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 If we center array about z=0, and normalize Result: Steers a beam in direction 1/2 θ= θ0 that has amplitude N compared to single element “Array Gain”
Normalize input power with additional elements for θ= θ0, sin(Nx)/sin(x) goes to N
Antennas and Propagation 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 M Generates AF = (AF1)
Antennas and Propagation Slide 20 Chapter 4 Binomial Array (2)
2-element Array Δ 11 Element 1 3-element Array Δ Δ Idea: 2-element array 1 21 each element has pattern AF1 Element 2
Element 1 4-element Array
1 331
Can repeat indefinitely Element 2 This procedure is just binomial series!
Antennas and Propagation Slide 21 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