| # Diagram Shaping via SOCP SOCP > SOC inequalities | Standard Forms | Applications > Antenna Array Design > Back | Diagram shaping | Next Shaping the antenna diagram Normalization Sidelobe level constraint Thermal noise power constraint ## Shaping the antenna diagram The squared modulus of the antenna’s diagram, , 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 ’s) the magnitude diagram 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 , 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: These constraints are affine in the (real and imaginary parts of) the decision variable . ## Sidelobe level constraint Now define a ‘‘pass-band’’ , where 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 where 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: where are regularly spaced angles in the stop-band.
| Antenna array design: sidelobe level constraints. The magnitude diagram must go through the blue point (on the right) at , 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).
The above constraints are second-order cone constraints on the decision variables, since they involve magnitude constraints on a complex vector that depends affinely on the decision variables. ## 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 , that is: ( mbox{Thermal noise power} alpha |z|_2 = sqrt{sum_{i=1}^n |z_i|^2}.