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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
Chapter 4Antennas and Propagation Slide 3
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)
Chapter 4Antennas and Propagation Slide 4
General Array
Assume we have N elementspattern of ith antenna
Total pattern
Identical antenna elements
Pattern MultiplicationElement Factor Array Factor
Chapter 4Antennas and Propagation Slide 5
Uniform Linear Array (ULA)
Place N elements on the z-axisUniform spacing
Chapter 4Antennas and Propagation Slide 6
Uniform Excitation
Apply equal amplitude to elements(different phases only)
Recall:
Chapter 4Antennas and Propagation Slide 7
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
Chapter 4Antennas and Propagation Slide 8
Uniform Excitation: Examples
Example: N=8, =/2
Chapter 4Antennas and Propagation Slide 9
Grating Lobes
Problem for > /2Lobes with amplitude equal to main beam appearCalled grating lobesSimilar to aliasing in signal processing
Example
Chapter 4Antennas and Propagation Slide 10
ULA Beamwidth, Directivity
Note: Example values in (.) are for N=8, =/2
Chapter 4Antennas and Propagation Slide 11
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
Chapter 4Antennas and Propagation Slide 12
Hansen-Woodyard (2)
Consider smallMeans scan angle on main beam
Progressive phase shift
Chapter 4Antennas and Propagation Slide 13
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?
Chapter 4Antennas and Propagation Slide 14
Hansen-Woodyard (4)
Minimize
Find v, then can compute
Chapter 4Antennas and Propagation Slide 15
Hansen-Woodyard (5)
vmin = -1.46
Chapter 4Antennas and Propagation Slide 16
Hansen-Woodyard (6)
Directivity of HWA: Is there a cost to increased directivity?
Chapter 4Antennas and Propagation Slide 17
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
Chapter 4Antennas and Propagation Slide 18
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
Chapter 4Antennas and Propagation Slide 19
Symmetric Array (2)
Array factors are
Example MethodsBinomial array
Derive based on heuristic argument
Tschebyscheff arrayUse direct synthesis procedure
Chapter 4Antennas and Propagation Slide 20
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
Chapter 4Antennas and Propagation Slide 21
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
Chapter 4Antennas and Propagation Slide 22
Binomial Array (3)
Coefficients
Also given by Pascals triangle
Chapter 4Antennas and Propagation Slide 23
Binomial Array (4)
AdvantageNo side lobes
DisadvantagesWide main lobeHigh variation in weights
Chapter 4Antennas and Propagation Slide 24
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
Chapter 4Antennas and Propagation Slide 25
Tschebyscheff Array
Array factorEven number of antennas (M is twice # antennas)
Cosine Power Series
Chapter 4Antennas and Propagation Slide 26
Tschebyscheff Array (2)
Tschebyscheff Polynomials
Recursion
Direct Computation with cos/cosh
Chapter 4Antennas and Propagation Slide 27
Tschebyscheff Array (3)
Tschebyscheff Polynomials
Chapter 4Antennas and Propagation Slide 28
Tschebyscheff Example
M = 3 (6 antenna elements)
Chapter 4Antennas and Propagation Slide 29
Tschebyscheff Example (2)
OK, but
How do we map z to x?
Chapter 4Antennas and Propagation Slide 30
Tschebyscheff Example (3)
Main beam atx = 1 x = cos uz = z0Let z = z0 x
Chapter 4Antennas and Propagation Slide 31
Tschebyscheff Example (4)
Straightforward generalization for higher orders.
Chapter 4Antennas and Propagation Slide 32
Tschebyscheff Array (Generalized)
Chapter 4Antennas and Propagation Slide 33
Gen. Tschebyscheff Array (2)
Can find the am using the same recursive procedure as before.
Chapter 4Antennas and Propagation Slide 34
Comparison of Beamforming Methods
=/4, N=8, R0=10 (-20dB side lobes)
Chapter 4Antennas and Propagation Slide 35
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
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