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American Institute of Aeronautics and Astronautics092407

1

Antenna Pattern Design Based on

Minimax Algorithm

H. K. Hwang1

and Robert Grados2

The Aerospace Corporation, El Segundo, California 90009, United States

This paper is the first of two describing the antenna pattern design using the minimax

algorithm. By using the minimax signal processing algorithm to minimize the maximum

absolute value of the weighted cost function, we can emphasize the difference between the

ideal and realizable functions in some critical regions. This paper presents circular antenna

pattern designs using a 7-by-7 rectangular phased-array antenna, demonstrates an example

of using the minimax algorithm to approximate the exponential function by polynomials,

and discusses the effect of the weighting factor in the cost function to the antenna pattern

and other key algorithm parameters and corresponding design tradeoffs.

I. Introduction

O gain an understanding of the minimax algorithm as well as its application to phased-array antennas, we

started an effort to develop a minimax-based computer program to synthesize array antenna beam patterns. Thispaper is the first of two papers on this subject. This paper describes the minimax algorithm and its applications on

phased-array antenna pattern design. It describes the synthesis of boresight and off-boresight circular beams using

the unconstrained (amplitude and phase) minimax algorithm.

The paper starts with a brief description of the minimax algorithm in Section II. Section III presents the

application of the minimax algorithm to digital filter design. Section IV presents its application to phased-arrayantenna beam pattern design, with one-dimensional and two-dimensional antenna array examples respectively.

Representative antenna patterns from computer simulations are shown in Sections III and IV to demonstrate

algorithm parameters and performance trade-offs. Finally, Section V draws conclusions from this study and section

VI addresses suggested future work.

The minimax algorithm has been around for several decades and used in various engineering applications,

including digital filter design and antenna pattern design. It is now used for pattern design for phased-array antennas

as well.

II. Minimax Algorithm

The minimax algorithm [1] approximates a known (desired) function f 1(y) by using another function f 2(x; y),

where vector x = [x1, x2, ... xK]H

(the superscript H represents the complex conjugate transpose, or Hermitian,

matrix) represents the parameters to be estimated in f 2(x; y) such that the maximum absolute value of the difference

between f 1(y) and f 2

The minimax algorithm starts with an arbitrary initial state x

(x; y) is minimized. If this algorithm is used in filter design, x is the filter coefficient vector. If,

however, this algorithm is used in array antenna pattern design, x is the array weighting vector.

0 and iteratively minimizes the maximum absolute

value of the cost function f(y) = f 2(x; y) – f 1(y). The algorithm searches for ymax, where the cost function has its

maximum absolute value. If f 2(x; ymax) > f 1(ymax

x = x – α

), the parameter vector x is adjusted according to the following

equation:

f x∇ (2.1)

If f 2(x; ymax) < f 1(ymax

1 Consultant, Communication Systems Engineering Department, The Aerospace Corporation, P.O. Box 92957, Los

Angeles, CA 90009, M1-937

), the parameter vector is adjusted by the following equation:

2Advanced Degree Fellow, Communication Systems Engineering Department, The Aerospace Corporation, P.O.

Box 92957, Los Angeles, CA 90009, M1-937

T




2

x = x + α f x

∇ (2.2)

where α is the step size, and f x∇ =

T

K21 x

f ,..,

x

f ,

x

f

∂∂

∂∂

∂∂

is the gradient vector of the cost function. This

parameter vector updating process continues until the change in parameter vector is sufficiently small.

The minimax algorithm is summarized as follows:

1.

Choose any arbitrary state x0 and compute the cost function f(y) = f 2(x; y) – f 1(y) over the predefinedsample points y1, y2, . . , yN

2. Find sample point y

. In general N is larger than K, where K is the dimension of vector x.

max

3. Compute the gradient vector

, where the absolute value of f(y) is maximum.

f x

∇ at y = ymax

4. Replace the parameter vector by its new value x

.

new = xold f x∇– α if f(ymax) > 0 and xnew = xold

f x∇

+

α otherwise, where α is the step size.

5. Go to step 1 and repeat the process until ||xnew – xold

Consider a simple case of approximating the exponential function f

|| < ε.

1(y) = ey

over the interval [–1, 1] by

polynomial function f 2 ∑=

n

0k

k

k yx(x; y) = , where the parameter vector x = [x0, x1, . . , xn]T

The cost function in this case is f(y) =

. The sample point goes

from y = –1 to y = 1, with an increment of 0.1.

∑=

n

0k

k

k yx – ey

. The gradient vector of the cost function is

f x∇ = [ ]Tn

T

n10

yy1x

f

x

f

x

f =

∂∂

∂∂

∂∂

(2.3)




3

The curves of approximating the exponential function by polynomials of the first, second, and third order using

the minimax algorithm are shown in Figure 2.1. Note that the curve for the third-order approximation is overlaid

onto the curve of the ideal function.

Figure 2.1 shows that the approximation improves as the order of the polynomial increases. The optimum

parameter vectors for first-, second-, and third-order approximations are [1.2647, 1.1752]T, [0.9892, 1.1305,

0.5539]T, and [0.9945, 0.9958, 0.5431, 0.1794]

T

Figure 2.2 shows the error between the ideal function f

, respectively.

1(y) = ey

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

y

Ideal Function

First Order Approximation

Second Order Approximation

Third Order Approximation

and the various polynomial approximations.

Figure 2.1. Approximating the Exponential Function by Polynomials Using the Minimax

Algorithm




4

A unique feature of the minimax algorithm is that the error exhibits equal ripple properties. That is, the

maximum positive error and maximum negative error have the same absolute value.

For comparison, let us consider the least squares approximation [2], which is another well known estimation

method. This method approximates the known function f 1(y) by another function f 2

f

(x; y) in the sense that the sum

of squared errors is minimized. The method of least squares has a particularly simple mathematical expression if the

approximation function can be expressed in the form of a power series, as in Equation (2.4).

2 ∑=

K

0k

k

k yx(x; y) = (2.4)

If we want to force f 2(x; y) = f 1(y) at sample point y1, y2, . . , yN

, we simply set up the following set of

equations:

∑=

K

0k

k nk yx = f 1(yn

where N is the number of sample points, and K+1 is the dimension of the parameter vector. This system of equations

can be expressed in a compact matrix form as

) n = 1, 2, . . , N (2.5)

Ax = b (2.6)

where x = [x0, x1, . . , xK]T

b = [f

(2.7)

1(y1), f 1(y2), . . , f 1(yN)]T

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

y

E r r o r

First Order Approximation

Second Order Approximation

Third Order Approximation

(2.8)

Figure 2.2. Estimation Errors of Various Polynomial Approximations




5

A =

K

NN

K

22

K

11

y..y1

:::::

y..y1

y..y1

(2.9)

In general, the number of samples N is larger than the dimension of the parameter vector, in which case Equation

(2.6) is an over-determined system for which there is no solution. The least squares method finds a particularsolution of Equation (2.6) such that the sum of the squared error is minimized. The error at each sample point is

e(yn ∑=

K

0k

k

nk yx) = – f 1(yn

E =

). The sum of the squared error E is

∑=

N

1n

2

n |)e(y| (2.10)

The least squares solution has a simple matrix form given by

x = (ATA)–1AT

b (2.11)

Figure 2.3 shows the estimation errors of the third-order approximation using the minimax and least squares

methods. The maximum absolute error of the least squares method is larger than that of the minimax method.

The parameter vectors of the least squares method of first-, second-, and third-order polynomials are [1.1937,

1.1140]T, [0.9892, 1.1305, 0.5539]T, and [0.9958, 0.9945, 0.5431, 0.1794]T

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-8

-6

-4

-2

0

2

4

6x 10

-3

y

E r r o r

Least Square

Minimax

, respectively.

Figure 2.3. Estimation Errors of Least Squares and Minimax Methods




6

III. Applications to Filter Design

A digital finite impulse response (FIR) filter consists of a tapped delay line with a proper filter coefficient at the

output of each delay element. The FIR filter output is the weighted sum of signals at the tapped delay line. Figure

3.1 is a block diagram of an FIR filter where u(n) is the filter input, y(n) is the filter output, and*

k w represents the

filter coefficients, where k = 0, 1, . . , N–1. The FIR filter input/output relation is

y(n) = k)u(nw1N

0k

*

k −∑−=

(3.1)

The input vector u(n) and filter-weighting vector w are defined as

u(n) = [u(n) u(n–1) . . . , u(n–N+1)]T

w = [w

(3.2)

0, w1, . . . , wN–1]T

Equation (3.1) can be expressed in compact matrix form as

(3.3)

y(n) = wH

The frequency response of the FIR filter is given by the following equation:

u(n) (3.4)

H(f) = wH

where v = [1 e

v (3.5)

–j2πf

. . . e–j2π(Ν−1)f

]T

The frequency response H(f) in Equation (3.5) depends on the filter-weighting vector w. A celebrated algorithm

to compute the weighting vector w to satisfy Equation (3.5) is the Parks-McClellan [3]. This algorithm is based on

the Chebyshev approximation criterion and implemented with the Remez exchange algorithm.

(3.6)

The program that we have developed is based on the minimax algorithm as well. Rather than starting from the

frequency response of Eq. (3.5), we start our filter design from the absolute square of the frequency response. The

absolute square of the frequency response is given by the following equation:

|H(f)|2 = wHvv

Hw = w

H

z–1

z–1

z–1 u(n)

y(n)

*

0w *

1w *

2-Nw *

1-Nw

Rw (3.7)

Figure 3.1. Block Diagram of FIR Filter




7

where R = vvH

−−−−

−−

−

1ee

e1e

ee1

2)f j2π(N1)f j2π(N

2)f j2π(N j2πf

1)f j2π(N j2πf

. (3.8)

The cost function C(f) is defined as the difference between the function |H(f)| 2 and desired function |Hd(f)|2

C(f) = |H(f)|

, as

shown in Equation (3.9).

2 – |Hd(f)|2 = wHRw – |Hd(f)|2

Because the desired function H

(3.9)

d

(f) is usually independent of the filter-weighting vector w, the gradient vector of

the cost function C(f) is

Cw

∇ = RwwH

w∇ = 2Rw (3.10)

For a given desired frequency response, the filter coefficients can start at any arbitrary state wo

w(n+1) = w(n)

. By using the

minimax algorithm stated in section 2, the filter-weighting vector is updated according to the following equation:

± 2αRw (3.11)

The choice of sign in Equation (3.11) depends on the sign of the cost function at its absolute maximum value:

negative if the cost function at the absolute maximum has positive value, and positive otherwise.

To guarantee that the algorithm converges [4], step size α has to satisfy the following equation:

α <

maxλ

2(3.12)

where λmax

λ

is the maximum eigenvalue of matrix R evaluated at the frequency where the cost function has itsmaximum absolute value. Since matrix R is a rank 1 matrix, there is only one nonzero eigenvalue.

max ∑k

k λ = = trace(R) = N (3.13)

In our algorithm, we choose step size α <N

1.

A. Filter Design ExampleThe following example demonstrates our minimax filter design algorithm. Suppose we want to design a square

root raised cosine filter with a roll-off factor of 0.5. Assuming the sampling frequency is 4 times the symbol rate andthe FIR filter has 31 filter weights, Figure 3.2 shows the frequency response of this filter.

The green curve of Figure 3.2 is the ideal (desired) filter frequency response. The blue curve is the frequency

response of the filter designed by the minimax algorithm. The minimax frequency response of the filter in the

passband closely matches the ideal frequency response. The stop band exhibits an equiripple property with the

maximum stop band ripples about –25 dB. Expanding the frequency response in the passband, Figure 3.3 shows that

the passband ripple is very small, with the maximum value around 0.013 dB.




8

Further stopband suppression can be achieved by using the weighted cost function, which is defined as

C(f) = W(f)[ |H(f)|2

– |Hd(f)|2

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Normalized Frequency

F i l t e r G a i n

( d B )

Minimax

Ideal

] (3.14)

Figure 3.2. Frequency Response of Square Root Raised Cosine Filter with Roll-off

Factor of 0.5 and Stop Band to Pass Band Cost Function Weight Ratio of Unity

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

-0.03

-0.02

-0.01

0

0.01

0.02


F i l t e r G a i n ( d B )

Minimax

Ideal

Figure 3.3. Passband Frequency Response of Square Root Raised Cosine Filter with

Roll-off Factor of 0.5 and Stopband to Passband Cost Function Weight Ratio of Unity




9

where W(f) is the weighting function. By assigning a larger value to W(f) in the stopband region, greater stopband

attenuation can be achieved. Figure 3.4 shows the frequency response of the same filter when the weight of the cost

function in the stop band is 50 times the weight of the cost function in the passband.

Figure 3.4 shows that by emphasizing the cost function in the stop band, maximum stop band ripple can be

suppressed to –35 dB. However, a heavier weighting to the cost function in the stop band creates a larger ripple in

the passband. Figure 3.5 shows that the maximum ripple in the passband increases to 0.063 dB, which is still quite

acceptable for most filter design applications.

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-50

-40

-30

-20

-10

0


F

i l t e r G a i n ( d B )

Minimax

Ideal

Figure 3.4. Frequency Response of Square Root Raised Cosine Filter with Roll-off

Factor of 0.5 and Stop Band to Pass Band Cost Function Weight Ratio of 50




10

IV. Application to Array Antenna Pattern Design

Signals transmitted over an antenna system generally have narrowband waveforms. A narrowband condition

exists when the signal bandwidth is small compared to c/D, where c is the speed of light and D is the diameter of a

circular antenna, or the length of a linear array antenna. A narrowband signal can be modeled by the following

equation:

s(t) = m(t)t j2πf ce (4.1)

where m(t) is the baseband waveform and f c

A. Uniformly Spaced Linear Array (ULA)

is the center frequency.

A uniformly spaced linear array (ULA) antenna consists of M equally spaced antenna elements, as shown in

Figure 4.1. The spacing between elements is represented by d. If this ULA antenna is used to receive waveform

coming from an angle θ, the signal picked up by each element sk

s

(t) is

k (t) = s1(t–τk ) = m(t–τk )exp[j2πf c(t–τk

where

)], k = 1, 2, . . , M (4.2)

( )1 cos

k

k d

c

θ τ

−=

is the relative time delay of the k th

element with respect to the first element. The speed of light c can be expressed as

f cλ, where λ is the wavelength at frequency f ck τ c-j2πf

e. The term can be written as1)β- j(k

e where β = –2πdcosθ / λ is

the electrical angle of the ULA.

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06


F i l t e r G a i n ( d B )

Minimax

Ideal

Figure 3.5. Passband Frequency Response of Square Root Raised Cosine Filter with

Roll-off Factor of 0.5 and Stop Band to Pass Band Cost Function Weight Ratio of 50




11

With the narrowband assumption, m(t–τk ≈) m(t), and exp[j2πf c(t–τk t j2πf ce)] =

1)β- j(k e , the signal at the k

th

s

element can be expressed as

k (t) = s11)β- j(k

e(t) k = 2, 3, . . , M (4.3)

where s1t j2πf ce(t) = m(t) is the reference signal at the first antenna element.

If the ULA antenna is used as the transmitting antenna, the antenna output is the sum of the products of element

weights and a corresponding element input signal. Using different element weights, we can make the antenna pattern

match the shape of the intended service region. The ULA antenna pattern at a given angle θ is given by the following equation:

G(θ) = wHvv

H

where w = [w

w (4.4)

1, w2, . . , wM]T jβeis the element-weighting vector, and v = [1, , . . ,

1)β- j(Me ]

T

The antenna pattern of a ULA antenna with 20 elements of constant weight is shown in Figure 4.2. This pattern

has a mainlobe at the antenna boresight (θ = 90

is the array manifold

vector. This equation has the same mathematical form as Equation (3.7). Thus the filter design technique can be

applied to the design of the ULA antenna pattern.

o

d

θdcosθ . . . . . . .

Wavefront

), and the peak sidelobe gain is –13 dB. To avoid the lobe grating,

the inter-element spacing d is assumed to be one-half of the wavelength.

Figure 4.1. Uniformly Spaced Linear Array Antenna

0 20 40 60 80 100 120 140 160 180-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Angle (Degree)

A n

t e n n a G a i n ( d B )

Figure 4.2. 20-Element ULA Antenna Pattern with Uniform Element Weights




12

Any desired (ideal) antenna pattern can be realized using the minimax algorithm. The desired antenna pattern is

typically defined by one or several mainlobes with predefined beamwidth, mainlobe ripple, desired sidelobe gain

levels, and transition region. Usually, the pattern in the transition region is not important. Thus, we often do not

specify the pattern in the transition bands.

Assume that the desired antenna pattern has a unit gain in the interval 85o

< θ < 95o

and zero sidelobe gains in

the interval θ > 100o

and θ < 80o

Figure 4.3 shows that the sidelobe attenuation is 18.2 dB.

. The antenna pattern of this ULA antenna with 20 elements is derived from theminimax algorithm and shown in Figure 4.3.

B. If we allow for larger ripples in the mainlobe, we can further suppress the gains in the sidelobe. This can be

achieved by assigning a heavier weight to the cost function in the sidelobe region. Figure 4.4 shows that if the

weight in the sidelobe region is 10 times larger than the weight in the mainlobe region, the sidelobe can be

suppressed to –21.3 dB.

Sidelobe Attenuation and Mainlobe Ripple Tradeoff

0 20 40 60 80 100 120 140 160 180-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Angle (Degree)

A n t e n n a G a i n ( d B )

Figure 4.3. 20-Element ULA Antenna Pattern with Sidelobe-to-Mainlobe Cost FunctionWeight Ratio of Unity




13

The mainlobe ripples and sidelobe attenuation for the two cases are summarized in Table 4.1.

With 20 antenna elements and inter-element spacing of one-half of a wavelength, the mainlobe beamwidth of a

conventional dish antenna is about λ /D = 0.1 radians, or 5.73o. If the desired mainlobe beamwidth is less than this

value, the antenna pattern will be considerably worse. Figures 4.5 shows that if the desired antenna pattern has unit

gain in the interval 89o ≤ θ ≤ 91

oand 3

o

By assigning a heavier weighting to the cost function in the sidelobe region, the sidelobe gain can be reduced.Figure 4.6 shows the antenna pattern when the cost function in the sidelobe is weighted 10 times more than the cost

function in the mainlobe. The sidelobe amplitude is reduced to –17.2 dB. Note, however, that the mainlobe also

drops to –0.7 dB. The antenna patterns shown in Figures 4.5 and 4.6 for a smaller beamwidth are considerably worse

than the corresponding antenna patterns in Figures 4.3 and 4.4.

roll-off on each side, the sidelobe amplitude can be as high as –13.5 dB.

0 20 40 60 80 100 120 140 160 180-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Angle (Degree)

A n t e n n a G a i n ( d

B )

Figure 4.4. 20-Element ULA Antenna Pattern with Sidelobe-to-Mainlobe Cost Function

Weight Ratio of 10

Table 4.1. Mainlobe Ripple and Sidelobe Attenuation of 20-Element ULA Antenna with

Sidelobe-to-Mainlobe Cost Function Weight Ratios of Unity and 10

Cost FunctionWeight Ratio

(Sidelobe/Mainlobe)

Mainlobe Ripple(dB)

Sidelobe Attenuation(dB)

1 0.06 18.2

10 0.32 21.3




14

0 20 40 60 80 100 120 140 160 180-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Angle (Degree)


Figure 4.5. 20-Element ULA Antenna Pattern with 2

o

0 20 40 60 80 100 120 140 160 180-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Angle (Degree)


Mainlobe Beamwidth and

Sidelobe-to-Mainlobe Cost Function Weight Ratio of Unity

Figure 4.6. 20-Element ULA Antenna Pattern with 2o

Mainlobe Beamwidth and

Sidelobe-to-Mainlobe Cost Function Weight Ratio of 10




15

Let us return to the assumed mainlobe beamwidth with a coverage region 85o

< θ < 95o

Increasing the number of elements from 20 to 30 improves sidelobe suppression. Figure 4.7 shows that by increasing to 30 the number of

elements in the ULA antenna, we can achieve a 25.5 dB sidelobe suppression. The sidelobe level can be further

suppressed by assigning a heavier weight to the cost function in the sidelobe region. Figure 4.8 shows that if the cost

function in the sidelobe region is 10 times more than the weight in the mainlobe region, the sidelobe suppression

increases to 29.9 dB.

0 20 40 60 80 100 120 140 160 180-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Angle (Degree)


Figure 4.7. 30-Element ULA Antenna Pattern with Sidelobe-to-Mainlobe

Cost Function Weight Ratio of Unity

0 20 40 60 80 100 120 140 160 180-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Angle (Degree)

A

n t e n n a G a i n ( d B )

Figure 4.8. 30-Element ULA Antenna Pattern with Sidelobe-to-Mainlobe

Cost Function Weight Ratio of 10




16

The mainlobe ripples and sidelobe attenuation are summarized in Table 4.2.

Thus far, we have considered only the boresight case in which the mainlobe is centered at θ = 90º. The antenna

mainlobe can be placed at any arbitrary angle θ. In the following example, the mainlobe is assigned in the interval

between 40o

and 50o

and sidelobes in the interval θ > 55o

and θ < 35o

Figure 4.9 shows that shifting the antenna mainlobe 45º off-boresight reduces the sidelobe suppression to 11.4

dB. The sidelobe can be suppressed by increasing the relative weight of the cost function in the sidelobe region. If

the relative weight of the cost function in the sidelobe is 10 times more than the cost function in the mainlobe, the

sidelobe attenuation increases to 17.1 dB. This antenna pattern is shown in Figure 4.10.

. The antenna pattern of a ULA antenna with

20 elements is shown in Figure 4.9.

Table 4.2. Mainlobe Ripple and Sidelobe Attenuation of 30-Element ULA

Antenna with Sidelobe-to-Mainlobe Cost Function Weight Ratios of Unity and 10

Cost FunctionWeight Ratio

(Sidelobe/Mainlobe)

Mainlobe Ripple

(dB)

Sidelobe Attenuation

(dB)

1 0.01 25.510 0.04 29.9

0 20 40 60 80 100 120 140 160 180-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Angle (Degree)

A n t e n n

a G a i n ( d B )

Figure 4.9. 20-Element ULA Antenna Pattern for Off-Boresight Beam with





17

The mainlobe ripples and the sidelobe attenuation are summarized in Table 4.3.

As illustrated previously, increasing the antenna size to 30 elements produces a better antenna pattern. Figure

4.11 shows the 30-element ULA antenna pattern with mainlobe between 40o and 50o

. The sidelobe attenuation

increases to 22.2 dB, while the ripples in the mainlobe region are only 0.25 dB.

0 20 40 60 80 100 120 140 160 180-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Angle (Degree)


B )

Figure 4.10. 20-Element ULA Antenna Pattern for Off-Boresight Beam with


Table 4.3. Mainlobe Ripple and Sidelobe Attenuation of 20-Element Off-Boresight

Antenna with Sidelobe-to-Mainlobe Cost Function Weight Ratios of Unity and 10

Cost Function

Weight Ratio

(Sidelobe/Mainlobe)

Mainlobe Ripple

(dB)


(dB)

1 0.3 11.4

10 0.9 17.1




18

Finally, by properly adjusting the element weights, multiple beams can be created from ULA antennas.

Assuming that we want two mainlobes centered at θ = 35o, and θ = 105

o, each with a beamwidth of 10

o, and a 10

o

0 20 40 60 80 100 120 140 160 180-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Angle (Degree)


transition band at each side of the mainlobe. Figure 4.12 and Figure 4.13 show the antenna pattern of a 30-element

ULA antenna, with equally and unequally weighted cost functions, respectively. The mainlobe ripple and sidelobe

attenuation for both cases are summarized in Table 4.4.

Figure 4.12. 30-Element ULA Antenna Pattern for Multiple Beams with

Sidelobe-to-Mainlobe Cost Function Weight Ratios of Unity

0 20 40 60 80 100 120 140 160 180-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Angle (Degree)


Figure 4.11. 30-Element ULA Off-Boresight Antenna Pattern with

Sidelobe-to-Mainlobe Cost Function Weight Ratio of 10




19

C. Two-Dimensional Array AntennaThe ULA antenna design can be extended to a two-dimensional array antenna with antenna elements uniformly

placed on the x-y plane. The antenna output is the weighted sum of the signals from each element. The antenna

pattern equation is the same as Equation (4.4) except that the array manifold vector is

v = [ 1 jβe 2 jβ

e . . . M jβe ]T

where the electrical angle β

(4.5)

k

β

is

k ( )sinθsinφysinθcosφxλ

2πk k +−= (4.6)

and xk , yk are the coordinates of the k th

0 20 40 60 80 100 120 140 160 180-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Angle (Degree)


element, and θ and φ are the boresight elevation and azimuth angles. Figure

4.14 shows the configuration of the two-dimensional array antenna system.

Figure 4.13. 30-Element ULA Antenna Pattern for Multiple Beams with


Table 4.4. Mainlobe Ripple and Sidelobe Attenuation of 30-Element

Antenna for Multiple Beams with Sidelobe-to-Mainlobe Cost Function

Weight Ratios of Unity and 10

Cost Function

Weight Ratio

(Sidelobe/Mainlobe)

Mainlobe Ripple

(dB)


(dB)

1 0.04 20

10 0.14 25




20

Consider a simple two-dimensional array consisting of 49 elements arranged in a 7 ×7 square grid with inter-

element spacing equal to half of a wavelength. Figure 4.15 shows the antenna pattern of this array in which all

elements have constant weights.

The desired antenna pattern is defined as the following:

1. Gain = 0 dB at the boresight at which elevation angle θ = 0

2. Gain = –0.5 dB at θ = 5

o

sensor ϕ

θ

ir

k

x

Wavefront 0

1 2

3

4 5

6 y

o

Figure 4.14. Two-Dimensional Array Antenna System

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120

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150

330

180

0

-120

-110

-100

-90

-80

-70

-60

-50

-40

-30

-20

Figure 4.15. Antenna Pattern of 7×7 Array Antenna with Constant

Weights




21

3. Gain = –6 dB at θ = 10

4. Sidelobe level as small as possible for θ > 20

o

5. Symmetric antenna pattern with respect to azimuth angle.

o

Many factors, such as step size and the relative weight of the cost function in the mainlobe and sidelobe regions,

will influence the final antenna pattern. Figure 4.16 shows the antenna pattern for the uniformly weighted cost

function and fixed step size α = 0.015. After 100,000 iterations, the antenna pattern shown in Figure 4.16 has yet to

satisfy the design specifications.

To guarantee that the adaptive process converges smoothly, we can use the “gear shift” step size. As the iteration

process goes on, step size is progressively reduced to avoid granular noise. This is referred to as “gear shift.” If we

introduce exponentially decaying step size such that

α(n) = α(0)(1 – ε)n

where α(0) is the initial step size and ε is a small number, an antenna pattern can be obtained in much fewer

iterations. The antenna pattern for α(0) = 0.015 and ε = 5

(4.7)

×10–5

is shown in Figure 4.17.

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90 270

120

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150

330

180

0

-55

-50

-45

-40

-35

-30

-25

-20

-15

-10

Figure 4.16. Antenna Pattern of 7×7 Array Antenna with Constant

Step Size and Uniformly Weighted Cost Function




23

Increasing the relative weight in the sidelobe region by a factor of 30 improves sidelobe attenuation to 23 dB.

The antenna plot and sliced antenna plots are shown in Figure 4.21 and Figure 4.22, respectively.

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90 270

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150

330

180

0

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-100

-90

-80

-70

-60

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-40

-30

-20

Figure 4.19. 7×7 Array Antenna Pattern with Exponentially Decaying

Step Size and Sidelobe-to-Mainlobe Cost Function Weight Ratios of 10

0 10 20 30 40 50 60 70 80 90-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10

Elevation Angle (Degree)


Azimuth Angle = 90 Degree


Figure 4.20. Sliced Antenna Plot of Figure 4.19 along Azimuth Angles of 90

o

and 240o




24

0 10 20 30 40 50 60 70 80 90-80

-70

-60

-50

-40

-30

-20

-10

0


A

n t e n n a G a i n ( d B )



Figure 4.22. Sliced Antenna Plot of Figure 4.21 along Azimuth Angles of 90

oand 240

D. Phase Array Antenna Example

o

For this design example we chose to study a two dimensional phase array antenna of 312 elements uniformlyplaced on a plane. The inter-element spacing used is d=1.5792 λ, where λ corresponds to the signal wavelength. The

antenna pattern for elevation angles between 0o

and 10o

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90 270

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150

330

180

0

-90

-80

-70

-60

-50

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with the same weight for every element is shown in Figure

4.23.

Figure 4.21. Antenna Pattern of 7×7 Array Antenna with Exponentially

Decaying Step Size and Sidelobe-to-Mainlobe Cost Function Weight Ratios of 30




25

Sliced along the azimuth angle of 90o

and 225o

Suppose the antenna pattern specification is set as follows:

, the sliced antenna plot is shown in Figure 4.24.

1. Gain at the boresight (θ = 0o

2. Gain at θ = 0.5

) = 0 dBo

3. Gain at θ = 1

= –0.1 dBo

4. Gain at θ > 3

= –3 dBo

5. Symmetry antenna pattern in azimuth angle

be as small as possible

Figure 4.25 shows the antenna pattern for elevation angle between 0o and 10o

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-90

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-60

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-20

using the minimax algorithm and

the relative cost function weight in the sidelobe 10 times heavier than the cost function weight in the mainlobe

Figure 4.23. Antenna Pattern (Identical Element Weights)

0 1 2 3 4 5 6 7 8 9 10-80

-70

-60

-50

-40

-30

-20

-10

0


A n t e n n a G a i n (

d B )



Figure 4.24. Sliced Antenna Plot of Figure 4.23




26

The “gear shift” step size is used in this case. Since our chosen design antenna has 312 elements, the initial step

size is α(0) = 0.003. To prevent the step size from becoming excessively small, the minimum step size is set to be

2×10–5

To identify their sidelobe levels, the antenna plot can be sliced at azimuth angle of 180.

oand 280

o

. The sliced

antenna plots are shown in Figure 4.26. From this sliced antenna plot, it shows that the maximum sidelobe level is –

23 dB.

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-60

-50

-40

-30

-20

-10

Figure 4.25. Antenna Pattern with Relative Sidelobe CostWeighting 10 Times More Than the Mainlobe Cost Weighting

0 1 2 3 4 5 6 7 8 9 10-60

-50

-40

-30

-20

-10

0









27

Figure 4.27 shows the antenna plot if the relative cost function weighting in the sidelobe region is increased to

30.

Figure 4.28 shows the sliced antenna plot at azimuth angles of 180o

and 280o

Figure 4.29 shows the beam pattern generated by the minimax algorithm when the antenna mainlobe moves to

(5

. This sliced antenna plot shows that

the maximum sidelobe level improves to –27 dB.

o, 30o), assuming the same circular beam pattern with a 3 dB beamwidth equal to 2o

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-120

-110

-100

-90

-80

-70

-60

-50

-40

-30

-20

. The relative weights of the

cost function in the sidelobe and mainlobe in this simulation is 30. This result is obtained with 5000 iterations, step

Figure 4.27. Antenna Pattern with Relative Sidelobe Cost

Weighting 30 Times More than the Mainlobe Cost

0 1 2 3 4 5 6 7 8 9 10-70

-60

-50

-40

-30

-20

-10

0









28

size is α = 0.003×0.99(n–1)

. To prevent the step size from becoming too small, the minimum step size αmin is set to

be 10–5

The antenna pattern shown in Figure 4.29 seems to have a relatively high gain in the region [8

.o

< θ < 10o] and

[285o

< φ < 300o]. Figure 4.30 shows the sliced antenna plot along azimuth angles of 30

o, 180

o, and 300

o. Figure

4.31 shows the sliced antenna plot along elevation angles of 5o, 9o, and 10o

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300

150

330

180

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-100

-90

-80

-70

-60

-50

-40

-30

-20

. Those figures show that the worst

sidelobe amplitude is around –24dB.

Figure 4.29. Antenna Pattern with Relative Weighting of Sidelobe Cost

to Mainlobe Cost Equals 30

0 1 2 3 4 5 6 7 8 9 10-70

-60

-50

-40

-30

-20

-10

0










29

If the relative cost function weight in the sidelobe region increases to 60, better sidelobe suppression can be

achieved. Figure 4.32 shows the antenna pattern when the relative cost function weight in the sidelobe region is 60

times higher than the corresponding weight in the mainlobe. Figures 4.33 and 4.34 contain the sliced antenna plot

along the azimuth and elevation angles. The worst sidelobe amplitude in this case is reduced to –26 dB.

0 50 100 150 200 250 300 350 400-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Azimuth Angle (Degree)


Elevation Angle = 5 Degree




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90 270

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150

330

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0

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-100

-90

-80

-70

-60

-50

-40

-30

-20

Figure 4.32. Antenna Pattern with Relative Weighting of Sidelobe

Cost to Mainlobe Cost Equals 60




30

Further increasing the relative cost function weight in the sidelobe region to a factor of 100 reduces the

maximum sidelobe amplitude to –27 dB. Figure 4.35 shows the antenna plot when the relative cost function weight

in the sidelobe is 100 times higher than the corresponding weight in the mainlobe. Figures 4.36 and 4.37 are the

sliced antenna plots along azimuth and elevation angles.

0 1 2 3 4 5 6 7 8 9 10-80

-70

-60

-50

-40

-30

-20

-10

0



B )




Figure 4.33. Sliced Antenna Plots of Figure 4.32

0 50 100 150 200 250 300 350 400-120

-100

-80

-60

-40

-20

0


A n t e n n a G a i n ( d B

)








31

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90 270

120

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150

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0

-110

-100

-90

-80

-70

-60

-50

-40

-30

-20

Figure 4.35. Antenna Pattern with Relative Weighting of Sidelobe Cost to

Mainlobe Cost Equals 100

0 1 2 3 4 5 6 7 8 9 10-120

-100

-80

-60

-40

-20

0










32

V. Conclusions

The application of the minimax algorithm to digital filter design was well documented in the literatures. In this

study, we modified the digital filter design approach and extended the application to antenna beam design. In the

digital filter design application, Parks and McClellan use the Chebyshev approximation and Remez

exchange algorithm to design a linear phase finite impulse response filter. Our study shows that similar results can

be achieved by starting with the arbitrary initial filter coefficient vector and recursively updating it by using the

minimax algorithm. Chebyshev approximation is no longer necessary. Our study shows that the minimax algorithm

exhibits equal ripples in passband and stopband. Sidelobe suppression in the stopband can be further enhanced byassigning a heavier weighting to the cost function in the sidelobe regions. This will introduce slight larger ripples in

the mainlobe (passband).

In the antenna pattern design application, rather than adjusting the gain and phase independently , the gain and

phase can be adjusted simultaneously by treating the weighting vector as a complex vector. Several important

conclusions of this study are:

1. Compared with the fixed antenna, an antenna pattern design based on the minimax algorithm providesmuch better sidelobe suppression.

2. It is difficult to achieve a good antenna pattern if the desired antenna mainlobe beamwidth is smaller than

λ /D, where λ is the wavelength and D is the diameter of the antenna.

3. To avoid the granular noise, a proper forgetting factor has to be applied to the step size in the weighting

vector updating equation.

4. Converging time is closely related to the forgetting factor. A conservative forgetting factor requires a longiteration time to make the algorithm converge.

5. Similar to the results for the filter design application, the minimax algorithm exhibits equal ripples in

antenna mainlobe and sidelobes.

6. Given a design specification, the optimum relative weighting of the cost functions can be experimentally

defined by extensive computer simulations.

VI. Suggested Future Work

The application of the minimax algorithm to the antenna pattern design presented in this paper can be extended

to cover the following topics.

1. To improve the coding efficiency by using matrix computation rather than loop operation.

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2. To evaluate the performance of multibeam design. This has important applications if the service regions are

separated by long distances, for example, the continental United States and Hawaii.

3. To guarantee the service quality, it is desired to have uniform gain over the intended service region. We

proposed studying the antenna beam shaping such that the antenna mainlobe follows the physical contour

of the service region.

4. The element weights are stored in memory with finite register size. Thus the antenna pattern due to the

quantization effect should be studied.

5. During the service lifetime, one or several antenna elements may cease to function. A statistical analysis

should be carried out to evaluate whether a satisfactory antenna pattern can be maintained even if a fewelements are not functioning properly.

VII. References1Hald Jorgen and Madsen Kaj, 1981 “Combined LP and Quasi-Newton Methods for Minimax Optimization” Mathematical

Programming 20 p. 49-622Gilbert Strang, 2006 “Linear Algebra and its Applications, 4th Ed” Thomsen Brooks/Cole, ISBN 0-03-010567-63Parks T. W. and McClellan J. H, 1972 “Chebyshev Approximation for Nonrecursive Digital Filter with Linear Phase” IEEE

Trans. Circuit Theory, p.189-1944Haykin Simon, 2002 “Adaptive Filter Theory, 4 th

Acknowledgement

Ed” Prentice Hall, ISBN 0-13-090126-1

We would like to express our sincere appreciation to Dr. David Scholler, senior project engineer of TheAerospace Corpoaration, for providing us with the opportunity to explore the exciting field of applications of the

minimax algorithm. We would also like to thank Dr. James Yoh for providing many valuable comments and

suggestions during the course of this work.

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