Smart Antennas and Space-Time Processing

Synchronization and Detection in MIMO Systems

Introduction

The first approach to carrying out space-time processing of data sampled at an array of sensors dates back to the second world-war (the Bartlett beamformer), and was an application of Fourier-based spectral analysis to spatio-temporally sampled data. Space-time processing emerged as a field of interest, because, at the time, source localization was an imminent problem to radar and sonar applications.

Nowadays, the ever growing demand for mobile communications is constantly increasing the need for better coverage, improved capacity and higher quality service. Space-time processing promises to improve mobile radio performance significantly. The principles of space-time processing can be used to develop 'smart' antennas that use adaptive arrays of antenna sensors. Therefore, the 'smart' antenna concept has become very interesting to the mobile communications industry.

This tutorial gives an overview of the basic principles and the current state of research in 'smart' antennas. Note, that throughout the report narrowband communication systems are assumed. Wideband communication systems like CDMA are not discussed here!

The document is organized in the following manner. Section 1 is a brief introduction to the subject of 'smart' antennas. Firstly, the basic principle behind 'smart' antennas is explained using a simple, but insightful example. Afterwards, the advantages of using 'smart' antennas in wireless communication systems are discussed. Some mathematical preliminaries concerning the treatment of narrowband signals are introduced in Subsection 1.3. The presentation of a fading channel model appropriate for 'smart' antenna systems follows in Section 2. Section 3 presents signal processing algorithms that are used in the Uplink (mobile to base station) of a wireless communication system. Similarly, Section 4 presents algorithms that are used in the Downlink (base station to mobile radio).

1.1 Basic principle

The diagram of Figure 1 shows the principal system elements of a 'smart' antenna system. The 'smart' antenna consists of the sensor array, the patternforming network and the adaptive processor:

Sensor Array

The sensor array consists of N sensors designed to receive (and transmit) signals. The physical arrangement of the array (linear, circular, etc.) is arbitrary, but places fundamental limitations on the capability of the 'smart' antenna

Patternforming Network

The output of each of the N sensor elements is fed into the patternforming network, where the outputs are processed by linear time-variant (LTV) filters. These filters determine the directional pattern1 of the 'smart' antenna. The outputs of the LTV filters are then summed to form the overall output y(t). The complex weights of the LTV filters are determined by the adaptive processor

Adaptive processor

The adaptive processor determines the complex weights of the patternforming network. The signals and known system properties used to compute the weights include the following items, details of which are discussed in Sections 3 and 4 about Uplink and Downlink Processing:

  • The signals received by the sensor array, i.e. xk(t) for k = 1 ¼N
  • The output of the 'smart' antenna, i.e. y(t)
  • The spatial structure of the sensor array
  • The temporal structure of the received signal
  • Feedback signals from the mobiles
  • Network topology

 

The working principle of a 'smart' antenna is now explained using a simple example. In the example, the sensor array is assumed to be a uniform linear array (ULA) consisting of 2 identical omnidirectional sensors as shown in Figure 2.

 

Example

Assuming that a signal s(t) is generated by a mobile radio located in the far-field of the 'smart' antenna, the electromagnetic wave arriving at the sensor array is approximately plane (see Figure 2). If the direction q is different from zero, then sensor 2 experiences a time delay with respect to element 1 of

t = dsinq
v
,
(1)
where d is the sensor spacing and v is the velocity of the plane wave. If s(t) is a narrowband signal with carrier frequency f0, then the time delay t corresponds to a phase shift of
f = 2pd
l0
sinq
(2)
where l0 is the wavelength corresponding to the carrier frequency, i.e.
l0 = v
f0
(3)
Now assume that a second interfering signal n(t) with the same carrier frequency impinges on the array. The directions of s(t) and n(t) are 0 radians and [(p)/6] degrees, respectively. The task of the 'smart' antenna is to null out the interfering signal such that the output becomes s(t).

In this example, the patternforming network is reduced to two complex weights, w1 = w1,1 + jw1,2 and w2 = w2,1 +jw2,2. Then the 'smart' antenna output due to s(t) is
s(t){[w1,1 + w2,1] + j[w2,1 + w2,2]}
(4)
For a sensor spacing d = l0/2, the interfering signal n(t) exhibits a phase lag of p/2 at sensor 2 with respect to sensor 1. Hence the output of the 'smart' antenna due to n(t) can be written as
n(t)exp(jp/4)[w1,1 + jw1,2] +n(t)exp(-jp/4)[w2,1 + jw2,2]
(5)
So for the array output to be equal to s(t), it is necessary that
w1,1 + w2,1
=
1
w1,2 + w2,2
=
0
w1,1 - w1,2 + w2,1 + w2,2
=
0
w1,1 + w1,2 - w2,1 + w2,2
=
0
(6)
Solving equation (6) yields:
w1,1 = 1
2
,     w1,2 = 1
2
,    w2,1 = 1
2
,     w2,2 = - 1
2
(7)
The normalized directional pattern in decibels for an N-element ULA with single complex co-efficient LTV filters is given by
G(q) = 10log10ì
ï
í
ï
î
ê
ê
ê
ê
ê
N-1
å
k=0 
wkexp([(j2pkdsin(q))/(λ0)])

wHw
ê
ê
ê
ê
ê
2


 
ü
ï
ý
ï
þ
(8)
Figure 3 shows the normalized directional pattern of the two-sensor 'smart' antenna without any weighting in the patternforming network. Figure 4 shows how the directional pattern changes, when the weights of equation (7) are used instead. It is seen that now a null is placed exactly at an azimuth of [(p)/6] radians, the direction of the interferer. Additionally, there is no signal attenuation at 0 radians, the direction of the desired signal. Therefore it can be said, that the 'smart' antenna is capable of separating the desired signal s(t) and the interfering signal n(t).

Figure
Figure 3: Normalized directional pattern of the non-weighted sensor array

Figure
Figure 4: Normalized directional pattern of the weighted sensor array

1.2  Performance improvements

Figure
Figure 5: Channel Re-use via Angular Separation
Figure
Figure 6: Channel Re-use via Spatial Separation

A space-time processor ('smart'antenna') is capable of forming transmit/receive beams towards the mobile of interest. At the same time it is possible to place spatial nulls in the direction of unwanted interferences. This capability can be used to improve the performance of a mobile communication system:

Increased antenna gain
The 'smart' antenna forms transmit and receive beams as shown in Figure 5. Therefore, the 'smart' antenna has a higher gain than a conventional omni-directional antenna. The higher gain can be used to either increase the effective coverage, or to increase the receiver sensitivity, which in turn can be exploited to reduce transmit power and electromagnetic radiation in the network.
Decreased inter-symbol-interference (ISI)
Multipath propagation in mobile radio environments leads to ISI. Using transmit and receive beams that are directed towards the mobile of interest reduces the amount of multipaths and ISI.
Decreased co-channel-interference (CCI)
'Smart' antenna transmitters emit less interference by only sending RF power in the desired directions. Futhermore, 'smart' antenna receivers can reject interference by looking only in the direction of the desired source. Consequently 'smart' antennas are capable of decreasing CCI. A significantly reduced CCI can be taken advantage of by Spatial Division Multiple Access (SDMA):

  • The same frequency band can be re-used in more cells, i.e. the so-called frequency re-use distance can be decreased. This technique is called Channel Re-use via Spatial Separation. See also Figure 6 where the cells using the same frequency band are always separated by two cells using different frequencies. Channel Re-use via Spatial Separation can reduce the necessary amount of seperating cells without increasing CCI.
  • Several mobiles can share the same frequency within a cell. Multiple signals arriving at the base station can be separated by the base station receiver as long as their angular separation is bigger than the transmit/receive beamwidths. This is shown in Figure 5. The beams that are hatched identically use the same frequency band. This technique is called Channel Re-use via Angular Seperation

1.3  Narrowband Signals

For any communication system, a linearly modulated bandpass signal z(t) with bandwidth B, symbol rate T, complex envelope s(t), complex data symbols b(k), and pulseform g(t) can be written in complex notation as

z(t)
=
s(t)exp(j2pfct)
(9)
s(t)
=

å
k=-¥ 
b(k)g(t-kT)
(10)
The signal z(t) delayed by a time constant τ is
z(t-t) = s(t-t)exp(-j2pfct)exp(j2pfct)
(11)
Therefore, the complex envelope of z(t-t) is s(t-t)exp(-j2pfct). Let S(f) be the Fourier transform of s(t). Then
s(t-t) = ó
õ
B/2

-B/2 
S(f)exp(-j2pft)exp(j2pft)df
(12)
If |2pft| << 1 " |f| £ [B/2], then exp(-j2pft) » 1. This inequality can also be rewritten as Bt << 1. Hence
s(t-t) » ó
õ
B/2

-B/2 
S(f)exp(j2pft)df = s(t)
(13)
Therefore the complex envelope of z(t-t) can be approximated as
s(t-t)exp(-j2pfct) » s(t)exp(-j2pfct)
(14)
Thus it is seen that for narrowband signals small time delays may be represented as phase shifts of the complex envelope. The signal z(t) delayed by a time constant τ is
z(t-t) = s(t-t)exp(-j2pfct)exp(j2pfct)
(11)
Therefore, the complex envelope of z(t-t) is s(t-t)exp(-j2pfct). Let S(f) be the Fourier transform of s(t). Then
s(t-t) = ó
õ
B/2

-B/2 
S(f)exp(-j2pft)exp(j2pft)df
(12)
If |2pft| << 1 " |f| £ [B/2], then exp(-j2pft) » 1. This inequality can also be rewritten as Bt << 1. Hence
s(t-t) » ó
õ
B/2

-B/2 
S(f)exp(j2pft)df = s(t)
(13)
Therefore the complex envelope of z(t-t) can be approximated as
s(t-t)exp(-j2pfct) » s(t)exp(-j2pfct)
(14)
Thus it is seen that for narrowband signals small time delays may be represented as phase shifts of the complex envelope.

2  Propagation modelling for fading radio channels

In this Section a model for the vector channel of 'smart' antenna systems is derived. The derivation begins with first principles based on Maxwell's equations, so that the connection between channel model and physical wave propagation becomes clear. Good channel modelling is essential for the more sophisticated parametric signal processing methods that exploit the structure of the underlying model.

In empty space Maxwell's equations are given by [27]

Ñ·E
=
0
(15)
Ñ·B
=
0
(16)
Ñ×E
=
- B
t
(17)
Ñ×B
=
e0m0E
t
(18)
where Ñ·, and Ñ×, respectively, denote the "divergence" and "curl". Furthermore, B is the magnetic induction, E is the electric field, and m0 and e0 are the magnetic and dielectric constants. The above equations can be combined easily to derive the fundamental wave equation
Ñ2E - 1
c2
2
t2
E = 0
(19)
The constant c is the speed of propagation, and for electromagnetic waves in free space we have c = 1/ö{e0m0} = 3×108 m/s. If r is the position vector, then any scalar field of the form
E(r,t) = f(t - rTz)
(20)
satisfies equation (19) and can be interpreted as a wave travelling in the direction z with speed of propagation c = 1/|z| [12]. Note, that an important assumptions was made: The solution to E(r,t) has only one scalar component E(r,t) in the direction of propagation. Therefore the electromagnetic wave is plane and hence this model is only valid in the far-field of the transmitter [27,12,15].

The electromagnetic wave at position r due to a modulated bandpass signal source s(t) at the origin with bandwidth B is
E(r,t) = s(t - rTz)exp(j2pfc(t-rTz))
(21)
If we use the definition of the wave vector k = 2pfcz then equation (21) can be put into the following form [15]
E(r,t) = s(t - rTz)exp(j(2pfct - rTk))
(22)
The wave vector k points in the direction of propagation (i.e. the source). The height difference between source and receiver is usually much smaller than the distance between the two. Therefore, a two-dimensional model (see [15]) is used here and elevation is ignored. In that case, the wave vector is defined in the xy-plane to be
k = k(cosq    sinq)T,
(23)
where q is the azimuthal direction of propagation, defined clockwise relative to the array normal.

Empty space is a lossless propagation medium and k is a real valued scalar. In a lossy propagation medium, however, an augmented wave equation signifies that k can be a complex number, that varies with frequency [27,12], i.e.
kH k = f(w)
(24)
If k is complex, then the electromagnetic wave is attenuated depending on the position vector r. Furthermore, different frequency components of s(t) will experience different time delays through the frequency dependency of k. This phenomenon is called dispersion. The above equation is therefore called the dispersion relation. A typical mobile radio system operates in an inhomogenous, lossy and time-varying environment. Finding a solution to the wave equation for a particular mobile radio environment is a hard, if not impossible, task. We will see in the following that modelling this type of environment is still possible, though, if certain assumptions are made.

If the signal envelope s(t) is sufficiently narrowband, then the assumption BrTz << 1 holds true. Therefore the delay experienced by the complex envelope of the transmitted signal can be approximated as a phase shift only.
s(t - rTz)exp(-j2pfcrTz) » s(t)exp(-j2pfcrTz)
(25)
Furthermore, for sensor n of an N-dimensional antenna array, the position vector is
rn = (xn     yn)T,
(26)
Let us assume again a free space propagation environment which is homegenous, lossless and time-invariant. In that case we have that k = 2p/l0. It will be seen in the following that the introduction of an additional time varying complex fading envelope c(t) can help to overcome this restricting assumption. Combining all of the above, the solution to the wave equation now becomes
E(r,t) = c(t)s(t)exp æ
ç
è
j2p æ
ç
è
fct - 1
l


0 
(xncosq+ynsinq) ö
÷
ø
ö
÷
ø
(27)
If a flat frequency response is assumed for sensor n over the signal bandwidth B, then the sensor output will be proportional to the field at position rn. Dropping the carrier term exp(j2pfc) for convenience we arrive at equivalent lowpass representation and the output of sensor n due to a single source s(t) becomes
xn(t)
=
exp æ
ç
è
-j 2p
l


0 
(xncosq+ynsinq) ö
÷
ø
cn(t)s(t)
=
an(q)cn(t)s(t)
(28)
The output vector of an N-element antenna array is thus obtained as
x(t) = diag{c1(t),c2(t),¼,cN(t)}a(q)s(t)
(29)
The vector field a(q) is called the array response vector or steering vector. The steering vector represents the response of the antenna elements relative to the first sensor element for a wavefront arriving at the carrier frequency from a direction q. The curve that a(q) describes in the N-dimensional complex vector space CN when q is varied over its feasible set is called the array manifold.

Most radio channels are characterized by multipath propagation [21,16,25]. Multipath propagation occurs, if the channel consists of the superposition of a number of reflected or scattered radio rays. Hence, it is now assumed that L multipath signals impinge on the N-dimensional array. The array output vector then becomes
x(t) = L
å
l=1 
diag{c1,l(t),c2,l(t),¼,cN,l(t)}a(ql)s(t-tl)
(30)
The tl's in equation (33) are due to the different time delays experienced by the multipaths. They are not absolute values, but are defined with respect to the path with the shortest propagation delay. Hence, the maximum integer delay spread can now be defined as
DtT = é
max
l 
tl -
min
l 
tlù
(31)
If the integer delay spread DtT > 0 then the channel is said to be frequency selective. Furthermore, the fading envelopes cn,l(t) are assumed to be equal at each sensor for the lth multipath, i.e.
c1,l(t) = c2,l(t) = ¼ = cN,l(t)
(32)
This assumption is valid, because usually the sensor spacing is chosen small enough to avoid spatial aliasing ( £l0/2), and as such any time delays can be ignored. Therefore, equation (30) simplifies to
x(t) = L
å
l=1 
a(ql)cl(t)s(t-tl)
(33)
In a multi-user system Q sources emit signals in the same frequency-time slot, and in such an environment the received signal vector is given by
x(t) = Q-1
å
q=0 
Lq
å
l=1 
a(qq,l)cq,l(t)sq(t-tq,l)
(34)

2.1  The Discretized Vector Channel

When designing communication systems for fading channels, it is important to be able to assess and verify system performance during the entire design phase, long before actually implementing the system in hardware. Therefore, it becomes necessary to simulate the fading channel with a software model. A realization of a channel may then be reproduced arbitrarily often, whereby a comparison between different receivers is possible, even if simulation time is limited, and it is possible to emulate a wide range of well-defined channel conditions, in particular worst-case conditions that occur very rarely in nature. Statistical channel models are particularly suited for the design of receivers, as they present a mathematical framework that allows the derivation of receiver receiver algorithms in a systematic manner.

The complex envelope of a linearly modulated signal is given by

s(t) = +¥
å
k=-¥ 
b(k)g(t-kT)
(35)
The impulse response of the transmitter filter g(t) is of infinite duration in the time domain, because its spectrum G(f) has to be bandlimited. However, pulses such as the root-raised cosine pulse consist of a dominant mainlobe and sidelobes that die out fairly quickly. This is illustrated in Figure 7.

Figure
Figure 7: root-raised cosine pulse

Now, denote the mainlobe duration with dT. If we assume that the only significant contribution of the impulse response to the channel memory is due to its mainlobe duration, then the maximum channel memory is given by PT = (Dt + 2d)T, where Dt is the integer delay spread mentioned above. If x(t) is sampled at the twice the symbol rate, i.e. Ts = [T/2], then the T-spaced partial discrete array output vector can be written as

x(i)(k) = Q-1
å
q=0 
Hq(i)(k)bq(k) + n(i)(k)     (i=0,1)
(36)
The superscipts i = 0,1 denote the samples taken at timing instants kT (integer multiples of T) and kT + T/2 (half-integer multiples of T), respectively. Hq(i)(k) is the N ×P T-spaced partial channel matrix for user q which is given by
Hq(i)(k) = Lq
å
l=1 
cq,l(i)(k)a(qq,l)gq,l(i)T
(37)
It represents the combined effect of the transmitter filter and channel for the entire array. The data symbols are stored in the P ×1 column vector bq(k)
bq(k) = é
ê
ê
ê
ë
bq(k + 1 - Dt - d)
:
bq(k + P - Dt - d)
ù
ú
ú
ú
û
(38)
The first symbols that are stored in bq(k) are the symbols with a ``past'' time index. These symbols' contribution is due to the delay spread DtT and the mainlobe duration dT of the transmitter filter impulse response g(t). The last symbols stored in bq(k) have a ``future'' time index and they contribute to the received signal only because of the mainlobe duration dT. The appropriately delayed and sampled T-spaced partial transmitter filter impulse response is stored in the P ×q vector gq,l(i):
gq,l(i)
=
é
ê
ë
g æ
ç
è
(Dt+d-1- i
2
)T -eqT - tq,l ö
÷
ø
¼
g æ
ç
è
(Dt+d-P- i
2
)T -eqT - tq,l ö
÷
ø
ù
ú
û
T

 
(39)
The fractional timing error in the above equation is eq and is due to the asynchronous receiver and transmitter clocks of the q-th user. Figure 8 illustrates how the vector gq,l(0) is assembled for the case of Dt = d = 1.

Figure
Figure 8: Sampling the transmitter filter impulse response

In practice this sampling can be achieved by storing a highly oversampled impulse response in an array and then calculating the appropriate array indexes as needed.

In the context of channel simulation, it is convenient to first generate a realization of the sequence of Ts = (T/2)-spaced channel matrices at sample rate 1/Ts = 2/T and then split the matrices so obtained into the partial channel model of equations (37) and (36). Therefore the (T/2)-spaced N ×2P channel matrix Hq(k) is expressed as the ``sum'' (å) of the two partial channel matrices Hq(0)(k) and Hq(1)(k):

Hq(k) = Hq(0)(k)åHq(1)(k)
(40)
The ``sum'' (å) is defined such that the channel matrix Hq(k) is given by
Hq(k) = é
ê
ê
ê
ë
hq,0,0(0)(k)
hq,0,0(1)(k)
¼
hq,0,P-1(1)(k)
:
···
:
hq,N-1,0(0)(k)
hq,N-1,0(1)(k)
¼
hq,N-1,P-1(1)(k)
ù
ú
ú
ú
û
(41)
Likewise, we define
gq,l = gq,l(0)ågq,l(1)
(42)
With these definitions in mind,the (T/2)-spaced discrete array output vector can be written as
x( ~
k
 
) = Q-1
å
q=0 
Hq(k)sq( ~
k
 
) + n( ~
k
 
)
(43)
where the 2P ×1 data vector sq([k\tilde]) is defined differently depending on whether [k\tilde] is even or odd, i.e. we have
sq( ~
k
 
) = é
ê
ê
ê
ê
ê
ê
ë
bq(k + 1 - Dt - d)
0
:
bq(k + P - Dt - d)
0
ù
ú
ú
ú
ú
ú
ú
û
    for     ~
k
 
    even
(44)
or
sq( ~
k
 
) = é
ê
ê
ê
ê
ê
ê
ë
0
bq(k + 1 - Dt - d)
:
0
bq(k + P - Dt - d)
ù
ú
ú
ú
ú
ú
ú
û
    for     ~
k
 
    odd,
(45)
The N ×2P channel matrix Hq(k) is given by
Hq(k) = Lq
å
l=1 
a(qq,l)gq,lTCq,l(k)
(46)
with the diagonal fading coefficient matrix Cq,l(k) defined as
Cq,l(k) = diag{cq,l(1)(k), cq,l(0)(k), ¼, cq,l(0)(k)}
(47)
Note that the matrices Hq(k) are only updated every k-th time-step. However, for all practical purposes the diagonal elements of Cq,l(k) are approximately the same, and hence Cq,l(k) » cq,l([k\tilde])I is valid. Using this approximation, the simulator now has to update one fading coefficient cq,l([k\tilde]) every [k\tilde]-th time-step, instead of updating the two fading coefficients cq,l(0)(k) and cq,l(1)(k) every k-th time-step. Thus, equations (43) and (46) can now be reduced to the convenient T/2-spaced channel model
x( ~
k
 
)
=
Q-1
å
q=0 
Hq( ~
k
 
)sq( ~
k
 
) + n( ~
k
 
)
Hq( ~
k
 
)
=
Lq
å
l=1 
cq,l( ~
k
 
)a(qq,l)gq,lT
(48)

2.2  Statistical Characterization of the Vector Channel

The fading coefficients cq,l([k\tilde]) of equation (48) can be used to model the time selective fading inherent in the channel. There are two types of time selective fading:

  • Fast (Rayleigh) fading - caused by mobile motion
  • Slow (log-normal) fading - caused by shadowing
Slow signal variations which are often modelled as lognormal fading determine the outage probability and thus strongly affects the choice of transmission protocols and, to some lesser extend the error control coding scheme. From the viewpoint of receiver design, however, it is sufficient to focus on the fast signal fading. Both, diffuse scattering, and specular reflections or LOS connections contribute to the fast signal fading. How to generate the fading coefficients in each of the two cases is discussed below:

In the case of diffuse scattering, cq,l([k\tilde]) is a Rayleigh fading complex-Gaussian random process [21]. Each process cq,l([k\tilde]) is simulated by applying complex white Gaussian noise to an appropriate digital filter with z-Transform Tq,l(z). It should be designed such that it is a unit-energy filter, matches the shape of the desired Doppler spectrum, and introduces no Doppler shift, but some Doppler spread much larger than could occur in practice [21]. The Doppler spread can then be scaled down by linearly interpolating the output of the digital filter by a factor Iq,l = sq,0/sDq,l, where sq,0 is the fixed Doppler spread introduced by the filter, and sDq,l is the desired Doppler spread of the l-th multipath. Because a unit-energy filter is used, the power of the multipath component is then equal to the power of the complex white Gaussian noise process rl. Finally, the desired Doppler shift yq,l = Lq,lTs is achieved by multiplying with a rotating phasor exp(jLq,l[k\tilde]). Under the common assumption of isotropic scattering from all directions the Doppler spectrum becomes the so-called ``classical'' or ``Jakes'' spectrum [33]. The ``classical'' Doppler spectrum is ``U''-shaped and has sharp cut-off frequencies at the Doppler frequency yq,l [33]. Therefore in this special case yq,l = sDq,l. The spectrum is given by
Tq,l(f) = 1

ö

1 - (f/yq,l)2
(49)
The Doppler frequency yq,l is given by
yq,l = Lq,lTs = fcvq
c
(50)
where vq is the velocity of the q-th user. A widely used digital filter which approximates the ``classical'' Doppler spectrum is an eighth-order IIR filter with co-efficients given in Table 1.

zeroes
0.99015456438065 ±j0.04500919952989
0.98048448562622 ±j0.01875760592520
0.99652880430222 ±j0.05493839457631
0.99827980995178 ±j0.05666938796639
poles
0.99835836887360 ±j0.05727641656995
0.99744373559952 ±j0.07145611196756
0.99440407752991 ±j0.10564350336790
0.96530824899673 ±j0.26111298799515
Table 1: Set of co-efficients for the digital filter approximating the ``classical'' Doppler spectrum

The poles of this filter are very close to the unit circle in the z-plane and stability problems might occur due to finite wordlength effects. In such a case stability can be improved by using a IIR filter cascade form realization instead of the IIR direct form. This filter has a normalized Doppler frequency of

2pyq,lTs = 0.05686
(51)
Take, for example, a carrier frequency fc = 900 MHz and assume the sampling rate for the GSM system, i.e. 1/Ts = 2/T = 541.666 kHz. Then the normalized Doppler frequency corresponds to a fixed velocity of
v = 0.05686c
2pfcTs
= 1633.94 m/s = 5882.2 km/h
(52)
If the q-th mobile user moves, say, with a speed of vq = 13.89 m/s = 50.0 km/h then this results in an interpolation ratio of Iq,l = 5882.2/50.0 » 117. Figure 9 shows an example of the interpolation process for a ratio of Iq,l = 4. Both time indeces k and [k\tilde] are shown along the horizontal axis of the graph. The values at [k\tilde] = 0, 4, 8 are the outputs of the digital filter Tq,l(z) at these time instants. The other values are obtained by linear interpolation. For simplicity only the real part of the fading coefficients cq,l([k\tilde]) is plotted. However, the interpolation process of the imaginary parts is carried out in an entirely analogous fashion.

Figure
Figure 9: Interpolation of the real part of the fading envelopes

Note, that the interpolation process described above can be used to create the fading coefficients for both the partial T-spaced channel model of equations (36) and (37) or the T/2-spaced channel model of equation (48) via the relation

cq,l( ~
k
 
) = ì
ï
ï
í
ï
ï
î
cq,l(0)(k)    for     ~
k
 
    even
cq,l(1)(k)     for     ~
k
 
    odd
(53)


The fading processes of the LOS ray or specularly reflected rays can be modeled directly by rotating phasors cq,l([k\tilde]) = cq,lexp(jLq,l[k\tilde]) with constant amplitudes cq,l and Doppler shifts yq,l = Lq,lTs. It is seen from the above discussion that the generation of the fading envelopes is a type of quadrature amplitude modulation in the case of diffuse scattering, whereas it is a type of phase modulation otherwise.

In order to be able to simulate the channel, knowledge about the power-, direction-of-arrival- (DOA-), and the delay profiles is necessary. The power- and delay-profiles of many channels have been widely investigated and are well known with sufficient accuracy for a wide range of different channels. However, researchers have only recently started to investigate the DOAs of multipath rays which determine the amount of space selectivity of the channel. One good recent work is [20], where the author determines the channel parameters for a densely built-up urban environment, and also fits distribution functions to the powers, delays and DOAs. Table 2, which was extracted from [24], gives typical delay and DOA spreads for different environments at a carrier frequency of 1.8 MHz.

Environment Delay Spread DOA Spread
Rural 0.5 ms 1°
Urban 5 ms 20°
Hilly 20 ms 30°
Microcell 0.3 ms 120°
Picocell 0.1 ms 360°
Table 2: Typical delay and DOA spreads



The structure of the vector channel simulator for Q users each with a total of Lq LOS and diffuse multipaths with distinct delays tq,l is visualized in Figures 10 and 11. Figure 10 shows the generation of the channel matrix Hq. The Ts-spaced fading envelope processes cq,l([k\tilde]) are used for weighting the elements of the N ×2P matrix a(qq,l)gq,lT. All Lq such weighted matrices are then superponed to form the Ts-spaced channel matrix Hq.

The data path of the vector channel simulator can be seen in Figure 11. Firstly, the channel symbols bq(k) are converted to the Ts-spaced signal sq([k\tilde]). Then each signal sq([k\tilde]) is convolved in N different linear time-variant (LTV) FIR filter. The coefficients of the first of the N FIR filters are given by the first row of the channel matrix Hq([k\tilde]), the coefficients of the second FIR filter by the second row of the same matrix, and so on. If present, a global oscillator frequency shift Lq is simulated for each user q by multiplying the resulting output signals with the rotating phasor exp(jLq[k\tilde]). For simplicity this phasor is left out of the derivations given above. Afterwards, the signals of all Q users are summed for each of the N channels, and, finally, the N ×1 noise process vector n([k\tilde]) is added to yield the received signal vector x([k\tilde]).

Figure
Figure 10: Generating the channel matrix Hq

Figure
Figure 11: Data path of the vector channel simulator

3  Uplink Processing

The uplink is the communication link from the mobile user to the base-station. It is assumed, that a 'smart' antenna is only employed at the base station and not the mobile radio. The mobile radios transmit using omni-directional antennas. Therefore, it is the base-stations task to employ spatially selective reception, in order to separate the desired signals from interferences. This task is called uplink processing. When receiving communication signals at an antenna array, it is neccessary to differentiate between two different scenarios

Single input-single output (SISO)
scenarios, in which only one user is allocated to each carrier frequency. The objective of uplink processing could be, for example, spatio-temporal equalization of the channel or direction of arrival (DOA) estimation ( Spatio-temporal equalization directly separates the desired signal from interferences, whereas DOA estimation subsequently uses the DOAs in a beamformer in order to separate the desired signal).
A multi input-single/multiple output (MIxO)
scenarios, in which several users are allocated to each frequency. The objective of uplink processing in this scenario is, for example, to separate the signals and equalize the vector channel, or to simultaneously estimate the DOAs of several signals for subsequent use in a beamformer.
The above mentioned objectives in either a SISO or MIxO scenario can be achieved via signal processing. The appropriate signal processing methods can be grouped in three main categories
Spatial Structure Methods
that exploit the steering vector information to achieve the signal processing objective
Temporal Structure Methods
that exploit temporal structure information of the transmitted signals, such as constant modulus (CM), finite alphabet (FA) or cyclostationarity to achieve the signal processing objective
Training Signal Mehods
that use a known training signal or code to achieve the signal processing objective
The methods available in each of the three main categories mentioned above can again be grouped into
Conventional methods
that only use the received data to achieve the desired signal processing objective. The structure of the channel/data model is not used explicitly.
Parametric methods
that use both, the received data and knowledge of the channel/data model to achieve the desired signal processing objective.
Because parametric methods exploit the knowlege of the underlying model, their performance depends strongly on the validity of the model. However, if the model is valid, then the parametric methods easily outperform the conventional methods. Most modern signal processing methods are parametric as are most of the signal processing methods presented in this report. Exceptions are the methods titled 'Conventional Methods'.

This Section of the report gives an overview about the different signal processing methods that are used in the Uplink. Current research trends are indicated and papers are cited in which the different signal processing methods are used. Note, that the LTV filters in the patternforming network of the 'smart' antenna are assumed to be single complex co-efficients, if only signal seperation without temporal equalization is considered. Otherwise the LTV filters are assumed to be finite impulse response (FIR) filters with the number of taps being equal to the channel memory P. This is standard practice if narrowband signals are used with 'smart' antennas [22]. Furthermore, note that in this Section a T-spaced channel model is used. This implies that the received signal does not represent a sufficient statistic for some of the necessary synchronisation tasks, because the signal envelopes are not striclty bandlimited to B = 1/T. However, for simplicity, this assumption is made for the algorithms presented here.

3.1  Spatial Structure Methods

As mentioned before, spatial structure methods exploit the information in the steering vectora(q). The spatial structure is used to estimate the direction of arrivals (DOAs) of the signals impinging on the sensor array. The estimated direction of arrivals are then used to determine the weights in the patternforming network. This is called beamforming. Spatial structure methods only exploit spatial structure and training signals and the temporal structure of the signals is ignored. In the following an overview will be given about the three main spatial structure methods, namely conventional beamforming methods, maximum likelihood estimation and the so-called subspace-based methods.

For simplicity, the vector channel model used here (and everywhere in the array processing literature for spatial structure methods) is a spatial-only vector channel

x(t) = A(q )s(t) + n(t),
(54)
where the N ×L steering matrix A(q ) is defined as
A(q ) = [a(q1), ¼, a(qL)]
(55)
Note, that knowledge about the number of impinging multipath signals L is assumed in the models that make use of spatial structure.

3.1.1  Conventional Methods

The first attempt to automatically localize signal sources using antenna arrays was through beamforming methods. The idea is to ßteer" the array in one direction at a time and measure the output power. The steering locations which result in maximum power yield the DOA estimates. The output of the 'smart' antenna is given by

y(t) = wHx(t)
(56)
Given M samples {y(1), y(2), ¼, y(M)}, the output power is given by
P(w) = 1
M
M
å
k=1 
|y(k)|2 = 1
M
M
å
k=1 
wHx(k)xH(k)w = wH ^
R
 

xx 
w
(57)
where [^(R)]xx is an estimate of the covariance matrix. Different beamforming approaches correspond to different choices of the weighting vector w. A simple and widely used approach is the Mean Square Error (MSE) performance measure, which is formulated as

min
w 
E{(d(k) - wHx(k))2}
(58)
where d(k) is the desired response of the 'smart' antenna output. The solution to the above stated minimization problem is the well known Wiener-Hopf solution and is given by [22]
w = Rxx-1rxd
(59)
The crosscorrelation vector rxd is given by
rxd = E{ x(k)·d*(k)} = a(q)E{s(k)d*(k)}
(60)
Of course, ideally the desired signal is given by s(k). Setting d(k) = s(k), the above equation becomes
rxd = a(q)E{s(k)s*(k)} = aa(q)
(61)
The constant a is the power of the transmitted signal s(k), but basically it justs scales the output of the 'smart' antenna. Setting a = 1, the solution for the patterforming network weights is simply
w = Rxx-1a(q)
(62)
Inserting equation (62) into equation (57) and using the autocorrelation matrix estimate [^(R)]xx, the classical spatial spectrum is obtained as
P(q) = aH(q) ^
R
 

xx 
a(q)
(63)
Other choices for the weight vector w are possible and are based on other performance measure such as

  • Signal to Noise Ratio (SNR) performance measure
  • Maximum Noise Variance (MNV) performance measure
However, for conventional methods, the solutions are all basically the same. For a more detailed review of such beamforming methods refer to [31]. The conventional methods obtain P(q) as the spatial analogue of the classical periodogram in temporal time series analysis. The classical periodogram suffers from the fact, that its standard deviation is approximately as large as the quantity to be estimated. Therefore in general it can be said, that the resolution of these methods is poor, because it is simply an extension of classical Fourier-based spectral analysis to sensor array data.

3.1.2  Maximum Likelihood (ML) Method

The essentials of maximum likelihood (ML) estimation are assumed to be known by the reader. For an excellent introduction to ML estimation refer to [13]. Given M samples {x(1), x(2), ¼, x(M)}, the likelihood function for the vector channel model assumed in Subsection 3.1 is given by [15]

p(x(k); q , s(k), s2) = M

k=1 
(ps2)-Nexp æ
ç
è
- 1
s2
||x(k)-As(k)||2 ö
÷
ø
(64)
where q is the directional information, s(k) is the transmitted signal and s2 is the variance of the noise process. The ML estimates of these unknowns are calculated as the maximising arguments of p(x(k); q , s(k),s2), the rationale being that these values make the probability of the observations as large as possible. Alternatively it is possible to minimize the negative log-likelihood function which is given by
-ln(p(x(k); q , s(k), s2)) = Nlns2+ 1
s2M
M
å
k=1 
||x(k) - As(k)||2
(65)
Obviously, the estimate for the signal waveform is
^
s
 
(k) = A+x(k)
(66)
where A+ is the pseudo-inverse of A. To calculate [^(s)]2, it is necessary to take the derivative of the log-likelihood function and set the result equal to zero, i.e.
^
s
 
2
 
= 1
NM
M
å
k=1 
||x(k) - As(k)||2
(67)
If [^(s)](k) is inserted in the above equation, then [^(s)]2 becomes
^
s
 
2
 
= 1
NM
M
å
k=1 
||x(k) - AA+x(k)||2 = 1
N
trì
í
î
PA^ ^
R
 

xx 
ü
ý
þ
,
(68)
because the orthogonal projection matrix PA^ = I - AA+ is idempotent and hermitian. Inserting equations (66) and (68) into equation (65), the following non-linear optimization problem is obtained as an estimator for q :
^
q
 
= arg
min
q  
  trì
í
î
PA^ ^
R
 

xx 
ü
ý
þ
(69)
Maximum likelihood estimation is a parametric method and hence its resolution is not limited as is the case for the conventional beamformer. However, a multidimensional search is required to find the estimates, resulting in a high computational complexity. The ML estimator presented here can be classified as a deterministic ML estimator, because the impinging multipath rays of both, the desired signal and the interferers, are modelled deterministically. It is also possible to model the interfering sources as coloured Gaussian noise. In Subsection 3.2 such a stochastic ML estimator is introduced for training signals. According to [15], the stochastic ML estimator has been shown to have a better large sample accuracy than the corresponding deterministic ML estimates. Furthermore, for Gaussian signal sources, the stochastic ML estimator attains the Cramer-Rao lower bound (CRB), since all unknowns in the stochastic model are estimated consistently. For the deterministic model, the number of signal waveform parameters grows as the number of samples increases, implying that they cannot be estimated consistently.

3.1.3  Subspace-Based Methods

All the subspace based methods are based on the eigenvector decomposition of the covariance matrix

Rxx = E(x(t)xH(t))
(70)
Dropping the index of the steering matrixA(q ), we get for the covariance matrix
Rxx = AE(s(t)sH(t))A +E(n(t)nH(t))
(71)
Denote the covariance matrix of s(t) with Rss. Assuming the noise is i.i.d. Gaussian, the covariance matrix of n(t) is s2I. Therefore Rxx can now be written as
Rxx = ARssAH + s2I
(72)
Because Rxx is a positive definite, hermitian matrix, we can use singular value decomposition (SVD) to get
Rxx = ULUH
(73)
with U unitary and L = diag{l1,l2, ¼, lN} a diagonal matrix of real eigenvalues ordered such that l1 ³ l2 ³ ¼ ³ lN > 0. If a vector x is orthogonal to AH, then it is an eigenvector of Rxx with eigenvalue s2, because then
Rxxx = ARssAHx +s2x = s2x
(74)
The eigenvector of Rxx with eigenvalue s2 lies in N[AH], the nullspace of AH. If and only if L < N, then
N[AH] = Â[Q],     QîCN ×(N - L),     rank(Q) = (N - L),
(75)
where Â[Q] is the range of Q. It is concluded, that the smallest (N - L) eigenvalues are
lL+1 = lL+2 = ¼ = lN = s2
(76)
Therefore it is possible to partition the eigenvectors into noise eigenvectors and signal eigenvectors and the covariance matrix Rxx can be written as
Rxx = UsLsUsH + UnLnUnH
(77)
Furthermore, the range of Q is the orthogonal complement to the range of A, because
Â[Q] = N[AH] = ^Â[A]
(78)
and thus we have
Â[Us]
=
Â[A]
(79)
Â[Un]
=
^Â[AH]
(80)
Â[Us] is called the signal subspace, and Â[Un] is called the noise subspace. The projection operators onto these signal and noise subspaces are defined as
PA
=
AA+ = Us(UsHUs)-1UsH = UsUsH
(81)
PA^
=
I - AA+ = Un(UnHUn)-1UnH = UnUnH
(82)
A+ is the pseudo-inverse of A.

Multiple SIgnal Classification (MUSIC) Algorithm

The simplest of the algorithms that are based on the above stated subspace decomposition is the MUSIC (Multiple SIgnal Classification) algorithm: Assume L signals impinging on the sensor array. Now a(q) is projected onto the noise subspace R[Un]. The projection gives the vector

z = PA^a(q)
(83)
The magnitude squared of z can be written as
f(q) = zHz = aH(q)PA^HPA^a(q) = aH(q)UnUnHa(q)
(84)
Obviously, f(q) = 0, if q î {q1,q2, ¼, qL}. Therefore, we search the array manifold, i.e. f(q) is evaluated for all q and we select as the DOA estimates the points which satisfy f(q) = 0. Note, that for coherent or correlated signals the signal autocorrelation matrix Rss is not full rank. Therefore equation (79) has to be replaced with the following relationship
Â[Us] îÂ[A]
(85)
The above constitutes the major drawback of the MUSIC algorithm is, i.e. it breaks down for correlated or coherent signals.

Subspace based approximations of ML estimators

There exist a number of ML-like algorithms that are also based on the subspace decomposition described beforehand. The most important, perhaps, is the Subspace Fitting (SSF) approach. This approach does not use the orthogonality between noise subspace and steering vector directly. Instead it tries to fit an estimate of the signal subspace to the parameters that are of interest using a ML-like minimization. Therefore the SSF approach does not break down completely for coherent signals as MUSIC does. Coherent and strongly correlated signals still pose a problem for such methods, however [1,2]. The MUSIC algorithm and the Weighted Subspace Fitting (WSF) approach are compared in [14] by simulation for a flat fading scenario. It is found that the MUSIC algorithm performs almost as well as the WSF algorithm, and at the same time is computationally much more attractive. Another problem with subspace based algorithms is that they require knowledge about the number of impinging signals, so that the noise and signal subspaces can be estimated [15]. If the sensor array is uniform and linear (a ULA), then some special forms of the SSF approach are the ESPRIT algorithm, the root-MUSIC algorithm, 4×S algorithm [2], VIASS algorithm [8], etc. The 4×S and the VIASS merit special mentioning, because they only use 1 single snapshot of x(k) to estimate the DOAs of the impinging wavefronts. Therefore, these algorithms are suited for coherent multipath signals, too.

3.1.4  Receiver for Spatial Structure Methods

A possible receiver structure for spatial structure methods is depicted below in Figure 12 [3]. The block called 'DOA estimation' uses one or several snapshots of x(k) and knowledge of the steering vectora(q) to estimate the DOAs of all impinging wavefronts, as described previously. The complex envelopes of the impinging multipaths are then estimated by the block called 'signal waveform estimation'. This block is a beamformer that selects the weights of the patternforming network accordingly. The complex envelopes transmitted by the same mobile radio have to be optimally combined [3] by the block called 'select signals'. The difficulty here is to decide which multipaths have to be combined to a signal corresponding to one source. Finally, to reconstruct the original sequences, some type of sequence estimator is needed (i.e. linear, decision feedback, or maximum likelihood sequence estimation equalization). See [3,4], for example, for a derivation of the maximum likelihood sequence estimator.

Figure
Figure 12: Receiver for Spatial Structure Methods

3.1.5  Discussion of Spatial Structure Methods

Spatial structure methods directly estimate the DOAs of the impinging wavefronts. Once the DOAs are found, the weight vector necessary to separate the wavefronts can be determined via beamforming methods. The available beamforming methods can be grouped into conventional methods and superresolution methods. For conventional beamformers, the resolution is, through Rxx a function of the signal-to-noise ratio (SNR). For superresolution methods, the resolution is independent of the SNR. If the number of signals is smaller than the antenna elements, then superresolution beamforming methods can result in complete interference cancellation [19]. A conventional beamformer, for example, is the Wiener-Hopf solution as given in Subsubsection 3.1.1. A popular superresolution beamformer is, for example, the maximum likelihood (ML) beamformer, which is given by [^(s)](k) = A+x(k) as derived in Subsubsection 3.1.2. After the wavefronts are separated using their known DOAs, they have to be combined corresponding to the source of the wavefronts. The number of impinging multipath signals has to be estimated. Another difficulty lies in identifying which wavefronts belong to which signal source. This might be especially difficult, when angular spread is large. Spatial structure methods exploit the information contained in the steering vectora(q) but ignore training signals and the temporal structure of the signals. Therefore, spatial structure methods are only capable of estimating the signal waveforms but not the vector channel. Hence sequence estimation has to follow spatial structure methods in a receiver. Below follows a summary of important points that have to be kept in mind when spatial structure methods are to be used.

Coherent multipath signals

For coherent multipath signals, the subspace based methods do not work properly, because the signal subspace and the subspace spanned by the steering matrix are not equivalent in that case. There exists a technique called spatial smoothing [15] that is able to mitigate this disadvantage. Spatial smoothing means that the array is split into identical subarrays, the covariances of which are averaged. The rank of the averaged covariance matrix can be shown to increase by 1 with probability 1 for each additional subarray [15]. The drawback of spatial smoothing is that the effective aperture of the array is reduced, since the subarrays are smaller than the original array. The other possibility is to use single snapshot methods or the computationally more complex ML estimation method, both of which do not have any problems with coherent signals. Coherent multipath signals do not pose serious problems, when the angular spread is small, i.e. the multipath source is a cluster of scatteres located closely around the mobile. Then the so-called point source model is valid and hence only one DOA has to be estimated for each cluster of scatteres. According to [24], the point source model is valid in flat rural environments, whereas in many urban or hilly rural areas it is not.

Number of DOAs that can be estimated

The number of DOAs that can be estimated is smaller than the number of antenna elements. This is a major disadvantage in environments suffering from large angle spread. If large angle spread is present, then the point source model is not valid and inevitably many different DOAs correspond to a single signal source. In that case spatial structure methods require more antenna elements than the total number of impinging signals and their multipaths. This may not be feasible in many applications. The number of base station antenna elements is to be kept down to a minimum for cost reasons.

Array calibration

Throughout this section, the antenna elements of the antenna array are assumed to be identical and without any mutual coupling between them. In reality, however, the antenna elements are not be identical and they are mutually coupled. spatial structure methods explicitly exploit the knowledge of the steering vectora(q). Therefore, mutual coupling and difference of antenna elements have to be included into the steering vector, if spatial structure methods are to work properly. Because usually this data is not known beforehand, it has to be estimated very accurately. This is called array calibration.

3.2  Training Signal Methods

In many mobile communication systems such as GSM and IS-54, explicit training signals are inserted into the data bursts. These training signals can be used to estimate the beamformer or the channel for each transmitted signal. There are several different approaches that may be taken when training signals are available. Conventional methods, for example, use the training signal and the received signal vector x(k) to determine a beamformer that separates the impinging signals. Maximum likelihood estimation can be used to jointly estimate DOAs and the channels, an interesting special case being the type of single snapshot algorithm described below. Maximum likelihood estimation can also be used to estimate the channels alone, ignoring any knowledge of the steering vectora(q).

3.2.1  Conventional Methods

If a desired response, d(t), is given for the output of the 'smart' antenna, it can be used to calculate the weight vector w. All introductory books [22,11] give a thourough discussion of these methods. Similarly to Subsection 3.1.1 the MSE (Wiener-Hopf) solution for the weight vector w is stated here:

w = Rxx-1rxd
(86)
The weight vector w can then be used to separate the transmitted signal from interferences. Again, other choices for the weight vector w are possible and are based on performance measure such as the SNR or the MNV. However, the conventional beamformer does not take into account the impulse response of the channel and therefore is not appropriate as a stand-alone for most mobile communication problems. Especially in the case of fading channels, the channel must be estimated and its effects reversed.

3.2.2  Maximum Likelihood Method Ignoring Spatial Structure

It is usually assumed [3] that the training signals from the interfering mobiles are temporally white. The interferers can then be modelled not deterministically, but stochastically, as noise which is spatially coloured. No assumption is made about the number of interferers or their channels. Thus the T-spaced vector channel model used here reduces to

x(k) = H0b0(k) + n(k)
(87)
n(k) ~ CN(0, Q)
(88)
During the training period, the data bs(k) is known. Dropping the subscripts for convenience, the estimation problem is then to jointly estimate H and Q. Using, as in Subsubsection 3.1.2, the negative log-likelihood function, the estimates are given by
é
ë
^
H
 
, ^
Q
 
ù
û
= arg 
min
H, Q 
 ln æ
è
det
(Q) ö
ø
+ 1
M
M
å
k=1 
(x(k) - Hb(k))HQ-1(x(k) - Hb(k))
(89)
Using the trace property xHy = tr(yxH), the above minimization problem can be rewritten as
é
ë
^
H
 
, ^
Q
 
ù
û
=
arg 
min
H, Q 
 ln æ
è
det
(Q) ö
ø
+tr æ
è
Q-1 ^
R
 

xx 
ö
ø
- tr æ
è
Q-1H ^
R
 

bx 
ö
ø
- tr æ
è
HHQ-1 ^
R
 

xb 
ö
ø
+tr æ
è
HHQ-1H ^
R
 

bb 
ö
ø
(90)
Differentiating this with respect to H and Q and setting both equations to zero, the following two estimators are derived
^
H
 
=
^
R
 

xb 
^
R
 
-1
bb 
(91)
^
Q
 
=
^
R
 

xx 
- ^
R
 

xb 
^
R
 
-1
bb 
^
R
 
H
xy 
(92)
The estimates [^(H)] and [^(Q)] can then be used in a sequence estimator. See [3,4], for example, for a derivation of the maximum likelihood sequence estimator.

3.2.3  Maximum Likelihood Method Using Spatial Structure

If spatial structure is incorporated into the maximum likelihood method described in the previous Subsubsection, then the vector channel model has to be adapted to the following form in order to incorporate the steering matrix A(q )

x(k) = A(q )H0b0(k) + n(k)
(93)
n(k) ~ CN(0, Q)
(94)
Note, that then A is a N ×L matrix and Hs is a L ×P matrix. As for spatial structure methods, the number of impinging multipath signals L is assumed to be known. Dropping the subscripts for convenience, the ML minimization problem becomes
é
ë
^
q
 
, ^
H
 
, ^
Q
 
ù
û
= arg 
min
q , H, Q 
  æ
ç
è
ln æ
è
det
(Q) ö
ø
+
1
M
M
å
k=1 
(x(k) - A(q )Hb(k))Q-1(x(k) - A(q )Hb(k)) ö
÷
ø
(95)
See, for example, [32] for algebraeic solutions to the above minimization problem. The estimates [^(q )], [^(H)] and [^(Q)] can then be used in a sequence estimator, where A([^(q )])[^(H)] is used as the channel estimate. Furthermore, some important observations were made about this type of algorithm:

  • Due to the use of training signals, the DOAs that are estimated belong to the multipaths of the desired signals. The DOAs of the interferers are not estimated.
  • Knowledge about the number of multipaths is needed, so that A([^(q )])[^(H)] can model the channel adequately.
  • This type of approach can estimate more DOAs than the number of antenna elements used. This is due to the fact that for each training sequence belonging to a desired signal, N-1 multipath directions can be estimated.

Single Snapshot Algorithm using Spatial Structure

In [6], it is proposed to combine single snapshot algorithms [8,2,1], with the use of training signals. The approach taken is basically the same as for the ML estimator described above. However, this approach is computationally much more attractive than direct ML estimation. Further investigation concerning the capabilities of this type of uplink processing is needed.

3.2.4  Receivers for Training Signal Methods

Depending on whether knowledge of spatial structure was used to estimate the parameters, the resulting receiver structure differs slightly. Figure 13 shows the receiver for the training signal methods that ignore spatial structure. It is seen, that the resulting receiver structure is fairly simple. After demodulation, the received signal vector x(k) is used to estimate the unknown parameters [^(H)] and [^(Q)]. These two parameters can be used subsequently in a sequence estimator. Figure 14 shows the receiver for the training signal methods that use spatial structure. The receiver has basically the same structure. There are two differences:

  • The DOAs of the impinging wavefronts are estimated as well.
  • The estimated channel is assumed to be A([^(q )])[^(H)], instead of [^(H)] only.

Figure
Figure 13: Receiver for Training Signal Methods Ignoring Spatial Structure

Figure
Figure 14: Receiver for Training Signal Methods Using Spatial Structure

3.2.5  Discussion of Training Signal Methods

If training signals are transmitted by the mobile stations, additional information apart from spatial structure is available. It was seen that there are two types of training signal ML methods

ML method using spatial structure
This method estimates both, the vector channel and the DOAs. The combination A([^(q )])[^(H)] is used in the sequence estimator as the channel estimate.
ML method ignoring spatial structure
This method only estimates the vector channel [^(H)], which is used in the sequence estimator as the channel estimate
Both methods can handle scenarios that have a larger number of impinging wavefronts than antenna elements. The ML method that ignores spatial structure does not have any limitation in terms of the number of impinging signals. The ML method that uses spatial structure can estimate (N-1) DOAs for each training sequence. As is the case for spatial structure methods, computationally attractive versions of the ML estimator exist for uniform linear arrays (ULA) [24]. One such version is the combination of single snapshot algorithms with training signals as described before. Note, that the mentioned ML methods model other mobile users as coloured noise. Another possibility would be to model other mobile users and interferes as deterministic sources (see also the ML estimator presented in Section 3.1). This approach has not been found in the literature, possibly because it would be a computationally very complex approach. Furhter investigation is necessary to determine the feasibility of a purely deterministic ML estimator for training signals. In general it can be said, that using training sequences is a robust approach to spatio-temporal processing, because it utilizes more information to estimate the unknown parameters than spatial structure methods. However, the training sequences consume spectrum resource. In GSM, for example, 20% of the bits are dedicated to training. Below follows a summary of important points that should be kept in mind when training signal methods are used.

Synchronization

The training signal methods using ML estimators can have problems related to frame synchronization, and symbol and carrier recovery. Prior synchronization is necessary, if the training sequence is to be exploited. The whole subject of synchronization and training signal methods has not been investigated extensively, yet, and further research is necessary to assess the feasibility of training signal methods in multi-user environments.

Choice of training signals

In a MIMO scenario, training sequences have to be assigned to each user. The multiple training sequences should be designed to have low cross correlation properties (i.e. orthogonal training sequences) so as to minimize cross coupling in the vector channel estimate. It is noticed, that orthogonal training sequences sound very much like CDMA in which not the training signals are orthogonal but the each user transmits burst that use an orthogonal code. The relationship between CDMA systems and SDMA systems using orthogonal training sequences and performance and feasibility comparisons of both approaches is an interesting area for future research.

Substituting Training Signal with Blindly Estimated Copy

Instead of using a training signal, it is also possible to use a blindly estimated copy of the signal. temporal structure methods, such as the constant modulus algorithm, can be used, for example, to solve the blind estimation problem. In [30] an approach called Joint Angle and Delay Estimation (JADE) is presented which is basically nothing more than the combination of temporal structure methods and ML estimation using spatial structure. It is therefore closely related to the ML estimator presented in this Subsection. In [3] the ML estimator using spatial structure is compared via simulation to other approaches. However the potential of the approach to estimate more DOAs than number of antenna elements was not examined and remains an area open to future research.

3.3  Temporal Structure Methods

A signal s(t) transmitted by a mobile radio has a rich temporal structure that can be used to improve the estimator performance. There are different types of temporal structure inherent in the transmitted signal, for example

  • Finite Alphabet (FA)
  • Constant Modulus (CM)
  • Cyclostationarity
Temporal structure methods rely on this type of information to separate and equalize desired signals and interferers. Unlike Spatial Structure methods, the information contained in the array manifold is not used.

3.3.1  Finite Alphabet

This approach is based on the finite alphabet (FA) property of the transmitted signals. The FA approach tries to fit the received data to the unknown channel and multi-user data. The T-spaced partial channel model from Subsection 2.1 is used, which is given by

x(0)(k) = Q-1
å
q=0 
Hq(0)(k)bq(k) + n(0)(k)
(96)
If there are M snapshots of the received signal, the channel model can be put as
X = ~
H
 
B + N
(97)
where now X and N are N ×M matrices, [(H)\tilde] is a N ×PQ matrix and B is a PQ ×M matrix. Remember that P is the maximum memory of the channel.

Obviously, the joint ML estimates for the channel and the data matrix are then given by the following minimization problem where the feasible set of the data matrix B is constrained to the known finite alphabet

min
[(H)\tilde], BîFA 
 ||X - ~
H
 
B||F2
(98)
The method of alternating projections [24,18], that makes use of the FA property of the transmitted signals, can be used to solve the above minimization problem. The idea is to alternatingly estimate the channel and the transmitted symbols via least squares. The estimated symbols are then projected onto the finite alphabet, which removes ambiguities in the solution. The FA approach is fairly involved mathematically and the detailed solution is therefore not presented here. For details refer to [18,28]. In [28,29,17] an FA approach for identifying frequency selective FIR channels carrying multiple signals is presented. The approach uses oversampling of the received signal by a factor h. In that case it is possible to use an h vector channel representation where each individual channel 'sees' only a stationary signal. In other words, this approach exploits the cyclostationarity inherent in digitally modulated signals, and therefore is capable of estimating non-minimum phase channels using second order statistics only. The FA approach is computationally fairly complex, because a multidimensional least squares approach has to be employed to find a solution.

3.3.2  Cyclostationary Statistics

A linearly modulated bandpass signal is given by

z(t)
=
s(t)exp(j2pfct)
(99)
s(t)
=
+¥
å
k=-¥ 
b(k)g(t-kT)
(100)
Consider first the bandlimited baseband component s(t). The spectrum of s(t) is shown in Figure 15.

Figure
Figure 15: Spectrum of bandlimited baseband process s(t)

Due to the random process b(k), the spectrum does not contain any spectral lines, i.e. the Fourier co-efficients at all frequencies are zero and s(t) does not contain any first-order periodicities [9]. If the bandwidth of the pulse g(t) is < 1/T, then s(t) can be squared to obtain

s2(t) = +¥å
k=-¥ 
b2(k)g2(t-kT)
(101)
Assume now, that BPSK modulation is used, i.e. b(k) is the asynchronous random telegraph signal that switches between +1 and -1. Then b2(t) = 1 and s2(t) is therefore the periodic signal
s2(t) = +¥
å
k=-¥ 
g2(t-kT)
(102)
This signal will have spectral lines at the harmonics m/T of the symbol rate 1/T. Thus the hidden periodicity is converted into first-order periodicity by using a quadratic transformation. The signal z(t) is the baseband signal s(t) shifted via a sinusoidal carrier to bandpass. As shown in [9], the carrier also introduces hidden periodicity. Squaring z(t) reveals more spectral lines at a = ±2fc as well as at a = 0. The frequencies at which spectral lines are produced by the quadratic transformation are called cycle frequencies. They are denoted with a in order to differentiate with f which denotes spectral frequencies. In [9] the above example is generalized, and the cyclic autocorrelation function and its Fourier transform, the spectral-correlation density (SCD) function, are derived. The SCD is a two-dimensional function of both spectral frequencies and cycle frequencies. Figure 16 shows the magnitude of the SCD function for an amplitude modulated (AM) signal that has hidden periodicity only due to the sinusoidal carrier.

Figure
Figure 16: Magnitude of the SCD function for an AM signal

It is seen, that hidden periodicities are equivalent to spectral redundancy. Futhermore, for a linearly modulated signal, the parameters that determine the cycle frequency are the frequency of the carrier and the symbol rate. Signals that use either a different carrier frequency or different symbol rates, will have different SCDs even though they might occupy the same bandwidth in the bandpass. The spectral redundancy can be exploited, also in the case of spatial filtering. Assuming that several signals with different SCDs impinge on the array, it is possible to construct a linear combiner that nulls out the unwanted signals, using only the signal selectivity contained in the cycle frequencies [9]. One major drawback of the approach is the fact, that different symbol rates and/or different carrier frequencies are needed for seperating multiple superposed signals. In most mobile communication systems however, the symbol rates are equal for all users.

3.3.3  Constant Modulus

The constant modulus (CM) algorithm [11] has its origins in temporal (SISO) equalization techniques. The idea is to penalize deviations of the equalizer output y(k) from a constant modulus. Therefore the CM algorithm minimizes a cost function of the form

J(n) = E[(|y(k)|p -Rp)2]
(103)
with respect to the weight vector w of the equalizer. In the above equation, p is a positive integer, and Rp is a positive real constant defined by
E[|s(k)|2p]
E[|s(k)|p]
(104)
This minimization can be solved with a stochastic gradient algorithm [11]. When using 'smart' antenna systems, the MIMO case becomes interesting. In the MIMO case, the output of the equalizer is given by
y(k) = WH(k)vec(X(k))
(105)
where for M snapshots and Q users W(k) is a NM×Q matrix and X(k) is a N ×M matrix. For more than 1 user, it is convenient to add a term to the cost function, that penalizes the correlation between the equalized outputs [24]. Therefore the CM algorithm is now minimizes the following cost function with respect to the weight matrix W
J(n) = E é
ë
Q
å
j=1 
(|yj(k)|p -Rp)2 ù
û
+ 2Q
å
l,n = 1 ; l ¹ n 
  d = (N+P)-1
å
d = -(N+P)+1 
|rln(d)|2
(106)
The cross-correlation function between users l and n is defined as
rln(d) = E[yl(k)yn*(k-d)]
(107)
and P is the maximum channel memory as defined in 2.1. For the corresponding stochastic gradient algorithm that solves the minimization problem refer to [24,5]. In [5] a CM approach for identifying frequency selective FIR channels carrying multiple signals is presented. It is shown, that depending on the terms included in the cost function, this approach is capable of seperating some or all of the transmitted user signals. The class of CM algorithms have the lowest computational complexity of the temporal structure methods. Furthermore, they do not require prior synchronization as this is achieved by the algorithms themselves. However, CM algorithms make indirect use of higher order statistics, unlike the FA algorithms using oversampling.

3.3.4  Receiver for Temporal Structure Methods

Figure 17 shows the receiver for the temporal structure methods. Temporal structure methods ignore spatial structure when estimating the channel and hence DOAs are not estimated. Furthermore, the interferers are modelled deterministically, so the noise covariance matrix does not have to be estimated. After demodulation, the received signal vector x(k) is used to estimate the unknown channel [^(H)] making use of the transmitted signals known temporal structure. The estimated channel can be used subsequently in a sequence estimator.

Figure
Figure 17: Receiver for Temporal Structure Methods

3.3.5  Discussion of Temporal Structure Methods

Temporal structure methods rely on the temporal structure contained in the transmitted signals. It was seen, that there are different types of temporal structure. Manmade signals transmit symbols from a finite alphabet, they are cyclostationary and usually the symbols have a constant modulus. The methods presented in this Subsection all exploit one of the three mentioned structures. According to [24], the FA property is stronger than the CM property. However, the FA methods are also computationally much more complex. Neither a quantitative nor a qualitative comparison of FA and CM methods to methods exploiting cyclostationary is available. Generally, temporal structure methods do not make explicit use of available knowledge about spatial structure. Not using the knowledge about spatial structure has both positive and negative effects:

  • No calibration of the array is needed (+)
  • Temporal structure methods are not restricted by a maximum number of DOAs that can be estimated (+)
  • The approach works better in scenarios with large angle spread (+)
  • In scenarios with small angle spread, spatial structure methods could outperform temporal structure methods, because available information is not used in order to estimate the unknown parameters (-)
Note, that other temporal structure methods exploiting higher order statistics (4th order or higher) are not mentioned here. These methods are usually computationally quite complex.

Temporal Structure Methods vs Training Signal Methods

In general, temporal structure methods are quite similar to the ML approach ignoring spatial structure as described in the previous Subsection 3.2. The advantages of temporal structure methods over training signal methods are
  • No training sequence is needed, which saves available spectrum
  • The channel is tracked during the whole duration of a burst
  • No synchronization is required
However, temporal structure methods cannot be as robust as training signal methods. MLSE based on a blindly estimated vector channel can suffer from error propagation and cannot be guaranteed to converge.

Methods exploiting cyclostationary statistics

The temporal structure methods that exploit the cyclostationary statistics of the transmitted signals differ in one important aspect from both FA and CM based methods: They only work, if spectral redundancy is created among the multi-user signals by employing either different symbol rates or different carrier frequencies for each signal. Most modern standards, though, are based on using the same symbol rates for each user. Using different carrier frequencies for signals that are superposed in the same frequency spectrum (as in spectrum re-use via angular seperation SDMA) is not a viable alternative either. Therefore the methods exploiting cyclostationary statistics do not seem to be a promising vector channel estimation method for usage in modern wireless communication systems.

4  Downlink Processing

It was seen before, that a 'smart' antenna can be used to separate several co-channel signals arriving from different angles. It is assumed, that a 'smart' antennna is only employed at the base station and not in the mobile radio. Therefore, it is the base station's task to employ spatially selective transmission while the mobile radio's reception system is not spatially selective. The task of spatially selective transmission is called downlink processing. The type of downlink processing used strongly depends on whether the communication system uses frequency division duplex (FDD) or time division duplex (TDD). This is, because in most FDD systems the uplink and downlink fading may be considered independent, whereas in TDD systems the uplink and downlink channels can be considered reciprocal [18]. Hence, in TDD systems uplink channel information may be used to achieve spatially selective transmission. In FDD systems, the uplink channel information cannot be used directly and other types of downlink processing have to be considered.

4.1  Time Division Duplex method

As mentioned beforehand, in TDD systems the uplink and downlink channels can be considered reciprocal [18]. Hence, in TDD systems uplink channel information may be used to achieve spatially selective transmission. In order to model the transmission problem, it is convenient to restate the vector channel as follows. Consider the single-user case. If the basestation transmits z(k), then the user receives

r(k) = h(k) Äz(k) = P
å
l=0 
zH(k-l)h(l)
(108)
where h(k) is the channel that was estimated during uplink processing. The goal is then to find a vector FIR filter g(k) such that
z(k) = g(k)Är*(k) = På
l=0 
r*(k-l)g(l)
(109)
which is simply a set of linear equations to be solved. The model is extended easily to the multi-user case. Mobile user q receives
rq(k) = hq(k) Äz(k)
(110)
Hence, the signal transmitted by the base-station has to fulfil
z(k) = Q-1
å
q=0 
gq(k)Ärq*(k)
(111)
Note, that the conditions under which the above sets of linear equations are solvable are not examined here. For furhter details refer to [18], where this type of downlink processing was used in an experimental set-up. The signal constellation received at the mobile station was a degraded version of the transmitted signal constellation. No quantitative results are stated about the performance of TDD downlink processing and further research is necessary.

4.2  Feedback Signal

While the downlink channel is the reciprocal of the uplink channel for TDD systems, this is not valid for FDD systems. The model from Subsection 4.1 can still be used though, i.e.

rq(k)
=
hq(k) Äz(k)
z(k)
=
Q-1
å
q=0 
gq(k)Ärq*(k)
(112)
The vector channel hq(k) is, unlike in Subsection 4.1, unknown. Feedback signals from the mobile, however, can be used to estimate hq(k). Once the estimate is obtained, it is possible to proceed as described in Subsection 4.1. In [10] such an approach is described, assuming no delay spread in the downlink. If no delay spread is present, the above vector channel model reduces to
rq(k)
=
hqHz(k)
z(k)
=
Q-1
å
q=0 
gqrq*(k)
(113)
In order to estimate hq, the concept of probing is introduced. In the probing mode, the base station transmits probing signals, whose amplitude reponses at the mobiles are measured and fed back to the base. Let R orthogonal probing signals be
p1(t), p2(t), ¼, pR(t)
(114)
Each probing signal is transmitted using its own probing vector vj as a weight in the patternforming network. The response at the qth mobile due to the jth probing signal is then given by
rq(k) = pj(k)bj,q = pj(k)hqHvj
(115)
where bj,q is the amplitude received. Let [B]j,q = bj,q and H = [h1, ¼,hQ]. We then have
B = VHH
(116)
Since the probing signals are orthogonal, each mobile can measure a column of B. Each mobile can then feed back its estimated column of B to the base station. Then the transmitter can estimate H according to
^
H
 
= VH+B
(117)
if the condition R ³ N prevails.

4.3  Downlink Channel Estimation from Uplink Channel Data

In [26], a downlink processing method for FDD is proposed, that uses uplink channel information to form the transmit beam. It is argued, that while there is no correlation between the instantaneous values of the transmit and receive vector channels, there is a strong relationship between the average receive channel vector subspace and the average transmit channel vector subspace. The vector channel model is the same as the flat fading (no delay spread) single user model used in the previous Subsections, i.e. the signal received by the mobile user is

r(k) = hTH(k)w(k)sT(k) + nT(k)
(118)
where the subscript T denotes transmission. In this approach, the transmit weight vector w(k) of the patternforming network is calculated such that it maximizes the average signal power received by the desired mobile, subject to a maximum transmitter power constraint, i.e.
wopt = arg
max
w 
[E{|wHhT(k)sT(t)|2}]
(119)
subject to
||w||2 = PT
(120)
In [26] it is examined how the received power degrades, when the transmit channel hT in equation (119) is substituted with the receive channel. It is shown, that for small frequency shifts between transmit and receive channel and small multipath angle spread, the power degradation is approximately 0.5 dB. The same paper also addresses the problem of Channel Re-use via Spatial Seperation for which a multiuser solution is presented. For details refer to [26]. Note, that this approach only seems to work well under quite strict assumptions, namely the small frequency shift between transmit and receive channel and the small angle spread. If the transmit and receive channel have to be closely spaced in frequency, one could use a more robust TDD approach directly. Furthermore, small angle spreads and flat fading allow to use point-source models and high resolution direction finding. In that case it would be convenient to use direction of arrival beamforming as described in Subsection 4.4 to form the transmit beams. It would be interesting to find out, however, if this type of approach can improve the previously described TDD method. After all, in a TDD system uplink and downlink channel will be strongly correlated but not absolutely identical.

4.4  Direction of Arrival Beamforming

According to [7], the period from uplink reception to downlink transmission has to be shorter than 100 ms. It is assumed that during this period the number of multipaths as well as their DOAs are constant. Hence, if this information was estimated during uplink processing, it can now be used to calculate the weights in the patternforming network for spatially selective transmission. The calculation of the weights can be expressed in a similar fashion as the constrained optimization problem presented in the previous Subsection. Now, however, the DOAs are known and the directional information can be incorporated into the vector channel model. This type of downlink processing is presented, for example, in [7].

5  Conclusion

This report gives an overview about the different methods of uplink and downlink processing available for use in 'smart' antenna systems. The following three short paragraphs summarize and conclude the most important findings.

Uplink processing

The different concepts that are presented in this report are summarized as

  • training signal, spatial structure and temporal structure methods each individually
  • training signal and spatial structure methods simultaneously
  • temporal structure and spatial structure methods simultaneously
Of course, each conceptually different approach can be implemented in many different ways, and by no means have all the different combinations been reported already in the literature. In that sense there is still space for more investigation. Temporal structure and training signal methods simultaneously are not included in the summary list, because really these two methods exclude each other. This is so, because the training signal is known exactly and thus is its temporal structure. A hybrid solution employing these two methods together is still feasible, though, with the temporal structure method tracking the channel while the training signal is absent.

A fundamental question in uplink processing is whether to include the knowledge about spatial structure in the signal processing methods or not, because ignoring spatial structure implies some important advantages

  • no array calibration is needed
  • it works better in scenarios with large angle and delay spreads

Downlink processing

The type of downlink processing used depends strongly on the type of multiplexer used, namely FDD or TDD.

For FDD systems, as GSM, uplink and downlink channel can be considered independent. The most robust approach to estimate the downlink channel seems to be to employ feedback signals from the mobiles. This has the disadvantage that existing mobile communication systems cannot be adapted easily to incorporate feedback signals, because it requires a complete re-design of protocols. Additionaly it has to be kept in mind that feedback signals require spectrum resource. The other feasible approach is DOA beamforming, the disadvantage being that most DOA estimators only work well in scenarios with small delay and angle spread.

For TDD systems, uplink and downlink channels can be considered reciprocal and it is very easy to determine the filter weights of the patternforming network. In [18] a TDD approach was investigated experimentally. It was found, that the signal constellation received at the mobile station was a degraded version of the transmitted signal constellation. This is an indication, that uplink and downlink channel are not completely reciprocal, but strongly correlated. An area for future research could be to investigate, whether maximising the average signal power received by the mobile, as presented in Subsection 4.3 could improve TDD downlink processing.

Protocolling aspects

If a Channel Re-use via Angular Seperation SDMA system is considered, then it is possible to separate multiple signals arriving at the base station as long as their angular separation is bigger than the transmit/receive beamwidths. If the angular seperation becomes too small, then the carrier frequency has to be switched, or CCI occurs. If an uplink processing method using spatial structure is employed, then the DOAs of the received signals are estimated. The DOAs give an indication of the spatial location of the mobile users. If the DOAs corresponding to two users sharing the same frequency band get too close, a handover to a different carrier frequency can be initiated. If, however, an uplink processing method ignoring spatial structure is employed, then the DOAs of the received signals are not estimated. Consequently, a different handover strategy has to be used, such as measuring the received power levels. Another idea to achieve correct handover is to combine macro-diversity with 'smart' antenna systems. In macro-diversity mobile radio, the network architecture uses overlapping cells [23]. If, for example, the network architecture is chosen such that a mobile is always located within the range of two different base stations, then it is also possible to deduce information about the spatial locations. Further research is necessary in order to determine the possibilities of such a communication system.

Subjects not covered at all in this report

Important subjects not covered at all in this report include multi-user sequence estimation, polarization diversity, antenna topology and array calibration methods.

 

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Footnotes:

Smart Antennas and Space-Time Processing1The relative sensitivity of response to signals for a specified frequency from various directions
 

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