antenna arrays - guc · no minor lobes for the arrays with spacing of λ/4 and λ/2 between the...

ADAPTIVE ANTENNAS. PROF. A.M.ALLAM

1

ANTENNA ARRAYS

ADAPTIVE

ANTENNAS


2 2/3/2015 LECTURES 2

Why antenna arrays?-1

1.The element pattern

2.The amplitude excitation of the individual element

3.The phase excitation of the individual element

4.The relative displacement between elements

5.The geometrical distribution of the array, linear,

rectangular, circular, spherical, …

The array has 5 degrees of freedom which help in

configuring some parameters


3 2/3/2015 3

N Elements linear arrays: Uniform amplitude and spacing

An array of identical elements fed with identical magnitude and a progressive

phase. The array factor can be obtained by considering the elements to be point

sources then multiplied times the element pattern

cos

:

.....1

1

)1(

1

cos)1(

cos)1(cos2cos

kdwhere

eAF

aswrtittenbecanwhich

eAF

eeeAF

N

n

nj

N

n

kdnj

kdNjkdjkdj


4 2/3/2015 LECTURES 4

2

1sin

2sin

1

1

)1()1(

);2()1(

2

1

2

1

2

1

222

1

N

e

ee

eee

e

eAF

eeAF

gSubtractin

Nj

jj

Nj

Nj

Nj

j

jN

jNj

If the reference point is the physical center of the array, the

array factor reduces to:

2

1sin

2sin

N

AF

)2(.....

;

)1(.....1

)1(32

)1(2

jNNjjjjj

j

Njjj

eeeeeAFe

ebysidesbothgMultiplyin

eeeAF


5 5

2

1

2sin

N

AF

The maximum value of the array factor is equal to (N), to normalize the array

factor, so that the maximum value of each is equal to unity,

For small value of ψ it

is approximated by:

For small values of ψ , the above expression can be approximated by:

2

2sin

N

N

AFn

2

1sin

2sin

N

N

AFn

coskd


6

Nulls

The maximum values occur when:

Maximum

2

12

22

..3,2,12

12

2sin

2

..3,2,12

12

2cos

2

12cos

2

12

sin

1

1

s

d

dfor

ss

dor

ss

d

skd

Nor

N

s

s

s

s

Maxima of minor lobes

They occur approximately when:

The null values of the array occur when:

m=0 for main lobe

m=1,2,… for grating lobes

...2,1,022

cos

cos2

1

2

1

mmd

mkd

m

n

...3,2,

...3,2,12

2cos

2

02

sin

1

NNNn

n

N

n

d

nN

N

n

n


7 LECTURES 7

The maximum radiation of any array is directed normal to the axis of the

array (broadside θo=90o) which occurs when:

valueanyofbecand

putkd o

oo

;0

900cos

(a) Broadside array

dnLL

Do )1(...............2

2

1sin

2sin

N

N

AFn

coskd

The total phase difference ψ:

1- current excitation phase β=0

2-path difference phase (kdcosθ)=0,

ψ =0, i.e.,

constructive normal to the array axis θ=90o


8

N=10 N=10 Only the maximum in

broadside direction

One more

maximum appears

( grating lobes) due

to large spacing

If the spacing between the elements is chosen between λ < d < 2λ, then the

maximum toward θ0 = 0◦ shifts toward the angular region 0◦ < θ < 90◦

while the maximum toward θ0 = 180◦ shifts toward 90◦ < θ < 180◦

GRATING LOBES d > λ/2

NO GRATING LOBES d ≤ λ/2

The separation is an important constraint you have to

mind to avoid the onset of the grating lobes in

broadside arrays

d=λ/4 d=λ

If d = 2λ, there are maxima toward 0◦, 60◦, 90◦, 120◦ and 180◦


9 9

(b) Endfire array

The maximum radiation of any array is directed along to the axis of the array

(θo=0o or θo=180o) which occurs when:

dnLL

Do )1(...............4

2

1sin

2sin

N

N

AFn

kdkd cos


1- current excitation phase β=-kd

2-path difference phase (kdcosθ)=kd,

ψ =0, i.e.,

constructive along the array axis θ =0

kdkdkd

kdkdkd

o

o

0cos

0cos

180

0


10

N=10 d=λ/4

β = -kd=-π/2 θo= 0o β = kd =π/2 θo=180o

θo=0o

θo=180o


11 11

(c) Phased (scanning) array

The maximum radiation of the array is required to be oriented at an angle

θ0(0◦ ≤ θ0 ≤ 180◦). To accomplish this, the phase excitation β between the

elements must be adjusted so that:

oo kdkdkdo

coscoscos

Thus by controlling the progressive phase difference between the elements,

the maximum radiation can be squinted in any desired direction to form a

scanning array

2

1sin

2sin

N

N

AFn okdkd coscos

nNN

n

2..............0

2sin

mkd

n

cos2

1

2

m=0,1,2,…

Maxima

Nulls

n=1,2,…

n≠N,2N,…



1- current excitation phase β=-kdcosθo

2-path difference phase (kdcosθ)= kdcosθo

ψ =0, i.e.,

constructive along the direction of scanning to maximum at θ = θo

It is clear that the general array can be scanned to any direction within the visible region

(-kd to kd) . The control parameter is the phase progressive of the excitation current; β

0 β kd -kd βo

The visible region

To avoid the onset of the grating lobe while scanning :

βo-2π

-kd > βo- 2π cosθo>1 - λ/d for an array along z-axis

The

main

lobe

The

grating

lobe


13 2/3/2015 LECTURES 13

(d) Different array orientations

•Linear array oriented along z-axis

2

1sin

2sin

N

N

AFn

coskd


14 2/3/2015 LECTURES 14

•Linear array oriented along x-axis

2

1sin

2sin

N

N

AFn coskd

cossinkd


15 15

•Linear array oriented along y-axis

2

1sin

2sin

N

N

AFn coskd

sinsinkd


16

Odd Even

N Elements linear arrays: Nonuniform amplitude and uniform spacing

Broadside arrays β=0 , Array along z-axis


17 2/3/2015 LECTURES 17

•Array factor for even number


18 2/3/2015 LECTURES 18

•Array factor for odd number


19 19

Binomial broadside array

Binomial expansion

s triangle’Pascal

5Example N=

10Example N=

β = 0


20 2/3/2015 LECTURES 20

•It’s observed that there are

no minor lobes for the arrays

with spacing of λ/4 and λ/2

between the elements

•Binomial arrays have very

low level minor lobes nearly

zeros, but they exhibit large

beamwidth

•A major disadvantage of

binomial arrays is the wider

variation between the

amplitudes excitation of the

different elements of an array

•Example

N=10

One more maximum appears

( grating lobes) due to large

spacing d=3λ/4 , d=λ


21

Dolph Tschebyscheff broadside array

The order of the polynomial should be one less than the total number of elements

of the array ( array of 10 elements means Chebyshev polynomial of order 9)

-Since the array factor of an even or odd number of elements is a summation of cosine terms, whose

form is a polynomial as the Dolph-Chebyscheff polynomials, then the unknown coefficients of the

array factor can be determined by equating the series representing the cosine terms of the array

factor to the appropriate Chebyscheff polynomial

Tm(z) = cos[mcos-1(z)] for -1≤z≤1 represents the region of the side lobes

Tm(z) = cosh[mcosh-1(z)] for z<-1, z>1 represents the region of the main beam

Tm(z)= 2zTm-1(z)- Tm-2(z) is the recursive formula for Chebyscheff polynomial

1yln(y)cosh 21- yNote:


1- All polynomial pass through the point (1,1)

2- For -1≤z≤1, Tm(z) have values within -1 to +1

3- All roots occur within -1≤z≤1 , all maxima and minima have values +1 and

-1 respectively

Tm(z)

z


2/3/2015 LECTURES 23

1- Select the appropriate array factor (even or odd)

2- Expand the array factor {cosmu=…..}

3- Determine the point zo, such that Tm(zo)=Ro

* The order (m) of the Chebyscheff polynomial is always one less than

the total number of the element

4- Substitute cos u=z/zo

5- Equate the array factor to Tm(z)

6- Determine an

•Design procedures


24

Example:

Design a broadside Tschebyscheff array of 10 elements with spacing (d) between

the elements. The major to minor lobe level is 26 dB. Find the excitation

coefficients and form the array factor.

Solution:

9

1

29

1

2

1

1

9

5432110

5

1

2

112

1

0851.120cosh9

1cosh

cosh9cosh20)(

20log20263

exp9cos,7cos,5cos,3cos,cos

9cos7cos5cos3coscos2

cos,12cos1

ooooo

o

oo

ooo

n

nM

RRRRzor

z

zzT

RRdBR

ansionseriesbyuuuuuforsubstitute

uauauauauaAF

duunaAF


2/3/2015 LECTURES 25

uuuuuAFthen

aaaaa

or

aaaaa

formnormalizedin

aaaaa

zzzzz

az

zaa

z

zaaa

z

z

aaaaz

zaaaaa

z

zAF

zTwithstepCompare

stepAFtheinz

z

zuSubistute

ooo

oo

o

9cos7cos357.15cos974.13cos496.2cos798.2

189.0706.0485.0357.0

798.2496.2974.1357.11

:

8377.52073.51184.48308.2086.2

2565764321209

256576443211216

120552049753

)()4(5

)2(0851.1

cos4

10

12345

12345

12345

9753

5

9

54

7

543

5

5432

3

5432110

9


26 26

MXN Elements planner arrays: Nonuniform amplitude and uniform spacing

where

The array factor for this planar array with its maximum along θ0, φ0 , for an even

number of elements in each direction can be written as

wmn is the amplitude excitation of the

individual element

For separable distributions wmn = wmwn

Number of excitation values need to

be computed are M+N

For nonseparable distributions wmn ≠ wmwn

Number of excitation values need to be

computed are MxN

i.e., wmn is a 1 x N vector for a

linear array and a M x N matrix

for a planar array


27

•The amplitude coefficients an or wn control primarily the shape of the

pattern and the major-to-minor lobe level

•Tapered amplitude distributions exhibit wider beamwidth but lower

sidelobe

•The phase excitations control primarily the scanning capabilities of the

array

Therefore, an antenna designer can choose different amplitude

distributions and different phase excitation to conform to the

application specifications

In smart antenna systems the objectives of a DSP are to estimate:

1-The direction of arrival (DOA) of all impinging signals using DOA

algorithms, and

2. The appropriate weights using the adaptive beam forming algorithms to

ideally steer the maximum radiation of the antenna pattern toward the

SOI and to place nulls toward the SNOI


28

antenna arrays - guc · no minor lobes for the arrays with spacing of λ/4 and λ/2 between the...

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