Wireless Communication

Chapter: Wireless Channels
Section: Vector Channel Models for DSP


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.

 

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

                                                         
h(t,t,W)= [
h1(t,t,W) ··· hM(t,t,W)
]
T
 
 
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
                                                        
h(t,t,q)= [
h1(t,t,q) ··· hM(t,t,q)
]
T
 
 
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. 

1   The derivation of vector channel model

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
                                                         
a(q)= é
ê
ê
ê
ê
ê
ê
ê
ë
1
e
j2pDsin(q)
 
·
·
·
e
j2pD(M-1)sin(q)
 
ù
ú
ú
ú
ú
ú
ú
ú
û
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
                                                        
x (t)=
N
å
i=1
aia(q0+qi)s(t-ti)
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=2p fcti (fc is carrier frequency), which is uniformly distributed in the interval [0,2p). This leads to the received signal vector
                                                       x(t)=a's(t)

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

                                                       a'~ N(0,R),     R:=E[a'a'H].

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

                                                   
x (t)=
r
å
i=1
a'is(t-ti),
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
                                                   
x (t)=
r
å
i=1
a(qi)his(t-ti),
where {hi=hi(t)} are the time-varying fading amplitudes with a complex Gaussian distribution.

Generally, the vector channel model can be described as
                                                   
h (t)=
r
å
i=1
a(qi)hig(t-ti),
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.

2   Discrete Multipath Model

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.,

                                              h(t)=0,   tÏ[0,L),
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
                                                
g(t)= é
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ë
g(0-t)
g(
1
P
-t)
·
·
·
g(L-
1
P
-t)
ù
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ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
û
 












LP× 1
,    0£t£maxti
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
                                                    
h= é
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ë
h(0)
h(
1
P
)
·
·
·
h(L-
1
P
)
ù
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ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
û
 












MPL× 1
Now sampled channel model is written as
                                                   h =
r
å
i=1
(giÄai)hi=[g1Äa1, ···, grÄar] é
ê
ê
ê
ê
ë
h1
·
·
·
hr
ù
ú
ú
ú
ú
û
    (1)
gi = g(ti),   ai=a(qi),
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
 
q
= [a(q1), ···, a(qr)],
G
 
t
= [g(t1),···,g(tr)],
                                                            B = diag[h1,···,hr]
G
 
t
°A
 
q
= [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,
                                                        
H=[h1,···,hd]=(Gt°A
 
q
)BJ,
where the size of the matrix H is MPL× d and the selection matrix J is defined as
                                                           J =
é
ê
ê
ê
ê
ê
ê
ë
1
 
r1
  0
  ·  
 · 
  ·
 
0  
1
 
rd
ù
ú
ú
ú
ú
ú
ú
û
.
    (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.



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