The nominal signal waveform is known, but the actual signal
present in the observed data can be corrupted slightly. Using
the minimax approach, we
seek the worst possible signal that could be present consistent
with constraints on the corruption. Once we know what signal
that is, our detector should consist of a filter matched to that
worst-case signal. Let the observed signal be of the form
sl=
s
o
l+cl
s
l
s
o
l
c
l
:
s
o
l
s
o
l
is the nominal signal and
cl
c
l
the corruption in the observed signal. The nominal-signal energy is
E
o
E
o
and the signal corruption is assumed to have an energy
that is less than
E
c
E
c
. Spectral techniques are assumed to be applicable so
that the covariance matrix of the Fourier transformed noise is
diagonal. The signal-to-noise ratio is given by
∑k|Sk|2
σ
k
2
k
S
k
2
σ
k
2
. What corruption yields the smallest signal-to-noise
ratio? The worst-case signal will have the largest amount of
corruption possible. This constrained minimization problem -
minimize the signal-to-noise ratio while forcing the energy of
the corruption to be less than
E
c
E
c
- can then be solved using Lagrange multipliers.
min
S
k
{∑k|Sk|2
σ
k
2+λ∑k|Ck|2-
E
c
}
S
k
k
S
k
2
σ
k
2
λ
k
C
k
2
E
c
By evaluating the appropriate derivatives, we find the spectrum
of the worst-case signal to be a frequency-weighted version of
the nominal signal's spectrum.
S
w
k=λ
σ
k
21+λ
σ
k
2
S
o
k
S
w
k
λ
σ
k
2
1
λ
σ
k
2
S
o
k
where
∑k11+λ
σ
k
22|
S
o
k|2=
E
c
k
1
1
λ
σ
k
2
2
S
o
k
2
E
c
The only unspecifies parameter is the Lagrange multiplier
λλ, with the latter equation
providing an implicit solution in most cases.
If the noise is white, the
σ
k
2
σ
k
2
equal a constant, implying that the worst-case signal
is a scaled version of the nominal, equaling
S
w
k=1-
E
c
E
o
S
o
k
S
w
k
1
E
c
E
o
S
o
k
. The robust decision rule derived from the likelihood
ratio is given by
ℜ∑k=0L-1Rk¯
S
o
k
≷
ℳ
0
ℳ
1
γ
k
L
1
0
R
k
S
o
k
≷
ℳ
0
ℳ
1
γ
By incorporating the scaling constant
1-
E
c
E
o
1
E
c
E
o
into the threshold
γγ, we find that
the matched filter used in the white noise,
known-signal case is robust with respect to signal
uncertainties. The threshold value is identical to
that derived using the nominal signal as model
ℳ
0
ℳ
0
does not depend on the uncertainties in the signal
model. Thus, the detector used in the known-signal, white noise
case can also be used when the signal is partially
corrupted. Note that in solving the general signal corruption
case, the imprecise signal amplitude situation was also solved.
If the noise is not white, the proportion between the nominal
and worst-case signal spectral components is not a constant. The
decision rule expressed in term of the frequency-domain
sufficient statistic becomes
ℜ∑k=0L-1λ
σ
k
21+λ
σ
k
2Rk¯
S
o
k
≷
ℳ
0
ℳ
1
γ
k
L
1
0
λ
σ
k
2
1
λ
σ
k
2
R
k
S
o
k
≷
ℳ
0
ℳ
1
γ
Thus, the detector derived for colored noise problems is
not robust. The threshold depends on the
noise spectrum and the energy of the corruption. Furthermore,
calculating the value of the Lagrange multiplier in the colored
noise problem is quite difficult, with multiple solutions to its
constraint equation quite possible. Only one of these solutions
will correspond to the worst-case signal.
