EE263
S. Lall
2011.05.16.02
Review Session 6
1.
Estimating a signal with interference.
46075
This problem concerns three proposed methods for estimating a signal, based on a mea
surement that is corrupted by a small noise and also by an interference, that need not be
small. We have
y
=
Ax
+
Bv
+
w,
where
A
∈
R
m
×
n
and
B
∈
R
m
×
p
are known. Here
y
∈
R
m
is the measurement (which is
known),
x
∈
R
n
is the signal that we want to estimate,
v
∈
R
p
is the interference, and
w
is a noise. The noise is unknown, and can be assumed to be small. The interference
is unknown, but cannot be assumed to be small. You can assume that the matrices
A
and
B
are skinny and full rank (
i.e.
,
m > n
,
m > p
), and that the ranges of
A
and
B
intersect only at 0. (If this last condition does not hold, then there is no hope of finding
x
, even when
w
= 0, since a nonzero interference can masquerade as a signal.) Each of
the EE263 TAs proposes a method for estimating
x
.
These methods, along with some
informal justification from their proposers, are given below. Nikola proposes the
ignore
and estimate method.
He describes it as follows:
We don’t know the interference, so we might as well treat it as noise, and just
ignore it during the estimation process.
We can use the usual leastsquares
method, for the model
y
=
Ax
+
z
(with
z
a noise) to estimate
x
. (Here we
have
z
=
Bv
+
w
, but that doesn’t matter.)
Almir proposes the
estimate and ignore method.
He describes it as follows:
We should simultaneously estimate both the signal
x
and
the interference
v
,
based on
y
, using a standard leastsquares method to estimate [
x
T
v
T
]
T
given
y
. Once we’ve estimated
x
and
v
, we simply ignore our estimate of
v
, and use
our estimate of
x
.
Miki proposes the
estimate and cancel method.
He describes it as follows:
Almir’s method makes sense to me, but I can improve it. We should simulta
neously estimate both the signal
x
and the interference
v
, based on
y
, using
a standard leastsquares method, exactly as in Almir’s method.
In Almir’s
method, we then throw away ˆ
v
, our estimate of the interference, but I think
we should use it. We can form the “pseudomeasurement” ˜
y
=
y
−
B
ˆ
v
, which
is our measurement, with the effect of the estimated interference subtracted
off. Then, we use standard leastsquares to estimate
x
from ˜
y
, from the simple
model ˜
y
=
Ax
+
z
. (This is exactly as in Nikola’s method, but here we have
subtracted off or cancelled the effect of the estimated interference.)
These descriptions are a little vague; part of the problem is to translate their descriptions
into more precise algorithms.
(a) Give an explicit formula for each of the three estimates. (That is, for each method
give a formula for the estimate ˆ
x
in terms of
A
,
B
,
y
, and the dimensions
n, m, p
.)
(b) Are the methods really different? Identify any pairs of the methods that coincide
(
i.e.
, always give exactly the same results).
If they are all three the same, or all
three different, say so.
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 Summer '08
 BOYD,S
 Least Squares, Method, Linear least squares, Xie, S. Lall

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