From the results presented in the previous sections, the colored noise problem was found to be
pervasive, but required a computationally difficult detector.
The simplest detector structure occurs when the additive noise
is white; this notion leads to the idea of whitening the
observations, thereby transforming the data into a simpler form
(as far as detection theory is concerned). However, the
required whitening filter is often time-varying and can have a
long-duration unit-sample response. Other, more computationally
expedient, approaches to whitening are worth considering. An
only slightly more complicated detection problem occurs when we
have a diagonal noise covariance matrix, as in the white noise
case, but unequal values on the diagonal. In terms of the
observations, this situation means that they are contaminated by
noise having statistically independent, but unequal variance
components: the noise would thus be non-stationary. Few
problems fall directly into this category; however, the colored
noise problem can be recast into the white, unequal-variance
problem by calculating the discrete Fourier Transform (DFT) of
the observations and basing the detector on the resulting
spectrum. The resulting spectral detectors greatly
simplify detector structures for discrete-time problems
if the qualifying assumptions described in
sequel hold.
Let WW be the
so-called
L×L
×
L
L
"DFT matrix"
W=(
111…1
1WW2…WL−1
1W2W4…W2(L−1)
⋮⋮⋮⋮⋮
1WL−1W2(L−1)…W(L−1)(L−1)
)
W
1
1
1
…
1
1
W
W
2
…
W
L
1
1
W
2
W
4
…
W
2
L
1
⋮
⋮
⋮
⋮
⋮
1
W
L
1
W
2
L
1
…
W
L
1
L
1
where WW is the elementary complex
exponential
e−(i2πL)
2
L
. The discrete Fourier Transform of the sequence
rl
r
l
, usually written as
Rk=∑l=0L−1rle−(i2πlkL)
R
k
l
0
L
1
r
l
2
l
k
L
, can be written in matrix form as
R=Wr
R
W
r
. To analyze the effect of evaluating the DFT of the
observations, we describe the computations in matrix form for
analytic simplicity. The first critical assumption has been
made: take special note that the length of the transform
equals the duration of the observations.
In many signal processing applications, the transform length can
differ from the data length, being either longer or shorter.
The statistical properties developed in the following discussion
are critically sensitive to the equality of these lengths. The
covariance matrix
K R
K R of
RR is given by
W
K
r
WH
W
K
r
W
. Symmetries of these matrices - the Vandermonde form
of WW and the
Hermitian, Toeplitz form of K r
K r
- leads to many simplifications in
evaluating this product. The entries no the main diagonal are
given by
K
R
K,K=∑l=−(L−1)L−1(L−|l|)
K
1
,
|
l
|
+
1
r
e−(i2πlkL)
K
R
K
K
l
L
1
L
1
L
l
K
1
,
|
l
|
+
1
r
2
l
k
L
The variance of the
k th
k th
term in the discrete Fourier Transform of the noise thus equals
the discrete Fourier Transform of the windowed
covariance function. This window has a triangular shape;
colloquially termed the "rooftop" window, its technical name is
the Bartlett window and it occurs frequently in
array processing and spectral estimation. We have found that
the variance equals the smoothed noise power spectrum evaluated
at a particular frequency. The off-diagonal terms of
K
R
K
R
are not easily written; the complicated result is
∀
k
1
,
k
2
,
k
1
≠
k
2
:
K
R
k
1
,
k
2
=∑l=0L−1
K
1
,
l
+
1
r
-1
k
1
−
k
2
+1sinπl(
k
1
−
k
2
)Lsinπ(
k
1
−
k
2
)L(ei2πl
k
1
L+e−(i2πl
k
2
L))
k
1
k
2
k
1
k
2
K
R
k
1
k
2
l
0
L
1
K
1
,
l
+
1
r
-1
k
1
k
2
1
l
k
1
k
2
L
k
1
k
2
L
2
l
k
1
L
2
l
k
2
L
(1)
The complex exponential terms indicate that each off-diagonal
term consists of the sum of two Fourier Transforms: one at the
frequency index
k
2
k
2
and the other negative index
−
k
1
k
1
. In addition, the transform is evaluated only
over non-negative lags. The transformed quantity again equals a
windowed version of the noise covariance function, but with a
sinusoidal window whose frequency depends
on the indices
k
1 k
1 and
k 2
k 2 . This window can be negative-valued! In contrast to
the Bartlett window encountered in evaluating the on-diagonal
terms, the maximum value achieved by the window is not large
(
1sinπ(
k
1
−
k
2
)L
1
k
1
k
2
L
compared to
LL).
Furthermore, this window is
always zero at
the origin, the location of the maximum value of any covariance
function. The largest magnitudes of the off-diagonal terms tend
to occur when the indices
k
1 k
1 and
k 2
k 2 are nearly equal. Let their difference be one; if the
covariance function of the noise tends toward zero well within
the number of observations,
LL,
then the Bartlett window has little effect on the covariance
function while the sinusoidal window greatly reduces it. This
condition on the covariance function can be interpreted
physically: the noise in this case is wideband and any
correlation between noise values does not extend over
significant portion of the observation record. On the other
hand, if the width of the covariance function is comparable to
LL, the off-diagonal terms will be
significant. This situation occurs when the noise bandwidth is
smaller than or comparable to the reciprocal of the observation
interval's duration. This condition on the duration of the
observation interval relative to the width of the noise
correlation function forms the second critical assumption of
spectral detection. The off-diagonal terms will thus be much
smaller than corresponding terms on the main diagonal
|
K
R
k
1
,
k
2
|2≪
K
R
k
1
,
k
1
K
R
k
2
,
k
2
≪
K
R
k
1
k
2
2
K
R
k
1
k
1
K
R
k
2
k
2
.
In the simplest case, the covariance matrix of the discrete
Fourier Transform of the observations can be well approximated
by a diagonal matrix.
K
R
=(
σ
0
20…0
0
σ
1
20⋮
⋮0⋱0
0…0
σ
L
-
1
2
)
K
R
σ
0
2
0
…
0
0
σ
1
2
0
⋮
⋮
0
⋱
0
0
…
0
σ
L
-
1
2
The non-zero components
σ
k
2
σ
k
2
of this matrix constitute the noise power spectrum at
the various frequencies. The signal component of the
transformed observations RR is represented by S i
S i
, the DFT of the signal
s i
s i , while the noise component has this diagonal
covariance matrix structure.
In the
frequency domain, the colored noise problem can be approximately
converted to a white noise problem where the components of the
noise have unequal variances.
To recap, the critical
assumptions of spectral detection are
- The transform length equals that of the observations.
In particular, the observations cannot be "padded" to force
the transform length to equal a "nice" number (like a power of
two).
- The noise's correlation structure should be much less
than the duration of the observations. Equivalently, a narrow
correlation function means the corresponding power spectrum
varies slowly with frequency. If either condition fails to
hold, calculating the Fourier Transform of the observations
will not necessarily yield a simpler noise covariance matrix.
The optimum spectral detector computes, for each possible
signal, the quantity
ℜRH
K
R
-1
S
i
−
S
i
H
K
R
-1
S
i
2
R
K
R
S
i
S
i
K
R
S
i
2
. Because of the covariance
matrix's simple form, this sufficient statistic for the spectral
detection problem has the simple form
ℜRH
K
R
-1
S
i
−12
S
i
H
K
R
-1
S
i
=∑k=0L−1ℜRk¯
S
i
k
σ
k
2−12|
S
i
k|2
σ
k
2
R
K
R
S
i
1
2
S
i
K
R
S
i
k
0
L
1
R
k
S
i
k
σ
k
2
1
2
S
i
k
2
σ
k
2
(2)
Each term in the dot product between the discrete Fourier
Transform of the observations and the signal is weighted by the
reciprocal of the noise power spectrum at that frequency. This
computation is much simpler than the equivalent time domain
version and, because of algorithms such as the fast Fourier
Transform, the initial transformation (the multiplication by
WW or the discrete
Fourier Transform) can be evaluated expeditiously.
Sinusoidal signals are particularly well-suited to the spectral
detection approach. If the signal's
frequency equals one of the analysis frequencies in the Fourier
Transform (
ω
0
=2πkL
ω
0
2
k
L
for some kk), then the
sequence
S
i
k
S
i
k
is non-zero only at this frequency index, only one
term in the sufficient statistic's summation need be computed,
and the noise power is no longer explicitly needed by the
detector (it can be merged into the threshold).
ℜRH
K
R
-1
S
i
−12
S
i
H
K
R
-1
S
i
=ℜRk¯
S
i
k
σ
k
2−12|
S
i
k|2
σ
k
2
R
K
R
S
i
1
2
S
i
K
R
S
i
R
k
S
i
k
σ
k
2
1
2
S
i
k
2
σ
k
2
If the signal's frequency does not correspond to one of the
analysis frequencies, spectral energy will be maximal at the
nearest analysis frequency but will extend to nearby frequencies
also. This effect is termed "leakage" and has been well
studied. Exact formulation of the signal's DFT is usually
complicated in this case; approximations which utilize only the
maximal-energy frequency component will be sub-optimal
(i.e., yield a smaller detection
probability). The performance reduction may be small, however,
justifying the reduced amount of computation.
