Optimization Models
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1. Introduction
1.1. Overview
1.2. Optimization Models
2. Linear Algebra
2.1. Vectors
2.2. Matrices
2.3. Linear Equations
2.4.  Least-Squares
2.5. Eigenvalues
2.6  Singular Values
3. Convex Models
3.1. Convexity
3.2. LP and QP
3.3. SOCP
3.4. Robust LP
3.5. GP
3.6. SDP
3.7. Non-Convex Models
4. Duality
4.1. Weak Duality
4.2. Strong Duality
4.3.  Applications
5. Case Studies
5.1. Senate Voting
5.2. Antenna Arrays
5.3. Localization
5.4. Circuit Design

Diagram Shaping: Constraints and Trade-Offs

  • Shaping the antenna diagram

  • Normalization

  • Sidelobe level constraint

  • Thermal noise power constraint

  • Examples of trade-offs

In this section we discuss basic constraints and trade-offs involved in the design of antenna arrays.

Shaping the antenna diagram

The squared modulus of the antenna's diagram, |D_z(phi)|^2, turns out to be proportional to the directional density of electromagnetic energy sent by the antenna. Hence, it is of interest to ‘‘shape’’ (by choice of the z's) the magnitude diagram |D_z(cdot)| in order to satisfy some directional requirements.

A typical requirement is that the antenna transmits well along a desired direction (on or near a given angle), and not for other angles. That way, the energy sent is concentrated around a given “target” direction, say phi_{rm target} =0^o, and small outside that band. Another type of requirement involves the thermal noise power generated by the antenna.

Normalization

First, we normalize the energy sent along the target direction. When multiplying all weights by a common nonzero complex number, we do not vary the directional distribution of energy; therefore we lose nothing by normalizing the weights as follows:

mbox{bf Re}(D_z(0)) ge 1.

These constraints are affine in the (real and imaginary parts of) the decision variable z in mathbf{C}^n.

Sidelobe level constraint

Now define a ‘‘pass-band’’ [-Phi,Phi], where Phi>0 is given, inside which we wish the energy to be concentrated; the corresponding ‘‘stop-band’’ is the outside of that interval.

To enforce the concentration of energy requirement, we require

forall : phi, ;; |phi| ge Phi ~:~ |D_z(phi)| le delta ,

where delta is a desired attenuation level on the ‘‘stop-band’’ (this is sometimes referred to as the sidelobe level).

The sidelobe level constraint is actually an infinite number of constraints. We can simply discretize these constraints:

|D_z(phi_i)| le delta, ;; i=1,ldots,N,

where phi_1,ldots,phi_N are regularly spaced angles in the stop-band.

alt text 

Antenna array design: sidelobe level constraints. The magnitude diagram must go through the blue point (on the right) at phi = 0^o, and be contained in the white area otherwise. To simplify the design problem we can replace the sidelobe constraints by a finite number of constraints at given angles (in blue).

Thermal noise power constraint

It is often desirable to control the thermal noise power generated by the emitting antennas. It turns out that this power is proportional to the Euclidean norm of the (complex) vector z, that is:

mbox{Thermal noise power } alpha |z|_2 = sqrt{sum_{i=1}^n |z_i|^2}.

Trade-off between sidelobe attenuation level and thermal noise power

A typical design problem would involve

  • A normalization constraint which assigns a unit value to the magnitude diagram in a specific direction. The constraint is of the form

mbox{bf Re}(D_z(0)) ge 1.

  • A constraint on the sidelobe attenuation level, which is of the form

|D_z(phi_i)| le delta, ;; i=1,ldots,N.

  • A constraint on the thermal noise power, |z|_2 le gamma.

A typical trade-off curve would plot for example the best achievable thermal noise level gamma for a given value of sidelobe attenuation level delta. Each point on the (delta,gamma) curve is obtained by solving the optimization problem

min_z : |z|_2 ~:~ mbox{bf Re}(D_z(0)) ge 1, ;; |D_z(phi_i)| le delta, ;; i=1,ldots,N.