antennas and propagation chapter 5: antenna · pdf filecombine multiple antennas more...

Chapter 5: Antenna Arrays
Antennas and Propagation

Chapter 4Antennas and Propagation Slide 2
5 Antenna Arrays
AdvantageCombine multiple antennasMore flexibility in transmitting / receiving signalsSpatial filtering
BeamformingExcite elements coherently (phase/amp shifts)Steer main lobes and nulls
Super-Resolution MethodsNon-linear techniquesAllow very high resolution for direction finding

5 Antenna Arrays (2)
DiversityRedundant signals on multiple antennasReduce effects due to channel fading
Spatial Multiplexing (MIMO)Different information on multiple antennasIncrease system throughput (capacity)

General Array
Assume we have N elementspattern of ith antenna
Total pattern
Identical antenna elements
Pattern MultiplicationElement Factor Array Factor

Uniform Linear Array (ULA)
Place N elements on the z-axisUniform spacing

Uniform Excitation
Apply equal amplitude to elements(different phases only)
Recall:

Uniform Excitation (2)
Note: sin(Nx)/sin(x) behaves like Nsinc(x)
Maximum occurs for = 0If we center array about z=0, and normalize
Normalize input power with additional elements for = 0, sin(Nx)/sin(x) goes to N
Result: Steers a beam in direction = 0 that has amplitude N1/2compared to single element
Array Gain

Uniform Excitation: Examples
Example: N=8, =/2

Grating Lobes
Problem for > /2Lobes with amplitude equal to main beam appearCalled grating lobesSimilar to aliasing in signal processing
Example

ULA Beamwidth, Directivity
Note: Example values in (.) are for N=8, =/2

Hansen-Woodyard (HWA)
IdeaEnd-fire excitation has a fat main lobeSimple coherent excitation not optimal solution for directivityHWA: do direct maximization
AnalysisArray factor for N elements and progressive phase shift
Max max AF = 1

Hansen-Woodyard (2)
Consider smallMeans scan angle on main beam
Progressive phase shift

Hansen-Woodyard (3)
Radiation intensity: proportional to |AF|2
In beam direction, =0, U() is
Normalize U to make unity at =0. Call new function U()
Directivity found as D0=4Umax/Prad = Umax/U0, with
How do we maximize D0?

Hansen-Woodyard (4)
Minimize
Find v, then can compute

Hansen-Woodyard (5)
vmin = -1.46

Hansen-Woodyard (6)
Directivity of HWA: Is there a cost to increased directivity?

Non-Uniform Excitation
Increased FlexibilityWeights are generalSimilar to a filter synthesis problem
Example methodsBinomial Array
Similar to maximally flat filterNo side lobes for < /2
Tschebyscheff ArraySimilar to equiripple filterProduces smallest beamwidth
for given sidelobe level

Symmetric Array
Antennas placed symmetrically on z axis(Also same excitation)
Odd number of elements:put two copies of center element (for two sides)
Amplitude on true center element is 2a1

Symmetric Array (2)
Array factors are
Example MethodsBinomial array
Derive based on heuristic argument
Tschebyscheff arrayUse direct synthesis procedure

Binomial Array
2-element Array
Plot of AF1 = 1 + x
Has no side-lobes for < /2
Idea to make more dir.Successively superimpose
pairs of arraysGenerates AF = (AF1)M

Binomial Array (2)
2-element Array
3-element ArrayIdea: 2-element array
each element has pattern AF1
4-element Array
Can repeat indefinitelyThis procedure is just binomial series!
Element 1
Element 2
1 2 1
1 1
1 3 3 1
Element 1
Element 2

Binomial Array (3)
Coefficients
Also given by Pascals triangle

Binomial Array (4)
AdvantageNo side lobes
DisadvantagesWide main lobeHigh variation in weights

General Array Synthesis
ProcedureExpand AF in a (cosine) power seriesAF is a polynomial in x, where x=cos uChoose a desired pattern shape
(polynomial of same order)Equate coefficients of polynomials yields weights on arrays
ExampleDolph-Tschebyscheff ArraySolves: Minimum beamwidth for a prescribed max. sidelobe level

Tschebyscheff Array
Array factorEven number of antennas (M is twice # antennas)
Cosine Power Series

Tschebyscheff Array (2)
Tschebyscheff Polynomials
Recursion
Direct Computation with cos/cosh

Tschebyscheff Array (3)
Tschebyscheff Polynomials

Tschebyscheff Example
M = 3 (6 antenna elements)

Tschebyscheff Example (2)
OK, but
How do we map z to x?

Main beam atx = 1 x = cos uz = z0Let z = z0 x

Straightforward generalization for higher orders.

Tschebyscheff Array (Generalized)

Gen. Tschebyscheff Array (2)
Can find the am using the same recursive procedure as before.

Comparison of Beamforming Methods
=/4, N=8, R0=10 (-20dB side lobes)

Summary
Antenna ArraysOffer flexibility over single antenna elementsArray factor / Element FactorDirect synthesis methods for designing AF
BeamformingConsidered mainly ULAUniform excitation (change phases)Non-uniform: Binomial array, Tschebyscheff
Other possibilitiesNon-ULA: circular array, rectangular, sparse arraysNon-symmetric excitationNon-linear processing

antennas and propagation chapter 5: antenna · pdf filecombine multiple antennas more...

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