Antennas and Propagation
Chapter 5: Antenna Arrays
5 Antenna Arrays
Advantage
Combine multiple antennas
More flexibility in transmitting / receiving signals
Spatial filtering
Beamforming
Excite elements coherently (phase/amp shifts)
Steer main lobes and nulls
Super-Resolution Methods
Non-linear techniques
Allow very high resolution for direction finding
Antennas and Propagation
Slide 2
Chapter 4
5 Antenna Arrays (2)
Diversity
Redundant signals on multiple antennas
Reduce effects due to channel fading
Spatial Multiplexing (MIMO)
Different information on multiple antennas
Increase system throughput (capacity)
Antennas and Propagation
Slide 3
Chapter 4
General Array
Assume we have N elements
pattern of ith antenna
Total pattern
Identical antenna elements
Element Factor
Array Factor
“Pattern Multiplication”
Antennas and Propagation
Slide 4
Chapter 4
Uniform Linear Array (ULA)
Place N elements on the z-axis
Uniform spacing Δ
Antennas and Propagation
Slide 5
Chapter 4
Uniform Excitation
Apply equal amplitude to elements
(different phases only)
Recall:
Antennas and Propagation
Slide 6
Chapter 4
Uniform Excitation (2)
Note: sin(Nx)/sin(x) behaves
like Nsinc(x)
Maximum occurs for θ= θ0
Result: Steers a beam in direction
θ= θ0 that has amplitude N1/2
compared to single element
If we center array about z=0, and normalize
“Array Gain”
Normalize input power with
additional elements
Antennas and Propagation
for θ= θ0, sin(Nx)/sin(x) goes to N
Slide 7
Chapter 4
Uniform Excitation: Examples
Example: N=8, Δ=λ/2
Antennas and Propagation
Slide 8
Chapter 4
Grating Lobes
Problem for Δ > λ/2
Lobes with amplitude equal to main beam appear
Called “grating lobes”
Similar to aliasing in signal processing
Example
Antennas and Propagation
Slide 9
Chapter 4
ULA Beamwidth, Directivity
Note: Example values in (.) are for N=8, Δ=λ/2
Antennas and Propagation
Slide 10
Chapter 4
Hansen-Woodyard (HWA)
Idea
End-fire excitation has a fat main lobe
Simple coherent excitation not optimal solution for directivity
HWA: do direct maximization
Analysis
Array factor for N elements and progressive phase shift β
Max max AF = 1
Antennas and Propagation
Slide 11
Chapter 4
Hansen-Woodyard (2)
Consider
small
Means scan angle on “main beam”
Progressive phase shift
Antennas and Propagation
Slide 12
Chapter 4
Hansen-Woodyard (3)
Radiation intensity: proportional to |AF|2
In beam direction, θ=0, U(θ) is
Normalize U to make unity at θ=0. Call new function U′(θ)
Directivity found as D0=4πUmax/Prad = Umax/U0, with
How do we maximize D0?
Antennas and Propagation
Slide 13
Chapter 4
Hansen-Woodyard (4)
Minimize
Find v, then can compute β
Antennas and Propagation
Slide 14
Chapter 4
Hansen-Woodyard (5)
vmin = -1.46
Antennas and Propagation
Slide 15
Chapter 4
Hansen-Woodyard (6)
Directivity of HWA:
Is there a cost to increased directivity?
Antennas and Propagation
Slide 16
Chapter 4
Non-Uniform Excitation
Increased Flexibility
Weights are general
Similar to a filter synthesis problem
Example methods
Binomial Array
Similar to “maximally flat” filter
No side lobes for Δ < λ/2
Tschebyscheff Array
Similar to “equiripple” filter
Produces smallest beamwidth
for given sidelobe level
Antennas and Propagation
Slide 17
Chapter 4
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
Antennas and Propagation
Slide 18
Chapter 4
Symmetric Array (2)
Array factors are
Example Methods
Binomial array
Derive based on heuristic argument
Tschebyscheff array
Use direct synthesis procedure
Antennas and Propagation
Slide 19
Chapter 4
Binomial Array
2-element Array
Δ
Plot of AF1 = 1 + x
Has no side-lobes for Δ < λ/2
Idea to make more dir.
Successively superimpose
pairs of arrays
Generates AF = (AF1)M
Antennas and Propagation
Slide 20
Chapter 4
Binomial Array (2)
2-element Array
Δ
1
1
Element 1
3-element Array
Δ
Idea: 2-element array
each element has pattern AF1
1
2
1
Element 2
Element 1
4-element Array
1
Can repeat indefinitely
This procedure is just binomial series!
Antennas and Propagation
Δ
Slide 21
3
3
1
Element 2
Chapter 4
Binomial Array (3)
Coefficients
Also given by Pascal’s triangle
Antennas and Propagation
Slide 22
Chapter 4
Binomial Array (4)
Advantage
No side lobes
Disadvantages
Wide main lobe
High variation in weights
Antennas and Propagation
Slide 23
Chapter 4
General Array Synthesis
Procedure
Expand AF in a (cosine) power series
AF is a polynomial in x, where x=cos u
Choose a desired pattern shape
(polynomial of same order)
Equate coefficients of polynomials
⇒ yields weights on arrays
Example
Dolph-Tschebyscheff Array
Solves: Minimum beamwidth for a prescribed max. sidelobe level
Antennas and Propagation
Slide 24
Chapter 4
Tschebyscheff Array
Array factor
Even number of antennas (M is twice # antennas)
Cosine Power Series
Antennas and Propagation
Slide 25
Chapter 4
Tschebyscheff Array (2)
Tschebyscheff Polynomials
Recursion
Direct Computation with cos/cosh
Antennas and Propagation
Slide 26
Chapter 4
Tschebyscheff Array (3)
Tschebyscheff Polynomials
Antennas and Propagation
Slide 27
Chapter 4
Tschebyscheff Example
M = 3 (6 antenna elements)
Antennas and Propagation
Slide 28
Chapter 4
Tschebyscheff Example (2)
OK, but
How do we map z to x?
Antennas and Propagation
Slide 29
Chapter 4
Tschebyscheff Example (3)
Main beam at
x=1
x = cos u
z = z0
Let z = z0 x
Antennas and Propagation
Slide 30
Chapter 4
Tschebyscheff Example (4)
Straightforward generalization for higher orders.
Antennas and Propagation
Slide 31
Chapter 4
Tschebyscheff Array (Generalized)
Antennas and Propagation
Slide 32
Chapter 4
Gen. Tschebyscheff Array (2)
Can find the am using the same recursive procedure as before.
Antennas and Propagation
Slide 33
Chapter 4
Comparison of Beamforming Methods
Δ=π/4, N=8, R0=10 (-20dB side lobes)
Antennas and Propagation
Slide 34
Chapter 4
Summary
Antenna Arrays
Offer flexibility over single antenna elements
Array factor / Element Factor
Direct synthesis methods for designing AF
Beamforming
Considered mainly ULA
Uniform excitation (change phases)
Non-uniform: Binomial array, Tschebyscheff
Other possibilities
Non-ULA: circular array, rectangular, sparse arrays
Non-symmetric excitation
Non-linear processing
Antennas and Propagation
Slide 35
Chapter 4