 0.005. Tests with the small amplitude equal to 0.001 and 0.0001 suggest that this threshold scales with the amplitude of the small signal.

182 Applications of Digital Signal Processing

**(b)**  Fig. 7. Standard deviation of the errors in frequency and amplitude of sinusoids mixed by a mixing matrix ) with dimensions *M* x 1024 recovered using OMP/NLS as a function of the small dimension *M* of the mixing matrix ) for V = 10-2. The results are obtained from the

Fig. 8. Standard deviation of the frequency and amplitude errors, Vf (lower red curve) and V<sup>a</sup> (upper green curve), as a function of V averaged over 20 different 4x1024 mixing matrices.

Noise also affects the ability of our algorithm to recover a small signal in the presence of a large signal. Figs. 9 and 10 showVf and Va for a test case in which the amplitudes are given by {1.0, 0.01}, *M* = 10, *N* =1024 and the frequencies are well separated. These results are for a single realization of the mixing matrix and averaged over 20 realizations of the noise. Note that as expected, the frequency and amplitude of the large-amplitude component are much better recovered than those of the small-amplitude component. Knowledge of the parameters of the small component essentially disappears for V greater than about 0.005. Tests with the small amplitude equal to 0.001 and 0.0001 suggest that this threshold scales

**5.2 Signal composed of 2 sinusoids with 100:1 dynamic range** 

with the amplitude of the small signal.

average of 100 independent calculations. (a) Frequency, (b) amplitude error.

Fig. 9. Standard deviation of the error in the recovered frequency Vf as a function of noise standard deviation V for an input signal that consists of two complex sinusoids with amplitudes 1 and 0.01. The green, short dashed curve corresponds to the strong signal; the red, long dashed, to the weak signal. Each curve is averaged over 20 realizations of the noise.

Fig. 10. Standard deviation of the amplitude error VD as a function of noise standard deviation V for an input signal that consists of two complex sinusoids with amplitudes 1 and 0.01. The green, short dashed curve corresponds to the strong signal; red, long dashed, to the weak signal. Each curve is averaged over 20 realizations of the noise.

#### **5.3 Signal composed of 2 sinusoids with closely spaced frequencies in noise**

We have also investigated the ability of our algorithm to separate two closely spaced frequencies in the presence of noise. Fig. 11 shows Vf and Va for the case with input frequencies {0.3389, 0.3390}, unity amplitude and a 16x1024 mixing matrix. Note that significant amplitude error occurs at V > 10-4 compared to the single frequency results. The frequencies are roughly correct but are not separated for V > 10-2.

Applications of the Orthogonal Matching Pursuit/ Nonlinear

effect. 

frequencies.

drop linearly proportional to V for V < 0.1.

**6.3 Convex optimization** 

Least Squares Algorithm to Compressive Sensing Recovery 185

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

Method\ Nf 1 2 4 8

OMP with NLS 95 41 11 6

OMP 96 35 11 6 OMP with window 93 19 9 10

Method\ Nf 1 2 4 8

**(a)** 

OMP with NLS 3.9 10^-15 3.9 10^-15 3.5 10^-15 3.7 10^-15

OMP 0.000150 0.000136 0.000085 0.000060 OMP with window 0.000168 0.000141 0.000084 0.000059

**(b)** 

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

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

We have performed the same calculations with a penalized ell-1 norm code (Loris, 2008). None of these calculations returns reliable estimates of frequencies off the grid. Windowing helps recover frequencies slightly off the grid but not arbitrary frequencies. Subdividing the frequency grid allows finer resolution in the recovery but only up to the fine frequency grid.

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 mixing matrix.
