FIELD OF THE INVENTION
-
The present invention relates to the field of wireless telecommunications networks, and more precisely to a method for optimizing the spacing between receiving antennas of an array usable for counteracting both interference and fading in cellular systems. The invention is suitable to be employed in the Base Station receivers of multi-cell wireless systems based on frequency reuse in adjacent cells and, if needed, employing the SDMA technique in the same cell. The invention could find particular application in cellular systems based on different types of radio access, either narrowband or broadband, such GSM, UMTS, WiMAX IEEE 802.16-2004, WiMAX IEEE 802.16e, HiperMAN ETSI. TS 102 177, etc. The invention could also be applied, with obvious modifications anyway covered by the claims, to receivers belonging to Subscriber Stations and/or point-to-point links.
BACKGROUND ART
-
As known, the multipath fading together with co-channel interference from subscriber stations in the same or adjacent cells, are the major sources of SINR degradation at the output of the receivers. When using a Base Station (BS) with multiple antennas, the multi-cell interference is accounted by a spatial covariance matrix (or noise power) that is assumed as constant, and applying a spatial filter on the received signals (and a pre-filter at the transmitter whenever available) to improve the SINR at the output.
-
Multiple antennas (SIMO/MIMO) are a known manner to obtain larger values of SINR. When designing the antenna array, diversity and beamforming are two different strategies typically adopted depending on the specific impairment, either fading or interference, that has to be contrasted. There are some freedoms in the design of the arrays so as in the design of the reception signal processing.
- Diversity-oriented schemes take advantage of the spatial redundancy over uncorrelated fading for reducing the fading margin. Large antenna spacing compared to the carrier wavelength λ is used, say larger than 5-8 λ, so that signals are uncorrelated at the different antennas and can be processed by diversity-oriented algorithms, such as Selection Combining or MRC. These algorithms need the knowledge of channel responses at all antennas.
- Beamforming-oriented schemes for interference rejection are based on small spacing up to λ/2 so that signals are completely correlated at different antennas and
- Beamforming-oriented schemes for interference rejection are based on small spacing up to λ/2 so that signals are completely correlated at different antennas and beamforming techniques (e.g. MVDR) are adopted to filter out the interferences. Used algorithms require the knowledge of both channel response and the spatial features of the interference power.
-
Usually the spacing between adjacent antennas is not optimized to the channel/interference parameters and to the receiver scheme. Small spacing is typically adopted in LOS environments where beamforming is more effective in filtering the interference. On the other hand, NLOS applications call for diversity-oriented approaches. Still, most of the environments are characterized by mixed LOS/NLOS conditions and they need an optimized spacing whose value is between the two extreme cases mentioned above.
-
In the paper of
S. Savazzi, O. Simeone, U. Spagnolini, titled: "Optimal design of linear arrays in a TDMA cellular system with Gaussian interference", IEEE Proc. SPAWC 2005 [SPAWC], an optimal design of a non-uniform symmetric array is carried out by maximizing the channel capacity. Numerical simulations showed that substantial capacity improvements can be achieved by an exhaustive search over the optimization domain. The capacity maximization proposed in [SPAWC] is shown to be difficult in practice even when considering the uniform linear array case, under specific assumptions on the structure of the cellular planning.
-
As known, a spacing larger than λ/2 introduces a certain degree of angular equivocation in the directivity function of the ULA, aimed to induce the latter to see the interferers (all or a certain number depending on the degree of freedom of the directivity function) as grouped together along an unique apparent direction. In other words, the minimization of the spread of the wave numbers (spatial pulsations of the array directivity function) results in the maximization of the array interference suppression capability. This should allow to release some degrees of freedom of the directivity function, so that a corresponding number of zeroes could be placed at the angular positions they mostly were needed, for example in correspondence of the common direction of the interferers for a deeper attenuation. The optimal theoretical spacing Δopt is calculable in known way under the following restrictive hypotheses: a) null angular spread on the channel; b) interfering terminals are located in fixed positions with main DOAs symmetric with respect to the broadside direction; c) all terminals transmit with maximum power.
-
The theoretical approach of above is only valid for an unreal scenario of null angular spread which does not exist at all in the real radio communications, especially cellular. With that, the equivocation expression for Δopt only accounts for simple geometrical relations.
OBJECT AND SUMMARY OF THE INVENTION
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Object of the invention is that of obtaining the optimum spacing in closed mathematical form when real radio paths are taken into account without needing of demanding simulations, limitedly to an uniform linear array under some specific assumptions on the structure of the cellular planning and the interference .
-
In view of above, the invention provides a method for optimizing the spacing between the antennas of a receiving uniform array usable in a receiving station, either fixed or mobile, of a cellular communication system, as disclosed in claim 1.
-
According to the method of the invention, a closed form expression for the optimal spacing, which is function of the positions of the interfering cells and the angular power density of the multipath is proposed as solution of the spatial spread minimization problem.
-
According to the method of the invention, there is not the need of explicitly recurring to any minimization algorithm, but instead the theoretical expression known for the unreal scenario is adapted to the real case. The spacing so calculated is automatically optimized to the channel/interference parameters, and could be larger than the canonical λ/2 for trading between large spacing to take advantage of diversity and small spacing to have accurate interference rejection capability. This is suitable for either LOS, NLOS or mixed LOS/NLOS propagation environments.
-
According to the method of the invention, some significant restrictive hypotheses mentioned above are neglected, and the new ones are considered: A) the channel gathering the complex gains at the NR receiving antennas is modelled as a space-time multipath power profile characterized by angular-delay dispersion; B) terminals are randomly distributed within the respective cells; C) fast-fading and shadowing power fluctuations are considered; D) path-loss attenuation is considered conforming, for example, the Hata-Okomura model; E) all terminals regulate the transmission power according to an adaptive modulation, based on the channel state, so as to satisfy a fixed bit error rate (BER = 10-6);
-
The only pre-condition for the method is the symmetry of the main interfering paths with respect to the broadside direction of the array, to say, we are considering a planning by which the normal to the line crossing all antennas of the array is crossed by an interfering cell and the other interfering cells are symmetrically disposed with respect to the broadside direction. This is a reasonably assumption valid for all the most popular planning, e.g. square, hexagonal, etc. To match with the symmetry condition, rigorously, the interferer should be placed at the centre o the cell, in such a case the motion of the transmitter should not be accounted. The method of the invention overcomes this limitation by considering the motion of the interfering user as it happen in correspondence of the points of a grid used to spread the multipath over the complete cell.
-
The adaptation of the theoretical expression of the equivocation to the real multipath is carried out by introducing the concept of barycentric DOAs of the
NI interfering cells. Accordingly, the multipath generated by the
ith interferer station ideally collapses in a single path characterized by a barycentric DOA
and a barycentric received power
Referring to the
ith interfering cell, with
i=1,...,
NI , the
ith barycentric DOA, is calculated from the power-angle profile of the channel between the interfering station located in the
ith cell and the receiving station located in the cell of interest, when the spatial location of the interfering station is randomly distributed within its cell. Assuming a multipath channel with an arbitrary number of paths
Np, the
ith barycentric DOA is calculated by executing a weighted average extended to the
Np ×
S directions of arrival of the
Np paths by the
S points of a grid indicative of the positions spanned by the
ith interfering station inside its cell, weighting each DOA by the power received on that path.
-
Once the barycentres have been calculated, the successive step is that to find a an angular separation between the interferers to be introduced into the theoretical expression to get the maximum equivocation, and hence the optimal spacing Δopt. According to the method of the invention, this is done calculating a weighted average barycentric DOA separation ΔθB and introducing it into the theoretical expression of the maximum equivocation. Operating as said above on the barycentres, the calculated optimal spacing Δopt is the spacing that minimizes the spread between the NI wave numbers associated to the barycentric DOAs of the NI interfering cells. The advantage is that a brute-force minimization is avoided.
-
In a scenario with square cell planning and receiving array with 90° aperture, the value Δopt =1.8λ is obtained as an optimum trade-off between beamforming and diversity. Other types of cellular planning have been investigated too.
BRIEF DESCRIPTION OF THE DRAWINGS
-
The features of the present invention which are considered to be novel are set forth with particularity in the appended claims. The invention and its advantages may be understood with reference to the following detailed description of an embodiment thereof taken in conjunction with the accompanying drawings given for purely non-limiting explanatory purposes and wherein:
- fig.1 shows the frame structure for the uplink SIMO channel compliant with IEEE 802.16-2004 specification for (fixed) WiMAX;
- fig.2 shows an exemplary frame structure for uplink SIMO channel compliant with IEEE OFDMA -TDMA specification for (mobile) WiMAX 802.16e;
- fig.3 shows a typical uplink interferer scenario in a wireless cellular system for the reception by a Base Station BS0 of the signal transmitted by the user SS0;
- fig.4 shows a diagram useful to evaluate the interference power;
- fig.5 shows the interfering scenario of fig.3 with assumed initial positions of the fixed interferers;
- fig.6 shows an optimally spaced ULA for a given system planning;
DETAILED DESCRIPTION OF AN EMBODIMENT OF THE INVENTION
-
In order to simplify the radio channel description and the whole calculations and in meanwhile achieving a satisfactorily generalization, the embodiment is referred to a SIMO configuration for WiMAX-compliant systems, either fixed or mobile, but the same concepts and results are applicable to GSM, UMTS, etc., without changing the guidelines of the method.
-
As known, in each cell of a WiMAX-compliant system the multiple access is handled by a combination of time, frequency, and/or space division. With reference for example to fig.1 for fixed/nomadic WiMAX and fig.2 for mobile WiMAX, within the available bandwidth, composed of N subcarriers, the transmission is organized in L time-frequency resource units, called blocks (or bursts), each containing K < N subcarriers and a time window of Ls OFDM symbols. Each block includes both coded data and pilot symbols. Pilot subcarriers are distributed over the block to allow the estimation of the channel/interference parameters. In addition to these distributed pilot subcarriers, a preamble containing only known training symbols could also be included in the block, as shown in the example of fig.1. In this case, the preamble is used for the estimation of the channel/interference parameters, while the other pilot subcarriers are used to update the parameters' estimate along the block. Note that the same time-frequency unit may be allocated to more users in case SDMA is adopted. Every OFDM symbols of each block includes a subset of pilot subcarriers used to track the channel estimation in fast time-varying channels.
-
With reference to
fig.3, let us consider a subscriber station among those simultaneously active in the cell, say SS
0, transmitting (or receiving) signals to (or from) its own base station BS
0 (the communication can be either in the uplink or in the downlink). The transmitter is assumed to employ a single antenna, while the receiver has
N R antennas. The scenario of interest is exemplified for a squared cellular layout with frequency reuse factor
F=4. This example refers to an uplink communication, where the transmission by SS
0 to BS
0 is impaired by the interference from N
I = 3 out-of-cell terminal stations
that employ the same subcarriers as SS
0. In the figure,
di denotes the distance of the
ith terminal from its base station for
i=0,...,
NI , while
di0 is the distance of the interferer SS
i (with
i ≠ 0) from the base station BS
0 of the user of interest.
SIMO SYSTEM AND SIGNAL MODEL
-
Focusing on uplink communications, the transmitter at the station SS
0 maps the sequence of data to be transmitted to BS
0 into a sequence of blocks, indexed by ℓ=1,2,...L. (figures 1 and 2). In the following the symbol ℓ is disregarded because the given expressions are valid for every ℓ. The signals received by the
N R receiving antennas on the
k th subcarrier can be written as:
where:
is the spatial vector containing the
N R complex channel gains for the
N R receiving antennas, while
xk denotes the symbol (either pilot or data) transmitted on the
k th subcarriers.
-
Data symbols can be generated according to an adaptive modulation-coding scheme where the transmission mode is selected based on the channel state (see, as an example, the transmission modes for IEEE-802.16-2004 in
TABLE 1 (APPENDIX A). The
N R × 1 vector
n k , modelling both the background noise and the out-of-cell interference, is assumed to be zero-mean complex (circularly symmetric) Gaussian, temporally uncorrelated but spatially correlated with spatial covariance
Q, where:
Here δ(·) denotes the Dirac delta, while the index n spans the subcarriers. The channel vector
h k is assumed to be constant within the block. It varies from block to block in case of mobile applications (fast-varying channels), while it can be considered as constant over several blocks for fixed/nomadic systems (slow-varying channels). Active interferers may be indeed different in each block, as the access is uncoordinated (not synchronized) between cells. In
fig.3, for instance, the interferer SS
1 may stop at any given time and a new terminal may become active in the cell, generating an abrupt change in the signal interfering on user SS
0. Furthermore, in case of frame structures as the one exemplified in
fig.1, the interference covariance could also vary within the block, on each OFDM symbol. By gathering signals (1) for the
K useful subcarriers into the
NR ×
K matrix
Y=[
y 1···
y K ], the signal model can be rewritten in the standard matrix notation:
where
H=
[h 1···
h K ] is the
NR ×
K space-frequency channel matrix, whose element (
m,
k) represents the channel gain for the
m th receiving antenna on the
kth subcarrier. The
K×
K diagonal matrix
X=diag{
x 1,...,
xK } contains the transmitted symbols.
The model above is general and it applies to several cases such as:
- Frame structures where the block is composed by a preamble and a data field containing pilots (see example in fig.1), such as in the physical layer of WiMAX IEEE 802.16-2004 systems. Being these systems suitable for fixed applications, the preamble can be used to estimate the channel/interferer parameters, while the pilots are exploited to track the non-stationary interference parameters.
- Frame structures where the block does not contain the preamble and pilot subcarriers are inserted in data-bearing OFDM symbols (see example in fig.2), such as in the physical layer of WiMAX IEEE 802.16e systems. These systems are suitable for mobile applications, thus the pilots can be used to track both the channel and interference parameters.
- Decision feedback receivers where estimated data symbols are used as pilot symbols for the estimation of channel/interference parameters.
SIMO CHANNEL MODEL
-
In order to model the space-frequency matrix
H, it is useful to write it in terms of the
NR ×
W space-time channel matrix
H̃ that gathers by columns the
W taps of the discrete-time channel impulse response in the time-domain:
The element (
k,w) of the
NR ×
W matrix
F , for
k=1,...,
K and
w=1,...,
W, is defined as:
with
nk ∈ {0,...,
N-1} denoting the frequency index for the
kth useful subcarrier and
N the total number of subcarriers. The multiplication by
F in (3) performs the DFT transformation of the matrix
H̃ by rows.
The propagation channel between SS
0 and BS
0 (
fig.3), the space-time matrix
H̃ is assumed to be the superposition of
Np paths' contributions. Each path, say the
rth , is described by a direction of arrival (DOA) at the receiving array (θ
0,r ), a delay (τ
0,r ) and a complex fading amplitude (α
0,r ):
The
Np ×1 vector
a(θ
0,r ) denotes the array response to the direction of arrival θ
0,r (θ
0,r =0 denotes the broadside), while the
W×1 vector
g(τ
0,r ) collects the symbol-spaced samples of the waveform
g(
t-τ
0,r ), that is the cascade of transmitter and receiver filters shifted by the delay τ
0,r . The fading
amplitudes are assumed to be uncorrelated and to have normalized power-delay-angle-profile Λ
0,r =
E[|α
0,r |
2] so that
The matrices
S=[
a(θ
0,1)···
a(θ
0,Np )],
G=└
g(τ
0,1),...,
g(τ
0,Np )┘, and
A=diag(α
0,1,...,α
0,Np ) in (7) gather the channel parameters for the whole multipath set.
Possible values of the parameters indicated by expression (7) are given by known multipath models, for example the temporal ones called SUI enriched with a characterization of the spatial interference (SUI-ST); see also
Table 2 (APPENDIX A).
-
The received power
[dBm] in (7) is given by:
and it depends on: the transmitted power
[dBm]; the transmitter-receiver antenna gain
G=
G (T)+
G (R) [dB]; the power loss
L(
d 0) [dB] experienced over the distance
d 0 between SS
0 and BS
0; the
random fluctuations due to shadowing. As recommended in IEEE 802.16-2004, the path-loss is herein modelled according to the Hata-Okamura model:
with λ denoting the wavelength, γ the path-loss exponent,
dref a reference distance, and
fc the carrier frequency [GHz]. Note also that
is limited by the maximum power available at the SSs, i.e.
The power random fluctuations described above have to be ascribed to variations of the user position inside the cell.
SIMO INTERFERENCE MODEL
-
The inter-cell interference is the limiting factor in the performance on estimating the channel, and hence of the system. It is assumed to be spatially correlated with covariance:
sum of the background noise matrix
and the contribution
Q I from the
NI out-of-cell active interferers.
We assume that the signal from each interferer SS
i, placed in the spatial location
si within the
ith cell, with
i=1,...,
NI , is received by BS
0 through a multipath channel with the same characteristics as in (7). It follows that the
ith interferer spatial covariance (averaged with respect to fast fading) depends on the DOA's
(evaluated with respect to the broadside) the normalized power-angle-profile
and the received power
[dBm], according to:
As in (8), the received power is obtained from the power
transmitted by SS
i, taking into account the power loss due to propagation over the distance
d i0 and the shadowing effect
over the link SS
i - BS
0:
Since adaptive modulation and coding is adopted to satisfy a fixed bit error rate (BER = 10
-6), the transmission mode selected (from those listed in
Table 1) by the
ith user (i ≠ 0) and the corresponding transmitted power will be functions of the path loss over the distance
di and the shadowing over the link SS
i - BS
i.
For a simple AWGN scenario (without shadowing),
fig.4 shows the transmission mode T(d) (in dotted spaced scale) and the corresponding power
P (T)(
d) required by a SS at distance
d from its own BS. The power can be written as:
where
d max(
T) is the maximum distance at which the transmission mode T is supported. In our framework, the power transmitted by the
ith interferer has to be increased with respect to (13) to compensate the shadowing fluctuations
among SS
i and BS
i. As this is possible only up to the maximum available power
we can equivalently write the transmitted power as:
OPTIMIZATION OF THE SPACING BETWEEN THE RECEIVER ANTENNAS
-
In fig.5 a possible interference scenario is based on planning Q4 (Q is meaning square cells and 4 is the frequency reuse factor). The considered BS0 receives the useful signal from SS0 and three first-ring interfering signals from SS1, SS2, and SS3 located in the centre of their cells and transmitting on the same frequency as SS0. The useful and interfering signals are transmitted with the following characteristics:
- Carrier frequency 3.5 GHz;
- Channel bandwidth 4 MHz;
- Number of subcarriers N = 256;
- Number of useful subcarriers NU = 200;
- Channel length W (cyclic prefix length) = (32/N)×Tb, where Tb is the symbol duration;
-
To the only aim of simplifying the problem, the following assumptions are introduced:
- SS0, SS1, SS2, and SS3 have a single omnidirectional antenna;
- SSi omnidirectional antenna gain = 2 dBi;
- BS0 is equipped with an ULA of 4 antennas having 90° aperture;
- the wavefront impinging the BS0's array is supposed to be plane;
- BS0 directional antenna gain = 16 dBi (broadside);
- the received signal is narrowband;
- the NI interfering terminals are located in NI fixed positions with main DOAs symmetric with respect to the broadside line: according to fig.5 only three line-of-sight interferers are considered;
- null angular spread and fading for both useful and interferers;
- shadowing is not considered;
- fading uncorrelated over the subcarriers.
-
These simplifications shall be considered as an useful preliminary expedient to the only aim of introducing some theoretic arguments, but will be soon removed to achieve a better trade-off between beamforming and diversity. Successively, the channel and the interference will be modelled as said for expressions (7) and (11), and Table 2 (APPENDIX A).
-
The spatial covariance expression (11) of the
ith interferer includes
Np steering vectors
a(θ
i,r (
si )) (
N p=1 under the above restrictive hypothesis), each of them denoting the response of receiving array to the direction of arrival
θi,r (
si ). The response of the ULA illustrated in
fig.6 to the useful signal from SS
0 is:
where Δ is the spacing between adjacent antennas.
-
The response (15) is a spatial sinusoidal signal whose wave number is:
-
The DOA of the interferer impinging on the array is thus discriminated without equivocation (alias), if for every angle
θs a different wave number ω
s is obtained. The maximum admissible spacing Δ
max is the value which associates the greatest angular deviation θ
max in respect of the normal to the antennas' array at the Nyquist wave number. For every spacing Δ if the following bound is satisfied:
the equivocation does not turn up. When the DOAs are unknown it shall suppose
and the corresponding spacing is
value that does not introduce alias for every DOA. For the examined planning covering a
sector the maximum deviation from the normal is
From the equation (17):
The angular resolution is proportional to the reciprocal of the length
L =
NR Δ of the array. When Δ=Δ
max the incoming interfering signals are associated to distinct wave numbers maximally separated, for planning Q4 they are equal to ± 2.46 radiant.
-
Let ω
i be the wave number of the signal coming from the generic subscriber station SS
i : the (19) is:
it is possible select Δ in a way that the wave numbers {ω
i }
i=1,2,3 of the steering vectors in the direction of the interferers coming from {SS
i }
i=1,2,3 are coincident to each other. This happens when:
where Δθ is the angular separation of the LOS interferers visible in
fig.5. Being the (19) true, it results:
and the three interferers are seen as superimposed by the array, so that a fictitious reduction of the interferers from three to one takes place. The angular separation is bound to 33.7°, limiting to
n=1, Δ
opt =1.8λ is the best uniform spacing from (19). The length of a 4-element array for 3.5 GHz carrier used in WiMAX is 0.46 m (λ=8.6 cm). Shorter arrays more suitable for mobile applications are possible with highest carrier (WiMAX licensed bands cover up to 11 GHz). As known, a good suppression of the interference is achieved by placing at least a zero of the directivity function in correspondence of the direction of each interferer. In absence of alias three degrees of freedom (given by the number of antennas minus 1) are needed. In presence of alias only one degree of freedom is necessary, the remaining two can be used to superimpose two other zeros in the same direction of the first one, obtaining a larger attenuation.
-
The same considerations as for planning Q4 are extendible to other types of planning characterized by their parameters: AAP (Antenna Array Pattern), Δθ
max, Δ
max, Δθ, and Δ
opt. Considering for example the planning Q1 (square cells with unitary reuse factor where the three interfering cells are adjacent to the considered one), it results:
Δ
max=0.71λ,
ωs =±2.46π, Δθ=26.6° , Δ
opt=
n·2.24λ. Considering the planning E3 (Hexagonal cells with reuse factor equal 3 and AAP = 120°), it results:
Δ
max=0.58λ,
ωs =±2.61π, Δθ=46.10°, Δ
opt=
n·1.39λ. With certain types of planning, e.g. E1, the superimposition of the wave numbers {ω
i }
i=1,2,3 can be incomplete; this is valid in general using arrays with limited length, in such a case the best uniform spacing Δ
opt is calculated minimizing the maximum number of the interferers distinguishable by the array.
-
Now the above limitations about the channel and the interference leading to the theoretical value Δopt are removed for extending the (19) to a more realistic value of the optimum separation. To this aim the ULA is momentarily maintained but the channel and the interference are characterized as previously said for respective equations (7) and (11) with further reference to the space-time variable fading parameters listed in TABLE 2.
-
When the initial restrictive hypotheses are neglected, to say when: A) all terminals regulate the transmission power according to an adaptive modulation, based on the channel state, so as to satisfy a fixed bit error rate (BER = 10-6); B) the channel gathering the complex gains from the NR receiving antennas is modelled by means of a space-time multipath propagation, characterized by angular-delay dispersion; C) fast-fading and shadowing power fluctuations are considered; D) path-loss attenuation conforming for example the Hata-Okomura model is taken into account; E) terminals are randomly distributed within their respective cells, the optimal spacing Δopt is calculable as the spacing that minimizes the spread between the NI wave numbers associated to the barycentric DOAs of the NI interfering cells. The ith barycentric DOA refers to the ith interfering cell, with i=1,...,NI , and it is calculated from the power-angle profile of the channel between the interfering station located in the ith cell and the receiving station located in the cell of interest, when the spatial location of the interfering station is randomly distributed within its cell. Assuming a multipath channel with an arbitrary number of paths Np, the i th barycentric DOA is calculated by executing a weighted average extended to the Np ×S directions of arrival of the Np paths by the S points of a grid indicative of the positions spanned by the ith interfering station inside its cell, weighting each DOA by the power received on that path.
-
The following is a down-top description of the operations said above, starting from the calculation of the barycentric DOAs. Accordingly, the multipath generated by the
ith interferer station ideally collapses in a single path characterized by a barycentric DOA
and a barycentric received power
-
Term barycentric is appropriate indeed because of the strict resemblance between the (21) with the mathematical formula used to calculate the barycentric position of an ensemble of material points distributed as the
S iterated multipath. Introducing
(21) in the (18) we obtain the barycentric wave number
-
For a generic cellular planning, the optimal spacings for the ULA case can be found by minimizing the spread of the barycentric wave numbers weighted with respect to the barycentric received powers, therefore it is the solution to the following problem:
where:
(the objective function) is the wave number associated to the
ith barycentric DOA as a function of the spacing Δ, and
is the weighted average wave number.
-
Closed form solution can be dealt with when considering a planning with
NI interfering cells with non-negligible received interference power coming from the broadside: by defining a weighted average barycentric DOAs separation Δ
θB as:
For
NI = 3:
where interfering cell for
i =1 is in broadside direction. The (26) and (27) are right in all effects because of the symmetry of the main DOAs with respect to the broadside
θBS = 0 degrees and the fact they are evaluated with respect to the broadside, so that it results
The final optimum spacing reads:
-
The value Δ
opt =1.8λ is substantially confirmed by the unrestricted approach as a good trade-off between beamforming and diversity. The following wave numbers
of the steering vectors of the array in the barycentric directions
are coincident to each other, so that the maximum number of barycentric interferers distinguishable by the array is minimized.
-
Although the invention has been described with particular reference to a preferred embodiment, it will be evident to those skilled in the art, that the present invention is not limited thereto, but further variations and modifications may be applied without departing from the scope thereof. For example, expressions equivalent to the (19) valid for the ULA are easily obtainable for different arrays with equally spaced antennas arranged according to other geometrical shapes, for example circular, on the basis on the best suitability to cope with the cellular scenario.