Optimization of waveforms for UWB circular and disk antenna arrays

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Signal Processing 85 (2005) 657–665 www.elsevier.com/locate/sigpro

Optimization of waveforms for UWB circular and disk antenna arrays Yongfu Huang, Guangguo Bi, Xiangning Fan National Mobile Communication Research Laboratory, Department of Radio Engineering, Southeast University, Nanjing 210096, PR China Received 27 June 2003; received in revised form 11 November 2004

Abstract The basic pulse waveform is an important factor that affects the angular resolution of UltraWideBand (UWB) antenna array. The optimization of these basic pulse waveform for circular and disk array is considered; the optimization problem can be simplified and translated into a simple form, which can be solved effectively. Several optimized waveforms are given, and the main lobes of energy pattern of these optimized waveforms are much narrower than those of corresponding Gaussian waveforms. Numerical results show that optimized waveforms achieve considerable improvement in angular resolution. r 2004 Elsevier B.V. All rights reserved. Keywords: UWB; Non-sinusoidal radar; Waveform optimization

1. Introduction UWB systems and non-sinusoidal systems have attracted much attention recently. These systems use extreme short pulse as basic waveform, the time domain durations of these pulses are usually less than 1 ns and the bandwidths of them are larger than 1 GHz. Due to their large bandwidth, UWB signals can achieve high time domain Corresponding author. Tel.:+86 25 379 3267; fax:+86 25 771 2719. E-mail addresses: [email protected] (Y. Huang), [email protected] (G. Bi), [email protected] (X. Fan).

resolution and range resolution, this property makes UWB suitable for radar applications. Besides this, UWB can be used to transmit high data rate low power communication applications [1]. Since UWB systems do not use conventional sinusoidal waveforms, one naturally considers that the pulse waveform may have important effects on the system performance, and the optimal waveforms may be different for different systems and applications. Here we will consider the waveform optimization for direction finding application. Waveform optimization for antenna and radar has been studied for a long time. For impulse radar, the range resolution can be improved by

0165-1684/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2004.11.006

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Nomenclature r O P f(t) D f; y fm Rm

the radius of the array the center of the array the position of the target the basic pulse waveform the distance between O and P the angles of the target as Fig. 1 the azimuth of the mth array element ¼ Rðy; f; rm ; fm Þ; the distance between the target and the mth element V ðt; f; yÞ the normalized output signal of the array Gðf; yÞ the energy pattern, i.e., the energy of V ðt; f; yÞ

optimization of code sequences and pulse compression filters [2–5]. For smart antenna arrays in sinusoidal carrier-based systems, the main lobe of the array response pattern is narrowed and the side lobe is suppressed by adjusting the weighting coefficients of the array elements [6–9]. In this way, the interference arising from out of main beam direction can be null off or greatly suppressed. In [10–14], the average power pattern and peak power pattern of non-sinusoidal impulse radars that use Gaussian pulse are defined and analyzed. It is shown that the energy patterns produced by Gaussian waveforms are better than those produced by sinusoidal waveforms, i.e., the main lobe of the patterns produced by Gaussian waveforms are narrower than those produced by sinusoidal waveforms. In this paper, we will consider the optimization of the basic waveforms of UWB antenna arrays. We will show that in general the optimized waveforms are very different from Gaussian waveforms. At non-main beam direction, i.e., ‘‘side lobe direction’’, the normalized response patterns of optimized waveform arrays are lower than those of Gaussian waveform array by more than 26 dB for disk arrays with radius rX0:6 m: This improvement is important in location and direction finding application. In Section 2, the signal model is described, the optimization problem and corresponding solution

W ðf; yÞ the weighting function U the cost function of the optimization problem cn ðtÞ ¼ sin cð2W ½t  ðn=2W ÞÞ; the nth basic function in the sampling theorem an ¼ f ðn=2W Þ a ¼ ½a1 ; a2 ; . . . ; aN T RT Ak;l ¼ ck ðo0 ; tÞcl ðo0 ; tÞdt T

J 0 ðzÞ; J 1 ðzÞ the Bessel functions of the first kind with order 0 and 1, respectively ðÞT transpose

method are then presented in Section 3, several optimized waveforms and their energy patterns are presented in Section 4, and finally Section 5 concludes the paper.

2. Signal model We will use the signal model similar to that in [13], two kinds of antenna arrays are considered: circular array and disk array. A circular array is composed by M omnidirectional antenna elements that are uniformly distributed on a circular ring with radius r, as in Fig. 1. The ring is in x–y plane, and its center O is the common origin of x, y and z-axes. A basic pulse f (t) is scattered back by the target PðD; f; yÞ that is far away from O, D is the distance between P and O, f; y are angles of the target as in Fig. 1. Denote the azimuth of the mth element as fm ; the distance between the target and the mth element is Rm ¼ Rðy; f; r; fm Þ: The received signal at the mth element is f ðt  Rm =cÞ (the accurate form of the received signal should be kðRm Þ  Sðt  Rm =cÞ; kðRm Þ is an Rm-dependent attenuation term. But when Rm br; the attenuation kðRm Þ will be nearly the same for all elements. So for simplicity, we assume that the received signal can be expressed as f ðt  Rm =cÞ; where c is the speed of light. The normalized output signal of the

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array is V ðt; f; yÞ ¼

M 1 X f ðt  Rm =cÞ: M m¼1

(3)

When M ! 1; the normalized output can be approximated as Z r 1 V ðt; f; yÞ 2 r0 dr0 pr 0 Z 2p f ðt  Rðy; f; r0 ; f0 Þ=cÞ df0 : ð4Þ 0

An explanation of approximations (2) and (4) is given in the appendix. For both arrays, the energy pattern is defined as the energy received from direction f; y: Z Gðf; yÞ ¼ jV ðt; f; yÞj2 dt: (5)

3. The formulation, simplification and solution of the optimization problem

Fig. 1. The geometric and relative notations of (a) circular array, (b) disk array.

array is V ðt; f; yÞ ¼

M 1 X f ðt  Rm =cÞ: M m¼1

To achieve good angular resolution, the main lobe of Gðf; yÞ should be as narrow as possible, i.e., if G(0,0) is normalized to be 1, Gðf; yÞ should be made as little as possible for any other direction (f; y). The cost function of our optimization is defined as Z p=2 Z 2p U¼ df Gðf; yÞW ðf; yÞ dy; (6) 0

(1)

By a similar method as [1], when the element number M is very large, i.e., M ! 1; the normalized output can be approximated as Z 1 2p V ðt; f; yÞ

f ðt  Rðy; f; r; f0 Þ=cÞdf0 : (2) 2p 0 For disk array, M omnidirectional antenna elements are uniformly distributed over a disk with radius r. Using similar notations, Rm ¼ Rðy; f; rm ; fm Þ; the normalized output of the

0

where W ðf; yÞ is the weighting function, which can be flexibly chosen according to particular applications. In the following numerical examples, we use W ðf; yÞ ¼ y2 : Now the optimization problem can be formulated as: find a band-limited signal waveform f ðtÞ; so that min U

R

2

R T=2

(7) 2

subject to (i) jf ðtÞj dt ¼ 1; (ii) T=2 jf ðtÞj dtX E 0 ; (iii) F^ ðoÞ is band-limited within frequency band ½o0 ; o0 ; where F^ ðoÞ is the Fourier transform of f ðtÞ: The first constraint is to normalize the waveform f ðtÞ; the second constraint requires the

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energy of the waveform to be concentrated within a limited time duration. According to the sampling theorem, we know that f ðtÞ can be expanded as  h  n i X  n  f ðtÞ ¼ f sin c 2W t  2W 2W n X D an cn ðtÞ; ð8Þ ¼ n

We can see that U is a quadratic function of a ¼ ½a1 ; a2 ; . . . ; aN T ; where Z

f ðtÞ ¼

an cn ðtÞ:

(9)

Substituting (9) into (2) and (4), we have V ðt; f; yÞ ¼

N X

an jn ðf; y; tÞ;

(10)

n¼1

where for circular array jk ðf; y; tÞ

and Z jk ðf; y; tÞjl ðf; y; tÞ dt Z 1 ^ k ðf; y; oÞj ^ l ðf; y; oÞ do: ¼ j 2p

1 2p

Z

2p

 Gð1=2Þ 2 4pW  Z 2pW kl exp jo 2W 2pW o 2 r1 cos y do: J0 c

Gk;l ðf; yÞ 2

ck ðt  Rðy; f; f0 Þ=cÞ df0 :

0

For disk array, Z

jk ðf; y; tÞ

1 pr2 Z

r

r0 dr0 ck ðt  Rðy; f; r0 ; f0 Þ=cÞ df0 : ð12Þ



Gð1=2Þ 2 2W pr2 Z 2pW kl ejo 2W

Gk;l ðf; yÞ 2

Substituting (9) into (6), we have Z

W ðf; yÞ dy 0

jk ðf; y; tÞjl ðf; y; tÞ dt;

N X N X k¼1 l¼1

¼

h

2p

df 0

Z

¼

Z

p=2

ak al

k¼1 l¼1

N X N X k¼1 l¼1

Z ak al

ð16Þ

For disk array,

0 2p

0

N X N X

ð15Þ

It is very time consuming to directly evaluate Gk;l ðf; yÞ because: (a) it is relative to ck ðtÞ; but ck ðtÞ couples with R as in (11) and (12); (b) it is involved with multi-layer integration. Fortunately, we can decouple Rðy; f; f0 Þ from ck ðtÞ by Fourier transform, and then the simplification can be carried out in frequency domain. The simplification of G k;l ðf; yÞ is detailed in the appendix.For circular array,

(11)



(14)

0

Gk;l ðf; yÞ ¼

n¼1

U¼

2p

Gk;l ðf; yÞW ðf; yÞ dy

df 0

where W ¼ ðo0 =2pÞ is the frequency bound in Hz. Due to the constraint (ii), we will only consider those signals f ðtÞ that have following form: N X

Z

p=2

C k;l ¼

Z

p=2

2p

W ðf; yÞG k;l ðf; yÞ dy;

df 0

0 T

ak al Ck;l ¼ a Ca:

2pW

o i2 cr J1 r sin y do: ð17Þ o sin y c RT Denote Ak;l ¼ T ck ðo0 ; tÞcl ðo0 ; tÞ dt; with these elements, we can construct a matrix A. Now problem (7) has been translated as: Find optimal coefficients a, so that min aT Ca

ð13Þ

a

subject to (i) aT a ¼ 1; (ii) aT AaXE 0 :

(18)

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Problem (18) can be solved with the standard Matlab function ‘fmincon’. Once the coefficient vector a is obtained, the corresponding optimized waveform can be determined with (8). If constraint (ii) is removed, the optimal coefficients a of (18) is the eigenvector of C that correspond to the minimum eigenvalue.

4. Numerical examples Using the above method, we have optimized waveforms for six arrays: (1) circular array with r ¼ 0:1 m; (2) circular array with r ¼ 0:6 m; (3) circular array with r ¼ 1:2 m; (4) disk array with r ¼ 0:1 m; (5) disk array with r ¼ 0:6 m; (6) disk array with r ¼ 1:2 m: For all these six cases, the constraint parameters are W ¼ 10 GHz; E 0 ¼ 0:96; T ¼ 2 ns; i.e., the bandwidth is 10 GHz, and the waveform concentrates 96% of its energy within a duration of 2 ns. We use the weighting function

661

W ðf; yÞ ¼ y2 : This weighting function is an increasing function of y; which implies that an energy pattern Gðf; yÞ with higher sidelobe will be subject to higher cost. In other words, the optimal solution of the minimum problem should correspond to an energy pattern Gðf; yÞ with low sidelobe. This is confirmed by the numerical results: the optimized waveforms, together with the ‘narrowest’ Gaussian waveforms that concentrate 96% of their energy within the frequency band [10, 10] GHz are shown in Fig. 2. The square of their Fourier transforms, i.e., jF ðoÞj2 are shown in Fig. 3. Their normalized energy patterns received Gðf; yÞ are shown in Fig. 4. For jyj ¼ p=24; in the above six arrays, the side lobes of optimized waveforms are lower than those of Gaussian waveform by 13.24, 9.40, 8.47, 5.05, 26.46 and 32.75 dB, respectively. This means that arrays with optimized waveforms can get better interference suppression performance, i.e., better angular resolution than those with Gaussian waveform.

Fig. 2. Optimized waveforms and corresponding Gaussian waveforms. Solid: optimized, dot: Gaussian.

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Fig. 3. jF ðoÞj2 of optimized waveforms and Gaussian waveform. Solid: optimized, dot: Gaussian.

This improvement can be explained as follows: For Gaussian waveform f Gaussian ðtÞ ¼ expðt2 =2s2 Þ; there is only one tunable parameter s; while for optimization problem (18), the number of tunable parameters is in fact equal to [length(a)–2], where –2 represents the effect of two constraints. More tunable parameters correspond to ‘‘more candidate waveforms’’, and the optimal one among them should be very likely ‘‘better’’ than Gaussian waveform in terms of angular resolution. It can be seen that the angular resolutions become better for larger arrays, and angular resolutions of disk arrays are better than that of circular arrays. This can be understood as follows: remember that each element produces a delayed version of the basic waveform, and the array output is the average of all these delayed versions. When the element number is increased or the size

of array is enlarged, all delayed versions become more different for signal coming from not-mainlobe direction. Then an optimal waveform can be elaborated such that its delayed versions are ‘‘largely cancelled out’’ for all side-lobe directions. On the contrary, if the array size or the element number is very small, then all elements tend to produce nearly the same waveform even for a signal coming from side-lobe direction. And their average (the output of the array) cannot be ‘‘cancelled out’’. This means a high side-lobe for the array.

5. Conclusion As shown above, waveform optimization is an effective way to improve the angular resolution of circular arrays and disk arrays. The optimized

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Fig. 4. The normalized energy patterns of optimized waveforms and Gaussian waveforms. Solid: optimized, dot: Gaussian.

waveforms in general are quite different from Gaussian waveforms. The optimized waveforms in our numerical examples have longer duration and higher frequency band than corresponding Gaussian waveforms, but have narrower bandwidth and lower peak amplitude than corresponding Gaussian waveforms. These properties make these optimized waveforms suitable for direction finding applications that do not require a very high pulse repetition rate. The numerical examples here imply that waveform optimization may be a good way to improve the performance of systems that use nonsinusoidal pulses.

Acknowledgements This work is supported by Chinese 863 project under grant number 2001AA123042, and part of this work has been accepted by VTC fall 2003.

Appendix A. The justification of approximations (2) and (4): By definition, the definite integration of hðxÞ in an interval [a,b] satisfies: Zb hðxÞ dx ¼ lim

N!1

a

¼ lim

N!1

N X

hðxn Þg Dxn

n¼1 N X n¼1

hðxn Þg

ba ; N

ðA:1Þ

where [a,b] is divided into N small sub-intervals: [an,bn] ¼ [a+(n1)g((ba)/N), a+ng((ba)/N)], each with width Dxn ¼ ððb  aÞ=NÞ; and xn 2 ½an ; bn : In the case of circular disk, for a given r, f ðt  Rm =cÞ can be viewed as a function of f0 ; f0 2

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½0; 2p: Denote F ðf0m Þ ¼ f ðt  Rðy; f; r; f0 Þ=cÞ; then when M is very large

The following properties of Bessel functions will be applied:  Z 2p pffiffiffi 1 ejz cos f df ¼ 2 pG (B.1) J 0 ðzÞ; 2 0

M 1 X V ðt; f; yÞ ¼ f ðt  Rm =cÞ M m¼1 M 1 X f ðt  Rm =cÞ M!1 M m¼1

lim

Z

M 1 X 2p F ðf0m Þ ¼ lim M!1 2p M m¼1 Z 1 2p ¼ F ðf0 Þ df0 2p 0 Z 1 2p ¼ f ðt  Rðy; f; r; f0 Þ=cÞ df0 : 2p 0

J 0 ðzrÞr dr ¼

¼ lim

N!1

n¼1 N X n¼1

S ; N

M X

1 V ðt; f; yÞ lim f ðt  Rm =cÞ M!1 M m¼1 M 1 X pr2 hðxm ; ym Þ 2 M!1 pr M m¼1 ZZ 1 ¼ 2 hðx; yÞ dx dy pr

¼ lim

1 ¼ 2 pr Z

ðx;yÞ2O Z r 0

r dr0

0 2p

f ðt  Rðy; f; r; f0 Þ=cÞ df0 ; ðA:4Þ

0

this explains Eq. (4).

Rðy; f; r1 ; f1 Þ  Rðy; f; r2 ; f2 Þ

 sin y½r1 cosðf  f1 Þ  r2 cosðf  f2 Þ ðB:4Þ with (B.3), (B.4) and (B.1), we have Z2p df1

ðA:3Þ

where O is divided into N small sub- regions On ; each with area S/N. In the case of disk array, f ðt  Rm =cÞ can be viewed as a function of coordinate of the mth element ðxm ; ym Þ 2 O; O is the region of the disk. Denote hðxm ; ym Þ ¼ f ðt  Rm =cÞ; then

ðB:3Þ

So,

Z2p hðxn ; yn Þ

(B.2)

D  r0 sin y cosðf  f0 Þ:

This explains Eq. (2). Similarly, by definition, the definite integration of hðxÞ in a region O satisfies ZZ N X hðx; yÞ dx dy ¼ lim hðxn ; yn Þg Dsn ðx;yÞ2O

r J 1 ðzrÞ z

and note that when Dbr qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ D2  2Dr0 sin y cosðf  f0 Þ þ r0 2

ðA:2Þ

N!1

Appendix B. The simplification of G k;l ðf; yÞ

0

efðjo=cÞ½Rðy;f;r;f1 ÞRðy;f;r;f2 Þg df2

0

Z2p

eðjo=cÞr1

sin y cosðff1 Þ

df1

0

Z2p

eðjo=cÞr2

sin y cosðff2 Þ

df2

0

 2    o  1 o r1 cos y J 0  r2 cos y ¼ 4pG J0 2 c c  2   2 1 o r1 cos y : ¼ 4pG J0 ðB:5Þ 2 c To decouple Rðy; f; f0 Þ from ck ðtÞ; the simplification is carried out in frequency domain. For circular array, the Fourier transform of (11) is Z 1 2p I f2pW pop2pW g ^ k ðf; y; oÞ ¼ j 2p 0 2W   k Rðy; f; r0 ; f0 Þ þ df0 exp jo 2W c

ðB:6Þ

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with (15), (B.6) and (B.5), we have Z G k;l ðf; yÞ ¼ jk ðf; y; tÞjl ðf; y; tÞ dt Z 1 ¼ j^ k ðf; y; oÞj^ l ðf; y; oÞ do 2p Z 2pW Z 2p 1 ¼ do df1 32p3 W 2 2pW 0  Z 2p kl exp jo 2W 0

Rðy; f; r; f1 Þ  Rðy; f; r; f2 Þ þ df2 c  Z 

Gð1=2Þ 2 2pW kl

2 exp jo 4pW 2W 2pW o 2  r1 cos y J0 do: ðB:7Þ c For disk array, the Fourier transform of (12) is Z r Z 2p 1 I f2pW pop2pW g ^ k ðf; y; oÞ ¼ 2 r0 dr0 j pr 0 2W 0 0

0

ejo½ðk=2W ÞþRðy;f;r ;f Þ=c df0

ðB:8Þ

by a similar method as above, and with the help of (B.2) Z G k;l ðf; yÞ ¼ jk ðf; y; tÞjl ðf; y; tÞ dt Z 1 ¼ j^ k ðf; y; oÞj^ l ðf; y; oÞ do 2p  Z Gð1=2Þ 2 2pW joðkl=2W Þ e

2 2W pr2 2pW h cr o i2 J1 r sin y do: ðB:9Þ o sin y c

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