This effect is the result of many
propagation paths, each of which delays and attenuates the
input signal.
Additionally, the channel is time-varying due
to movement of the source, receiver, and/or
scattors. Therefore, the channel is acting like a linear time
varying filter. Due to the time-varying nature of the channel,
a sinusoidal input does not produce a pure sinusoid at the
output Figure 2.
Instead, the output is a narrowband process. Assuming the
channel is relatively slowly varying (compared to the
frequency of the input) we can view the input sinusoids as
being amplitdue modulated by the time-varying channel. This
effect is referred to as
fading and such channels
are called
fading multipath channels. If we
sample the output of the channel, then a very good model is
the low-pass tapped delay line model:
yn=∑k=0p-1
h
n
kvn-k
y
n
k
0
p
1
h
n
k
v
n
k
(1)
Where
yn
y
n
is the output,
h
n
k
h
n
k
is the impulse response depending on time
nn, and
vn-k
v
n
k
is the input. This is simply an FIR filter with
time-varying coefficients. In practice we wouldn't observe
this perfect output, but rather a noise-corrupted version of
it:
xn=∑k=0p-1
h
n
kvn-k+wn
x
n
k
0
p
1
h
n
k
v
n
k
w
n
(2)
Where
wn
w
n
is observation noise. The goal of channel estimation
is to determine the linear time-varying filter
h
n
k
h
n
k
based on the input
vn
v
n
and measured output
xn
x
n
. Is this possible?
Assume
vn=0
v
n
0
for
n<0
n
0
. Then
x0=
h
0
0v0+
h
0
1v-1+w0=
h
0
0v0+w0
x
0
h
0
0
v
0
h
0
1
v
-1
w
0
h
0
0
v
0
w
0
x1=
h
1
0v1+
h
1
1v0+w1
x
1
h
1
0
v
1
h
1
1
v
0
w
1
(3)
x2=
h
2
0v2+
h
2
1v1+w2
x
2
h
2
0
v
2
h
2
1
v
1
w
2
⋮
⋮
For each
n≥1
n
1
we have two new parameters we must estimate!
Even in the absence of measurement noise we
have more unknowns than equations and we can't determine the
filter. What can we do?
Well, suppose that the filter weights are not
changing too rapidly from sample to sample. This is known as a
slow-fading channel model. Probabilistically, we
can view the slowly varying channel as a vector-valued
Gauss-Markov process:
hn+1=Ahn+un
h
n
1
A
h
n
u
n
(4)
Where
hn=
h
n
0…
h
n
p-1T
h
n
h
n
0
…
h
n
p
1
,
A A is a
ppx
pp
matrix designed to reflect the correlation expected between
filter weights at different time samples, and
un
u
n
is a white Gaussian noise vector process with covariance
Q
Q. That is,
…un-1unun+1…
…
u
n
1
u
n
u
n
1
…
are iid vectors and
un∼0Q
u
n
0
Q
. A standard simplifying assumption is to assume
that
A
A and
Q Q are diagonal which
implies that the filter weights are uncorrelated with each
other. This is called an
uncorrelated scattering
model.
The measurement/observation model in vector
form is
x
n
=vnThn+
w
n
x
n
v
n
h
n
w
n
(5)
Where
vn=vnvn-1…vn-p+1T
v
n
v
n
v
n
1
…
v
n
p
1
. With this notation and our Gauss-Markov model for
the time-varying filter, we can now devise a Kalman filter to
estimate and
track the channel.
In the case we have the state equation:
∀n,n≥0:hn+1=Ahn+un
n
n
0
h
n
1
A
h
n
u
n
(6)
(with
hn
h
n
in place of
yn
y
n
now). Furthermore, assume that
h0∼0R0
h
0
0
R
0
with
R0
R
0
also diagonal. The measurement equation:
x
n
=vnThn+
w
n
x
n
v
n
h
n
w
n
(7)
Note that
vn
v
n
is
known, but
time-varying. In our earlier discussion
the
C C vector of the
observation model was
constant. The
Kalman filter still is applicable here, we just replace
C
C with
vn
v
n
.
h
̂
n
|
n
=
h
̂
n
|
n
-
1
+Kn
x
n
-vnT
h
̂
n
|
n
-
1
h
̂
n
|
n
h
̂
n
|
n
-
1
K
n
x
n
v
n
h
̂
n
|
n
-
1
(8)
Where
h
̂
n
|
n
h
̂
n
|
n
is the best estimate of channel given measurements up
to time
nn.
h
̂
n
|
n
-
1
=A
h
̂
n
-
1
|
n
-
1
h
̂
n
|
n
-
1
A
h
̂
n
-
1
|
n
-
1
(9)
Kn=P
n
|
n
-
1
vn
γ
n
-1
K
n
P
n
|
n
-
1
v
n
γ
n
(10)
γ
n
-1=vnTP
n
|
n
-
1
vn+
σ
w
2
γ
n
v
n
P
n
|
n
-
1
v
n
σ
w
2
(11)
P
n
|
n
=P
n
|
n
-
1
-
γ
n
KnKnT
P
n
|
n
P
n
|
n
-
1
γ
n
K
n
K
n
(12)
P
n
|
n
-
1
=AP
n
-
1
|
n
-
1
AT+Q
P
n
|
n
-
1
A
P
n
-
1
|
n
-
1
A
Q
(13)
Q
Q is the covariance of vector
un
u
n
process instead of
σ
u
2bbT
σ
u
2
b
b
(which is a special case with
un=b
u
n
u
n
b
u
n
, where
u
n
u
n
is a scalar).
