Image Processing Reference
In-Depth Information
shown in detail here, the closest frequency pairs in the signal are {0.2663, 0.2689} and
{0.7715, 0.7736}, but while signals with nearly the same frequency are difficult cases, here the
combined OMP/NLS recovers all the sinusoids to machine precision. Fig. 2 shows the initial
calculation of
G
(
f
,
y
)
for a 128x1024 mixing matrix and 8192 frequency points (
N
f
= 8). Note
that most, but not all of the frequencies have peaks in the initial scan of
G
(
f
,
y
)
. Fig. 3 shows
G
(
f
,
r
19
)
during the 20
th
iteration of the Do loop in the algorithm shown in Table 1. After
refining the frequencies by finding the minimum of
R
(
f
)
in (10), the frequency errors are
reduced to less than 10
-16
and the amplitude errors are reduced to 4x10
-14
. Our results
compare favorably to those obtained using other recovery methods for a test problem with
20 arbitrary frequency complex sinusoids,
N
= 1024, and variable numbers of measurements
M
(Duarte and Baraniuk, 2010).
Fig. 2. The initial calculation of
G
(
f
,
y
) for a signal with 20 input frequencies mixed with a
128 x 1024 matrix. The red dots indicate the input frequencies.
Fig. 3. The next to last calculation of
G
(
f
,
r
19
) for a signal with 20 input frequencies mixed
with a 128x1024 matrix showing a large peak near the frequency of the only remaining
sinusoid.