Useful Vector Channel Model for Antenna Array Signal Processing
Contributed by Haibing Yang, inspired by lecture notes
taken from Alle-Jan van der Veen
Summary |
The propagation of signals through the wireless channel is fairly
complicated to model. A less sophisticated channel, i.e., vector channel
model, is introduced in the first part. Based on the vector
channel model, a discrete channel model is developed, where the channel
is modeled as an FIR filter. The discrete channel model is quite useful
for the establishment of I-MIMO (Multi-Input Multi-Output)and FIR-MIMO
data model.
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Over the past decades, mobile communications has experienced
an exponential growth. By employing
antenna arrays at both transmitter and receiver, we can reduce interference
and enhance the desired signal so that more subscribers can be supported and
higher data rates and better quality serviced can be obtained.
Multipath propagation results in the spreading
of the signal in various dimensions: delay
spread in time domain, Doppler spread
in frequency domain and angle spread in space domain. Hence, the dispersion
of the channel in the temporal and angular domain can be described by vector
channel
where
M is the number of antennas in the receiver,
t
is the time delay and
W is the direction of arrival
(DOA) in azimuth and elevation angle. A lot of field measurement reveals that
the multipath components are usually not uniformly distributed in delay and
angular domain but arrive in clusters. If the spatial extent of the cluster
is small enough so that all time delays can be modeled as phase shifts, we
may project the two-dimensional angular distribution onto a one-dimensional
space. Thus, the channel is rewritten as
where
q is the DOA in elevation angle.
The vector channel model
and the discrete multipath model
for array antennas are developed as follows. For the convenience
of notation, we denote h(t) as vector channel impulse response
in the following part.
Usually, multiple independent rays are received by antenna
array and have widely separated angles. A single ray is thought to be composed
by a large collection of individual
local scatter
contributions, where it's assumed that the spherical waves produced by
scatterers are approximately planar at the receiving array. Let
a(
q)
be the complex array response vector for a plane wave with DOA
q.
For an uniform linear array, the array response can be written as
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a(q)= |
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where
D is the element spacing in wavelengths. Meanwhile,
let
s(
t) be the transmitted signal and
q0
is the centered angle of the scatters. This gives the vector valued channel
impulse
where
q0+
qi
is the incident angle of the
i-th scatterer,
ai
is the attenuated amplitude,
ti
is the time delay and
N is the number of scatterers.
If the time delays
ti
are much smaller than the inverse bandwidth of the signal, i.e.,
Bti<<1,
where
B is the bandwidth of the signal, then we can approximate the
delays
ti by
phase shift
fi=2
p
fcti
(
fc is carrier frequency), which
is uniformly distributed in the interval [0,2
p).
This leads to the received signal vector
where a'=åi=1Nbia(q0+qi)
is the resulting vector channel response and bi=a
e-j2p fcti.
With the typical assumptions of a Rayleigh channel,
the real and imaginary parts of each entry of a' are Gaussian distributed,
which is specified by
Here we can see that the Doppler spread
is not described in this model. But it can be incorporated by specifying
that a' is not stationary but time varying.
If r rays are considered, then the collection of the received rays
can be written as
where
ti is
the nominal delay of the
i-th ray. If the angle spread for one ray
is negligible, then this gives a simplified expression
where {
hi=
hi(
t)}
are the time-varying fading amplitudes with a complex Gaussian distribution.
Generally, the vector channel model can be described as
where
g(
t) is the collection of the impulse responses by the
pulse shaping, transmit and receive filtering. We can see that this model
is only valid for specular multipath with small cluster angle spread.
In antenna receiver, the received signal is sampled before
it is proceeded. Thus, discrete expression of signal is more useful for the
purpose of signal processing. Some simplified data models have been proposed
because of their structral properties, such as
I-MIMO
and
FIR-MIMO data model, where a
discrete multipath channel is based on. Here we develop
the vector channel model in the previous part into the discrete multipath model.
Suppose that the vector channel h(t) has finite duration and is zero outside the
time duration of L symbols, i.e.,
where the symbol duration
T is set to 1. By collecting
LP samples
of
g(
t-
t), a parametric ``time manifold"
vector function
g(
t) is obtained
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g(t)= |
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where
g(
t) is the collection of the impulse responses by the
pulse shaping, transmit and receive filtering. Then the vector
h with samples of
h(
t) is written as
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h= |
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Now sampled channel model is written as
|
h |
= |
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(giÄai)hi=[g1Äa1,
···, grÄar] |
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(1) |
|
gi |
= |
g(ti),
ai=a(qi), |
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where `
Ä' denotes a Kronecker product. It can be
seen that the multiray channel vector is a weighted sum of vectors on the
space-time manifold g(
t)
Äa(
q).
To get a more compact structure, we define the parametric matrixes,
|
|
= |
[a(q1),
···, a(qr)], |
|
|
|
= |
[g(t1),···,g(tr)], |
|
|
B |
= |
diag[h1,···,hr] |
|
|
|
= |
[g1Äa1,···,grÄar] |
|
where (
Gt°Aq)
is a columnwise Kronecker product, which is called
Khatri-Rao product.
This gives
h=(
Gt°Aq)
B1r,
where
1r is defined as an
r×
1 column vector consisting of 1's.
Until now, only the channel for one source signal is considered. If
d
source signals are received, the discrete channel model for multiple sources
can be obtained as follows,
where the size of the matrix
H is
MPL×
d and the selection
matrix
J is defined as
|
J |
= |
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. |
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(2) |
The matrix
H can be rearranged into an
MP×
dL matrix,
which is equivalent to the discrete channel model and given in
FIR-MIMO data model.