Efficient Implementation of Broadband Beamformers Using Nested Hexagonal Arrays and Frustum Filters

4796 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 19, OCTOBER 1, 2013

Efficient Implementation of Broadband BeamformersUsing Nested Hexagonal Arrays and Frustum Filters

Iman Moazzen, Student Member, IEEE, and Panajotis Agathoklis, Senior Member, IEEE

Abstract—A broadband beamformer is proposed based onnested hexagonal arrays, hexagonal frustum filters and multiratetechniques. The nested hexagonal arrays used here consist ofseveral hexagonal arrays of increasing size in the x-y plane (eachone called subarray) where the distance between elements ineach subarray is two times larger than in the previous one. Theproposed beamformer consists of subarray beamformers, eachone using the signals obtained from one of the nested hexagonalarrays as the input. These signals are filtered and downsampledso that the Region of Support (ROS) of the resulting 3D signalsin the 3D frequency domain are the same for all subbands. Thesame hexagonal frustum filter design can therefore be used for allsubarray beamformers to pass the desired signal and eliminateinterferences. The use of nested arrays leads to larger effectiveaperture at low temporal frequencies and thus, better selectivityfor low frequencies. Further, hexagonal arrays are known torequire a lower sensor density for alias free sampling than rectan-gular arrays. Examples illustrate the performance of the proposedbeamformer with respect to beampattern and computationalcomplexity.

Index Terms—Broadband beamforming, frustum filters, multi-dimensional filters, nested arrays.

I. INTRODUCTION

B ROADBAND beamforming has many applications suchas communications, radio astronomy, sonar and micro-

phone arrays [1]. A review of wideband beamforming tech-niques is presented in [1]. In general, the approaches can be clas-sified into two groups: adaptive and fixed. In adaptive the beam-formers’ weights are updated for each new set of signal samplesto account for changing environment. Some well-known adap-tive techniques are the reference signal based beamformer, theLinearly Constrained Minimum Variance (LCMV) beamformer[2] and its derivations [3]–[5]. Fixed beamformers can be em-ployed in cases where the Direction Of Arrivals (DOAs) areknown and there is no need to update beamformer’s weights.The least squares approach [6], the eigenfilter approach [7],[8] and Multi-Dimensional filters (M-D Filters) [9]–[12] arethree examples of fixed beamformers. An important advantageof M-D filters is that they do not suffer from grating lobes [12].

Manuscript received February 20, 2013; revised July 04, 2013; accepted July10, 2013. Date of publication July 23, 2013; date of current version August 29,2013. The associate editor coordinating the review of this manuscript and ap-proving it for publicationwasDr. Antonio DeMaio. This workwas supported bythe Natural Sciences and Engineering Research Council of Canada (NSERC).The authors are with the University of Victoria, BC, Canada (e-mail:

[email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2013.2274279

To avoid aliasing in broadband beamforming, the distance be-tween elements must be less than where is thewavelength associated with the maximum frequency [13]. Fur-ther, the aperture size on the other hand depends on the ratio ofthe highest to the lowest frequency [13]. For broadband beam-forming with a large bandwidth this may lead to a large aperturewith a large number of sensors. To achieve the same aperturesize and significantly reduce the number of elements, a poten-tial option is to employ non-uniform arrays such as the onesconsidered here.In this paper, a broadband beamformer will be proposed

which combines nested hexagonal arrays, hexagonal Finite-du-ration-Impulse-Response (FIR) frustum filters, and multiratetechniques. The use of hexagonal arrays is motivated by theproperty that for alias free sampling hexagonal geometries forsensors are known to require less dense arrays than rectangulargeometries [14]. The proposed beamformer consists of subarraybeamformers, each of which uses the signals obtained from oneof the nested hexagonal arrays as the input. These signals arefiltered and downsampled so that the Region of Support (ROS)of the resulting 3-Dimensional (3D) signals in the 3D frequencydomain are the same for all subbands. This allows using thesame 3D hexagonal frustum filter design for all subbands. Theuse of nested arrays was proposed in [15] where 1-D nestedarrays, proposed by Chou [16], were combined with a General-ized Sidelobe Canceller (GSC) [3]. The adaptive beamformer(GSC) was replaced by a fixed beamformer realized using atrapezoidal filter in [17]. This approach was extended to nestedplanar arrays [18] in which the incoming Plane Wave (PW) issampled by nested rectangular arrays. In [18], a frustum filterwas used as the beamformer. Using nested arrays results in theeffective aperture for low temporal frequencies being largerthan in the case of using a Uniform Linear Array (ULA) withthe same number of array sensors. For low temporal frequen-cies this leads to a higher spatial selectivity for nested overuniform arrays. If the same aperture size is used, the nestedarray requires a much lower number of sensors than a uniformarray which leads to reduced implementation cost, reducedmutual coupling as well as fewer computations for the nestedarrays. It is shown by simulations that this comes at a cost ofonly a slight deterioration in the performance of the proposedbeamformer.The rest of this paper is organized as follows: In Section II,

a brief review of hexagonal and rectangular sampling patternsis provided. In Section III, the impulse response of hexagonalFIR frustum filter is obtained using the hexagonal Fourier trans-form. The proposed beamformer using hexagonal nested arraysis presented in Section IV. Examples in Section V graphically

1053-587X © 2013 IEEE

MOAZZEN AND AGATHOKLIS: EFFICIENT IMPLEMENTATION OF BROADBAND BEAMFORMERS 4797

Fig. 1. Rectangular and hexagonal sampling pattern (unit length is ). (a) Rect-angular 2D Array, (b) Hexagonal 2D Array.

illustrate how the method works and demonstrate the good per-formance of this beamformer.

II. REVIEW OF HEXAGONAL AND RECTANGULARSAMPLING PATTERNS

Consider a temporal continuous-time signal propa-gating through 3D space from a far-filed case with a givenDOA where andare zenith and azimuth angles in the spherical coordinate

system. According to [12], this signal can be approximated asa 4D PW given by:

(1a)

is the velocity of propagation in media. The 3D Fourier Trans-form (3DFT) of this PW received on the x-y plane (z is zero) isgiven by [10], [11]:

(1b)

where is a 1D unit impulse function, is equal to( and represent temporal and spatial frequencies, respec-tively), and is 1D FT of (called temporal intensityfunction of PW). From (1b), it is clear that the ROS of the 3DFTof the PW is located on a line. Considering all possible DOAs,i.e., and , one can conclude thatthe ROS is always confined within a 45 cone, which is usuallyreferred to as the light cone [19].The received continuous signal can be spatially sampled

using a rectangular or hexagonal 2D array, shown in Fig. 1,and uniformly sampled in time. The corresponding space/timesampling matrices for the rectangular and the hexagonal arrayscan be given using the following generating matrices for thecorresponding 3D lattices [14]:

(2)

Fig. 2. The repetition of light cone (from the top view) in the frequency domainusing (a) rectangular sampling, (b) hexagonal sampling.

The Fourier transform of the sampled signal is a periodic ver-sion of the continuous one with the periodicity matrix being thegenerating matrix of the reciprocal lattice [14]. They can be ob-tained using:

(3)

leading to:

(4)

The Voronoi cell of the reciprocal lattice [14] for rectangularsampling is a rectangular parallelepiped which when using nor-malized frequencies is equivalent to the Nyquist cube. In thecase of hexagonal sampling, the Voronoi cell of the reciprocallattice is a hexagonal prism with a regular hexagon for the spa-tial frequencies. Fig. 2 graphically illustrates the difference be-tween the ROS of the Fourier transforms of the sampled signals(for all DOAs) resulting from rectangular and hexagonal sam-pling. The periodicity (described by and ) in the direc-tion of the spatial frequencies is different while the periodicityin the temporal frequency direction is the same for both cases(not shown in Fig. 2). For PWs bandlimited with from allDOAs, aliasing can be avoided if the periodically repeated lightcones do not overlap as shown in Fig. 2. This implies that thetemporal sampling period must be less then . Fur-ther, using , follows that and

. Thus to avoid aliasing due to spatial sampling,must be less than . Setting and to these limiting

values, the sampling density [14] for rectangularand hexagonal sampling patterns is and ,respectively. This means that for a given aperture area and max-imal temporal frequency, a hexagonal arrangement of spatialsensors will lead to 13.4% fewer sensors than a rectangular ar-rangement [20], [21].


III. BROADBAND BEAMFORMING USING A HEXAGONALARRAY AND HEXAGONAL FIR FRUSTUM FILTER

The objective is to recover , the temporal intensity func-tion of the broadband PW (1a) received from the desired DOAand reject interference signals with different DOAs. Equation(1b) indicates that the ROS of the 3DFT of the PW is locatedon a line whose direction depends on the DOA. Disturbanceswith different DOAs will have ROS in 3D frequency domainon lines with different directions. Therefore, the broadband PWfrom a desired direction can be passed while distur-bances are attenuated using a frustum filter [10], [11] with spec-ifications given by (5), shown at the bottom of the page, where

and are the normalized spatial and temporal frequen-cies, and . The se-lectivity of the filter can be controlled using the parameter .The impulse response of the corresponding spa-tial-temporal domain 3D FIR filter can be obtained using thehexagonal Fourier transform pairs defined in (6) from [20].

(6a)

(6b)

Using (6b), is obtained in (7)

(7a)

(7b)

Otherwise:

(7c)

where is the first order Bessel function. The integral in (7c)can be numerically calculated using the Simpson method [22].In order to truncate the impulse response, a hexagonal window(with a similar geometry as the one shown in Fig. 1(b)) is em-ployed. It has (odd number) elements in the horizontal rowthrough the origin and the total number of elements is

[13]. The spatio-temporal window is defined as:

(8)

Finally, the hexagonal FIR frustum filter can be obtained as(o is Hadamard or element-wise product):

(9)

Example: Hexagonal FIR Frustum Filter Design: Using(7)–(9), a frustum filter was designed with

(normalized temporal frequencies)and . Fig. 3(a) shows the

dB surface of the amplitude response of the obtained filter.It should be noted here that the amplitude response shown inFig. 3 has been obtained using the hexagonal Fourier transformimplementation as in [23] and is repeated hexagonally using. Further, the amplitude response at two constant temporal

frequencies, and , is shown in Fig. 3(b) and (c)respectively. It is interesting to note that the amplitude responsewithin the passband area at high frequencies is almost 0 dB,and for low frequencies is almost dB which degrades theperformance. This amplitude attenuation at low frequencies isdue to the fact that the ratio of aperture to wavelength at lowfrequencies is much smaller than at high frequencies [1]. Totackle this drawback the aperture size can be increased at thecost of having more sensors and more computations. Anothersolution is using nested arrays which will be discussed in thenext section.

(5)


Fig. 3. (a) dB surface of the obtained hexagonal FIR frustum filter and the amplitude response of the obtained hexagonal FIR frustum filter at temporalfrequencies: (b) and (c) .

Fig. 4. The structure of NHA (different colors represent different subarrays).

IV. BROADBAND BEAMFORMING USING NESTED HEXAGONALARRAYS, FRUSTUM FILTERS AND MULTIRATE TECHNIQUES

In this section, a broadband beamformer will be presentedwhich is based on using nested hexagonal arrays, frustum fil-ters and multirate filtering techniques. A nested hexagonal array(NHA) consists of several hexagonal arrays in the x-y plane(each one called subarray). In the th subarray ,the distance between elements is times larger than that ofthe first subarray. An example of such an antenna with 4 subar-rays and 37 elements per subarray is shown in Fig. 4.Clearly, many elements (21) of the different subarrays are super-imposed leading to 127 sensors in total.

A. Proposed Beamformer

The structure of the proposed beamformer is shown inFig. 5. This beamformer is an extension of the broadbandbeamformer presented in [18] using rectangular arrays. Con-sider a PW with a temporal bandwidth given by

and . This implies that the proposed beam-former will consist of different subarrays. The impingingPW is received by NHA and the received signal at each arrayelement is temporally sampled with the rate of . The thsubarray is processing the th octave, namely

. Analysis filters, withappropriate passbands, i.e., , are used toextract the corresponding octave for each subarray. As a resultof downsampling in space and time, the ROS of the 3DFT of allsignals (see Fig. 5), become the same. Ideally, the ROS of all

is non-zero only within [18]. Thusa hexagonal frustum filter whose passband encloses only thetop half of ROS of the desired PW, i.e.,can be used as the beamformer. This filter can be designedas discussed in the Section III. The output of the beamformeris upsampled to the original sampling rate, i.e., and allreplicas of the signal spectrum generated by upsampling exceptfor the baseband copy are eliminated using a synthesis filter

. The outputs of the different subarrays are aligned usingappropriate delays and then added. The sum is the output of thebeamformer. This beamformer will be referred to as NHA-FF.A graphical example will be provided in Section V to illustratehow the proposed beamformer works.

B. Filter Bank Design

The analysis and synthesis filters, and respec-tively, have to satisfy the prescribed magnitude specificationsas well as Perfect Reconstruction (PR). Since the set of sam-pling rates of the proposed structure is not a compatible set [24],PR cannot be achieved. The approach used here is to design afilter bank satisfying near PR as well as the prescribed magni-tude specifications for the analysis filters. This is based on themethod discussed in [25]. The resulting analysis and synthesisfilters designed with this method may not have a linear phase,but the linear phase of the whole filter bank is guaranteed.The design approach is based on minimizing a performance

index, combining PR error and magnitude response error, with respect to analysis filters’ coefficients:

(10)


Fig. 5. The structure of the proposed beamformer.

The PR error is based on a formulation of the PR conditionsin the z domain leading to a set of linear equations [25]:

(11)

where is a matrix formed using the analysis filters’ coeffi-cients, is the vector containing the synthesis filters’ coeffi-cients, and is a vector whose elements are all zero expect the

th element which is 1 ( is the filter bank delay). Themagnitude response error, , is the difference between the ac-tual and the desired magnitude response of the analysis filters,

. This leads to:

(12)

where and are optional weights and is the -norm.The optimization parameters are the analysis filters’ coeffi-cients. After convergence, the synthesis filters are obtained asthe least square solution of (11).

C. Efficient Beamformer Implementation

The computational efficiency of the proposed beamformercan be improved using the Nobel identity and polyphase struc-tures [26] for the analysis and synthesis filters. To avoid com-puting unnecessary output samples which will be discarded bydownsampling, the order of analysis filters and downsamplerscan be changed. The th analysis filter given by:

(13)

can be replaced by its polyphase structure using:

(14)

as shown in Fig. 6(a). is the length of analysis/synthesis fil-ters. Then, by pulling the downsampling operation to the rightside of summation in Fig. 6(a) and applying the Nobel identity,

Fig. 6. (a) Replacing an analysis filter by its polyphase structure, (b) Finalstructure for downsampling part, and (c) Final structure for upsampling part.

the final structure, shown in Fig. 6(b), can be obtained. A com-muter can be employed to eliminate the downsamplers and de-liver successive samples to successive branches [26]. Note thatthe coefficients of are obtained by downsampling thecoefficients of by the rate starting from .The same procedure can be applied to the upsampling operationfollowed by synthesis filtering to avoid redundant computations.The final structure for this part is shown in Fig. 6(c). The majordifference between the downsampling and upsampling configu-ration is the path that the commuter is connected at the start. Theresulting beamformer is referred to as the modified NHA-FF.

V. ILLUSTRATIVE EXAMPLES

In this section, two examples will be provided to illustratethe functioning and evaluate the performance of the proposedmethod. The first example graphically illustrates how the pro-posed beamformer works. In the second example the perfor-mance of the proposed beamformer with the hexagonal frustumfilter is evaluated in terms of beampattern and computationalcomplexity.

A. Illustration of How the Method Works

Consider 5 broadband PWs propagating from the directionsgiven in Table I. The temporal Fourier transforms of the in-tensity functions are equal to one within

and zero elsewhere. Since, the beamformer will have four subbands.

The ROS of the signals in 3D frequency domain (obtained


Fig. 7. (for ) and dB surface of hexagonal FIR frustum filter.

in Matlab as the area where the most energy of the signal isconcentrated) is calculated using the hexagonal Fourier trans-form implementation of [23] and will be used to illustrate the

function of the beamformer. The ROS of the signals receivedby different subbands ( , see Fig. 5) are shownin Fig. 7(a)–(d). For the first subband, in (2) is set to


Fig. 8. (a) NHA-FF with 721 sensors, (b) HA-FF with 721 sensors, (c) HA-FF with 12481 sensors.

TABLE IFIVE PWS PROPAGATING FROM DIFFERENT DIRECTIONS

(alias-free condition [1]). For the th subarray ,the distance between elements is times larger than thatof the first subarray (Fig. 5), resulting in aliasing as seen inFig. 7(b)–(d). The th subarray is processingthe th octave, i.e., , and hasthe effect of extracting this octave. The ROS of the analysisfilters’ outputs for different subbands ( , seeFig. 5) are shown in Fig. 7(e)–(h). The aliased componentsin Fig. 7(b)–(d) are non-zero outside the passband area of

, and therefore are eliminated(Fig. 7(f)–(h)). Next, the th subband is downsampled by .The ROS of the downsampled signals for the four subbandsare shown in Fig. 7(i)–(l) ( , See Fig. 5). Itshould be noted that due to the same sub-sampling rate inspace and time, different frequency octaves are mapped intothe top octave and the ROS of all subband signals are thesame. Thus, the same frustum filter with passband in the topoctave can be used to pass the desired signal for all subbandsand reject the interferences. Thus the passband of the frustumfilter does not include the low frequencies with poor selectivity(Fig. 3(b)).The frustum filter with , andwas designed to pass the first PW and reject the others.

Fig. 7(i)–(l) also shows the dB surface of the amplituderesponse of this filter. After filtering the center sensor

is selected as the output [18], and upsampled by (togo back to the original sampling rate). To remove all replicasof the signal spectrum generated by upsampling except for thebaseband copy, a synthesis filter is used. Then, to alignthe outputs of the subband beamformers, appropriate delays areadded and the aligned signals are summed to form the output.

B. Performance evaluation of NHA-FF and HA-FF

In this example the performance of the proposed beam-former (NHA-FF, Section IV-A) and the hexagonal frustumfilter (HA-FF, Section III) are evaluated in terms of beam-pattern and computational complexity. The effect of antennacoupling is being ignored here. Two different scenarios will beconsidered:

The same number of sensors (721) for both NHA-FF andHA-FF are considered resulting in different aperture sizesas shown in Fig. 8(a) and (b).The same aperture size for both NHA-FF and HA-FF

is considered resulting in different number of sensors asshown in Fig. 8(a) and (c).

1) Evaluation of NHA-FF and HA-FF Beampatterns: Allbeamformers designed in this example are required to pass PWspropagating from and having temporalFourier transform of the intensity function equal to one within

and zero elsewhere. For thefirst scenario, the HA-FF was designed using (7)–(9) with

and resulting in 721 totalnumber of sensors. The NHA-FF has four subbands, i.e., .For all subbands the same frustum filter is being used. It hasbeen designed using the same parameters as the one used for theHA-FF expect for which was set to 17 for NHA-FF. Thisvalue for results in the same total number of sensors (721)for both the NHA-FF and the HA-FF. In Fig. 9(a) and (b) the 3Dbeampatterns versus and temporal frequency for the NHA-FFand the HA-FF for the first scenario are shown. For anotherview, the same 2D beampatterns versus for both methods areshown in Fig. 9(d) and (e) from to . It can be seen thatfor the same number of sensors, the selectivity of the NHA-FFis better than that of the HA-FF. This improvement achievedmainly because of the larger aperture size of the NHA-FF forlow frequencies compared to that of the HA-FF (Fig. 8(a) and(b)).For the second scenario, the same NHA-FF is used as in the

first scenario. The HA-FF is designed with the same parame-ters as in the first scenario with the only difference of beingequal to 129 leading to 12481 sensors and the same aperture asthe NHA-FF. In Fig. 9(c), the 3D beampattern of the HA-FF


Fig. 9. (a), (b), and (c) 3D beampatterns of NHA-FF and HA-FF, (d), (e), and (f) 2D beampatterns of NHA-FF and HA-FF. (a) NHA-FF, (b) HA-FF, First Scenario,(c) HA-FF, Second Scenario, (d) NHA-FF, (e) HA-FF, First Scenario, (f) HA-FF, Second Scenario.

as a function of and temporal frequency for the second sce-nario is shown. Further, the 2D beampattern versus fromto for this case is shown in Fig. 9(f). Comparing thesefigures with Fig. 9(a) and (d), one can conclude that the selec-tivity of the HA-FF is better than that of NHA-FF. However thiscomes at the cost of having almost 17.3 times greater number ofsensors which increase the cost and computational complexitysignificantly.2) Evaluation of the Computational Complexity of NHA-FF

and HA-FF: The computational complexity index (CCI), whichis the number of arithmetic operations (additions and multipli-cations) required to compute samples of the output, is sum-marized in Table II for HA-FF (Section III), NHA-FF (Fig. 5)and modified NHA-FF (Section IV-C).The CCI of HA-FF is straightforward. For NHA-FF, there areseparate frustum filters. If the sampling rate of the first sub-

band is , then the th frustum filter is working with asthe sampling rate. This implies that the first term of CCI of theNHA-FF is multiplied by (see Table II) rather than . Givenas the length of analysis (synthesis) filters, arith-

metic operations are required to compute the output of each in-dividual analysis (synthesis) filter. Assuming that each subarrayhas sensors, each connected to an analysis filter and fur-ther each subband has one synthesis filter, follows that the filterbank for each subband entails arithmeticoperations. Since the proposed beamformer is equipped withsubbands, the total computational cost due to the filter bank ismultiplied by (second term of CCI of the NHA-FF). The finaloutput of NHA-FF is formed as the summation of subbandoutputs. Thus, summations must be executed (last

TABLE IINUMBER OF ARITHMETIC OPERATIONS FOR EACH METHOD

term of CCI of the NHA-FF). For themodifiedNHA-FF, the firstand last terms of CCI are the same as those for the NHA-FF. Dueto the Nobel identity and polyphase structure used, the secondterm will decrease significantly. The use of the Nobel identity inthe th subband implies that instead of having an analysis filterwith length (Fig. 6(a), top), there are filters withlength (Fig. 6(b)). The computation saving of the modi-fied NHA-FF compared to NHA-FF comes from the fact that allfilters in Fig. 6(b) are working with as the sampling raterather than in Fig. 6(a).


Fig. 10. CCI for (a) First Scenario, (b) Second Scenario.

Table II shows that the CCI of NHA-FF depends on the orderof both the frustum filter and filter bank, while the CCI of HA-FFonly depends on the frustum filter order. In Fig. 10(a) the CCIof the modified NHA-FF and HA-FF is shown for equal to4, total number of sensors 721 (scenario 1), equal to 81 andthree different orders for (21, 61, and 101). Also, to illus-trate the improvement achieved by the modified NHA-FF, theCCI of NHA-FF (Fig. 5) is shown in Fig. 10(a) for .From Fig. 10(a), it can be seen that when is large, CCI of themodified NHA-FF is greater than that of HA-FF. However, themodified NHA-FF can be implemented as separate sub-beam-formers in parallel resulting in a much faster implementationthan HA-FF. In Fig. 10(b) the CCI of NHA-FF (just for equalto 101), the modified NHA-FF, and HA-FF is shown for thesecond scenario (equal aperture area) with the same , andas before. It can be seen that in this case, even if is large, the

HA-FF requires much more computations than the other two.

VI. CONCLUSION

A broadband beamformer is proposed based on nested hexag-onal arrays, hexagonal FIR frustum filters and, multirate tech-niques. The nested hexagonal arrays used here consist of sev-eral hexagonal arrays in the x-y plane where the distance be-tween elements in each subarray is two times larger than inthe previous one. The use of hexagonal arrays was motivatedfrom the fact that they are known to require a lower sensor den-sity for alias free sampling than rectangular arrays. The pro-posed beamformer consists of subarray beamformers, each oneusing the signals obtained from one of the nested hexagonalarrays as the input. These signals are filtered and downsam-pled so that the ROS of the resulting 3D signals in the 3D fre-quency domain are the same for all subbands. The same hexag-onal frustum filter design can therefore be used for all subarraybeamformers to pass the desired signal and eliminate interfer-ences. An efficient implementation of the proposed beamformerwas also proposed based on eliminating redundant computa-tions using the Nobel identity and polyphase structures. Exam-ples illustrate that beamformers using nested hexagonal arrayshave a larger effective aperture at low temporal frequencies andthus, better selectivity for low frequencies than beamformersusing simple hexagonal arrays. Further, for comparable selec-tivity the computational complexity of the hexagonal nestedarray beamformer is much lower that of a beamformer usinga simple hexagonal array.

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[24] P.-Q. Hoang and P. P. Vaidyanathan, “Non-uniform multirate filterbanks: Theory and design,” in Proc. IEEE Int. Symp. Circuits Syst.,May 8–11, 1989, vol. 1, pp. 371–374.

[25] I. Moazzen and P. Agathoklis, “A general approach for filter bankdesign using optimization,” Matlab toolbox [Online]. Available:http://www.mathworks.com/matlabcentral;fileexchange/40128, to besubmitted

[26] F. J. Harris, Multirate Signal Processing for Communication Sys-tems. Englewood Cliffs, NJ, USA: Prentice-Hall, 2004.

[27] D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital SignalProcessing. Englewood Cliffs, NJ, USA: Prentice-Hall, 1983.

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Iman Moazzen (S’13) was born in Tehran, Iran. Hereceived the B.Sc. degree in biomedical engineeringfrom the Science and Research Branch, Azad Uni-versity, Tehran, Iran, in 2006, the M.Sc. degree intelecommunication engineering from the IsfahanUniversity of Technology, Isfahan, Iran, in 2009, andthe Ph.D. degree in electrical engineering from theUniversity of Victoria, Victoria, Canada, in 2013.He is currently working as a Research Assistant

with the Faculty of Engineering, University ofVictoria, Canada. His research interest includes mul-

tidimensional signal processing, array signal processing and communicationtheory.Dr. Moazzen received the Best-Student Paper Award (WCSP 2009, China),

and the University of Victoria Fellowship and Andy Farquharson Teaching Ex-cellence Award.

Panajotis Agathoklis (M’81–SM’88) receivedthe Dipl. Ing. degree in electrical engineering andthe Dr.Sc.Tech. degree from the Swiss FederalInstitute of Technology, Zurich, in 1975 and 1980,respectively.From 1981 until 1983, he was with the University

of Calgary as a Post-Doctoral Fellow and part-timeInstructor. Since 1983, he has been with the Depart-ment of Electrical and Computer Engineering, Uni-versity of Victoria, B.C., Canada, where he is cur-rently a Professor. His fields of interest are in con-

trol, digital signal processing and their applications. He worked in the stabilityof multidimensional systems, applications of 2D and 3D filtering, image recon-struction from gradient data and its applications in adaptive optics.Dr. Agathoklis received a NSERC University Research Fellowship and Vis-

iting Fellowships from the Swiss Federal Institute of Technology, from the Aus-tralian National University, and the University of Perth, Australia. He has beenmember of the Technical ProgramCommittee ofmany international conferencesand has served as the Technical Program Chair of the 1991 IEEE PACRIMCon-ference, the 1998 IEEE Symposium on Advances in Digital Filtering and SignalProcessing, and the 2009 ISSPIT.

Documents

Efficient Implementation of Broadband Beamformers Using Nested Hexagonal Arrays and Frustum Filters