Boyd Convex Optimization book - SFU Wiki

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336 6 Approximation and fitting
1.5
ŷ(t), y(t)
0.5
−0.5
−1.5
0
0.2
0.4
t
0.6
0.8
1
0.05
y(t) − ŷ(t)
0
−0.05
0
0.2
0.4
Figure 6.22 Top. The original signal (solid line) and approximation ŷ obtained
by basis pursuit (dashed line) are almost indistinguishable. Bottom.
The approximation error y(t) − ŷ(t), with different vertical scale.
t
0.6
0.8
1
where
a(t) = 1 + 0.5 sin(11t),
θ(t) = 30 sin(5t).
(This signal is chosen only because it is simple to describe, and exhibits noticeable
changes in its spectral content over time.) We can interpret a(t) as the signal
amplitude, and θ(t) as its total phase. We can also interpret
ω(t) =
dθ
∣ dt ∣ = 150| cos(5t)|
as the instantaneous frequency of the signal at time t. The data are given as 501
uniformly spaced samples over the interval [0, 1], i.e., we are given 501 pairs (t k , y k )
with
t k = 0.005k, y k = y(t k ), k = 0, . . . , 500.
We first solve the l 1 -norm regularized least-squares problem (6.18), with γ =
1. The resulting optimal coefficient vector is very sparse, with only 42 nonzero
coefficients out of 30561. We then find the least-squares fit of the original signal
using these 42 basis vectors. The result ŷ is compared with the original signal y
in in figure 6.22. The top figure shows the approximated signal (in dashed line)
and, almost indistinguishable, the original signal y(t) (in solid line). The bottom
figure shows the error y(t) − ŷ(t). As is clear from the figure, we have obtained an