Wireless Communication

Chapter: Wireless Channels
Section: Vector Channel Models for DSP

Useful Data Model for Antenna Array Signal Processing

Contributed by Haibing Yang, inspired by lecture notes taken from Alle-Jan van der Veen

The properties of the physical channel for the communication system can be described by a channel model with a large set of parameters. Though useful for generating simulated data, it is not always a suitable model for the purposes of signal processing. For example, if the angle spreads within a cluster are large, thus the parametrization in terms of directions is not possible. Hence, some simplified data models may be meaningful and developed for the purpose of further array signal processing. Here two simplified data models are described: I-MIMO model and FIR-MIMO model.

1   I-MIMO model

Instantaneous Multi-Input Multi-Output model (I-MIMO) is a generic model for source separation, which is valid only when the delay spread of the dominant rays is much smaller than the inverse bandwidth of the signals, i.e., for narrowband signals, in line-of-sight situations or in scenarios where there is only local scattering.

In this model, it's assumed that d independent source signals s1(t), ···, sd(t) are transmitted from d independent sources at different locations. If the delay spread is small, then according to the vector channel model for antenna array,  the noise-free output signal of M antenna elements in the receiver is a simple linear combinations of these signals:
x(t)=a1(t)s1(t)+···+adsd(t),
which can be rewritten in matrix form:
x(t)=As(t),  A=[
a1 ··· ad
],  s(t)= é
ê
ê
ê
ê
ë
s1(t)
·
·
·
sd(t)
ù
ú
ú
ú
ú
û

Suppose we sample with a period T and set T=1. If N samples are considered, the data matrix is written as
X=AS,
where
X=[
x(0) ··· x(N-1)
]
and
S=[
s(0) ··· s(N-1)
].
This data model is called I-MIMO model.

For the multipath propagation model, each column vector ai in matrix A is the summation of the direction vector a(qij) by different fading amplitude hij, i.e.,
a i=
ri
å
j=1
a(qij)hij=[
a(qij) ···
a (q
 
iri
)
] é
ê
ê
ê
ê
ê
ë
hi1
·
·
·
h
 
iri
ù
ú
ú
ú
ú
ú
û
where ri is the number of rays for the i-th source.

To get a compact structure, we define
A
 
q
=
[
A(q11) ···
A (q
 
d,rd
)
],
B =
diag[
h11 ···
h
 
d,rd
],
J =
é
ê
ê
ê
ê
ê
ê
ë
1
 
r1
  0
  ·  
 · 
  ·		  
0  
1
 
rd
ù
ú
ú
ú
ú
ú
ú
û
 







r× d
    (1)
where r=åi=1dri is the total number of d sources and 1m denotes an m× 1 vector consisting of 1's. Finally, the noise-free I-MIMO data model can be written as
X=AS,     A=A
 
q
BJ.

2   FIR-MIMO model

Finite impulse response multi-input multi-output data model (FIR-MIMO) assumes that the channel impulse responses can be simply described by the multichannel finite impulse response (FIR) filters, which are time invariant. The derivation of FIR-MIMO data model is described as follows.

Assume again there are d independent source signals s1(t), ···, sd(t) and sj(t) is defined as the combination of infinite symbols, i.e.,
sj(t)=
¥
å
k=-¥
sj,kg(t-k),
where g(t) represents the pulse shape function and the symbol period T is set to 1 for the convenience of notation. If they are received by M antennas through a convolutive channel, then the received signal without noise is
x(t)=H(t)*s(t),
where `*' denotes the operation of convolution, and
x(t)= é
ê
ê
ê
ê
ë
x1(t)
·
·
·
xM(t)
ù
ú
ú
ú
ú
û		 ,  H(t)= é
ê
ê
ê
ê
ë
h11(t) ··· h1d(t)
·
·
·		   ·
·
·
hM1(t) ··· hMd(t)
ù
ú
ú
ú
ú
û		 ,  s(t)= é
ê
ê
ê
ê
ë
s1(t)
·
·
·
sM(t)
ù
ú
ú
ú
ú
û		 .

Here it's assumed that each channel hij(t) associated to the j-th source and the i-th antenna is a FIR filter with integer length at most Lj symbols, i.e.,
hij(t)=0,   tÏ[0,Lj)   (i=1, ···, M; j=1, ···, d).
With the assumption above, we notice that there are at most Lj consecutive symbols of signal sj(t) take effects in x(t) at any given moment. The maximal channel length among all sources is denoted by L=max Lj and for the simplicity of the exposition, we set the length of all the channels as L.

Suppose we sample the signal xi(t) during N symbol periods, at a rate of P times the symbol rate, then the data matrix X collecting all samples can be written as
X=[
x0 ··· xN-1
]= é
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ë
x(0) x(1) ··· x(N-1)
x(
1
P
)
x(1+
1
P
)
  ·
·
·
·		     ·
·
·
x(
P-1
P
· ···
x(N-1+
P-1
P
),
ù
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
û
 












MP× N
where the vector xk contains the MP spatial and temporal samples taken during the k-th symbol interval. Based on the FIR assumption above, the data matrix X has a factorization
X=H SL
where the sampled channel matrix
H =
é
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ë
H(0) H(1) ··· H(L-1)
H(
1
P
)
·   ·
·
·
·		     ·
·
·
H(
P-1
P
)
· ···
H(L-
1
P
ù
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
û
 












MP× dL
,
    (2)
and the transmitted symbol matrix
SL =
é
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ë
s0 s1 ·  
 · 
  ·		 sN-2 sN-1
s-1 s0 ·  
 · 
  ·		 ·  
 · 
  ·		 sN-2
·  
 · 
  ·		 ·  
 · 
  ·		 ·  
 · 
  ·		 ·  
 · 
  ·		 ·  
 · 
  ·
s-L+1 ·  
 · 
  ·		 ·  
 · 
  ·		 ·  
 · 
  ·		 sN-L
ù
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
û
 












dL× N
    (3)
where the data vector si=s(i) is d-dimensional. We observe that the transmitted symbol matrix SL has a block-Toeplitz structure, which is a consequence of the time invariance of the channel. (Note: The channel matrix H can be rearranged into the discrete channel matrix and they are equivalent).

By exploiting the data matrix above, a linear equalizer can be realized by multiplying a vector w to generate the output y=wHX. More generally, to develop a space-time equalizer with a length of m symbol periods, we write
Xm= é
ê
ê
ê
ê
ê
ê
ê
ë
x0 x1 ··· xN-m
x1 x2 ··· ·
·
·
·
·
·		 ·
·
·		 ·
·
·		 xN-2
xm-1 xN-2 ··· xN-1
ù
ú
ú
ú
ú
ú
ú
ú
û
 








mMP× (N-m+1)
By using this data matrix, a general space-time linear equalizer can be written as y=wH Xm, which combines mP snapshots of M antennas. Furthermore, the data matrix can be factorized into
Xm= Hm SL+m-1= é
ê
ê
ê
ê
ê
ë
0  
H
  ·
·
·		 ·
·
·
 
H
H
  0
ù
ú
ú
ú
ú
ú
û		 é
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ë
sm-1 ·  
 · 
  ·		 sN-2 sN-1
·  
 · 
  ·		 ·  
 · 
  ·		 ·  
 · 
  ·		 sN-2
s-L+2 s-L+3 ·  
 · 
  ·		 ·  
 · 
  ·
s-L+1 s-L+2 ·  
 · 
  ·		 sN-L-m+1
ù
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
û
where Hm has size mMP× d(L+m-1) and the m shifts of H to the left are each over d positions. It's observed that Hm has a block-Hankel structure and SL-m+1 block Toeplitz matrix. A necessary condition for space-time equalization is that the matrix Hm is tall, which results in
MP>d,     m³
d(L-1)
MP-d
.
where M is the number of antennas, P is the oversampling factor, d is the source number, m is the considered symbol number and L is the length of the channel impulse response.

3   Summation

Until now we have introduced two noise-free data models: I-MIMO and FIR MIMO. The I-MIMO model is generally valid for linear time-invariant channels, whereas the FIR MIMO modle is  the consequence of the vector channel model. These models have some structural properties and are very suitable for the purpose of array signal processing. More detailed introduction for these data models can be found in [1], [2], [3].

[1]  A.J. Paulraj and C.B. Papadias, "Space-time processing for wireless communications," IEEE Signal Proc. Mag., vol. 14, pp. 49-83, Nov. 1997.

[2] B. Ottersten, "Array processing for wireless communications," in Proc. IEEE workshop on Stat. Signal Array Proc., (Corfu), pp. 466-473,  June 1996.

[3] H. Liu and G.Xu, "Multiuser blind channel estimation and spatial channel pre-equalization," in Proc. IEEE ICASSP, (Detroit), pp. 1756-1759 vol. 3, May 1995.



Wireless Communication © Haibing Yang and Jean-Paul M.G. Linnartz, 2002.