Broadband Beamfoming using Nested Planar Arrays and 3D FIR Frustum Filters

Iman Moazzen and Panajotis Agathoklis Departmrnt of ECE, University of Victoria

Victoria, BC, Canada imanmoaz@uvic.ca, pan@ece.uvic.ca

Abstract— A topology of planar array called Nested Planar Arrays (NPAs) is used for broadband beamforming. The NPAs consist of several Uniform Planar Arrays (UPAs), each one with the double element distance of the previous array. The signals from these arrays are fed into different subbands which process different octaves of temporal frequency bands. The combination of NPAs and multirate techniques leads to the same 3D frustum filter frequency specifications for all subbands. The passband of these 3D frustum filters does not include the low temporal frequencies where it is difficult to achieve high selectivity. Simulation results indicate that with the same number of sensors, NPA can achieve longer aperture size compared to a UPA and thus higher selectivity particularly for lower temporal frequencies. For the same aperture size, NPA can be implemented with much less sensors and much less computations than a UPA with small deterioration in the performance.

I. INTRODUCTION Broadband beamforming has many applications and

several approaches have been proposed in the literature for it [8]. One of these techniques for broadband beamforming is based on multidimensional filtering [1-7]. The approach is based on designing 2D or 3D filters whose passbands encloses the Region of Support (ROS) of the desired broadband Plane Wave (PW) in the 2D or 3D frequency domain. All other PWs received by the antenna from a Direction of Arrival (DOA) different than that of the desired PW are attenuated. The obvious merit of this method is to have very low computational complexity compared to adaptive techniques [8]. Due to finite aperture, the filter selectivity of FIR 2D trapezoidal or FIR 3D cone filters at low frequencies is not as good as at high frequencies. One way to tackle this problem is to increase the spatial orders at the cost of larger computational complexity. An approach to improve selectivity at low frequencies without using as many sensors as required for a Uniform Linear Array (ULA) is based on using Nested Arrays and adaptive beamforming techniques presented in [9] and [11]. The resulting 2D beamformers require many computations due to the use of adaptive beamforming techniques. In order to reduce the computational cost, a method was proposed in [7] combining Nested Arrays (NA) [9, 11], multirate techniques, and Trapezoidal Filters (TF) [3]. The main motivation of using the NA and multirate techniques is that a large frequency range can be divided into smaller

frequency octaves using several subarrays. This results in the same passband specifications for the beamformer in all subbands which allows the use of the same TF for all subbands. The main advantage of NAs compared to ULAs is that with the same number of sensors, a larger aperture size can be achieved which leads to a better selectivity for low-frequencies. On the other hand for the same aperture size, fewer sensors are needed in a NA configuration than using ULA which leads to lower computational complexity and results in less mutual coupling between the elements (due to the greater distances between the elements).

In this paper, a new topology called Nested Planar Array (NPA) which is composed of several UPAs (each one called subarray here) is used for beamforming. The proposed beamformer is using NPAs, multirate techniques and 3D frustum filters [6]. The approach is based on extending the work of [7] from 2D to 3D. The use of NPAs corresponds to spatially sampling the signal by different rates. This combined with multirate temporal sampling techniques, results in the input signals to all subband beamformers having the same ROS in the 3D frequency domain. Further, this ROS contains only the upper half of the temporal bandwidth of the desired signal. Therefore the same frustum filter with a passband containing only the upper half of the temporal frequencies can be used for all subbands.

This paper is organized as follows: In section II a brief review of PW’s spectrum is provided and in section III the proposed method is outlined. The design specifications and design method for the filters used in the proposed beamformer are discussed in section IV. In section V, the performance of the proposed method is evaluated using simulations.

II. ROS OF THE PLANE WAVES’ SPECTRUM Consider a broadband PW propagating with the DOA

T]cossinsincossin[ θφθφθ −−−=a whereθ and φ are azimuth and zenith angles in the spherical coordinate system (the minus is due to the direction). Let’s assume the PW would be received by a UPA located on the xy plane. The 3-dimensional Fourier transform (3DFT) of the received PW is [1]:

)())sin()sin(())cos()sin((),,(

ctctyctx

ctyx

cfFffffcfff

ϕθδϕθδ −−

=F

where c is the velocity of propagation in media,δ is a 1D unit impulse function, ctf is equal to tfc 1− ( tf and yxf , represent temporal and spatial frequencies, respectively), and

)( ctcfF is 1D FT of )(tf (temporal intensity function of PW). Thus the Region of Support (ROS) of the 3DFT is a line with an angle of ))(sin(tan 1 θ− with the ctf axis as shown in Figure 1.a [1]. Let’s assume ],[ ss ϕθ is the DOA of the desired PW impinging on the antenna and the objective is to recover the broadband PW and attenuate interference with different DOA. A cone (or a frustum) filter whose passband encloses the ROS of the desired PW as shown in Figure 1.b can be used for this purpose. Methods to design such filters have been proposed in [1, 5-6, 10].

(a)

(b)

Figure 1. (a) ROS of the 3DFT of PW, (c) Cone filter passband

III. PROPOSED METHOD The objective is to recover )(tf , the temporal intensity

function of the broadband PW received from the desired DOA and reject interference signals with different DOAs and noise.

The receiver antenna consists of NPAs in the xy plane. An example of such an antenna with 4 subarrays and 55× elements per subarray is shown in Figure 2. Clearly, many elements of different subarrays are superimposed resulting to 73 total array elements.

Figure 2. A NPA with 4 subarrays and 55× elements for each subarray

The impinging PW is received by NPAs. Without loss of generality, let’s assume L

lu ff 2/ = . The structure of the proposed beamformer is shown in Figure 3. The received signal at each array element is temporally sampled by the rate of β/2 us ff = where 10 ≤<β . The proposed structure needs L different subarrays. The distance between elements in the first

subarray is d (to avoid aliasing it should be less than ufc 2/ ). This first subarray is processing the first octave, i.e. the frequency range uctu fcffc 11 2/ −− ≤≤ . The thl subarray ( Ll ...,,2,1= ) consists of receiver elements with distance kd, where k = 12 −l , and is processing the thl octave, namely

111 2/2/ −−− ≤≤ luct

lu fcffc .

Analysis filters, )(zH k with appropriate passbands are used to extract the appropriate octave for each subarray. As a result of downsampling in space and time, the ROS of the 3DFT of all signals kDf , become the same. Ideally, the ROS of all kDf is non-zero within uctu fcffc 11 2/ −− ≤≤ [7, 9]. Thus a frustum filter whose passband encloses the ROS of the desired PW form uctu fcffc 11 2/ −− ≤≤ can be used as a beamformer (Figure 4, left). The output of the beamformer is upsampled to the original sampling rate, i.e. sf and all replicas of the signal spectrum generated by upsampling except for the baseband copy are eliminated using a synthesis filters )(zGk . The outputs of the different subarrays are aligned using appropriate delays and then added. The sum is the output of the beamformer.

IV. FILTER DESIGN a) Analysis and Synthesis Filters

Since the set of sampling rates of the proposed structure is not a compatible set [13], the perfect reconstruction can not be achieved. In order to approach near perfect reconstruction, analysis and synthesis filters are designed using the following optimization problem:

( ) ( )

∑

∑∑

=

=

×Δ−∠+−

L

l

jl

jl

j

tjj

ttt

t

t

t

t

ll

eGeHeTFwhere

eTFeTF

1

,

)()()(

)(1)(min

ωωω

ω

ω

ω

ω ωgh

where Δ is a constant group delay, tω is frequency (rad/s), and )( tj

l eH ω and )( tjl eG ω are Fourier transform of

the analysis and synthesis filters. The sigma range is from ][ ult ωωω ∈ (corresponds to lf and uf ). The parameters of

the optimization problem are the coefficients of analysis and synthesis filters i.e. lh and lg . Any well-known techniques can be used to design initial FIR analysis and synthesis filters which then can be optimized. In the simulations discussed in the next session, the Kaiser Window method was used for the initial filter designs.

b) Frustum Filter The ideal frustum filter specification in the frequency

domain is given by:

⎪⎩

⎪⎨⎧ ∈≤−+−

=

otherwiseffffbffaff

fffF

ctctctyctx

ctyx

0],[,)tan()()(1

),,(

2122 α

Figure 3. The configuration of the proposed method

where )cos()sin( ssa ϕθ= and )sin()sin( ssb ϕθ= . Parameter α can control the selectivity of the filter. Ideally,

2/11 ufcf −= and ufcf 1

2−= . The FIR frustum filter can be

designed by the slight modification of the method described in [6]. An example of such a filter when spatial and temporal orders are 41, o20=θ , o80=ϕ , and o5.7≈α is shown in Figure 4 (right).

Figure 4. Frustum filter specification (left), Designed frustum filter (right)

V. SIMULATIONS In this section, the performance of the proposed method is

illustrated using simulations. The proposed method (denoted NPA-Frustum) is compared to a 3D cone filter using UPA (denoted UPA-Cone). The 3D Cone filter used is an FIR filter designed using the method of [6]. The following two scenarios are being considered: the same number of sensors (which results in different aperture size) and the same aperture size (which results in different number of sensors).

The PWs considered are sinc functions coming from ]9090[ oo−∈θ and o80=ϕ with temporal bandwidth within

[200 3200]. The signal is temporally sampled by β/2 us ff = where 8.0=β . The FIR frustum and cone filters are designed to pass signals from o

s 20=θ , os 80=ϕ , and o5.7≈α . In the

first scenario (the same number of sensors), the spatial and temporal orders of the cone (frustum) filters in UPA-Cone

(NPA-Frustum) method were chosen 213737 ×× ( 212121 ×× ) respectively. Ideally, the passband area of the frustum should be from =tf [1600 3200], but in order to keep the transition bands of the analysis and synthesis filters, it was designed from 1200 to 4000 [7]. The 1D DFT of the output signals versus different azimuth angles θ is shown in Figures 5.a-d. Ideally, the PWs coming from ]5.275.12[ oo∈θ should be passed by both cone and frustum filters (since o5.7≈α ) and other PWs should be rejected. Clearly, from Figures 5.c-d it can be seen that the UPA-Cone method leads to low selectivity at low temporal frequencies. This is due to the low selectivity of the 3D FIR cone filter for low-frequencies. On the other hand, NPA-Frustum has almost the same selectivity for all frequencies with some undesired bumps where the different octaves meet (Figures 5. a-b) due to lack of perfect reconstruction. From the point of view of computational effort, 3D filtering requires the most computations. Since the filter’s spatial order of the NPA-Frustum is lower than that of the UPA-Cone, the NPA-Frustum is working faster than the UPA-Cone.

In the second scenario (the same aperture size), the orders of the FIR cone filter (frustum) in UPA-Cone (NPA-Frustum) method were chosen 21161161 ×× ( 212121 ×× ), respectively. All other parameters are the same as before. The 1D DFT of the output (for the UPA-Cone method) versus different azimuth angles θ is shown in Figures 5.e-f. Comparing them with Figures 5.a-b, one can conclude that in this case the UPA-Cone has better selectivity than the NPA-Frustum. This is due to the fact that the spatial order of the frustum filter is much lower than that of the cone filter. On the other hand, in this scenario (the same aperture size), UPA-Cone requires 25921 sensors versus only 1401 for the NPA-Frustum. Therefore, the computational complexity and the cost of implementation of UPA-Cone are significantly larger than those of the NPA-Frustum.

VI. CONCLUSIONS

A method for broadband beamforming based on NPAs, multirate techniques, and 3D FIR frustum filters is proposed. The simulations indicate that with the same number of sensors, NPA can achieve better selectivity compared to UPA particularly for lower temporal frequencies. For the same aperture size, NPA can be implemented with much less

sensors and much less computations than UPA with small deterioration in the performance.

VII. ACKNOWLEDGEMENTS: The authors would like to thank Mr. Chamira Edussooriya

for the helpful discussions. Also, funding by NSERC, Canada is gratefully acknowledged.

(a) NPA-Frustum method, Isometric View

(c) UPA-Cone method, isometric view

(e) UPA-Cone method, isometric view

(b) NPA-Frustum method, top view

(d) UPA-Cone method, top view

(f) UPA-Cone method, top view

Figure 5. 1D DFT of the output versus different θ , Filter Orders: (a), (b) 212121 ×× , (c), (d) 213737 ×× , (e), (f) 21161161 ××

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