# Single and Multiple Scattering in UWB Bicone Arrays

Authors: Raffaele D'Errico; Alain Sibille
Journal: International Journal of Antennas and Propagation
 (2008-01-22)

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## Abstract

An analysis of interactions between radiators in a UWB biconical array, drawing attention to single and multiple scatterings, is carried out. The complementarity between electrical coupling and radiation scattering is argued. The point source approximation is discussed and shown to be insufficient. An approximation of radiation scattering based on angular averaging of the scattering coefficient is proposed. This approach yields a reduction of the problem complexity, which is especially interesting in UWB multiple antenna systems, because of the large bandwidth. Multiple scattering between radiators is shown to be a second-order effect. Finally, a time domain approach is used in order to investigate pulse distortion and quantify the exactness of  the proposed scattering model.


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## Body

## 1. Introduction

It is well known that multiple
antenna system (MAS) offers attractive applications in wireless
communications by means of MIMO techniques that give out
high channel capacities, by taking benefit of antenna diversity at
both ends of the link. Furthermore, ranging applications may take
advantage from MAS, through direction-of-arrival (DOA) estimation in
addition to time of arrival (TOA). The same MAS principles can be
applied to UWB technologies [1], in
order to improve the link robustness or range, and exploit the
high-multipath resolution.Usually, arrays
should be compact in order to be incorporated in radio devices.
Obviously, by reducing the single antenna size or the inter-antennas
distance, it is possible to minimize the overall system dimension.
On the other hand, the mutual coupling also increases for closely
spaced sensors, which can detrimentally modify the system
performances. Thus, radiation, phase, and distortion characteristics
of the single radiator are altered because of the presence of other
antennas. In the case of narrow-band arrays, the theory of active
element factor accounts for mutual coupling effects, including the
possibility of scan blindness [2, 3]. In UWB systems, these phenomena
are more difficult to investigate since the driven and the parasitic
radiators act in a different way according to the frequency.
Moreover, the electrical distance between the antennas depends on
the wavelength. Hence, the frequency dependence is an additional
parameter which introduces a supplementary complexity with respect
to the traditional narrow-band systems.Investigations
on UWB arrays [4–6], often neglect coupling effects.
Time-domain factor [6] frequently
employed for UWB impulse radio system should be a useful mathematic
tool, but it does not take into account that the single element
radiation characteristics are altered when placed within a MAS.Parametric
studies on scan impedance and mutual impedance in UWB arrays have
been presented in literature [7, 8]. Recently, pulse distortion
introduced by interelements interaction has been studied for a
particular UWB array [9], and a
time-domain approach has been used to investigate the pulse
distortion due to the antenna coupling [10].In this
paper, we present a general method, based on the definition of a
scattering coefficient, to predict the effects of the passive
radiators within MAS. The method has the advantage to avoid the use
of a full method of moments (MoM) computation of the array.In Section2, we present an electrical circuit analysis of MAS carried out in
the frequency domain, by distinguishing between coupling
and scattering effects. Point source approximation
is discussed in Section 3. In Section 4, we model the parasitic
antennas as scatters, by means of antenna cross section. We focus on
linear and circular arrays using UWB biconical antenna as simple
radiator. Multiple scattering is discussed and is shown to be a
second-order effect in Section 5. In Section 6, impulse distortion
due to interelements interaction is discussed.


## 2. An Electrical Network Analysis of Multiple Antennas Systems

The understanding of multiple
antenna systems operation is here based on electromagnetic theory
and electrical networks analysis as follows. All voltages,
impedances, and currents are frequency dependent. This dependence
can not be neglected in UWB multiple antennas. Let us consider a MAS
operating in transmission, with several source signals and impedance
loads on the antenna access port, as depicted in Figure1. We indicate with the
index i(i=1,2,…,N) the ith element of the
transmitting array and with r a receiving
antenna.Figure 1
Multiple antennas operating in transmission.Then, at a given frequency, the system
should be described by a(N+1)×(N+1) network
complex matrix, where N is the number of antennas
composing the transmitting MAS. For simplicity of discussion, let us
consider N=2.
(1)(E1−Z0I1E2−Z0I2Vr)=(V1V2−Z0Ir)=(Z11Z12Z13Z21Z22Z23Z31Z32Z33)(I1I2Ir).
Through
the superposition theorem, it is clear that a voltage
Vr at the receiving port can be obtained by summing the
received voltages, when V1 has its nominal value and
V2=0, and when V2 has its nominal value and
V1=0. Thus, in order to obtain Vr, it is
sufficient to have the knowledge of as many complex radiation
patterns as antenna ports, where each pattern requires 1 V applied to
one port, and 0 V (short circuit) on the others. The values of
V1 and V2 are simply obtained by solving the
electrical network equations of the array in transmission, given its
load and source voltage conditions.Furthermore, by neglecting couplingbetween
transmitting and receiving antennas, (1) can be simplified as
(2)(E1−Z0I1E2−Z0I2)=(V1V2)=(Z11Z12Z21Z22)(I1I2).
When the MAS operates in reception, the
theorem of reciprocity (see Figure 2) makes it straightforward to obtain the received
voltage over the loads of the array ports. Since the left and right
situations are reciprocal, the current in the load of array port 1
(a) is equal to the current in the load of the single antenna (b).
The second array port and its load can be considered to be part of a
unique port big antenna. Thus, I1 in the receiving array
can be obtained from the knowledge of the array operating in
transmission. Of course, what applies to port 1 applies to port 2 as
well, so that both I1 and I2 in the receiving
array can be computed from the complex gain patterns in
transmission. The computation of I1 requires to solve the
electrical network equations to obtain the voltages on each antenna
port when array port one is fed with E0. The same work
must be done again when E0 feeds array port 2. It is
important to note here that the conventions for currents signs
should be properly defined.Figure 2
(a) MAS operating in reception and (b)
single antenna in reception.Let us consider a two-antenna system with an
ideal voltage source on port 1, aZ0 load on port 2 (E2=0), and the receiving antenna is load with Zr. It is very easy to obtain
(3)E1=(Z11−Z122(Z22+Z0))I1,Ir=−(Z31Z33+Zr+Z32⋅Z12(Z33+Zr)⁢(Z22+Z0))I1.The last equation means that the
received signal is equal to the sum of two terms: the first is the
received signal due to the feeding of port 1
in the presence of port 2 open circuited; the second is
the received signal due to the current excited in port 2, due to the
finite loadZ0 and to the impedance coupling term
Z12, which causes radiation by antenna 2. Coming back to
the general equations of the three-antennas system, we have
(4)−(Z33+Zr)⋅Ir=Z31⋅I1+Z32⋅I2,
where Z33 is the
receiving antenna self-impedance. What is very interesting in this
expression is that it shows that the received signal is the
superposition of 2 terms: the first is due to antenna 1 excited by a
current I1in the presence
of antenna 2 with port 2 open circuited, and the second is
the equivalent of that by inverting indexes 1 and 2. In other words,
we can very rigorously compute the array effect on an excited
element as follows.(i)
Compute the radiation with one element excited and all others
open circuited.(ii)
Successively compute the radiation of each other element,in
the presence of all others open circuited, due to the
current which feeds it by proximity coupling from the first element.
This current is obtained from the electrical coupling network
equation.(iii)
Compute the sum of all received signals (as many ones as elements in
the array).Usually, an open-circuited element
provides little electromagnetic perturbation on the other elements.
This is due to the fact that a little current develops in such an
element (take as an example, a quarter wavelength monopole, which is
nonresonant when open circuited). Thus, the shadowing effect is very
weak in this case. Neglecting this contribution is the so
calledpoint source approximation [11], since only secondary currents induced
by coupling are responsible for radiation by the neighbouring
elements.However, if the element has a large
size, current can develop along it, producing scattered radiation
which we callscattering or diffraction, even if it
is open circuited. This effect adds to the secondary radiation due
to coupling.This approach precisely defines what
is meant byscattering and what is meant
by coupling.


## 3. Antenna Point Source Approximation

A nonspecific isolated
antenna radiates an electric field that can be written as
follows:(5)E→iso(r,θ,ϕ,f)=−jηe−jβr2πrE→0(θ,ϕ,f)=K(f,r)⋅Z11⁢iso(f)⋅I1⁢iso(f)⋅e→iso(θ,ϕ,f),
where I1 and Z11 represent, respectively, the current and self- impedance
at antenna port, and K is a constant depending on
the free space impedance η, the observation
distance r, and the wave vector in free
space β. The dependence on the
frequency f is now explicit. Let us consider
the N×N linear system [V(f)]=[Z(f)][I(f)],
where N is the number of antennas in the array. By
imposing that, one antenna is fed (Vi=1V), and other ones
are terminated with their loads; it is possible to calculate the
currents developed at the ports of each antenna. Then, the point
source approximation of the radiated field, when
the ith antenna is fed and other ones are loaded
with 50 Ohms in a linear array, is
(6)E→ri(r,θ,φ,f)=K(f,r)⋅Z11⁢iso(f)⋅[Ii(f)+∑n=1n≠iNIn(f)ejΔn(f)]⋅e→iso(θ,ϕ,f),
where Δn(f)=nd⋅(2πf/c)â•r^, d is the inter-antennas distance, â is the antennas alignment direction, and c the free space velocity. By means of this approximation, the only information
that one has to know is the radiation pattern and return loss of the
simple radiator, and the N×N impedance matrix of the
system. No explicit near field information is needed, which has the
advantage of a much simpler way to predict the multiple antenna
system behavior. In a thin dipole MAS, the current distribution on
the open-circuited dipole, when another one is fed, is very small;
thus the scattering effect does not affect the radiation
characteristics in a significant way. Let us consider two thin
dipoles (radius a=0.1 mm) resonant at 3 GHz in a
linear end-fire array configuration (d=50 mm). In
Figure 3, we compare the radiation
pattern, when the second antenna is loaded with 50 Ohms (black
line), calculated by the MoM, and the radiation pattern obtained by
point source approximation which neglects scattering
effects (blue line).Thin dipoles (d=50 mm): simulated expected radiation pattern (black line) and
point source approximation (blue line).Accordingly, the
point source approximation is satisfactory even in case of strong
mutual coupling in classical narrowband systems, where dipole is
commonly assumed as single radiator.Let us employ a
biconical omnidirectional antenna as simple radiator in order to
investigate UWB multiple antennas.This antenna is
derived from a previous work on the design of an UWB antenna for
channel measurements [12]. Antenna
dimensions are 31 mm × 37 mm (diameter × height), and its input bandwidth is
2.75–16 GHz; see Figure 4. The
inter-antennas distance d is taken with respect to
the bicone’s evolution axis.(a) ENSTA bicone antennas and (b) ENSTA
bicone return loss.
(a)(b)In Figure5, we show the
azimuth radiation pattern of two bicone antennas (d=50 mm), one driven and the other one loaded with 50 Ohms, together
with its point source approximation. It is easy to see that the
approximation does not agree with the expected radiation pattern.
This is due to the fact that, when the first antenna is excited, the
current density on the second open-circuited antenna is stronger
than that in the dipole case, and the scattering
effect is predominant over the coupling one. By increasing the
inter-antennas distance, radiator interactions become less
significant, and also scattering.Two ENSTA bicone antennas (d=50 mm): antenna 1 is excited; antenna 2 is 50 Ohms loaded. Simulated
expected radiation pattern (black line) and point source approximation
(blue line).
(a)(b)


## 4. Single Scattering between Radiators

As seen in the previous section, predicting
scattering turns out is essential to calculate the radiation pattern
of multiple antennas, especially in UWB applications as the antenna
size is not negligible with respect to the wavelength. Let us
consider the same two biconical antenna arrays as above, where open
circuit conditions are imposed on the second antenna port. We
consider that the total radiated field is due to the sum of two
contributions: the first is the suitably delayed field of the
excited isolated radiator; the second, which represents the
open-circuited antenna, is the same radiated field suitably delayed
and multiplied by a scattering coefficientSC(f,d,θ,φ).
In a two-antenna array, as depicted in Figure 4, this results for the field radiated by the array in
the expression:
(7)E→(r,θ,ϕ,f,d)=E→iso(r,ϑ,ϕ,f)⋅(ej(dπf/c)sin⁡θcos⁡ϕ+SC(f,d,θ,ϕ)⋅e−j(dπf/c)sin⁡θcos⁡ϕ).At this stage, there is no assumption as long
as SC is a free parameter. The formula is just
shaped in order to envision the open-circuited antenna as a scatter,
which has same geometry and physical characteristics as a single
element. Antennas have two possible modes of scattering: the first
is the scattering due to the fact that the antenna is of a given
shape, material and size; the second because the antennas have been
designed to transmit or receive RF energy at the relevant frequency
[13]. By means of a MoM tool
(Wipl-D), it is possible to compute the antenna radar cross section
and scattered field Es by the open circuited antenna, illuminated by
a field Ei of a plane wave incident from the direction
(θi,ϕi), where the excited radiator is
located in the array. We here limit ourselves to the main
polarization, that is, the θ-component. Once the scattered field by
the open-circuited antenna has been computed by MoM, the scattering
coefficient SC can be approximated for every
distance as follows:
(8)SC(f,d,θ,ϕ)≈1de−j(2πf/c)dEs(θ,ϕ,f)Ei(θi,ϕi,f).
Thus, the
scattering coefficient is approximated by a far field computation,
which is easier than a near field computation. In Figure 6, we show the result of
this SC approximation on the computation of the
azimuth radiation pattern. We observe a varying accuracy according
to the frequency; nevertheless the approximation is satisfactory
even at low inter-antennas distance (d=50 mm).
Naturally, the discrepancy increases at the higher frequencies, as
the wavelength becomes smaller than the radiator size. Using (7) and
by knowing the exact value of E→(r,θ,φ,f,d) and E→iso(r,θ,φ,f),
one can also obtain the exact value of the SC.Expected azimuth (X-Y plane) radiation pattern for two antennas (solid
line) and SC approximation (dashed line). Antenna 1 is excited; antenna
2 is open circuited (d=50 mm).
(a)(b)(c)(d)Looking at the
cylindrical antenna geometry, a natural but strong approximation is
to eliminate theϕ-dependency. Averaging over the azimuth angle
will introduce an inaccuracy in evaluating the SC
and consequently also on the radiation pattern. In Figures 7 and 8, we compare the
azimuth-averaged exact value of SC with that
computed with (8). A good agreement is shown for all frequencies and
elevation angles, with the exception of the lowest frequencies at
the zenith (z-axis). This should in part be explained by the
presence of a stronger side component of the scattered field, which
is not taken into account in the plane wave approximation.
Furthermore, it is interesting to see that from 10 to 16 GHz the
antenna presents a different scattering behavior in amplitude as
well in phase. This is due to the fact that an UWB antenna, even if
matched, does not present the same radiation behavior over its full
bandwidth. In particular, this bicone was designed for channel
measurement in the FCC band, that is, below 10.6 GHz [12].Azimuth-averaged SC modulus of
ENSTA bicone antenna (d=50 mm): (a) exact
computation and (b) computation by means of scattered fields.
(a)(b)Azimuth-averaged SC argument of ENSTA
bicone antenna (d=50 mm): (a) exact computation and
(b) computation by means of scattered field.
(a)(b)Obviously,
using a mean value in the azimuth dependence of
the SC yields an inaccuracy in the radiation pattern
computation, but it is interesting to observe (see Figure9) that this inaccuracy may be tolerable.
In particular, when the antennas are distant enough, the error
introduced by averaging can be neglected as depicted in Figure 9.
This allows a reduction of the problem complexity, and a reduction
of the number of parameters needed to describe the multiple antennas
behavior, which has some virtues due the large bandwidth in UWB. In
Figures 10 and 11, we show the expected radiation pattern
together with its scattering approximation on two elevation planes.
The SC model is less accurate on the X-Z plane (see
Figure 10) than on Y-Z plane (see Figure 11) where the array effect
is more significant.Figure 9
Expected azimuth (X-Y plane) radiation pattern for two antennas
(solid line); SC approximation (red line) with scattering
coefficient averaged over the azimuth angle. Antenna 1 is excited;
antenna 2 is open circuited.Figure 10
Expected radiation pattern in elevation (φ=0°, Z-X plane) (black
line) and SC azimuth-averaged approximation (red line). Antenna 1 is
excited; antenna 2 is open circuited.Figure 11
Expected radiation pattern in elevation
(φ=90°, Z-Y plane)
(black line) and SC azimuth-averaged approximation (red line). Antenna
1 is excited; antenna 2 is open circuited.The two antennas array
presented above is a simple case of interaction between radiators,
where the parasitic element can be easily modeled as a scatter.Let us now consider a
circular array, as depicted in Figure12 (radius a = 50 mm), composed by three
equally spaced bicone antennas. The driven radiator (antenna 1) is
placed along the y-axis, while the others are open circuited. This
is a generalization of the two bicones array presented above. Thus,
the total radiated field can be expressed as
(9)E→(r,θ,ϕ,f,d)=E→iso(r,θ,ϕ,f)⋅(ej(2π/λ)asin⁡θcos⁡(ϕ−ϕ1)+∑n=2NSCn1(f,d,θ,ϕ)⋅ej(2π/λ)asin⁡θcos⁡(ϕ−ϕn)),
where φn represents the azimuth position of the nth
radiator, and SCn1(f,d,θ,φ) is
the scattering coefficient calculated by (8), but properly turned in
order to take into account the nth antenna
position with respect to the fed antenna. Hence, we consider only
the scattered field by antennas 2 and 3 due to the excitation of
antenna 1. Multiple scattering between the two open-circuited
antennas is here neglected. Figure 12 shows that effectively multiple scattering should be
neglected without a significant degradation of the approximation
accuracy.Figure 12
Three-bicone circular array with
feeding on antenna 1 and open-circuit on antennas 2 and 3. Expected
azimuth (X-Y plane) radiation pattern (solid line) and SC approximation
(dot line).


## 5. Multiple Scattering between Radiators

Generally, by
increasing the number of antennas, multiple scattering effects have
to be taken into account. However, depending on the array
configuration this can be assumed as a second-order effect.Let us consider
a three bicones linear array along the x-axis: antenna 1 is placed
atx=d, antenna 2 at x=0, and antenna 3 at
x=−d.When the antenna 1 is fed, single and
multiple scatterings should be considered. According to (7) and (8),
the scattered field by antenna 2 can be computed as
(10)E→open⁢21(r,θ,ϕ,f,d)=E→iso(r,θ,ϕ,f)⋅SC21(f,d,θ,ϕ),
and the scattered field by
antenna 3 can be expressed as
(11)E→open⁢31(r,θ,ϕ,f,d)=E→iso(r,θ,ϕ,f)⋅(SC31(f,d,θ,ϕ)⋅e−j(2dπf/c)sin⁡θcos⁡ϕ).
Scattering coefficients are
computed according to (8):
(12)SC31(f,d,θ,ϕ)≈12⋅de−j(2πf/c)2⋅dEs(θ,ϕ,f)Ei(θ31,ϕ31,f),SC21(f,d,θ,ϕ)≈1de−j(2πf/c)dEs(θ,ϕ,f)Ei(θ21,ϕ21,f).
In addition,
multiple scattering between antenna 2 and antenna 3, which are
open circuited, can be expressed as
(13)E→open⁢32(r,θ,ϕ,f,d)=E→iso(r,θ,ϕ,f)⋅SC21(f,d,θ,ϕ)⋅(SC32(f,d,θ,ϕ)⋅e−j(2dπf/c)cos⁡θcos⁡ϕ),
where
(14)SC32(f,d,θ,ϕ)≈1de−j(2πf/c)dEs(θ,ϕ,f)Ei(θ32,ϕ32,f).
In (10),
(11), and (13), electric fields are opportunely delayed by taking
into account the antenna position along the x-axis of different
antennas. According to (8), the scattering coefficients in (12), and (14) are computed by simulating the single antenna cross
section. In the computation of a generic SCmn, the illuminating
radiated field is a plane wave, coming from the direction
(θmn,ϕmn)
of the mth antenna with respect to
the nth antenna. Practically, it is the same
scattering coefficient in (8), but suitably attenuated, delayed, and
oriented, in order to take into account the position of the antennas
within the array.Finally, the total radiated electric field can
be computed by means of the superposition theorem as
(15)E→(r,θ,ϕ,f,d)=E→iso(r,θ,ϕ,f)ej(2dπf/c)sin⁡θcos⁡ϕ+E→open⁢21+E→open⁢31+E→open⁢32.
This expression neglects
the 2nd-order scattered field by the fed antenna due to the
scattered fields by both antenna 2 and antenna 3. In Figures 13 and 14, we show the expected
radiation pattern (black line) and the electric field computed by
means of (15), (blue marker) for two different inter-antennas
distance d=5 cm and d=10 cm. The
electric field agrees quite well with the expected radiation pattern
especially at higher inter-antennas distance. However, multiple
scattering expressed in (14) is small comparing to the single
scattering terms, given that a little current distribution is
developed on open-circuited antenna 2.Figure 13
Three-bicone linear array (d=5 cm) with feeding on
antenna 1 and open circuit on antennas 2 and 3. Expected azimuth
(X-Y plane) radiation pattern (black line), SC approximation with
all single and multiple scattering terms (blue marker), and SC
approximation with only single scattering on antenna 2 (red
line).Figure 14
Three-bicone linear array (d=10 cm) with feeding
on antenna 1 and open circuit on antennas 2 and 3. Expected azimuth
(X-Y plane) radiation pattern (black line), and SC approximation with
all single and multiple scattering terms (blue marker), SC
approximation with only single scattering on antenna 2 (red
line).Furthermore, the
single scattering by antenna 3, computed by means of (11) should be
neglected because of the shadowing due to the presence of antenna
2.Consequently,
the total radiated electric field can be very simply approximated
as(16)E→(r,θ,ϕ,f,d)=E→iso(r,θ,ϕ,f)ej(2dπf/c)sin⁡θcos⁡ϕ+E→open⁢21.
Figures 13and 14 validate this assumption by showing that the
radiation pattern computed by means of (16) is nearly exactly
superimposed to the radiation pattern computed by means of (15),
which takes into account all the single and multiple scattering
terms. This means that, in this case, the three bicones array has
the same behavior as the two bicones array.Similarly, when antenna 2
is fed, only single scattering on antennas 1 and 3 can be taken into
account. Thus, the total radiated field can be written as(17)E→open(r,θ,ϕ,f,d)=E→iso(r,θ,ϕ,f)⋅(1+SC12(f,d,θ,ϕ)⋅ej(2dπf/c)sin⁡θcos⁡ϕ+SC32(f,d,θ,ϕ)⋅e−j(2dπf/c)sin⁡θcos⁡ϕ),
where SC12(f,d,θ,φ) and SC32(f,d,θ,φ) are the
scattering coefficients for antennas 1 and 3 calculated by (8), by
taking into account their position with respect to the fed antenna.
Figure 15 shows the azimuth pattern
together with its SC approximation for a linear
array, when the central antenna is fed and expressed in (17).
Multiple scattering between two open-circuited antennas can be
neglected while preserving the approximation accuracy.Figure 15
Three-bicone linear array with feeding on antenna 2 and open
circuit on antennas 1 and 3. Expected azimuth (X-Y plane) radiation
pattern (solid line) and SC approximation (red line).The model taking into account single and double scatterings,
between two parasitic elements, can be generalized for a
nonspecificN antennas array. Thus, the radiated
field by the mth fed element, within the array,
can be written as
(18)E→m(r,θ,ϕ,f,d)=E→iso(r,θ,ϕ,f)⋅(ejδm(f)+∑n=1n≠mN(SCnm+∑p=1p≠nN(SCnp⋅SCpm))ejδn(f)),
where SCnm is the scattering coefficient between the nth and mth antenna
computed by means of (8), omitting the (f,d,θ,ϕ)
dependence for the sake of briefness. The phase delay δn(f)=(2πf/c)a→n•r^ takes into account the position of the nth
antenna (indicated by the vector a→n) in the array. As far as double scattering can be considered
as a second-order effect, (18) can be simplified as
(19)E→m(r,θ,ϕ,f,d)=E→iso(r,θ,ϕ,f)⋅(ejδm(f)+∑n=1n≠mNSCnmejδn(f)).
Therefore,
once the single isolated antenna radiation and its antenna cross
section have been computed, one should easily predict the
performances of a generic array. The proposed approach is of
particular interest in compact MAS design, for which the radiation
pattern distortion is stronger. On the other hand, the proposed
model is still useful in large bidimensional arrays, but multiple
scattering should be considered, given that mutual coupling can
generate phase resonance among the arrays [14].


## 6. Pulse Distortion Due to Inter-Element Interaction

By means of time
domain approach, it is possible to access the waveform of the
radiated field as a function of the angular coordinates. When an
antenna is excited by an incident signalw(t), it
radiates an electric field, whose waveform, e(θ,ϕ,t) in the
(θ,ϕ) direction, is a distorted version of the
incident one. Distortion is due to dispersion, that is, the
frequency-dependence of the realized gain and the angular
frequency-deviation of the radiation pattern. A common feature in
UWB antenna characterization is a time domain approach with the
purpose to characterize the distortion introduced by the antenna. In
multiple antenna systems, the additional distortion due to the
interaction between radiators has to be taken into account. Even
when close antennas are open circuited, scattering effects are
responsible for some distortion, which is the time domain
counterpart of the alteration of the radiation pattern discussed in
the previous paragraphs. This is of practical importance in pulsed
schemes, where distortion may affect the overall system performance
by introducing inter-symbol interference.Here, the chosen
excitation signalw(t) is a Gaussian impulse with a
−10 dB power bandwidth of 3–10.0 GHz (see Figure 16). Our aim is to estimate the accuracy
of the SC approximation.Figure 16
Excitation signal at antenna port: Gaussian impulse with a−10 dB power
bandwidth of 3–10.0 GHz.In Figures17, 18, 19
we show, for different directions, the waveform e(θ,ϕ,t) radiated by a biconical
antenna in a two radiators linear array when the parasitic element
is open circuited, and we compare it with the radiated
waveform eSC(θ,ϕ,t) computed
by means of the SC. The inter-antennas distance is
5 cm. The computed waveform agrees quite well with the expected
one. A small degradation accuracy is shown for the highest elevation
angles and in the end-fire direction (ϕ=180°).Expected radiated waveform
(black line), SC approximation (red line) at (a)φ=0° and θ=90°, (b)
θ=30°.
(a)(b)Expected radiated waveform
(black line), SC approximation (red line) at (a)φ=90° and θ=90°, (b)
θ=30°.
(a)(b)Expected radiated waveform
(black line), SC approximation (red line) at (a)φ=180° and θ=90°, (b)
θ=30°.
(a)(b)Figures17, 18, 19 show a
visual agreement of the approximation. Let us consider a simple
pulse scheme in a simple AWGN channel (power spectral density equal
to N0). The receiver is a classical correlator, and the
reference template pulse is ref(t).Since the
receiver usually choosesτmax in
order to maximize SNR, then
(20)SNR=∫−∞+∞|rec(t)|2dtZ0N0×|∫−∞+∞rec(t)ref(t−τmax⁡)dt∫−∞+∞|rec(t)|2dt⋅∫−∞+∞|ref(t)|2dt|2,
where Z0 is the load on receiving antenna, and rec(t) is the received pulse which can be expressed as [15]
(21)rec(t,θ,ϕ)=∂−1∂t∗hTX(t,θ,φ)∗hRX(t,θ,φ)∗w(t),
where hTX(t,θ,φ) and hRX(t,θ,φ) are the transmitting impulse response of the
transmitting and receiving antennas, respectively. The reference
pulse is chosen to be identical to the received pulse when two ideal
isotropic antennas are used. If we consider an ideal antenna at the
transmitting side and an array on the receiving side, we can
estimate the effect of the SC approximation on
SNR:
(22)SNRD(θ,ϕ)=SNRSC(θ,ϕ)SNRexpected(θ,ϕ),
where SNRSC(θ,φ) and SNRexpected(θ,φ) are the SNR computed by using the SC approximated and the real antenna
impulse response.As depicted in
Figure20, the difference on SNR is
less then 0.7 dB in case of low inter-antennas distance
(d=5 cm) and about 0 dB for higher antenna
distances (d=10 cm, d=20 cm).(a) SNRD (dB) in azimuth and (b) elevation plane.d=5 cm (black line), 10 cm (open dots), and 20 cm (red line).
(a)(b)


## 7. Conclusions

A network
analysis of multiple antenna systems has been carried out in order
to identify the origins of inter-elements interaction and model
them. This approach discriminate precisely what
isscattering from what is coupling. When the first one is negligible, a point source approximation may
be applied, as commonly occurs with thin dipoles. In contrast with
UWB antennas, which generally occupy a larger volume, the
contribution to total electric field by the open-circuited elements
becomes more significant. Thus, scattering has been
modelled by a scattering coefficient
(SC). A practical way to obtain
this SC is to illuminate the open-circuited passive
element and compute the bistatic antenna radar cross section. This
approximation neglects the incident wave curvature and near field
effects, but the reasonable agreement with full computations
justifies its use, given its much greater simplicity. As a result,
only single radiator pattern and SC are sufficient
to predict the multiple antennas performance. The dependence on
azimuth angle can further be averaged in the bicone antennas case,
without significant loss in approximation validity. This means a
simplification of the problem of inter-elements interaction, and a
reduction of the number of parameters describing the phenomena.
Multiple scattering between antennas can be considered as a
second-order effect. Furthermore, single scattering can be neglected
for the most distant radiators, especially if shadowing by a closer
radiator occurs. As a consequence, a three bicones linear array has
the same behavior as the two bicones array when the driven radiator
is at the extremity. More generally, scattering due to the closest
elements is the major cause of radiation pattern alteration and
additional distortion. Finally, additional distortion due to
inter-elements interaction has been investigated. It has been shown
that the SC approximation is accurate enough to
predict the effect of the antenna in an UWB pulse scheme. The
generality of the proposed model simplifies the task of the antenna
designer, who does not have to simulate several array designs, and,
consequently, time consuming full-MoM simulations of the whole array
are not required.



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*Source: 129584-2008-03-19.xml*