Shaping the antenna diagram
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.
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.
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 .
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).
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:
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
A constraint on the sidelobe attenuation level, which is of the form
A constraint on the thermal noise power, .
A typical trade-off curve would plot for example the best achievable thermal noise level for a given value of sidelobe attenuation level . Each point on the curve is obtained by solving the optimization problem