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Abstract— This paper presents a generalized analytical

expression for directivity of phased array antennas (PAAs). It

includes important design consideration of PAAs like arbitrary

element type & array geometry, complex excitations, mutual

coupling, scan angle and embedded element pattern. We also

present an approximate expression of directivity, helpful for quick

analysis of large PAAs overcoming computing time constraints of

analytical expressions. It provides a quick PAA design tool for

predicting accurate locations of blind spots in desired scan volume

due to the chosen array geometry. The paper also presents study

on wide scan characteristics of periodic sparse phased array

antennas using the proposed expressions.

Index Terms— sparse array antennas, directivity, scan blindness, wide

scan

I. INTRODUCTION

hased array antennas (PAAs) consists of a radiating

aperture supporting regular or periodic arrangement

(equidistant) of radiating elements (REs). In general, to

avoid occurrence of grating lobes (GLs), the inter-element

separation, d/ ( is the operating wavelength) is kept small

(usually < /2) [1]. Thus, these type of PAAs referred to as

dense arrays. A thumb rule for choosing d is 𝑑/ ≤ 1/(𝑠𝑖𝑛𝜃𝑔 +

𝑠𝑖𝑛𝜃𝑜), where g is the angle at which the first GL shall occur

(usually kept 90o to keep the visible region clear from onset of

GLs) [1] and o, represents maximum desired scan angle.

Hence, for instance in order to achieve a desired scan volume

of ±60o, it is required to have 𝑑/ 0.535. However, these

smaller inter-element separations lead to another problem, i.e.,

stronger mutual coupling between elements, which leads to

scan blindness. The scan blindness implies that at particular

scan angles, input reflection coefficient of an array,|𝑖𝑛| → 1,

leading to total reflection or zero directivity [1]. These locations

referred as blind spots. Sparse phased array antennas (SPAAs)

provides an efficient solution for tackling the problem of GLs

due to array elements’ periodicity and scan blindness

simultaneously [2].

The directivity, D of PAAs, dense or sparse, may serve as a

figure of merit for its characterization and is the focus of the

present paper. Various attempts made in the past for computati-

A. Kedar is with Electronics & Radar Development Establishment (LRDE),

DRDO, Bangalore-560093, India (e-mail: [email protected]). L.P.

Ligthart is em. Prof. (retd.), Delft Univ. of Technology, The Netherlands.

on of D, which presents approximate or exact analytical

expressions for it. Tai [3], Bach [4] and Langston [5] proposed

an approximate expression for D of uniformly spaced linear

arrays. These expressions were valid for only isotropic and

dipole REs. Further, neither of these included effect of the scan

angle, arbitrary array geometry, arbitrary RE type, complex

excitations and mutual coupling. Hansen [6] proposed an

expression for D considering mutual resistance term, array

excitations and SLL. There were some more attempts for

computation of D for arrays with specific element type and

excitations [7-13]. Forman [14] proposed an accurate analytical

expression of D considering effect of arbitrary array geometry,

complex excitation, inter-element spacing and scan angle.

However, it did not consider mutual coupling and restricted

only to REs with 𝑐𝑜𝑠𝑛 patterns. Ligthart [15] proposed a

generalized accurate expression for G including effect of

arbitrary shape, complex excitations, mutual coupling, scan

angle and arbitrary RE type. However, the expressions are

computationally intensive especially in the case of large arrays.

Recently Das [16] has also presented a similar analysis for D.

The expressions proposed are computationally intensive,

restricted to only isotropic REs. The need for accurate

expressions for D and importance of incorporation of embedded

element pattern (EEP) in the expression for directivity is well

explained in references [17-19].

This paper proposes a generalized analytical expression of D

for PAAs, which includes most of the important design

considerations in it such as arbitrary array geometry, arbitrary

complex excitations, mutual coupling, scan angle and EEP. In

general, computation of D using such exact analytical

expressions is time consuming in case of large arrays (number

of elements, N>100). Hence, we also present an approximate

expression for D of PAAs assuming aperture with periodic

array grid of infinite extent, i.e. independent of N. Further,

assumption made is that the array comprises of isotropic REs.

These assumptions do not limit the effectiveness of the

proposed expression in providing an accurate estimate of D.

Further, it helps in assessing the scan performance of the chosen

array grid for PAAs. A comparison with earlier publications

[16, 23-24] made herein the paper validates the proposed

expressions. These expressions are accurate for dense as well

as sparse PAAs and used for studying wide scan characteristics

of SPAAs here with the aid of few case studies. The study

Wide Scanning Characteristics of Sparse Phased

Array Antennas Using an Analytical Expression

for Directivity

Ashutosh Kedar, Senior Member, IEEE and L. P. Ligthart, Fellow, IEEE

P



presented in this paper may serve as design considerations for

wide scan phased array antennas.

The remaining paper comprises of various sections; viz.,

Section II presents discussion on the exact expression of

directivity, Section III presents the approximate expression of

directivity, Section IV presents the results and discussion

followed by conclusion, references and appendices.

Fig. 1. Array geometry and its associated coordinate system.

II. EXACT ANALYTICAL EXPRESSION FOR DIRECTIVITY OF A FINITE

SIZED PHASED ARRAY ANTENNA

Fig. 1 shows a planar array with an array boundary, S and its

associated coordinate system. The array consists of REs, N in number,

placed in a quadrant of the xy- plane. An nth element is located at (xn,

yn). The elements are coherently phased to make the array look in a

direction given by (θo, φo). R is the distance of the observation point in

far-field from the origin, O. It is assumed here that the radiation occurs

only in the upper half-space. Directivity, D, of a planar aperture

antenna is defined as [1]:

𝐷(𝜃, 𝜑) = 4𝜋|𝐹(𝜃, 𝜑)𝑚𝑎𝑥|2 𝑃0⁄ (1)

In (1), Po is the total radiating power available from S, expressed as:

𝑃0 = ∫ ∫ |𝐹(𝜃, 𝜑)|2𝑠𝑖𝑛𝜃 𝑑𝜃 𝑑𝜑𝜋

0

2𝜋

0 (2)

In (2), 𝐹(𝜃, 𝜑)is the far-field pattern of the antenna, defined as:

𝐹(𝜃, 𝜑) = ∬ 𝐼(𝑥, 𝑦)𝑒[−𝑖𝑘(𝑥 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜑+ 𝑦𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜑)] 𝑑𝑥 𝑑𝑦 𝑆

(3)

In (3), 𝐼(𝑥, 𝑦) represents the current distribution over S.

In (1), |𝐹(𝜃, 𝜑)2| = 𝐹(𝜃, 𝜑) ∙ 𝐹(𝜃, 𝜑)∗, where ‘*’ denotes a

complex conjugate operator, where 𝐹(𝜃, 𝜑) is defined in (3) and

𝐹(𝜃, 𝜑)∗ is defined as:

𝐹∗(𝜃, 𝜑) = ∬ 𝐼(,) 𝑒[−𝑖𝑘( 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜑+ 𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜑)] 𝑑 𝑑 𝑆

(4)

In (4), (,)are the coordinates of an arbitrary point on the aperture

plane which plays the same role as (𝑥, 𝑦) coordinates. Therefore,

|𝐹(𝜃, 𝜑)|2 becomes:

|𝐹(𝜃, 𝜑)|2 = 𝐹(𝜃, 𝜑) ∙ 𝐹∗(𝜃, 𝜑) = ∬ 𝐼(𝑥, 𝑦)𝐼∗(𝑥 − , 𝑦 −

) 𝑒[𝑖𝑘{(𝑥−) 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜑+(𝑦−) 𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜑)]𝑑𝑥𝑑𝑦𝑑𝑑 (5)

An inspection of (5) reveals that the exponent represents the phase

dependence of an isotropic radiator in the direction, (,), at a

distance, 𝑑 = √(〖(𝑥 − )〗^2 +〖(𝑦 − )〗^2 ) which can be

written as, 𝑑 = √(^2 + ^2 ). Thus, the exponent is the only factor

having (,) dependence. By taking another isotropic radiator located

at = + , we get two-element array with isotropic radiators

having inter-element separation, d. The mutual impedance between

two isotropic radiators is estimated as [1]:

𝑅0 =𝑠𝑖𝑛𝑘√(𝑥−𝜉)2+(𝑦−𝜂)2

𝑘√(𝑥−𝜉)2+(𝑦−𝜂)2 (6)

Using (5) and (6), (2) gives an expression for 𝑃0 written as:

𝑃0 = 4𝜋∬ ∬ 𝐼(𝑥, 𝑦)𝐼∗(𝜉, 𝜂)𝑠𝑖𝑛𝑘√(𝑥 − 𝜉)2 + (𝑦 − 𝜂)2

𝑘√(𝑥 − 𝜉)2 + (𝑦 − 𝜂)2𝑑𝑥𝑑𝑦𝑑𝜂𝑑𝜉

= 4𝜋∬ 𝜒(𝜉, 𝜂)𝑠𝑖𝑛𝑘√𝜉2+𝜂2

𝑘√𝜉2+𝜂2 (7)

In (7), 𝜒(𝜉, 𝜂) is the convolution of the current distribution on the

aperture given as:

𝜒(𝜉, 𝜂) = ∬ 𝐼(𝑥, 𝑦)𝐼∗(𝑥 − 𝜉, 𝑦 − 𝜂)𝑑𝑥𝑑𝑦 (8)

The current distribution, I(x,y), is assumed as given below:

𝐼(𝑥, 𝑦) = 𝐼0(𝑥, 𝑦)𝑒−[−𝑖𝑘(𝑥𝑠𝑖𝑛𝜃0𝑐𝑜𝑠𝜑0+𝑦𝑠𝑖𝑛𝜃0𝑠𝑖𝑛𝜑0)] (9)

In (9), the exponential part represents phase of the current

distribution when a beam is scanned at an angle, (𝜃0, 𝜑0). Having

substituted (9) in (1) and after utilization of (7) and (8), an expression

of directivity is derived and given as:

𝐷(𝜃, 𝜑) =|∬𝑠 𝐼0(𝜂,𝜉)𝑑𝜉𝑑𝜂|

2

∬𝑠 𝜒0(𝜉,𝜂)𝑐𝑜𝑠 [𝑘𝑠𝑖𝑛𝜃0(𝜉𝑐𝑜𝑠𝜑0+𝜂𝑠𝑖𝑛𝜑0)]𝑠𝑖𝑛𝑘√𝜉2+𝜂2

𝑘√𝜉2+𝜂2𝑑𝜉𝑑𝜂

(10)

In (10), 𝜒0(𝜉, 𝜂) is the auto correlation function of the current

distribution at the aperture, defined as:

𝜒0 = 𝜒0(𝜉, 𝜂) = ∬𝑠𝐼0(𝑥, 𝑦)𝐼0∗(𝑥 − 𝜉, 𝑦 − 𝜂)𝑑𝑥𝑑𝑦 (11)

The denominator in (10), accounts for the coupling between two

finitely separated elements (isotropic) (through sinc (·)) and the phase

distribution along the array aperture (through cos(·) terms).

Considering a planar aperture consisting of point sources, the array’s

current distribution may be represented as:

𝐼(𝑥, 𝑦) = ∑ 𝛿(𝑥 − 𝑥𝑛)𝛿(𝑦 − 𝑦𝑛)𝐼𝑛𝑁𝑛=1 (12)

In (12), represents Kronecker’s delta function [21], and 𝐼𝑛 is defined

as:

𝐼𝑛 = 𝐼0𝑛𝑒𝑥𝑝 [−𝑖𝑘𝑠𝑖𝑛𝜃0(𝑥𝑛𝑐𝑜𝑠𝜑0 + 𝑦𝑛𝑠𝑖𝑛𝜑0)] (13)

Thus, (10) is re-written as (see detailed derivation in Appendix A)

𝐷(𝜃, 𝜑) =(∑ 𝐼0𝑚

𝑁𝑚=1 )

2

∑ 𝐼0𝑚𝐼0𝑛∗ 𝑐𝑜𝑠 [𝑘𝑟𝑛𝑚𝑠𝑖𝑛𝜃0𝑐𝑜𝑠(𝜑0−𝛼𝑛𝑚)]

𝑠𝑖𝑛𝑘𝑟𝑛𝑚𝑘𝑟𝑛𝑚

𝑁𝑚,𝑛=1

(14)

In (14), 𝛼𝑛𝑚 is defined by

𝑐𝑜𝑠𝛼𝑛𝑚 =𝑥𝑛−𝑥𝑚

√(𝑥𝑛−𝑥𝑚)2+(𝑦𝑛−𝑦𝑚)2 (15a)

𝑠𝑖𝑛𝛼𝑛𝑚 =𝑦𝑛−𝑦𝑚

√(𝑥𝑛−𝑥𝑚)2+(𝑦𝑛−𝑦𝑚)2 (15b)

and the distance 𝒓𝒏𝒎 between the elements with indices m and n is

given as:

𝑟𝑛𝑚 = √(𝑥𝑛 − 𝑥𝑚)2 + (𝑦𝑛 − 𝑦𝑚)2 (16)

Hence, (14) represents an exact analytical expression for D of a finite

sized arbitrarily shaped planar array. It is a generalized expression,

which accounts for (a) arbitrary array geometry and array lattice; (b)

mutual coupling (c) arbitrary current distribution on aperture and (d)

antenna look or scan angle. The proposed expression serves as an

efficient mean for characterization of dense as well as sparse PAAs.

The antenna arrays can be classified into different categories based on

the ‘choice of d’ as (a) dense arrays, d/<1; (b) moderately sparse

arrays, 1<d/<2; and (c) highly sparse arrays, d/>2. In the present

paper, the focus is on analysis of periodic or regular SPAAs having

large inter-element separations, d/.

In the case of SPAAs, due to large value of d (~ d/ >1), the mutual

coupling contribution between the elements is negligible and the

expression for D becomes:

𝐷(θ, φ) =(∑ 𝐼𝑜𝑛

𝑁𝑛=1 )

2

∑ 𝐼𝑜𝑚𝐼𝑜𝑛𝑁𝑚,𝑛=1

= 𝑁 (17)



Thus, D for SPAAs becomes equal to N in absence of the mutual

coupling. Whereas, in the case of dense array, due to dominance of

mutual coupling, D < N.

The effect of EEP [19-20] can be included in (14) as:

𝐺𝑎(θ, φ) = (1 − |Γ𝑎|2)휀𝑙𝑜𝑠𝑠𝐷(θ, φ) (18)

In (18), Ga is the gain including effect of EEP, Γ𝑎 is active reflection

coefficient [20] and 휀𝑙𝑜𝑠𝑠 is the dissipative loss due to antenna and the

associated transmission network.

A ‘factor of 2’ needs to be multiplied to (14) in case radiation is

assumed only in upper hemisphere. In this case, the maximum D limit

achievable by optimally designed sparse arrays is 𝐷𝑠𝑝 = 2𝑁. The

expression, (14) is time consuming for large arrays (~ N >100). In

order to facilitate quick design aid to PAAs designers to estimate D

and scan performance of PAAs, an approximate analytical expression

is proposed in the next section.

III. APPROXIMATE EXPRESSION FOR DIRECTIVITY OF AN INFINITE

PLANAR PHASED ARRAY ANTENNA

This section presents an approximate expression for D of PAAs,

which provides a quick design tool for its estimation. It considers all

possible interference modes or GLs that may exist in the visible region

due to chosen inter-element separations. The proposed expression

assumes an isophoric and periodic infinite planar PAA and do not

consider finite edge effects. It also considers the effect of mutual

coupling, scan angle and inter-element separations. Further, it helps in

predicting the scan performance of the PAAs by estimating accurately

the angular locations for the onset of blind spots. The total number of

elements in a planar array are assumed as, 𝑁 = 𝑀𝑥 × 𝑀𝑦; Mx and My

are the number of elements along x- and y- axes.

Here onwards, normalized directivity, 𝐷𝑛𝑜𝑟𝑚 is considered for

analysis and drawing various inferences from the study presented in

this paper and is expressed as (see detailed derivation in Appendix B):

𝐷𝑛𝑜𝑟𝑚 =𝐷(θ,φ)

𝑀𝑥𝑀𝑦=

2𝜋𝑑𝑥𝑑𝑦

𝜆2 ∑ [1−(𝜆

𝑑𝑥𝑝−𝑠𝑖𝑛𝜃0)

2−(

𝜆

𝑑𝑦𝑞)

2

]

−1/2

𝑝,𝑞

(19)

The indices, 𝑝 and 𝑞 corresponds to 𝑝𝑞𝑡ℎ GL entering into the real

(visible region) space by the virtue of the choice of dx and dy. A

particular pqth mode contributes towards the summation in

denominator of (19) iff it satisfies the condition given below:

1 − (𝜆

𝑑𝑥𝑝 − 𝑠𝑖𝑛𝜃0)

2− (

𝜆

𝑑𝑥𝑞)

2≥ 0 (20)

Thus, the choice for the number of pq modes is critical while

computing Dnorm to include effects of all possible GLs that may exist

in the visible region. The corresponding expression for the radiated

power is:

𝑃0 = ∑ ∑ 𝑐𝑜𝑠[𝑘𝑑𝑥𝑠𝑖𝑛𝜃0(𝑝1 − 𝑝2)]𝑠𝑖𝑛(𝑘𝑟𝑝1,𝑝2,𝑞1,𝑞2)

𝑘𝑟𝑝1,𝑝2,𝑞1,𝑞2

𝑀𝑦

𝑞1,𝑞2

𝑀𝑥𝑝1,𝑝2

(21)

In (21), 𝑟𝑝1,𝑝2,𝑞1,𝑞2 is given as:

𝑟𝑝1,𝑝2,𝑞1,𝑞2= √𝑑𝑥

2(𝑝1 − 𝑝2)2 + 𝑑𝑦2(𝑞1 − 𝑞2)2 (22)

After summation over two indices (21) becomes (see Appendix B]:

𝑃0 = 𝑀𝑥𝑀𝑦 ∑ ∑ (1 −|𝑝|

𝑀𝑥) (1 −

|𝑞|

𝑀𝑦)

𝑀𝑦−1

𝑞=−(𝑀𝑦−1)

𝑀𝑥−1

𝑝=−(𝑀𝑥−1)

∙

𝑠𝑖𝑛 (𝑘√𝑑𝑥2𝑝2 + 𝑑𝑦

2𝑞2)

𝑘√𝑑𝑥2𝑝2 + 𝑑𝑦

2𝑞2

. (𝑐𝑜𝑠(𝑘𝑑𝑥𝑚𝑠𝑖𝑛𝜃))

(23)

In (23), assuming a large array (𝑀𝑥 → ∞, 𝑀𝑦 → ∞) and considering

the “oscillating” nature of the 𝑐𝑜𝑠(𝑘𝑑𝑥𝑚𝑠𝑖𝑛𝜃), it is assumed that

(1 −|𝑝|

𝑀𝑥) (1 −

|𝑞|

𝑀𝑦) ≈ 1. The use of Poisson’s formula and the

properties of modified Bessel function [21], a simplified expression for

Po is obtained as:

𝑃0 =𝑀𝑥𝑀𝑦𝜋

𝑘𝑑𝑥

∑ ∑ 𝐼0 (𝑘𝑑𝑦𝑚√1 − (𝑐𝑜𝑠𝜃0 −𝑙𝜆

𝑑𝑥))𝑙

∞𝑚=−∞ (24)

Restricted to those values of 𝑙 for which 1 − (𝑐𝑜𝑠𝜃0 −𝑙𝜆

𝑑𝑥) ≥ 0.

In (24), Io denotes modified Bessel’s function of first kind of order

zero [21-22]. The expression of Po may be re-written as:

𝑃0 =𝑀𝑥𝑀𝑦𝜆


∑ [1 − (𝜆

𝑑𝑥𝑝 − 𝑠𝑖𝑛𝜃0)

2− (

𝜆

𝑑𝑦𝑞)

2

]

−1/2

𝑝,𝑞 (25)

for those values of p and q which satisfies the condition, (20).

Considering ‘dense’ arrays (𝑑𝑥/𝜆 and 𝑑𝑦/𝜆 < 1), (19) becomes:

𝐷𝑛𝑜𝑟𝑚 =2𝜋𝑑𝑥𝑑𝑦

𝜆𝑐𝑜𝑠𝜃0 (26)

which is valid at any angle except for those satisfying the condition

𝑐𝑜𝑠𝜃0 <3

4√2

𝜆

𝐿. (27)

(L denotes length of array). Hence, a conclusion may be drawn from

the above study that in case the distance between the elements in the

equidistant array is small, the directivity varies according the cosine

law, except in the sector where the main beam is near the aperture

plane following (27).

IV. RESULTS AND DISCUSSION

This section presents analyses carried out for PAAs utilizing the

proposed expressions with aid of few case studies.

A. Array Characterization Using Exact Expression (14)

This section presents the analysis of PAAs using exact expression

(14), carried out for linear and planar array configurations. In planar

arrays, square as well as rectangular array geometries are considered.

All the figures presented now onwards consider 𝐷𝑛𝑜𝑟𝑚 (with respect

to Dsp). Here onwards we assume only isotropic REs radiating only in

upper hemisphere.

Fig. 2 shows the variation of 𝐷𝑛𝑜𝑟𝑚 (in dB) vs. scan angle, o, for

different values of d, for a 100-elements linear array along 𝜑𝑐 =0𝑜 plane. It is observed that for d=0.25 (/4) spacing, Dnorm is

maximum in the end-fire direction and is nearly constant in the range

0o-70o. As d increases, Dnorm increases and few dips are observed in the

curves (d=0.7 and 1.2). Further, when d (in ) =0.5, 2 and 3, Dnorm

vs. scan angle is constant.

Fig. 2. Dnorm (in dB) vs. scan angle (in deg) for a linear array.

Fig. 3 shows the variation in 𝐷𝑛𝑜𝑟𝑚 (in dB) vs. o for a 2500-

(50x50) elements square planar array for 𝑑 (𝑖𝑛 𝜆) =0.5, 0.7, 1.2, 2 𝑎𝑛𝑑 3 along 𝜑𝑐=0o

plane. When, 𝑑 = 0.25, 𝐷𝑛𝑜𝑟𝑚



decreases with the increase in o, up to an angle, g (~60o), after which

it again rises. When, 𝑑 = 0.5, 𝐷𝑛𝑜𝑟𝑚 follows 𝑐𝑜𝑠𝜃 pattern and falls

down to half of its value at some angle, say g (~60o-65o). In the case

of 𝑑 > 1, 𝐷𝑛𝑜𝑟𝑚 remains almost constant with respect to o, except

for few dips in the pattern. These dips in the curves shows the onset of

GLs at specific values of o, representing blind spots due to the

mismatch in array input impedance. However, as d increases further

the depth of dips decreases showing negligible variation in Dnorm,

demonstrating wide scan performance (>80o) for highly sparse PAAs.

Fig. 3. Dnorm (in dB) vs. scan angle for a square planar array.

Fig. 4 shows the variation of Dnorm (in dB) vs. o for a 250- (5 x 50)

elements rectangular PAA. A similar behavior is shown as before

except for less shallow dips (slowly falling edges) compared to the

square array. The analysis is carried out along three -cuts, 0o, 45o and

90o. It is observed that d<1 (dense arrays) the performance is not same

along different -cuts (see curves for d=0.5 and 0.7). However, in

the case of SPAAs (d >1), variation in Dnorm is minimal with respect

to o across the -cuts. This clearly establishes the wide scan behavior

of SPAAs over the hemispherical volume of scan.

(a)

(b)

(c)

Fig. 4. Dnorm (in dB) vs. scan angle for a rectangular planar array.

Fig. 5 shows the variation of D (in dB) vs. d (in steps of 0.05) for

different array sizes at boresight. It shows that 𝐷𝑛𝑜𝑟𝑚increases as N

increases. A strong variation in Dnorm is observed for 𝑑 < 1, but as 𝑑

becomes larger (>1) the value stabilizes and becomes almost

constant. Thus, it establishes the fact that in SPAAs there is least

variation in Dnorm over a wide scan volume.

Table 1 presents comparison of D for a N-elements linear array

[taken from reference [23]] with computed values from (14) showing

a close agreement of results. This validates the accuracy of (14).

Fig. 5. Variation of D (in dB) vs. d for different array sizes.

TABLE 1:

ANALYSIS OF N-ELEMENTS LINEAR ARRAY (D: DIRECTIVITY AT BORESIGHT)

Case N Location

Amplitudes D [23] (dB)

D (dB)

1 14

0.26 0.82 1.36 2.05 2.87 3.72

4.46

0.90 0.78 0.96 0.97 1.00 0.91

0.75

11.6 11.9

2 18

0.28 0.82 1.42 2.04 2.69 3.47

4.32 5.16

5.90

0.91 0.88 0.86 1.00 0.95 0.85

0.84 0.74

0.69

12.7 13.69

3 22

0.24 0.74 1.24 1.75 2.25 2.81

3.36 4.04 4.86 5.70 6.44

0.78 0.95 1.00 0.89 0.83 0.95

0.92 0.93 0.81 0.76 0.73

13.58 14.04

4 26

0.28 0.81 1.37 1.92 2.49 3.08

3.66 4.33 4.98 5.66 6.43 7.25

7.99

0.97 1.00 0.91 0.90 0.90 0.85

0.93 0.90 0.78 0.70 0.73 0.75

0.68

14.30 14.9



B. Array Characterization Using Approximate Expression (19)

An isophoric 900- (30 x 30) elements PAA having separations, 𝑑𝑥/𝜆

and 𝑑𝑦/𝜆, is considered. The value of Dnorm is computed considering

multiple modes, p and q (satisfying (20)). The variation of Dnorm vs. o,

for different d is shown in Fig. 6, which shows the effect of GLs on the

scan performance of PAAs. The analysis assumes a square array

(d=𝑑𝑥= 𝑑𝑦), although it do not limit the applicability of the proposed

expressions for any arbitrary array arrangement.

Fig. 6 shows that for a dense array (d/=0.5), shown by blue curve,

the variation in Dnorm follows 𝑐𝑜𝑠𝜃𝑜 pattern, and there is a GL onset at

𝜃𝑜 = 90𝑜. Here, two degenerate modes exists at (𝑝, 𝑞) = (0,0) and (1,

0) due to square geometry. There is a 2dB decrease in Dnorm from 1.6

to 0. The expression, (19) simplifies to 𝐷𝑛𝑜𝑟𝑚 = (2𝜋𝑑𝑥𝑑𝑦)𝑐𝑜𝑠𝜃0 𝜆2⁄

for dense array. The condition for validity of the expression of Dnorm

in dense array is 𝑐𝑜𝑠𝜃0 > √𝜆/𝑀𝑥𝑑𝑥 (scanning in x – z plane).

Table 2 lists the various possible 𝑝𝑞 modes’ combinations for

different values of d. It also lists angles of onset for GLs corresponding

these modes. A similar study can be carried out for any other arbitrarily

shaped PAA using these expressions. It is seen that the behavior

observed for variation in Dnorm is similar to that in sub-section A. The

only difference from previous section is that deeper dips (blind spots)

as observed, due to infinite nature of array extent assumed in (19).

Fig. 6. Dnorm curves vs. scan angles for different d.

The curves for Dnorm of SPAAs (d/> 1) highlights the major fact

that the cos law is not at all valid here. SPAAs have distinct

advantageous feature of stabilized Dnorm over wide angular scan sector

due to the decrease in mutual coupling. The only drawback observed

in equidistant SPAAs is a sharp decrease in the directivity at some

specific angles due to several ‘interference’ beams, as is obvious due

to periodicity in array grid. In general, in the case of periodic SPAAs,

due to the number of grating lobes, the energy in the sidelobes get re-

distributed and confined in the regions other than the dips. This leads

to the higher values of directivity as compared to the denser

counterpart at angles other than boresight.

Along with increase in d, correspondingly number of GLs increases,

with respect to various 𝑝𝑞 modes (see Table 2 and Fig.6). It is observed

that for any value of d, the square root in (19) becomes zero at 𝜃𝑜 =90𝑜 resulting in a dip in the computed directivity curves. Further it is

observed that for 𝑑𝑥 > 1 and o< 60o more than 90% of the normalized

directivity is larger than 1, showing non-existence of mutual coupling

in the array. Thus, ultra-wide angle scanning (>60o) is possible by

utilizing large inter–element separations of sparse arrays up to

0~ 85𝑜.

Fig. 7 shows a saw tooth curve illustrating the behavior of, Dnorm vs.

d by varying d/ from 0.5 to 8 in a square planar array at

boresight 𝜃𝑜 = 0𝑜. The value of Dnorm is observed to be zero whenever

𝑑 = 𝑛 (n=1, 2, 3….) showing that the GL at 90o exactly nullifies the

main beam leading to 𝐷𝑛𝑜𝑟𝑚 → 0. For values of 𝑑 = 𝑛/2, Dnorm is

observed almost equal for all values of 𝑛. In the intermediate range

of 𝑑, i.e,𝑛

2< 𝑑 < 𝑛, the value of Dnorm is observed to shoot up but at

the cost of deep nulls in the curves (shown in Fig.6). Thus, the saw-

tooth curve provides a design tool for choosing inter-element

separation, dn, in order to avoid these sharp dips at boresight.

TABLE 2.

VARIOUS POSSIBLE 𝑝𝑞 COMBINATIONS AND CORRESPONDING GL LOCATIONS

FOR DIFFERENT D FOR A SQUARE LATTICE (dx=dy=d).

d (in ) (p, q) o (deg)

0.5 (0,0) and (1,0) 90

1.0 (-1,0), (0,1) and (1,0)

(1,1), (0,0) and (2,0)

0

90

1.2

(-1,0)

(1,1)

(0,1)

(2,0)

(0,0)

9.6

16.3

33.5

41.8

90

1.5

(-1,1)

(-1,0) and (2,0)

(0,1)

(0,0)

(3,0)

4.5

19.5

36

48

90

2.0

(-2,0), (0,2) and (2,0)

(1,2), (-1,0), (1,2) and (3,0)

(0,1)

(2,2)

(0,0), (2,2) and (4,0)

0

30

60

80

90

2.5

(-2,1)

(-1,2)

(3,1)

(-1,1)

(-1,0)

(0,2)

(3,2)

(4,0)

(4,1)

(0,1)

(1,2), (4,2), (0,0) and (5,0)

6.7

11.5

16.5

31

37

37

37

37

43

66

90

3.0

(-3,0), (0,3) and (3,0)

(3,1)

(-2,2)

(-2,1)

(1,3), (4,0) and (-2,0)

(4,1)

(-1,2)

(4,2)

(5,0), (2,3) and (-1,0)

(5,1)

(0,2)

(5,2)

(0,1)

(3,3) and (6,0)

0

3.3

4.5

-15.9

19.5

23

24.3

36.1

41.8

46.5

48.15

67.2

70.5

90

The condition (20) implies the formation of unwanted GLs due to

in-phase field summations, which leads to the “absorption” of a large

percentage of the radiated power (may be 100%). This absorbed power

may lead to the condition, 𝐷𝑛𝑜𝑟𝑚 →0, for infinite PAAs

(𝑀𝑥𝑑𝑥, 𝑀𝑦𝑑𝑦 → ∞), see Fig. 6. A method to reduce these effects is



usage of specific inter-element separations as will be shown in Fig. 8

and/or choice of aperiodic planar SPAAs [2], which is beyond the

scope of this paper.

Fig. 7. Dnorm with respect to the inter-element separation.

(a)

(b)

(c)

Fig. 8. Dnorm vs. scan angle curves for rectangular arrays.

Fig. 8 illustrates the variation of the Dnorm vs. o for different

combinations of dx and dy in a rectangular array, along different -cuts.

The results are compared with the behavior of a dense array (dx=0.5,

dx=dy). Fig. 8a shows the variation of Dnorm vs. o for 𝑑𝑥 = 0.5𝜆 and

𝑑𝑥 = 1.2𝜆 along 𝜑𝑐 =0o, 45o and 90o planes, computed by modifying

the bracketed term in denominator of (19) as

1 − (𝜆

𝑑𝑥𝑝 − 𝑠𝑖𝑛𝜃0𝑐𝑜𝑠𝜑𝑜)

2− (

𝜆

𝑑𝑥𝑞 − 𝑠𝑖𝑛𝜃0𝑠𝑖𝑛𝜑𝑜)

2 (28)

Thus, Fig.8 demonstrates the applicability of approximate expression

along any -planes. Fig. 8 shows that due to rectangular shape of array,

different scan behavior is observed along 𝜑𝑐 =0o and 90o cuts. Fig. 8b

shows the results corresponding to dx=0.5 and dy=2. It shows

appreciable difference in scan performance of array along different 𝜑𝑐

cuts. It is seen in Fig. 8c, that for particular combination of dx=1.2

and dy=2, a highly sparse array, a wide scan volume for SPAAs is

observed with least variation in Dnorm vs. o. It also shows a wide scan

performance along three -cuts. Thus, an important design

consideration is demonstrated.

A 279-elements concentric ring array, is presented in [24] with

computed directivity of 29.35dB. In [23] Das et.al computes the value

as 29.36dB in 23.72s. The values computed using (14) is 29.357dB,

which is in close agreement to the original estimate. The approximate

expression (19) estimates the value as 29.427dB in 14.6s considering

five GL modes (p=-5:5 and q=-5:5). The results are in agreement and

faster to obtain. Thus, the proposed approximate expression for

directivity provides a quick aid to design a wide scan dense as well as

sparse PAAs.

V. CONCLUSIONS

This paper presents an exact analytical expression for computation

of directivity of phased array antennas. It includes most of the

important design considerations in PAAs like arbitrary array geometry

& element type, arbitrary complex excitations, mutual coupling, scan

angle and embedded element pattern. Further, it presents an

approximate expression of directivity for quick analysis of infinite

PAAs assuming isotropic radiator. Although, the expressions don’t

consider array geometry but it serves as a preliminary design aid for

large arrays. The expressions are validated with available literature.

This helps to check the possible onset of GLs due to chosen inter-

element separations. With the aid of various demonstrations, the wide

scan characteristics of SPAAs, especially highly sparse antenna arrays

is established and shown in the paper.

APPENDIX A

DERIVATION OF (14)

In=Ion (the maximum value of In at (𝜃0, 𝜑0)= (0, 0)) is computed as

|∬𝑠𝐼0(𝜉, 𝜂)𝑑𝜉𝑑𝜂|

2= ∑ 𝐼𝑜𝑚

2𝑁𝑚=1 (A-1)

Referring to the denominator of (10)

∬ 𝜒0(𝜉, 𝜂) cos[𝑘𝑠𝑖𝑛𝜃0(𝜉𝑐𝑜𝑠𝜑0 + 𝜂𝑠𝑖𝑛𝜑0)] 𝑠𝑖𝑛𝑘√𝜉2+𝜂2

𝑘√𝜉2+𝜂2 𝑑𝜉𝑑𝜂 (A-2)

From (10), 𝜒0(𝜉, 𝜂) = ∬𝑠𝐼0(𝑥, 𝑦) 𝐼0∗(𝑥 − 𝜉, 𝑦 − 𝜂)𝑑𝑥𝑑𝑦 (A-3)

Now for a planar phased array antenna

𝐼𝑜(𝑥, 𝑦) = ∑ 𝐼𝑚 𝑁𝑚=1 𝛿(𝑥 − 𝑥𝑚) 𝛿(𝑦 − 𝑦𝑚) (A-4)

𝐼𝑜(𝑥 − 𝜉, 𝑦 − 𝜂) = ∑ 𝐼𝑛𝑁𝑛=1 𝛿(𝑥 − 𝜉 − 𝜉𝑛) 𝛿(𝑦 − 𝜂 − 𝜂𝑛) (A-5)

𝜒0 = ∑ {𝐼𝑚(𝑥 = 𝑥𝑚, 𝑦 = 𝑦𝑚) ∑ 𝐼𝑛∗(𝜉 = 𝑥𝑚 − 𝜉𝑛, 𝜂 =𝑁

𝑛=1𝑁𝑚=1

𝑦𝑚−𝜂𝑛)} = ∑ {𝐼0𝑚exp [−𝑖𝑘𝑠𝑖𝑛𝜃0(𝑥𝑚𝑐𝑜𝑠𝜑0 + 𝑦𝑚𝑠𝑖𝑛𝜑0)] ∙𝑁𝑚=1

∑ 𝐼𝑜𝑛∗ exp[𝑖𝑘𝑠𝑖𝑛𝜃0((𝑥𝑚 − 𝜉𝑛)𝑐𝑜𝑠𝜑0 + (𝑦𝑚−𝜂𝑛)𝑠𝑖𝑛𝜑0)]𝑁

𝑛=1 }



= ∑ 𝐼0𝑚𝑁𝑚,𝑛=1 𝐼𝑜𝑛

∗ exp[−𝑖𝑘𝑠𝑖𝑛𝜃0(𝜉𝑛𝑐𝑜𝑠𝜑0 + 𝜂𝑛𝑠𝑖𝑛𝜑0)] =

𝜒0(𝜉𝑛, 𝜂𝑛) ∙ cos [𝑘𝑠𝑖𝑛𝜃0(𝜉𝑛𝑐𝑜𝑠𝜑0 + 𝜂𝑛𝑠𝑖𝑛𝜑0)] (A-6)

Denominator of (10) becomes

∑ 𝐼𝑜𝑚𝐼𝑜𝑛∗

𝑁

𝑚,𝑛=1

cos[𝑘𝑠𝑖𝑛𝜃0{(𝑥𝑚 − 𝜉𝑛)𝑐𝑜𝑠𝜑0 + (𝑦𝑚−𝜂𝑛)𝑠𝑖𝑛𝜑0}]

∙𝑠𝑖𝑛𝑘√(𝑥𝑚 − 𝜉𝑛)2 + (𝑦𝑚−𝜂𝑛)2

𝑘√(𝑥𝑚 − 𝜉𝑛)2 + (𝑦𝑚−𝜂𝑛)2

Putting, 𝜉𝑛 = 𝑥𝑛; 𝜂𝑛 = 𝑦𝑛;

Denominator=

∑ 𝐼𝑜𝑚𝐼𝑜𝑛∗𝑁

𝑚,𝑛=1 . cos[𝑘𝑟𝑛𝑚. 𝑠𝑖𝑛𝜃0{𝑐𝑜𝑠𝛼𝑛𝑚𝑐𝑜𝑠𝜑0 +

𝑠𝑖𝑛𝛼𝑛𝑚𝑠𝑖𝑛𝜑0}] ∙𝑠𝑖𝑛𝑘𝑟𝑛𝑚

𝑘𝑟𝑛𝑚 (A-7)

𝑟𝑛𝑚 = √(𝑥𝑛 − 𝑥𝑚)2 + (𝑦𝑛 − 𝑦𝑚)2;

𝑐𝑜𝑠𝛼𝑛𝑚 =𝑥𝑛−𝑥𝑚

𝑟𝑛𝑚 ; 𝑠𝑖𝑛𝛼𝑛𝑚 =

𝑦𝑛−𝑦𝑚

𝑟𝑛𝑚

Hence, we get the expression

𝐷 =(∑ 𝐼0𝑚

𝑁𝑚=1 )

2

∑ 𝐼0𝑚𝐼0𝑛∗ cos [k𝑟𝑛𝑚𝑠𝑖𝑛𝜃0𝑐𝑜𝑠(𝜑0−𝛼𝑛𝑚)]


𝑁𝑚,𝑛=1

(A-8)

APPENDIX B

Referring to (10), it is possible to rewrite it as:

𝐷 =(∑ 𝐼0𝑚

𝑁𝑚=1 )

2

∑ 𝐼0𝑚𝐼0𝑛∗ cos [k𝑟𝑛𝑚𝑠𝑖𝑛𝜃0𝑐𝑜𝑠(𝜑0−𝛼𝑛𝑚)]


𝑁𝑚,𝑛=1

(B-1)

𝐼𝑛 = 𝐼𝑜𝑛 exp(−𝑖𝑘𝑥𝑛𝑠𝑖𝑛𝜃0) (∵ 𝜑0 = 0) (B-2)

𝐼(𝑥, 𝑦) = ∑ 𝐼𝑛𝑁𝑛=1 𝛿(𝑥 − 𝑥𝑛) 𝛿(𝑦 − 𝑦𝑛) (B-3)

∑ 𝐼𝑜𝑚2𝑁

𝑚=1 = 𝑁2(= 𝑀𝑥2𝑀𝑦

2) (B-4)

(B-1) may be re-written as

𝐷 =𝑁2(=𝑀𝑥

2𝑀𝑦2)

∑ cos [𝑘𝑟𝑛𝑚𝑠𝑖𝑛𝜃0𝑐𝑜𝑠𝛼𝑛𝑚]∙𝑁𝑚,𝑛=1


(B-5)

𝑟𝑛𝑚 = 𝑟𝑚1,𝑚2,𝑛1,𝑛2= √dx

2(𝑚1 − 𝑚2)2 + dy2(𝑛1 − 𝑛2)2 (B-6)

It is known that – 𝑀𝑥 + 1 < 𝑚1 − 𝑚2 < 𝑀𝑥 − 1 and – 𝑀𝑦 + 1 <

𝑛1 − 𝑛2 < 𝑀𝑦 − 1. Introducing new variables 𝑚 = 𝑚1 −

𝑚2 & 𝑛 = 𝑛1 − 𝑛2 , the number of equal 𝑟𝑛𝑚 terms for given 𝑚 & 𝑛

are 𝑀𝑥 ∙ (1 −|𝑚|

𝑀𝑥) and 𝑀𝑦 ∙ (1 −

|𝑛|

𝑀𝑦), respectively.

Hence, denominator of (B-5) becomes

𝑀𝑥 𝑀𝑦 ∑ ∑ (1 −|𝑚|

𝑀𝑥)(1 −

𝑀𝑦−1

𝑛=−(𝑀𝑦−1)𝑀𝑥−1𝑚=−(𝑀𝑥−1)

|𝑛|

𝑀𝑦)

sin (𝑘√𝑑𝑥2𝑚2+𝑑𝑦

2𝑛2)

𝑘√𝑑𝑥2𝑚2+𝑑𝑦

2𝑛2cos (𝑘𝑑𝑥𝑚𝑠𝑖𝑛𝜃0) (B-7)

For large arrays & considering “oscillating” character of

cos(𝑘𝑑𝑥𝑚𝑠𝑖𝑛𝜃0), it may be assumed that

(1 −|𝑚|

𝑀𝑥) (1 −

|𝑛|

𝑀𝑦) ≈ 1 (B-8)

, denominator in (B-5) becomes

∑ {∑sin(𝑘√𝑑𝑥

2𝑚2+𝑑𝑦2𝑛2)

𝑘√𝑑𝑥2𝑚2+𝑑𝑦

2𝑛2

𝑀𝑦−1

𝑛=−(𝑀𝑦−1)}

𝑀𝑥−1𝑚=−(𝑀𝑥−1) cos(𝑘𝑑𝑥𝑚𝑠𝑖𝑛𝜃0) (B-9)

Let 𝑚𝑑𝑥 = 𝑥 𝑎𝑛𝑑 𝑛𝑑𝑦 = 𝑦 ,⇒ 𝑑𝑥 =𝑚∙𝑑𝑥

𝑀𝑥⇒ Δ𝑚 = 1 =

𝑀𝑥

𝑑𝑥∙ 𝑑𝑥 (B-10a)

Similarly, Δ𝑛 = 1 =𝑀𝑦

𝑑𝑦∙ 𝑑𝑦

⇒ ∑ →𝑀𝑦−1

𝑛=−(𝑀𝑦−1)

𝑀𝑦

𝑑𝑦∙ ∫

sin(𝑘√𝑥2+𝑦2)

𝑘√𝑥2+𝑦2)

∞

−∞ 𝑑𝑦 (−∞ ≤ 𝑦 ≤ ∞) (B-10b)

Thus, the denominator of (B-5) may be re-written as

∑ {𝑀𝑦

𝑑𝑦∙ ∫


𝑘√𝑥2+𝑦2)

∞

−∞}

𝑀𝑥−1𝑚=−(𝑀𝑥−1) cos(𝑘𝑥𝑠𝑖𝑛𝜃0) 𝑑𝑦 =

𝑀𝑥𝑀𝑦

𝑑𝑥𝑑𝑦∫ ∫


𝑘√𝑥2+𝑦2)𝑑𝑦 cos(𝑘𝑥 𝑠𝑖𝑛𝜃0) 𝑑𝑥 (−∞ ≤ 𝑥 ≤

∞

𝑦=−∞

∞

𝑥=−∞

∞) (B-11)

Alternately, the expression in (B-11) may be approximated as given

below (making use of Poisson’s formula & properties of modified

Bessel functions [21]), for those values of l & p integers where √∙= 0.

≈ 𝜆𝑀𝑥𝑀𝑦

𝑘 𝑑𝑥𝑑𝑦

∑ [1 − (𝜆

𝑑𝑥𝑙 − 𝑠𝑖𝑛𝜃0)2 − (

𝜆

𝑑𝑦𝑝)

2

]

−1

2

𝑙,𝑝 (B-12)

The qualifying (𝑙, 𝑝) set gives effect on maximum Dnorm as function of

o.

𝑙 = 0, 𝑝 = 0 ⇒ 𝐷𝑛𝑜𝑟𝑚 =𝐷

𝑀𝑥𝑀𝑦=


𝜆2 1/𝑐𝑜𝑠𝜃0=


𝜆2 𝑐𝑜𝑠𝜃0 (B-13)

(B-13) is 𝜃0 dependent and it is valid for dense array, depicting

angular dependence of Dnorm . The condition for Dnorm in dense array

is:

𝑐𝑜𝑠𝜃0 > √𝜆

𝑀𝑥𝑑𝑥 (B-14)

where Mxdx is the length of the array (for scanning in xz plane).

w.r.t. (B-11), Integral 1 in (B-11) becomes:

𝐼𝑛𝑡1 = ∫sin(𝑘√𝑥2+𝑦2)

𝑘√𝑥2+𝑦2𝑑𝑦 =

∞

𝑦=0 ∫sin(𝑘√𝑥2+𝑦2)

𝑘√𝑥2+𝑦2𝑑𝑦

∞

0 (B-15)

In order to solve the integral let

𝑧 = 𝑘√𝑥2 + 𝑦2 ⇒ 𝑑𝑧 =𝑘𝑦

√𝑥2 + 𝑦2𝑑𝑦

𝑘2(𝑥2 + 𝑦2) = 𝑧2 ⇒ 𝑦 = √𝑧2

𝑘2 − 𝑥2 𝑦 > 0; 𝑧 > 𝑘𝑥

𝑑𝑦

𝑘√𝑥2 + 𝑦2=

1

𝑘2𝑦𝑑𝑧 =

1

𝑘√𝑧2 − 𝑘2𝑥2𝑑𝑧

∴ 𝐼𝑛𝑡1 = ∫sin(𝑧)

𝑘√𝑧2−𝑘2𝑥2𝑑𝑧

∞

𝑧=𝑘𝑥 (B-16)

Introducing 𝑡 =𝑧

𝑘𝑥 ⇒ 𝑑𝑡 ∙ 𝑘𝑥 = 𝑑𝑧

∴ 𝐼𝑛𝑡1 = ∫sin 𝑘𝑥𝑡

𝑘√𝑡2 − 1𝑑𝑡

∞

𝑡=1

=2

√𝜋 Γ(1/2)∙

1

𝑘𝐽0(𝑘𝑥) (𝐸𝑞. 9.1.24 [21])

Since Γ (1

2) = √𝜋 ⇒ 𝐼𝑛𝑡1 =

2

π∙

1

𝑘𝐽0(𝑘𝑥) (B-17)

Now

𝐼𝑛𝑡2 = ∫2

π∙

1

𝑘𝐽0(𝑘𝑥)

∞

𝑥=0

cos(𝑘𝑥 𝑠𝑖𝑛𝜃0) 𝑑𝑥 (𝐸𝑞. 11.4.37 [21])

=2

𝜋𝑘

1

√𝑘2 − 𝑘2𝑠𝑖𝑛𝜃0

=2

𝜋𝑘2𝑐𝑜𝑠𝜃0

(B-18)

∴ 𝐷𝑛𝑜𝑟𝑚 =𝑀𝑥

2𝑀𝑦2

𝑀𝑥𝑀𝑦

𝑑𝑥𝑑𝑦∙

2

𝜋𝑘2𝑐𝑜𝑠𝜃0

= 𝑀𝑥 𝑑𝑥𝑀𝑦𝑑𝑦 2𝜋

𝜆2 𝑐𝑜𝑠𝜃0 (B-19)

Comparing expressions (B-19) and (B-13), the dependence on the

number of elements is an extra factor as being shown. Further, we may

write in case that no GL are visible (𝑑𝑥 <𝜆

2𝑎𝑛𝑑 𝑑𝑦 < 𝜆),

𝐼𝑛𝑡2 =2

𝜋𝑘2√1−𝑠𝑖𝑛2𝜃0

(B-20)

On the contrary in the case of visible GLs, for the case of non-dense

array, where 𝑑𝑥 >𝜆

2, we may use the under-mentioned treatment to

assess the effect on the directivity. The array factor in xz-plane [1] may

be written in the alternate form as:

sin(

𝑁

2𝜓)

sin(𝜓

2)

𝑤ℎ𝑒𝑟𝑒 𝜓 = 𝑘𝑑𝑥𝑠𝑖𝑛𝜃 + 𝛽; (B-21)

In (B-21), 𝛽 = 𝑘𝑑𝑥𝑠𝑖𝑛𝜃0 is the phase difference between the

successive elements, and 𝜃0 is the beam-pointing or scan direction, 𝜃0.

Now the GL appear when 𝑘𝑑𝑥

2(𝑠𝑖𝑛𝜃 − 𝑠𝑖𝑛𝜃0) =

0, ±𝜋, ±2𝜋, … … , 𝑚𝜋

∴ (𝑠𝑖𝑛𝜃 − 𝑠𝑖𝑛𝜃0) = 0, ±𝜆

𝑑𝑥, ±

2𝜆

𝑑𝑥, … … , ±

𝑚𝜆

𝑑𝑥



The value above expression in brackets is equal to zero, iff 𝑑𝑥 <𝜆

2,

representing main lobe (desired peak direction). The undesired

“grating lobes” occur for

(𝑠𝑖𝑛𝜃 − (𝑠𝑖𝑛𝜃0 ±𝑚𝜆

𝑑𝑥)) = 0 (B-22)

Using (B-2, B-22), (B-20) becomes

𝐼𝑛𝑡2 =2

𝜋𝑘2 ∑ √1−(𝑚𝜆

𝑑𝑥−𝑠𝑖𝑛𝜃0)2

𝑚

𝑤ℎ𝑒𝑛 𝑑𝑦 < 𝜆 (B-23)

In a similar manner is derived, dependency for 𝑑𝑦 > 𝜆,

considering GLs for 𝑠𝑖𝑛𝜃0 ±𝑚𝜆

𝑑𝑥 in xz plane and ±

𝑛𝜆

𝑑𝑦 in yz plane (no

o dependence, scanning is only in xz plane)

Now, 𝐼𝑛𝑡2 =2

𝜋𝑘2 ∑ √1−(𝑚𝜆

𝑑𝑥−𝑠𝑖𝑛𝜃0)2−(

𝑛𝜆

𝑑𝑦)2

𝑚,𝑛

(B-24)

i.e., summation over all (m, n) values for which

1 − (𝑚𝜆

𝑑𝑥− 𝑠𝑖𝑛𝜃0)2 − (

𝑛𝜆

𝑑𝑦)2 ≥ 0

Hence the expression of normalized directivity becomes

∴ 𝐷𝑛𝑜𝑟𝑚 =𝐷

𝑀𝑥𝑀𝑦

=1

𝑀𝑥𝑀𝑦∙

(𝑀𝑥𝑀𝑦)2

𝑀𝑥𝑀𝑦

𝑑𝑥𝑑𝑦

2

𝜋𝑘2 [1 − (𝜆

𝑑𝑥𝑚 − 𝑠𝑖𝑛𝜃0)

2

− (𝜆

𝑑𝑦𝑛)

2

]

1/2

=2𝜋𝑑𝑥𝑑𝑦

𝜆2 ∑ [1 − (𝜆

𝑑𝑥𝑚 − 𝑠𝑖𝑛𝜃0)

2

− (𝜆

𝑑𝑦𝑛)

2

]

−1/2

𝑚,𝑛

(B-25)

ACKNOWLEDGMENT

The authors would like to thank the reviewers for their comments and

suggestions.

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Ashutosh Kedar (M’10–SM’13) was born in Delhi, India on

July 9, 1973. He received his Ph.D., M.Tech. and M.Sc. in Elect., Microwave Elect. and Physics in 2003, 1998 & 1995

respectively, from Univ. of Delhi, India.

He fulfilled Teaching & Research duties at Univ. of Delhi and is currently working as a Radar scientist in LRDE,

Bangalore since 2003. He gives various lectures on radar,

antennas and phased arrays and has published approximately 60 papers. He is reviewer of many international journals like IET, PIERS, ACES, IEEE, etc. He

is listed in Asia’s who’s who.

He received technology group award for 1st indigenous active phased array radar development in country. He received best paper award in ICMARS and

ATMS’18. His research interest includes CEM, phased array antennas & radars.

L.P. Ligthart born in The Netherlands, on September 15, 1946.

He received an Engineer's degree (cum laude) and Ph.D. degree

from Delft Univ. of Technology. He is Fellow of IET, IEEE, and Academician of the Russian Academy of Transport. He

received Honorary Doctorates at MSTUCA in Moscow, Tomsk

State University and MTA Romania. Since 1988, he held a chair on MW transmission, remote sensing, radar and

positioning and navigation at Delft Univ. of Technology. He supervised over

50 PhDs. He founded the IRCTR at Delft Univ. He is founding member of the

EuMA, chaired 1st EuMW in 1998 and initiated EuRAD in 2004. Currently he

is em. prof., elft University, guest professor at Universities in Indonesia and

China, Chairman of CONASENSE, Member BoG of IEEE-AESS. His areas of specialization include antennas & propagation, radar & remote

sensing, satellite, mobile & radio communications. He gives various courses on

radar, remote sensing & antennas and published over 650 papers, various book chapters and books.

Documents

Wide Scanning Characteristics of Sparse Phased Array ...static.tongtianta.site/paper_pdf/18c69b5e-f8f8-11e... · The directivity, D of PAAs, dense or sparse, may serve as a figure