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:

h
n+1
=A
h
n
+
u
n
h
n
1
A
h
n
u
n

(4)
Where

h
n
=
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

u
n
u
n
is a white Gaussian noise vector process with covariance

Q
Q. That is,

…
u
n−1
u
n
u
n+1
…
…
u
n
1
u
n
u
n
1
…
are iid vectors and

u
n
∼𝒩0Q
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
=
v
n
T
h
n
+
w
n
x
n
v
n
h
n
w
n

(5)
Where

v
n
=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:
h
n+1
=A
h
n
+
u
n
n
n
0
h
n
1
A
h
n
u
n

(6)
(with

h
n
h
n
in place of

y
n
y
n
now). Furthermore, assume that

h
0
∼𝒩0
R
0
h
0
0
R
0
with

R
0
R
0
also diagonal. The measurement equation:

x
n
=
v
n
T
h
n
+
w
n
x
n
v
n
h
n
w
n

(7)
Note that

v
n
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

v
n
v
n
.