v2009.01.01 - Convex Optimization

4.5. CONSTRAINING CARDINALITY 299
a television transmission over cable or the airwaves, or a typically ravaged
cell phone communication, etcetera.
Here we consider signal dropout in a discrete-time signal corrupted by
additive white noise assumed uncorrelated to the signal. The linear channel
is assumed to introduce no filtering. We create a discretized windowed
signal for this example by positively combining k randomly chosen vectors
from a discrete cosine transform (DCT) basis denoted Ψ∈ R n×n . Frequency
increases, in the Fourier sense, from DC toward Nyquist as column index of
basis Ψ increases. Otherwise, details of the basis are unimportant except
for its orthogonality Ψ T = Ψ −1 . Transmitted signal is denoted
s = Ψz ∈ R n (686)
whose upper bound on DCT basis coefficient cardinality cardz ≤ k is
assumed known; 4.31 hence a critical assumption that transmitted signal s is
sparsely represented (k < n) with respect to the DCT basis. Nonzero signal
coefficients in vector z are assumed to place each chosen basis vector above
the noise floor.
We also assume that the gap’s beginning and ending in time are precisely
localized to within a sample; id est, index l locates the last sample prior to
the gap’s onset, while index n−l+1 locates the first sample subsequent to
the gap: for rectangularly windowed received signal g possessing a time-gap
loss and additive noise η ∈ R n
⎡
⎤
s 1:l + η 1:l
g = ⎣ η l+1:n−l
⎦∈ R n (687)
+ η n−l+1:n
s n−l+1:n
The window is thereby centered on the gap and short enough so that the DCT
spectrum of signal s can be assumed invariant over the window’s duration n .
Signal to noise ratio within this window is defined
[
∥
SNR = ∆ 20 log
]∥
s 1:l ∥∥∥
s n−l+1:n
‖η‖
(688)
In absence of noise, knowing the signal DCT basis and having a
good estimate of basis coefficient cardinality makes perfectly reconstructing
4.31 This simplifies exposition, although it may be an unrealistic assumption in many
applications.