IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. 4, NOVEMBER 1988 513

Antenna Patterns of Nonsinusoidal Waves with the Time Variation of a Gaussian Pulse-Part I1

Abstract-In Part I of this series of papers, antenna patterns for nonsinusoidal waves with the time variation of a Gaussian pulse, received (or radiated) by a linear array of sensors (or radiators) were derived. Computer plots of the peak-amplitude pattern, peak-power pattern, energy pattern, and slope pattern were presented. In this paper, the above antenna patterns are derived for a planar array of sensors (or radiators) receiving (or radiating) Gaussian pulses. Based on the characteristics of the Gaussian pulse, which are presented in Part I, the principle of frequency-domain array beam forming with Gaussian pulses is described. An expression for the antenna energy pattern based on the frequency- domain analysis is derived and plotted for comparison to the one obtained in Part I from the time-domain analysis.

Key Words-Nonsinusoidal wave, planar array, antenna pattern, beam forming, Gaussian pulse.

Index Code-IlJh/c/d/g

I. INTRODUCTION HE PLANAR ARRAY antenna has found many T applications in radar and radio communications where

long range detection, high angular resolution, and three- dimensional space coverage are desired [ 11-[3]. The theory of planar array for nonsinusoidal waves with the time variation of a rectangular pulse has been developed [4], [5], and three- dimensional antenna patterns have been derived too [4]. In Section 11, antenna patterns for a planar array of sensors receiving Gaussian pulses are derived. Three-dimensional peak-amplitude pattern, peak-power pattern, energy pattern, and slope pattern are presented too. In Section 111, frequency- domain array-beam forming with Gaussian pulses is intro- duced. The principle of the frequency-domain beam forming technique is based on the fact that the auto-correlation function and the energy spectral density of a Gaussian pulse are of a Gaussian shape too [6]. An expression for the antenna energy pattern of a linear array is derived from the frequency-domain analysis. A computer plot of the antenna energy pattern is presented for comparison to that derived in Part I [6]. Conclusions are given in Section IV.

11. ANTENNA PATTERNS OF PLANAR ARRAY Consider the rectangular-shaped planar array with (2rn +

1) x (2n + 1) sensor elements shown in Fig. 1 . The planar array is composed of 2rn + 1 linear arrays arranged parallel to they axis, with equal spacing d, between any two adjacent

Manuscript received January 30, 1988; revised April 8, 1988. This research is supported by The Research Management Unit (RMU) of Kuwait University, Project EE 03 1.

The author is with the Department of Electrical and Computer Engineering, Kuwait University, P.O. Box 5969, Safat, 13060, Kuwait.

IEEE Log Number 8823349.

ones. Each of the linear arrays consists of 2n + 1 sensor elements with interelement spacing dy between adjacent sensors. The dimension of the planar array along the x axis is denoted L, = 2rn d,, and the dimension along the y axis is denoted Ly = 2n dy . Each sensor within the planar array can be identified by a pair of integers i = 0, + 1, f 2 , * . . , k m and j = 0, f 1 , f 2 , a , f n, where i and j indicate the row R, and the column Cj at which the sensor is located.

To derive a beam pattern for the planar array in Fig. 1, the voltages from the sensors must be added properly. A beam- Emning system for the planar array of Fig. 1 is shown in Fig. 2. Let a planar wavefront be received by the array sensors in Fig. 2. The sensor elements transform the received wavefront into voltages V j j ( t ) , i = 0, f 1, + 2 , * * e , + rn, and j = 0, f 1, + 2 , * * . , f n. According to Fig. 2, the voltage signals from the sensors of each linear array at row R j are summed in the summer SUM,, which yields the voltage signal V,(t)

+ n

V;( t )= V;j ( t+7yj ) , i=O, + I , + 2 , ..., f r n . (1)

In ( l ) , 7yj is the progressive propagation delay along each linear array parallel to they axis

j = - n

TYj=( jdy/c) sin 6 sin 4, j = O , k 1, f 2 , * e * , f n ( 2 )

where c is the propagation velocity. The voltages V j ( t ) from the summers SUMj’s are summed

in the summer SUM and the result is the voltage signal VT( t )

In (3), 7,; is the progressive propagation delay associated with the received wavefront along the x direction that affects the voltages Vi( t )

7,,=(idX/c)sin6cos4, i=O, k l , +2, * * e , f r n . ( 4 )

With the help of (I) , ( 2 ) , and (4), (3) can be written in terms of the voltage signal V,,(t) of each sensor element

0018-9375/88/1100-0513$01.00 O 1988 IEEE

5 14 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. 4, NOVEMBER 1988

e ........ e e

Fig. 1. The geometry of a rectangular planar array of (2m + 1) x (2n + 1 ) elements.

j - ........ c. ’ ’ ‘ ‘ ’ ’. .rl ’0 4 7” v 7 T T

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

SUM-, ,p ‘I Y Y

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

S U M O

‘I .............. . . . . . . . . . . . .

Fig. 2. Beam-forming system consisting of a rectangular planar array with (2m + 1) x (2n + 1 ) sensor elements and summers (SUM’S).

where

T ~ = T,;+ rYj= [ ( id , /c) cos 4 + ( j d y / c ) sin 41 sin 8. (6 )

For a square planar array with m = n = k , d, = dy = d , and L, = Ly = L

T i j = ( d / c ) sin f3[i cos 4 + j sin 41,

(7) i=O, +1, + 2 , . . a , + k j = o , + l , + 2 , e . . , +k.

Let the received wavefront at the sensor elements in Fig. 2 be planar with the time variation of a Gaussian pulse Q ( t ) having peak amplitude ( H A T ) and relative duration 2AT

Q ( t ) = ( H A T ) exp [ - a( t / A T ) 2 ] . (8 )

Each sensor element transforms the received wavefront into a voltage signal with the time variation of the Gaussian pulse Q ( t ) . Thus, the voltage signal V,,(t + T;,) in (5) can be

expressed as follows:

~ , ( ~ + T ~ ) = ( E / A T ) exp { - ? r [ ( t + ~ ; , ) / A T l ~ )

= ( E / A T ) exp [ - ? T ( ~ / A T ) ~ ]

exp [ - 274 t / A T) (q , /A T ) ]

. exp [ - ? ~ ( T ~ / A T ) ~ ] . (9)

Based on (6) the ratio (T; , /AT) in (9) can be written in the form

r i j /AT= (i/2m)(2mdX/cAT) cos 4 sin 0

+ (j /2n)(2ndy/cAT) sin 4 sin 0. (10)

With the help of the following substitutions (10) can be reduced to a simplified expression

(1 1 ) a =cos 4 sin 0, /3 = sin 4 sin f3

px = (2md,/cA T ) = L,/cA T , py = (2ndy/cA T ) = Ly/cA T.

(12)

HUSSAIN: ANTENNA PATTERNS OF NONSINUSOIDAL WAVES-PART II

The parameters px and py may be considered as design parameters since they consist of the planar array dimensions L, and Ly, and the duration AT which defines the nominal frequency bandwidth Af = l /AT. Note that the relative duration of Q( t) given in (8) is 2AT, which yields the nominal frequency bandwidth B = Af/2. For a high-resolution, line- of-sight, all-weather radar the duration 2AT of Q ( t ) in (8) should be in the order of 0.1 to 1 ns. Such pulse durations result in the frequency bandwidth in the range 1 GHz I B 5 10 GHz, over which the atmospheric attenuation is at its minimum. Insertion of ( 1 1 ) and (12) into (10) yields

rV/A T= (i/2m)pXa + (j/2n)pyP,

(13) i = O , f l , f 2 , - - e , f m , j = O , k1, k 2 , fn.

For a square planar array with m = n = k, L, = Ly = L, and p, = py = p

~ j j / A T= (p/2 k)[ ia +jP] ,

Insertion of (9) and ( 1 3) into (5) results in the voltage signal VT(t, a, 6) at the output of SUM in Fig. 2

VT( t , a , P ) = ( H A T ) exp [ - a( t /A T I2]

exp { - 274 t / A T)[( i/2m)pxa

For a planar array with a large number of sensors, m and n are large numbers, the summations in (15) can be approximated by integrals. Multiplication and division of (15) by the factors (2m) and (2n) , and by making the substitutions

i= -m+q= - 1/2 i= +m+q= + 1/2 q=(i/2m), dq=d(i /2m),

(16) j = -n+[= -1/2 j = + n - + [ = + 1 / 2 4 = ( j /2n) , d t = d(j /2n) ,

one obtains Vr(t , a, P ) in the following integral form:

= (4mn)(E/A T ) exp [ - a( t /A T ) 2 ]

If the Gaussian wavefront is received from the angular directions 0 and b with a = B = 0. the resulting voltage

~

Y The ratio U(a , P)/U(O, 0) is defined as the energy pattern

515

signal at the output of SUM in Fig. 2 is

VT( t , 0, 0) = (4mn)(E/A 7') exp [ - a( t / A T ) 2 ] ( 1 8 )

with the peak amplitude, at t = 0, VT(O, 0, 0) = (4mn) (E/ A T ) . According to (17), the amplitude of VT( t , a, 0) at t = 0 is

+1/2 +1/2

-1 /2 -1 /2 VT(O, a , P ) = (4mn)(E/AT) l s The ratio VT(O, a, P)/VT(O, 0, 0) yields the peak-amplitude pattern A (a, 0) for the planar array in Fig. 2

A ( a , P ) = 1 , for a = P = O

P + 1 / 2 P + l / 2

+ ( p y P ) [ ] 2 } dq dE, otherwise. (20)

The function A (a, P ) may be referred to as the array factor of planar array receiving (or radiating) nonsinusoidal waves with the time variation of a Gaussian pulse.

The peak-power pattern P(a, P ) is defined as the square of the peak-amplitude pattern A (a, 0)

P(a , P ) = [ A ( a , P ) I 2 = 1 , for a = P = O

+(pyP)EI2} dq d [ ) 2 , otherwise. (21)

The energy U(a, 0) of the voltage signal VT( I, a, 0) given in (17) can be calculated as follows:

=(4mn)2(E/AT)2 s + m -m exp [ - 2 ~ ( t / A T ) ~ l

For a = P = 0, one obtains

U(0, 0) = (4mn)2(E/A T ) 2 +m - m exp [ - 274 t /A T ) 2 ] dt.

516 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. 4 , NOVEMBER 1988

t/AT - (b)

Fig. 3 . The time variation of the voltage signal V,( t , a, 0) given in (17), for square planar arrays with (a) p = 3, and (b) p = 6; the plots correspond to different values of a and 0.

W(a, P ) of the planar array in Fig. 2

W ( a , @)= 1 , for a=@=O

+ m

= l exp [ - 274 t / A T ) 2 ] - m

+ m 1 exp [ - 274 t / A T ) 2 ] dt , otherwise. (24) - m

By letting px = py = p in (20), ( 2 1 ) , and (24) one obtains the antenna patterns of a square planar array.

A slope pattern g(a, @) can be derived for the planar array in Fig. 2 by following the numerical technique discussed in [4] and [6 ] . The time variation of the normalized voltage signal V,(t, a, @)/(4mn) ( H A T ) given in (17) is shown in Fig. 3

for different values of a and 0. According to Fig. 3 , V,(t, a, 0) is a Gaussian pulse whose duration and amplitude are functions of a and @ for a given value of p . A three- dimensional plot of the slope s' of the least-square-difference line [6, ( 3 1 ) ] that best fits the rising section of V,(t , a, 0) versus a and @ yields a slope pattern s'(a, @) [4] . The slope pattern is specifically known for nonsinusoidal waves, it does not exist for sinusoidal waves.

Computer plots of the peak-amplitude pattern A ( a , P ) , peak-power pattern P(a, P ) , energy pattern W(a, P ) , and slope pattern $(a, @) are shown in Figs. 4-11 for square planar arrays with p = 1, 3, 6, and 10. The main lobe of the antenna patterns narrows and the restlobes drop as the value of p is increased. Hence, the larger the value of the design parameter p the better the angular resolution of the antenna patterns. According to ( 1 2 ) , a large value of p can be obtained by either increasing the array size or by operating with pulses of a narrow duration AT, which means a large frequency bandwidth Af. The trade-off between array size and frequency bandwidth for good resolution is useful in practice. The peak- power pattern P(a, @) shown in Figs. 6 and 7 is most attractive for achieving good angular resolution since the width of its mainlobe and the magnitude of the restlobes are the smallest of all those of the other antenna patterns.

III. FREQUENCY-DOMAIN ARRAY BEAM FORMING The conventional method of array beam forming with zero-

bandwidth sinusoidal waves yields antenna patterns with [(sin x)/xI2 shape, and the classical equation E = kc / fL for the resolution angle: k is a constant, c is the propagation velocity, f is the frequency, and L is the array length. For achieving a smal l resolution angle either the frequency f or the array length L must be increased. It has been shown [8] that the principle of array beam forming with nonsinusoidal waves, having a triangular auto-correlation function, yields the equation E =

Kc/AfL F N f o r the resolution angle, where K is a constant, Af is the nominal frequency bandwidth, S/N is the signal-to- noise power ratio, and c and L are as defined above. The trade-off between signal power, frequency bandwidth, and array size for a small resolution angle is attractive in practice whenever the attenuation is an increasing function of fre- quency as in underwater acoustic beam forming [7] and high- resolution, all-weather radar.

The principle of frequency-domain array beam forming based on nonsinusoidal waves with the time variation of a Gaussian pulse is depicted in Fig. 12. The beam forming system consists of 2m + 1 omnidirectional sensors with interelement spacing d . The sensors transform the received wavefront into voltage signals V j ( t ) , i = 0, f 1 , & 2 , a ,

f m , for processing. Each voltage signal V,( t ) is first processed through a matched filter (MFi) to enhance the signal-to-noise power ratio, and second through the Fourier transform processor (FT;). The outputs of the FT's are summed by the summer circuit (SUM) to obtain a beam pattern.

Let a planar wavefront with the time variation of the Gaussian pulse Q ( t ) defined in (8 ) be received by the linear array of sensors in Fig. 12. For noise suppression the voltage

HUSSAIN: ANTENNA PATTERNS OF NONSINUSOIDAL WAVES-PART II 517

F 9 . m

8.75

8.25 J

- 1 AT-1 .a - 1 .e'-! .e (a) (b)

Fig. 4. Peak-amplitude pattern A ( a , 0) for square planar arrays with (a) p = 1, and @) p = 3, receiving a planar wavefront with the time variation of the Gaussian pulse Q(t ) defined in (8).

7.5

9.5

7 .§

6 .5

5.5 i I .5

k L 13.5

I::: I:::

3.5

2.5 2 .5

1.5 1.5

I . .e

- 1 a I - 1 .e -1 . e 5 .a

(a) (b) Fig. 5 . Peak-amplitude pattern A (a , (3) for square planar arrays with (a) p = 6, and (b) p = 10, receiving a planar wavefront with

the time variation of the Gaussian pulse Q(t) defined in (8).

-1 . e l 4 .e -1 . e 5 .e

(a) (b) Fig. 6 . Peak-power pattern P(a, (3) for square planar arrays with (a) p = 1, and (b) p = 3, receiving a planar wavefront with the

time variation of the Gaussian pulse Q ( t ) defined in (8).

5 1 8 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. 4, NOVEMBER 1988

-10.0

-9.0

-8.0

-7.0

-6.0

5.0 -5.0

t.0

(a) (b) Fig. 7. Peak-power pattern P(a, 6) for square planar arrays with (a) p = 6, and (b) p = 10, receiving a planar wavefront with the

time variation of the Gaussian pulse Q ( t ) defined in (8).

9.75

9.25

8.75

8.29

7.75

7.25

.0

-1.8'4 .e

(a) (b) Fig. 8. Energy pattern W(a, P ) for square planar arrays with (a) P = 1, and (b) p = 3, receiving a planar wavefront with the time

variation of the Gaussian pulse Q ( t ) defined in (8).

10.0

9.0

8.0 I 7.0

8.8

Fig. 9. Energy pattern W(a, 0) for square planar arrays with (a) p = 6, and (b) p = 10, receiving a planar wavefront with the time variation of the Gaussian pulse Q ( t ) defined in (8).

HUSSAIN: ANTENNA PATTERNS OF NONSINUSOIDAL WAVES-PART U 519

-1 .aT-1 .E -1 .E'-1 .E

(a) (b) Fig. 10. Slope pattern s(a, 0) for square planar arrays with (a) p = 1 , and (b) p = 3, receiving a planar wavefront with the time

variation of the Gaussian pulse Q ( t ) defined in (8).

-1.0'-1 .e (a) . .

Fig. 1 1 . Slope pattern s(a, /3) for square planar arrays with (a) p = 6, and (b) p = 10, receiving a planar wavefront with the time variation of the Gaussian pulse Q ( t ) defined in (8).

signal at the output of each sensor is passed through a matched filter MF; whose impulse response h ( t ) is matched to the Gaussian pulse O ( t )

h ( t ) = O( T, - t )

= ( H A T ) exp { - a [ ( T o - t ) / A T ] 2 } (25)

where To is a time constant to assure causality of the matched filter. For simplicity, one may set TO = 0. For off-axis reception 4 > 0, the time variation of the voltage signal V , ( t ) at the output of each sensor is

V , ( t ) = ( E / A T ) exp { - ? T [ ( ~ + ~ ; ) / A T ] ~ } ,

i=O, + 1 , f 2 , e - . , f m (26)

where 7; is the relative delay between the voltage signals V,( t ) and Vo(t)

r i = ( i d / c ) sin 4, i=O, f 1 , + 2 , * e , f m . (27)

The delay r; can be expressed in terms of the design parameter p as done in the preceding section

r; = (2md/cA T ) ( i /2m)A T sin 4 = ( i /2m)A Tp sin 4,

p=2md/cAT, i = O , f l , f 2 , e . . , +m. (28)

Based on (25) and (26), the output of each matched filter MF; in the array system of Fig. 12 is the following voltage signal, which represents the auto-correlation function of V , ( t ) given in (26):

R ; ( t ) = ( E 2 / A T & ) exp { -4 [ ( t + r j ) / A T I 2 } . (29)

The time variation of R ; ( t ) at the output of MF; equals that of the input voltage signal V;( t ) ; both are of a Gaussian shape. However, the duration of R j ( t ) is double that of Vj(t). In the case of beam forming with rectangular pulses [8 ] , the noise suppression process based on a sliding correlator (which is a

2

5 2 0 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. 4, NOVEMBER 1988

I .0’-3.0

Fig. 13. The function Q(u, 4) given in (32) as a function of u and p sin @.

sensors matched Fourier summer f i l ter transform With the help of the substitutions

Fig. 12. Principle of frequency-domain array beam forming based on nonsinusoidal waves with the time variation of a Gaussian pulse. q = i / 2 m , d q = d ( i / 2 m ) , i = -m+q= - 1 / 2 ,

matched filter) results in a triangular pulse whose duration is double that of the processed rectangular pulse. The increase in duration due to filtering is not desirable for achieving good resolution. For beam forming based on a rectangular pulses, a pulse shaping filter called a “slope processor” is employed after the sliding correlator to transform the wide triangular pulse into a narrower pulse (or pulses) [ 8 ] , [ 9 ] . The resolution of the resulting narrow pulse is much better than that of the wide triangular pulse [9] . The beam forming system in Fig. 12 transforms the voltage signal Rj( t ) to the frequency domain via the FT, to account for the increase in duration and to improve the resolution.

The Fourier transform of the voltage signal R;(t) is the energy spectral density S;(o) of V;(t) , which is of a Gaussian shape too [6]

and i = + m + q = + 1 / 2 one obtains

Q(o4) = ( 2 m E 2 ) exp [ - ( u A T ) ~ / ~ T ]

[ + ‘ i 2 - 1 / 2 exp [ j ( w A T ) ( p sin 4 ) q ] d q

= ( 2 m E 2 ) exp [ - (wA T ) 2 / 2 a ]

sin [ ( w A T / 2 ) p sin 41 ( w A T / 2 ) p sin

= ( 2 m E 2 ) exp [ - (wA T)2/27r]

sinc [ (wA T / 2 ) p sin 41. (32)

A three-dimensional plot of the function Q(o, I$), versus o A T and p sin 4, is shown in Fig. 13. For small values of p sin 4 Q(w, 4) is of a Gaussian shape.

+m The energy pattern W(4) of the beam forming system in Fig. 12 can be calculated as follows:

(30) W(4)= Q(o, 4) d ~ / l + ~ Q<o, 0) d w

S;(w) = 5 { Rj( t ) } = l R;( t ) exp ( - j u t ) d t - m

= E 2 exp ( jw7; ) exp [ - ( A T c 0 ) ~ / 2 a ] ,

- m - m

+ m

i = O , + l , +2, * e - , +m.

The sum of the Si(u)’s from the FT,’s yields the following function at the output of SUM in Fig. 12: exp [ - (wA T ) 2 / 2 a ] sinc [ (wA T / 2 ) p sin 41 d w

- m

+ m

Q(w, 4) = E 2 exp [ - ( o A T ) 2 / 2 a ] exp ( jw7; ) i = - m

= ( 2 m E 2 ) exp[ - (wA T ) 2 / 2 a ]

+ m * ( 2 m ) - ’ exp [ j ( w A T ) ( p sin 4 ) ( i / 2 m ) ]

I = - m

(31) where (28) is used for 7; in (31 ) . For large values of m, the summation in (31) can be approximated by an integration.

- - + m

[ exp [ - (wA T ) 2 / 2 a ] dw - m

(33)

Computer plots of W(4) are shown in Figs. 14 and 15. Fig. 14 shows “(4) as a function of p sin 4 while the plots of Fig. 15 correspond to W(4) as a function of 4 for p = 1, 3 , 6 , and 10, where the design parameter p is defined in (28) . Comparison of the energy patterns in Figs. 14 and 15 to the ones derived in Part I [ 6 , Figs. 9 and 131 of this series of papers, shows that they are almost equal. If one calculates the energy pattern based on the summation of the voltage signals Rj( t ) resulting

HUSSAIN: ANTENNA PATTERNS OF NONSINUSOIDAL WAVES-PART U 5 2 1

6

t

. . . . . . . . . . . . . . . . . . . -10 -

ps in@ - Fig. 14. The energy pattern W(q5), of the beam forming system in the Fig. 12, as a function of p sin 6.

. . . . . . . +' . . . . . . . . . . . . t " " " " " ' " " ' " " " ' ' ' ' ' 2" 40 0

0 x 1 0 - Fig. 15. The energy pattern W(+) as a function 9 for different values of the design parameter p defined in (28)

from the MFi's in Fig. 12, one obtains a beam pattern whose mainlobe is considerably wider than that of the patterns in Figs. 14 and 15. Hence, the frequency-domain array beam forming method depicted in Fig. 12, for Gaussian pulses, has the capability of improving the angular resolution, which is affected by the noise suppression process based on the matched filter.

V. CONCLUSIONS

Antenna peak-amplitude pattern, peak-power pattern, en- ergy pattern, and slope pattern are derived for planar array

of sensors receiving nonsinusoidal waves with the time variation of a Gaussian pulse. Three-dimensional computer plots of the antenna patterns are presented. A comparison of the computer plots shows that the antenna peak-power pattern is the most attractive for achieving good angular resolution because of its narrow mainlobe and low restlobes. The principle of frequency-domain array beam forming based on Gaussian pulses is described. It is based on the fact that the auto-correlation function and the energy spectral density of a Gaussian pulse are of a Gaussian shape too. The frequency- domain beam forming system circumvents the degradation in

5 2 2 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. 4, NOVEMBER 1988

angular resolution due to the process of noise suppression, which is done via a matched filter at each of the array sensors. Antenna energy pattern for the beam forming system is derived based on the reception of Gaussian pulses and a frequency-domain approach. Computer plots of the derived energy pattern are presented for comparison to the ones previously obtained [6 , Figs. 9 and 131, which are based on a time-domain approach. The beam patterns are almost equal and have desirable features for achieving good angular resolution.

REFERENCES [l]

[2]

T. C. Cheston and J. Frank, “Array antennas,” in Radar Handbook, ch. 11. M. I. Skolnik, Ed. R. E. Elliot, “Beam width and directivity of large scanning arrays, Part 2,” Microwave J . , vol. 7, pp. 74-82, Jan. 1964.

New York McGraw Hill, 1970.

W. H. Von Aulock, “Properties of phased arrays,” IRE Trans., vol.

M. G . M. Hussain, “Three-dimensional antenna patterns for nonsinu- soidal waves,” IEEE Trans. Electromagn. Compat., vol. EMC-28, no. 4, pp. 240-248, Nov. 1986. H. F. Harmuth, “Synthetic aperture radar based on nonsinusoidal functions. X . Array gain, planar array, multiple signals,” IEEE Trans. Electromagn. Compat., vol. EMC-23, no. 3, pp. 72-97, May 1981. M. G. M. Hussain, “Antenna patterns for nonsinusoidal waves with the time variation of a Gaussian pulse. Part I,” IEEE Trans. Electro- magn. Cornpat., vol. 30, no. 4, Nov. 1988. R. J . Urick, Principle of Underwater Sound for Engineers. New York: McGraw-Hill, 1967. H. F. Harmuth, “Synthetic aperture radar based on nonsinusoidal function: IX. Array beam forming,” IEEE Trans. Electromagn. Compat., vol. EMC-23, no. 2, pp. 20-27, Feb. 1981. M. G. M. Hussain, “Antenna energy patterns of nonsinusoidal waveforms,” IEEE Trans. Electromagn. Compat., vol. EMC-29, no. 1, pp. 24-31, Feb. 1987.

AP-9, pp. 1715-1725, Oct. 1960.