6
A new technique for designing a postbeamformer interference canceller M.H. Er School ofElectrical & Electronic Engineering, Nanyang Technological Institute, Nanyang Avenue, Singapore 2263 (Received 26 January1990; accepted for publication 10November 1990) This paper presents a newapproach for designing a postbeamformer interference canceller (PIC). The technique basically explores the use of the eigenvectors of a certain Q matrix asthe signal andinterference beamforming weights in the PIC. Numerical results show that the new PIC performs better, on the average, than the conventional PIC. Furthermore, the optimized new PIC achieves almostthe same performance as the optimumelement space processor without precise knowledge of the interference direction. PACS numbers: 43.60.Gk INTRODUCTION Postbeamformer interference cancellers (PIC) have been studied extensively in the literature. •-5These are essen- tially single-channel noisecancelling systems that process the signals derivedfrom an antennaarray by forming two beams employing fixed beamforming weights. The weighted output of one of the beams, referred to as the interference beam, is subtractedfrom the other beam, referred to as the signal beam, which has a specified response in the signal direction. The processing of the output of the interference beam requires theadjustment of a complex weight to achieve a certainperformance criterion. The mostcommonly used criterion is the minimization of the mean output power of PIC. In this paper,a new approach to the design of PIC is proposed andits performance investigated. The approach is based on a new technique for linear array synthesis recently proposed by theauthor. 6 It was found in Ref.6 that, when theeigenvectors of a certain Q matrixareused as a weighting vector,the eigenbeams corresponding to the eveneigenvec- torsalwayform nullsin the broadside direction whereas the eigenbeams corresponding to the oddeigenvectors achieve a certain gain. It is proposed in this paperthat the first two nonzerominimum eigenvectors of the Q matrix derivedin Ref. 6 canbeused asfixed beamforming weights for the PIC. The paper derives mathematical expression for the output signal-to-noise ratio (SNRo) of the new PIC and compares its performance with the conventional PIC. Numerical re- sults show that the new PIC performs better,on the average, than the conventionalPIC. Furthermore, the paper pro- poses a technique to optimizethe performance of the new PIC. Numerical resultsshow that the optimized new PIC achieves almostthe same performance as the optimumele- ment space processor. The paper is organized as follows: Sec.I gives a brief review of PIC. In Sec.II, a new approach for designing PIC ispresented. In Sec. III, the performances of the newPIC are illustrated usingcomputer simulation results. Section IV concludes the paper. I. A REVIEW OF POSTBEAMFORMER INTERFERENCE CANCELLER Figure 1 shows the structureof the PIC processor in which two beams are formed using fixed beamforming weights. The outputof the interference beam isprocessed by A x 1 R R A Y E L E E N T S ---1 I I , I SIGNAL BEAMFORMER i I I Y I (! t• I I I i I INTERFERENCE BEAMFO RIflER • g(•) FIG. 1. Basic structure of a post beamformer interference canceller. 1724 J. Acoust.Soc. Am. 89 (4), Pt. 1, April 1991 0001-4966/91/041724-06500.80 ¸ 1991 Acoustical Societyof America 1724 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.21.35.191 On: Fri, 19 Dec 2014 08:01:26

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Page 1: A new technique for designing a postbeamformer interference canceller

A new technique for designing a postbeamformer interference canceller

M.H. Er

School of Electrical & Electronic Engineering, Nanyang Technological Institute, Nanyang Avenue, Singapore 2263

(Received 26 January 1990; accepted for publication 10 November 1990)

This paper presents a new approach for designing a postbeamformer interference canceller (PIC). The technique basically explores the use of the eigenvectors of a certain Q matrix as the signal and interference beamforming weights in the PIC. Numerical results show that the new PIC performs better, on the average, than the conventional PIC. Furthermore, the optimized new PIC achieves almost the same performance as the optimum element space processor without precise knowledge of the interference direction.

PACS numbers: 43.60.Gk

INTRODUCTION

Postbeamformer interference cancellers (PIC) have been studied extensively in the literature. •-5 These are essen- tially single-channel noise cancelling systems that process the signals derived from an antenna array by forming two beams employing fixed beamforming weights. The weighted output of one of the beams, referred to as the interference beam, is subtracted from the other beam, referred to as the signal beam, which has a specified response in the signal direction. The processing of the output of the interference beam requires the adjustment of a complex weight to achieve a certain performance criterion. The most commonly used criterion is the minimization of the mean output power of PIC.

In this paper, a new approach to the design of PIC is proposed and its performance investigated. The approach is based on a new technique for linear array synthesis recently proposed by the author. 6 It was found in Ref. 6 that, when the eigenvectors of a certain Q matrix are used as a weighting vector, the eigenbeams corresponding to the even eigenvec- tors alway form nulls in the broadside direction whereas the eigenbeams corresponding to the odd eigenvectors achieve a certain gain. It is proposed in this paper that the first two nonzero minimum eigenvectors of the Q matrix derived in Ref. 6 can be used as fixed beamforming weights for the PIC. The paper derives mathematical expression for the output signal-to-noise ratio (SNRo) of the new PIC and compares its performance with the conventional PIC. Numerical re- sults show that the new PIC performs better, on the average, than the conventional PIC. Furthermore, the paper pro- poses a technique to optimize the performance of the new PIC. Numerical results show that the optimized new PIC achieves almost the same performance as the optimum ele- ment space processor.

The paper is organized as follows: Sec. I gives a brief review of PIC. In Sec. II, a new approach for designing PIC is presented. In Sec. III, the performances of the new PIC are

illustrated using computer simulation results. Section IV concludes the paper.

I. A REVIEW OF POSTBEAMFORMER INTERFERENCE CANCELLER

Figure 1 shows the structure of the PIC processor in which two beams are formed using fixed beamforming weights. The output of the interference beam is processed by

A x 1 R

R

A

Y

E

L E

E

N

T

S

---1

I

I

, I

SIGNAL BEAMFORMER

i

I

I

Y I (! t• I

I I

i I

INTERFERENCE BEAMFO RIflER

• g(•)

FIG. 1. Basic structure of a post beamformer interference canceller.

1724 J. Acoust. Soc. Am. 89 (4), Pt. 1, April 1991 0001-4966/91/041724-06500.80 ¸ 1991 Acoustical Society of America 1724

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Page 2: A new technique for designing a postbeamformer interference canceller

using a complex weight and is subtracted from the signal beam.

Let the L-dimensional complex vectors V and U repre- sent the fixed weights of the signal and the interference beamformers, respectively. It follows from Fig. 1 that the output of the signal beam Pt (t) and the output of the inter- ference beam y• (t) are given by, respectively,

the interference beamforming weights satisfy the following

ys(t) = V•X(t) (1)

linear constraints:

UnSs =0 (10) and

uHs• :•0. (11)

If the interference direction is precisely known, the weight- ing vector Uo that satisfies (10) and ( 11 ) is given by

and

y•(t) = UHX(t), (2)

where X (t) is the L-dimensional vector representing the L waveforms derived from L elements of the array, and the superscript H denotes conjugate transposition. The output g(t) of the PIC processor is formed by subtracting the weighted output of the interference beam from the output of the signal beam; that is,

g(t) = ys(t) -- wy•(t). (3)

For a given weight w, the mean output power P(w) of the PIC processor is given by

P(w) = E [g(t)g*(t) ]

= VHR V + w*wUHR U- w*VHR U- wUHR V,

(4)

Uo = PSi, (12)

where P is the projection matrix given by

P = I -- SsSsn/L. (13)

Using (6), (9), and (12), it can be shown 3 that the optimum weight • is given by

Ss"S,S,"Uo a

to -- , (14) L UgUo (a + )

where

a = U•SzS•HUo ?L U•Uo. (15)

Furthermore the output SNR of the optimum PIC 3 can be expressed as

psL [ a +o'2•/LP, ] SNR(•) = • 1 + a--/•-•)LP, ' (16) where

where E[.] denotes the expected value, the asterisk ß de- notes complex conjugation, and R is the L X L-dimensional array correlation matrix defined by

R --E[X(t)XH(t)]. (5)

In an environment consisting of two uncorrelated sinusoidal sources and uncorrelated white noise, the correlation matrix R can be expressed 3 as

R =psSsSs • +p,S,Sf + •I, (6)

whereps andpi represent the powers of the signal source and the interference, respectively, and • denotes the power of the white noise. $s and $i are the steering vectors in the signal and the interference directions, respectively.

The optimum weight • can be selected to minimize the mean output power of the PIC. It can be shown 3 that the optimum weight that minimizes the mean output power of the PIC is given by

• = VHR U/UHR U. (7)

The mean output power P(•) of the optimum PIC is given by

P(•) = VUR V -- UHR VVHR U/UHR U. (8)

•= 1 --I SSslVL (17) Substituting (12) into ( 15 ) and after some manipulation, it can be shown that a =/• and the expression for the SNR (16) reduces to

SNR(•) = psL [ a + ø'2•/LpI ] a2. 1 + •/Lp, ' (18) If the interference direction is not known, then Uo is given by 3

Uo = Pe, (19) where

e = [ 1,0,...,0] •. (20)

In this case, a %/•, in general.

II. A NEW APPROACH FOR DESIGNING PIC

This section presents a new approach for designing PIC and investigates its performance by comparing it to that of the conventional PIC described in Sec. I.

In an earlier paper, 6 a new technique for linear array synthesis was presented. The technique is to find a weight

The conventional PIC considered in the literature is usually designed using the conventional beamforming weights, that is, TABLE I. Source scenarios used in the computer studies.

V = Ss/L, (9)

where $s is the steering vector in the signal direction. The weighting vector defined by (9) ensures a unity response of the signal beam in the signal direction. The interference beam is designed such that it has a null in the signal direction and a nonzero response in the interference direction. That is,

Signal Scenario power

Signal InterferenceinterferenceWhite noise direction power direction power'

1 0dB 0 ø 0dB --90 ø to 90 ø --30dB

2 0 dB 0 ø 20 dB -- 90 ø to 90 ø -- 20 dB

3 0 dB 0 ø 20 dB -- 90 ø to 90 ø 0 dB

1725 J. A½oust. Soc. Am., Vol. 89, No. 4, Pt. 1, April 1991 M.H. Er: Technique for designing postbeamformer canceller 1725

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Page 3: A new technique for designing a postbeamformer interference canceller

(a)

(c)

,

MILl

(b)

(d)

•o o

-to.o

-]o o

-•o.o

.............

o.o

-1o.0

_

(e)

-lO o

- .o

. .

(f)

FIG. 2. Directional patterns corresponding to the ten eigen- vectors of the Q matrix for a 10- element array with interele- ment spacing set at 0.5 20.

(g) • -20 o

-dO o

...JO o

(h)

oo

• -20.0 -40.0

(i)

-Io o

_

-• o -,o o

leM.G .CL.[

(j)

-,o

400 -• o

1726 J. Acoust. Soc. Am., Vol. 89, No. 4, Pt. 1, April 1991 M.H. Er: Technique for designing postbeamformer canceller 1726

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Page 4: A new technique for designing a postbeamformer interference canceller

vector that minimizes the average integrated power response of the array over the spatial region ( - •r/2, •r/2) subject to a unity norm constraint. The resulting vector turns out to be the eigenvector corresponding to the nonzero minimum

50.0.

eigenvalue of a matrix Q arising from the integration. It was __established in RefL6 that theQ matrix is givenby

[Q ]S:,l = Yo [ 2,rr(k - l)d/,,1,o ], k,l = 1,2 ..... L, (21 )

where Jo (•/) is the Bessel function of the first kind of zero order, d is the interelement spacing, and Ao is the wavelength of interest. Furthermore, it was found in Ref. 6 that when the eigenvectors of the Q matrix are used a weighting vector, the eigenbeams corresponding to the even eigenvectors always form nulls in the broadside direction, whereas the eigen- beams corresponding to the odd eigenvectors achieve a cer- tain gain. Hence, it is proposed that the first two nonzero minimum eigenvectors of the Q matrix can be used as fixed beamforming weights for the PIC.

Let E1 and E2 denote the first two nonzero minimum eigenvectors of the matrix Q given by (21 ). Here, E1 and E2 are used as the fixed weights for the signal and the interfer- ence beamformers, respectively. As is derived in Appendix A, the optimum weight of the new PIC, denoted as •o, is given by

A

Wo = ß (22)

The mean output power and the output SNR of the new PIC using the optimum weight •o are given by

• I' ;; ', il ! ',i I

• •o,o • •11 • • I : 11/ !1 , ,•t I '

0.0 I I I I

I

-•o.o I

-•a'•so: : : : : : : S4•: : : : : : : :' : : : : : : : : : : : : : : : ' ' 0 45

IN•RF•R•NCE ANGL• ID•G)

FIG. 4. Output SNRs of the conventional PICs using Uo given by (12) (solid line) and (19) (broken line) and the new PIC (dash•ot line) for source scenario 2.

III. NUMERICAL RESULTS

To demonstrate the performance achievable with the new approach, computer studies involving a linear array have been carried out. The array was assumed to consist of 10 equally spaced isotropic elements. The interelement spac- ing was set at 0.5Ao. The source scenario was assumed to consist of two uncorrelated sinusoidal sources and uncorre-

lated white noise. The parameters of the signal and noise

(23) and

slESl 2 SNR (•o) =

X '1 -4-IE•S, IVIES,I + X,IES,I

respectively. (24)

field for three different scenarios are shown in Table I.

Figure 2 (a)-(j) shows the direction patterns when the ten eigenvectors of the Q matrix are used as beamforming weights. Notice that the eigenbeams corresponding to the even eigenvectors always form nulls in the broadside direc- tion, whereas the eigenbeams corresponding to the odd ei- genvectors achieve a certain gain.

Figure 3 compares the output SNRs of the conventional PICs using Uo given by (12) (solid line) and (19) (broken line) and the new PIC (dash-dot line) for scenario 1. Note that the PIC with Uo given by (12) requires precise knowl-

3o.o j • ....... 20.0

" I I •0.0 I I I I I I

0.1

-•D.0

.

-2o.•9a -45 INTERFEREN• ANGLE {OEG)

FIG. 3. Output SNRs of the conventional PICs using Uo given by (12) (solid line) and (19) (broken line) and the new PIC (dash-dot line) for source scenario 1.

FIG. 5. Output SNRs of the conventional PICs using Uo given by (12) (solid line) and (19) (broken line) and the new PIC (dash-dot line) for source scenario 3.

1727 J. Acoust. Soc. Am., Vol. 89, No. 4, Pt. 1, April 1991 M.H. Er: Technique for designing postbeamformer canceller 1727

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Page 5: A new technique for designing a postbeamformer interference canceller

20.0

FIG. 6. Output SNRs of the conventional PICs using Uo given by (12) (solid line) and (19) (broken line) and the new PIC with optimized E2 for source scenario 1.

20,0

10.0

0.0

-10.0

-20.0

FIG. 8. Output SNRs of the conventional PICs using Uo given by (12) (solid line) and (19) (broken line) and the new PIC with optimized E2 for source scenario 3.

edge of the interference direction. Hence, when comparing the new PIC with the conventional PIC, it is assumed that both are without precise knowledge of the interference direc- tion. It is clear that the new PIC performs better, on the average, than the conventional PIC with the U o given by (19).

Figures 4 and 5 are same as Fig. 2 except for scenarios 2 and 3, respectively. From Figs. 3-5, one can conclude that for high interference-to-white-noise ratio, the performance of the new PIC is comparable to that of the conventional PIC with Uo given by (19). However, under low interference-to- white-noise ratio, the new PIC performs better, on the aver- age, than the conventional PIC.

The performance of the new PIC can be improved further to achieve constant output SNR by doing the follow- ing optimization.

Recall from the Appendix that the total output interfer- ence plus noise power of the new PIC is given by

20.0

10.01

! ,I II I ,• ¾ I I :, ,, • I I i ,, '•, ,, ' I I , ,, • . •lll I, ,,

' 'll• • ,

ill, I ! ! ! I !

I!

-45 0 45 •0

FIG. 7. Output SNRs of the conventional PICs using U o given by (12) (solid line) and (19) (broken line) and the new PIC with optimized E• for source scenario 2.

= 4 + i)gi 3-4 . P•

It is clear from (25) that given E• for the signal beam, P• (•o) can be minimized by searching for an E 2 such that IE2•S•l 2 is maximum. In practice, this optimization can be carried out by using the even eigenvectors of the Q matrix and monitoring the total mean output power of the PIC. The one that gives rise to minimum total mean output power achieves maximum output SNR.

Figures 6 to 8 show the output SNRs of the new PIC using the optimum E 2 for scenarios 1-3, respectively. Super- imposed on Figs. 6-8 are the output SNRs of the convention- al PICs. It is clear that the new PIC with optimized E 2 achieves an output SNR which is very close to the conven- tional PIC with Uo given by (12). Note that it was estab- lished in Ref. 3 that under the condition a =/5, the perfor- mance of the optimum PIC is identical to the performance of the optimum element space processor. Hence, one can con- clude that the new PIC with optimized E 2 achieves almost the same performance as the element space processor with- out precise knowledge of the interference direction. How- ever, in the element space processor, the number of complex weights need to be adjusted is equal to the number of ele- ments in the array, whereas in the PIC processor, only one complex weight must be adjusted.

IV. CONCLUSION

The paper has presented a new approach for designing PIC. The technique basically involves using the eigenvectors of the Q matrix established in Ref. 6 as the signal and the interference beamforming weights. Numerical results show that when the first two nonzero minimum eigenvectors of the Q matrix are employed, the performance of the new PIC are better than the conventional PIC under low interference-

to-white-noise ratio. Furthermore, the new PIC with opti- mized interference beam achieves almost the same perfor- mance as the optimum element space processor without precise knowledge of the interference direction.

1728 d. Acoust. Soc. Am., Vol. 89, No. 4, Pt. 1, April 1991 M.H. Er: Technique for designing postbeamformer canceller 1728

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Page 6: A new technique for designing a postbeamformer interference canceller

APPENDIX

This appendix derives the expressions for the optimum weight, the mean output power, and the output SNR of the new PIC described in Sec. II.

Substituting E, and E2 for V and U, respectively, it fol- lows from (7•and (8) that the opumum weight andlhe mean output power of the new PIC are given by

and

•o -- E•R E2/E2UR E2 (A1)

P( •o ): E•R E, -- E2HR E l E•HR E 2/E•HR E 2 . (A2) Recall that in an environment consisting of two uncorrelated sinusoidal sources and uncorrelated white noise, the array correlation matrix R is given by

R -- psSsS• n + p,S,S• + oa, I. (A3) Now

E•R E, : Ps lg•Ss l :• -4-P, lg•S,I :• +

E•R E=: ps I g•Ss I :• + •, I g•S, I :• + • E•E=,

(A4)

(A5)

and

(A6)

But E, and E2 have the properties that

E,•E, -- E•E2 -- 1, (A7)

E,•Ea = E•E, = 0, (A8) and

WgSs = Ss"W• =0. It follows from (A7)-(A9) that

E•R E, -- ps IE•Ss I:' + p, [ E,'-'S, E•R E• = p, IE•S, I:' + •,

(A9)

(A10)

(All)

and

E,•R E2 =p,E,•S,S•E2 ß (A12) Substituting (A 10)-(A 12 ) into (A 1 ) and (A2) and after some manipulation, one obtains

E,•S,S•Ea •o -- (A13)

and

P(•o) =ps IE,•Ss [ • +p, IEgS, I • + •

-- p] IE,'-'S., I:•IEgS., I:•/ (p., IEgS., I :• + • )

=PslEgSsl • + •[1 +p, IE•&IV

x ( p, IE•S,7• •)1. (A14) It follows from (A14) that the output signal power and the output noise power of the new PIC are given by

Ps(•o) -ps IE•Ss I • CA15) and

P•(•o) =• [ • +p, IE•&IV(p, IE•S,I • + •) ]. (A16)

Hence the output SNR of the new PIC is given by

SNR(wo ) = • P•(•o) P•(•o)

ps IE,•Ss I •

1 + •/p., IE•S., I • x i =dn/p, I E•S, I =' (A17)

A. Cantoni and L. C. Godara, "Performance of a Postbeamformer Inter- ference Canceller in the Presence of Broadband Directional Signals," J. Acoust. Soc. Am. 76, 128-138 (1984). L. C. Godara and A. Cantoni, "The Effect of Bandwidth on the Perfor- mance of a Postbeamformer Interference Canceller," J. Acoust. Soc. Am. 80, 794-803 (1986). , r• r•_•a ..... ,• o ̂ u .... A•aptivc Array Processor," IEEE Trans. Cir- cuits Systems CAS-34(7), 721-730 (1987). L. C. Godara, "Analysis of Transient and Steady State Weight Covariance in Adaptive Postbeamformer Interference Cancellet," J. Acoust. Soc. Am. 85, 194-201 (1989). L. C. Godara, "Postbeamformer Interference Cancellet with Improved Performance," J. Acoust. Soc. Am. 85, 202-213 (1989). M. H. Er, "Antenna Array Synthesis Using Eigenvector Technique," Pro- ceedings International Conference on Signal Processing '90, Beijing, Chi- na, October 22-26 (1990).

1729 J. Acoust. Soc. Am., Vol. 89, No. 4, Pt. 1, April 1991 M.H. Er: Technique for designing postbeamformer canceller 1729

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