WO2009140998A1

WO2009140998A1 - A method for obtaining an array antenna

Info

Publication number: WO2009140998A1
Application number: PCT/EP2008/056173
Authority: WO
Inventors: Fredrik Athley
Original assignee: Telefonaktiebolaget L M Ericsson (Publ)
Priority date: 2008-05-20
Filing date: 2008-05-20
Publication date: 2009-11-26

Abstract

The invention discloses a method for obtaining a thinned antenna array (200) comprising a row of M antenna element positions (1, 2, M) spaced a predefined distance (c/) apart. The array comprises a plurality (1, 2', M) of N antenna elements and M-N empty (M-I ) element positions, and the method is used to determine which M-N of the M positions that should be left empty. The method comprises, for each combination of M-N antenna elements: -Determining the mean square error, MSE, of the antenna's performance in estimating a direction of arrival, DOA, of a signal, and determining a lower boundary for said performance, -Determining a point where the MSE has exceeded the lower boundary by a predefined level, a threshold, -Using the M-N combination which has the lowest threshold as the thinned array antenna (200).

Description

A METHOD FOR OBTAINING AN ARRAY ANTENNA

TECHNICAL FIELD The present invention discloses a method for obtaining a thinned antenna array comprising a row of M antenna element positions.

BACKGROUND

Array antennas, i.e. antennas which comprise a number of antenna elements can, for example, be used when determining the direction of arrival, DOA, of a received signal. The accuracy of the DOA estimate will to a great extent depend on the size of the array, i.e. the number of antenna elements comprised in the array antenna, so that a larger array will provide an increased accuracy.

However, as can be realized, a price will be paid for the increased accuracy of a larger array in the form of increased expenses when it comes to the implementation of the antenna, both in terms of hardware and computational complexity.

One way of reducing the complexity of a large array antenna is to remove, in a judicious manner, some of the elements from what would otherwise be a "fully populated array". A fully populated array could be defined as an array antenna with M antenna positions at a defined interval from each other, usually λ/2, where λ is the centre frequency of the operational wavelength of the antenna. An array antenna where some of the M antenna positions are not used for antenna elements, i.e. the elements of those positions have been "removed", is usually referred to as a thinned array antenna.

A thinned array antenna will, as compared to the fully populated corresponding antenna, have a reduced performance regarding such parameters as, for example, gain and side lobes. There are known methods for optimizing the design of thinned array antennas. However, these methods usually do not take into account the DOA estimation accuracy of the antenna, whilst other methods, which do provide for a good DOA accuracy will lead to an antenna which has large side lobes, thus causing an increased likelihood of, for example, ambiguity errors.

SUMMARY

As can be concluded from the description above, there is a need for a solution by means of which a thinned array antenna can be obtained which can provide good DOA measurement accuracy, whilst having low side lobes.

This need is addressed by the present invention in that it discloses a method for obtaining a thinned antenna array comprising a row of M antenna element positions which are spaced a predefined distance apart.

The array comprises a plurality of N antenna elements and M-N empty element positions, and the method of the invention can be used to determine which M-N of the M positions that should be left empty.

The method of the invention comprises, for each combination of M-N antenna elements:

• Determining the mean square error, MSE, of the antenna's performance in estimating a direction of arrival (DOA) of a signal, and determining a lower boundary for this performance, • Determining a point where the MSE has exceeded the lower boundary by a predefined level, a threshold,

• Using the M-N combination which has the lowest threshold as the thinned array antenna. In one embodiment of the invention, the lower boundary and the MSE are determined by means of a deterministic signal model, whilst, in another embodiment, they are determined by means of a stochastic signal model.

Also, in one embodiment, the threshold is determined as a function of the signal to noise ratio, SNR, of the signal, whilst, in another embodiment, the threshold is determined as a function of the number of "array snapshots", i.e. the number of time samples used from the M-N elements.

Suitably but not necessarily, the lower boundary is the Cramer-Rao Bound (CRB) of the antenna, a notion which will be explained in greater detail below.

Also, suitably but not necessarily, the distance between the antenna element positions is λ/2, where λ is the centre frequency of the antenna's operational bandwidth.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be described in more detail in the following, with reference to the appended drawings, in which

Fig 1 shows a symbolic antenna array, and Fig 2 shows a thinned array antenna, and Fig 3 shows MSE as a function of SNR, and Fig 4 shows a notion used in this text, and

Figs 5 and 6 show results obtained by means of the invention, and Fig 7 shows a flow chart of a method of the invention.

DETAILED DESCRIPTION Fig 1 shows a schematic view of an antenna array 100 with M possible antenna element positions spaced apart by a distance of d, exemplified as λ/2, where λ is the centre frequency of the antenna's operational frequency range. The spacing of the antenna element positions at a distance of λ/2 is a suitable value, but other values of the distance between the antenna element positions are also entirely possible in the invention.

As has also been stated above, a purpose of the present invention is to obtain a so called thinned array antenna with good characteristics when it comes to measuring the so called direction of arrival, DOA, of a received signal, as well as having low side lobe levels.

Fig 2 shows the principle behind the thinned line array antenna, by means of an array antenna 200 which is similar, although not equivalent, to the antenna 100 of fig 1. The antenna 200 has M possible antenna element positions, but of those, some are left unused, which can also be expressed as saying that N of the M antenna elements have been "removed", so that the thinned array antenna 200 of fig 2 only has M-N antenna elements, out of M possible antenna elements. In fig 2, antenna element M-1 is shown as being removed, in that it has dashed lines. Naturally, the amount of antenna elements that are unused or removed is a design parameter that can be varied within the scope of the invention.

A principle used in the present invention is to optimize the performance of a thinned array antenna when the maximum likelihood, ML, is used as the principle for DOA estimation. Fig 2 shows a typical behaviour of the mean square error, MSE, of the ML DOA estimates of an array antenna as a function of the signal to noise ratio, SNR:

At a high SNR, the MSE reaches a lower boundary of the estimation accuracy that can be achieved by any estimator, in this case the so called Cramer-Rao Bound, CRB. This can be seen in fig 3, in which the MSE vs. SNR is shown with a solid line, indicated as "1"; the CRB is shown with a dashed lined, indicated as "2". As can be seen, at a certain SNR level, here referred to as "the threshold" and marked as "T" in fig 3, the MSE increases rapidly and deviates from the CRB. The deviation of the MSE from the CRB at low SNR is caused by ambiguity errors which appear at those SNR levels. One aim of the present invention is to arrive at a thinned array antenna which has an optimally low threshold SNR with a given amount of antenna elements, i.e. the parameter N in M-N above is given.

The task of the thinned antenna array is to find the DOA, θ₀, of a source signal s(t) which is contaminated by zero-mean white Gaussian noise n(t). The complex baseband signals received by a linear array of K=M-N antenna elements or sensors can be modelled by the K ^* 1 complex vector:

X(O = a(θ_o)s(t) + nit), t = 1 L (1 )

where a(θo) is the K ^* 1 array steering vector that models the array response to a unit waveform from the DOA θo measured relative to the array boresight. If the signal source and the array are coplanar, θ₀ represents an azimuth angle, whilst if not, θ₀ represents a "cone" angle.

In addition, s(t) is the complex amplitude at baseband level of the impinging wave front, and n(t) is the K ^* 1 vector representing noise and interference. The noise is here assumed to be spatio-temporally white, with a variance denoted as σ ^Jln.-

The variable L in equation (1 ) above is the number of "snapshots" used, a notion which is illustrated in fig 4: a snapshot is a collection of samples from the array elements at a single time instant.

It is here assumed, for simplicity, that the antenna elements are unidirectional with unity gain. It is also assumed that the antenna elements are placed on a regular grid, but that not all grid points need to be occupied by an element. The steering vector can then be written as: α(β) = ([exp[-J-²ψ sinθ] [exp[-y-^ sir*]? (2) where dkΔ, k=1....K are the sensor positions along the axis of the array, either the fully populated array or the thinned array. . Typically, the spacing or separation Δ will be < λ/2. Below, sin θ will be replaced by the variable u, i.e. u = sin θ.

In the invention, two different versions of the ML DOA estimator can be used, either a deterministic or a stochastic estimator. The deterministic estimator is one that considers the signal samples as unknown deterministic parameters that need to be estimated along with the DOA, while the stochastic one models the signal samples as a random process with an unknown variance. In the stochastic estimator, the signal s(t) is modelled as a zero-mean white Gaussian process with the variance σ_s ².

It can be shown that an accurate approximation of the MSE of the ML DOA estimator is given by the expression:

JV JV

MSE = (1 " ∑ Pn) CRB + ∑ P_n (u_n - U₀)² (3) n=l n=l

CRB in (3) above is, as before, the Cramer-Rao Bound, uo is the true DOA, and U_n are the positions of the N side lobe peaks in the array beam pattern. P_n is the probability that noise and interference make the n-th side lobe peak have a larger magnitude than the main lobe peak. For the purposes of this invention, U₀ can be chosen to be equal to zero.

The CRB and the probabilities P_n will be different depending on which of the two signal models that is used, i.e. either the deterministic or the stochastic model, and will be shown below for both models. Deterministic signal model

In this model, the CRB can be shown to be as follows:

') where:

and:

In (5) and (6) above, S can be seen as the total SNR integrated over the spatial and temporal samples, and V as the variance of the element positions.

The probabilities P_n are given by:

P_n = Qφ (i - Vi - nTiJi C¹ +

e^~s/2

where: Q(a,β) = j te^J-^- I₀(at)dt (8) β is Marcum's Q function, and /_m(-) is the modified Bessel function of the first kind of order m.

In addition,

T_n ± . I I a(u_o ⁰)^JH _κa(^Ku_n ⁿ )⁾ I I (9)

which may be interpreted as the relative side lobe level of the n-th side lobe peak. It can be pointed out that x^H as used in (9) above means the transposed complex conjugate of x.

Stochastic signal model

In this model, the CRB is given by the following expression

CRB = 2π f²N^SKT²SNR'₂ ²V ( ^V10 ')

where SNR ± σ_s ⁿ/σ^ (11 )

The probabilities P_n are given by the expression:

where:

Turning now to the method of the invention, in one embodiment the following steps are followed. The method is applied when a choice has been made regarding the size of the fully populated array which is to be thinned in an optimal fashion by means of the invention, as well as how many of those elements which are to be populated, i.e. the parameters M and N as mentioned previously have been determined initially, where M is the amount of elements in a "fully populated" array, and N is the amount of "removed" elements, so that M-N elements are used.

The following steps are followed in a preferred embodiment of the invention in order to find which M-N positions should be populated in order to maximize the DOA measurements and minimize the side lobe levels of the array antenna:

• For each combination of M-N positions: o Determine the MSE and the CRB of the antenna as a DOA estimator, o Determine the "threshold" of the combination,

• The combination which has the lowest threshold is then chosen as the antenna array, with the M-N positions of the combination being used as the positions for the antenna elements in the array.

In the steps given above, a signal model is preferably used which is either stochastic or deterministic, as defined previously in this text. Thus, the CRB, and the MSE are preferably determined by means of a deterministic signal model, or by means of a stochastic signal model. Also, suitably, the MSE and CRB are determined as a function of the SNR or as a function of the number of array "snapshots".

In one embodiment of the invention, the "threshold" is determined as a function of the signal to noise ratio, SNR, of the signal, while in another embodiment, the "threshold" is determined as a function of the number of "array snapshots", i.e. the number of time samples used from the M-N elements.

It should also be mentioned that the use of the CRB as a lower boundary level is only a suitable way of determining this level; it is within the scope of the present invention to use other lower levels as well.

Regarding the distance between the M antenna element positions, one example of a suitable value in a preferred embodiment is λ/2 wavelengths, where λ is the centre frequency of the antenna's operational bandwidth.

However, other distances can also be used within the scope of this invention.

In order to illustrate the method of the invention, an example will now be given: In this example, the method is used to find the optimal thinned array with 8 antenna elements in an array with 24 antenna element positions spaced λ/2 wavelengths from each other. In other words, to use the notation above, M=24 and N=8, so that M-N=I 6. The number of possible thinned arrays in this case is ( ₆) instead of( ₈), since the outermost positions of the array will be used in order to maintain the end-to-end length of the antenna array. The numeric value of ( g) is 74613.

In this particular application, a stochastic signal model is chosen. The choice of signal model will of course depend on the application, but the deterministic signal model is suitable for applications in which the DOA estimation performance is expected to depend on the actual signal samples, whereas the stochastic signal model is suitable when the performance depends only on the signal power.

In this case, the method yields the result of [0, 1 , 7, 8, 9, 10, 12, 23], i.e. those are the positions which will be used in the antenna array. Fig 5 shows the MSE of the antenna as an ML estimator, indicated with the numeral "3", as well as the CRB for the array, indicated with the numeral "4". As comparison, the MSE for a uniform linear array, ULA, with 8 elements and the same element separation is also shown in fig 5, indicated with the numeral "5".

Fig 6 shows the beam patterns of the optimal thinned array, indicated with the reference numeral "6", and of the 8 element ULA, indicated with the reference numeral "7".

Fig 7 shows a generalized flow chart of a method 700 of the invention. Steps which are options or alternatives are indicated with dashed lines in fig 7.

Thus, as has also emerged above in this text, the method 700 is used for obtaining a thinned antenna array which comprises a row of M antenna element positions which are spaced a predefined distance apart. The array comprises a plurality of N antenna elements and M-N empty element positions. As shown in step 705, the method 700 is used to determine which M-N of the M positions that should be left empty; the method 700 comprises, for each combination of M-N antenna elements:

• Determining, step 710, the mean square error, MSE, of the antenna's performance in estimating a direction of arrival, DOA, of a signal, and determining a lower boundary for said performance, • Determining, 715, a point where the MSE has exceeded the lower boundary by a predefined level, a threshold, • Using, 720, the M-N combination which has the lowest threshold as the thinned array antenna.

As indicated in step 725, the lower boundary and the MSE can be determined by means of a deterministic signal model, or, as indicated in step 730, the lower boundary and the MSE can be determined by means of a stochastic signal model.

Step 735 indicates that the threshold, in one embodiment of the invention, can be determined as a function of the signal to noise ratio, SNR, of the signal.

In a further embodiment, as shown in step 740, the threshold is determined as a function of the number of "array snapshots", i.e. the number of time samples used from the M-N elements.

Step 745 shows that in one embodiment, the lower boundary is the Cramer- Rao Bound (CRB) of the antenna.

As indicated in step 750, in one version of the method, the distance between the antenna element positions is λ/2 wavelengths, where λ is the centre frequency of the antenna's operational bandwidth.

The invention is not limited to the examples of embodiments described above and shown in the drawings, but may be freely varied within the scope of the appended claims.

Claims

1. A method (700) for obtaining a thinned antenna array (200) comprising a row of M antenna element positions (1 , 2, M) spaced a predefined distance (d) apart, the array comprising a plurality (1 , 2', M) of N antenna elements and M-N empty (M-1 ) element positions, the method (700) being used to determine (705) which M-N of the M positions that should be left empty, the method (700) being characterized in that it comprises, for each combination of M-N antenna elements: • Determining (710) the mean square error, MSE, of the antenna's performance in estimating a direction of arrival, DOA, of a signal, and determining a lower boundary for said performance, • Determining (715) a point where the MSE has exceeded the lower boundary by a predefined level, a threshold, • Using (720) the M-N combination which has the lowest threshold as the thinned array antenna (200).

2. The method (700, 725) of claim 1 , according to which the lower boundary and the MSE are determined by means of a deterministic signal model.

3. The method (700, 730) of claim 1 , according to which the lower boundary and the MSE are determined by means of a stochastic signal model.

4. The method (700, 735) of any of claims 1 -3, according to which the threshold is determined as a function of the signal to noise ratio, SNR, of the signal.

5. The method (700, 740) of any of claims 1 -3, according to which the threshold is determined as a function of the number of "array snapshots", i.e. the number of time samples used from the M-N elements.

6. The method (700, 745) of any of the previous claims, according to which the lower boundary is the Cramer-Rao Bound (CRB) of the antenna.

7. The method (700, 750) of any of the previous claims, according to which the distance between the antenna element positions is λ/2 wavelengths, where λ is the centre frequency of the antenna's operational bandwidth.