Electronically Scanned Arrays MATLAB® Modeling and Simulation

Appendix B: Instantaneous Bandwidth (IBW) Derivation

In Chapter 1, the expression for IBW was shown to be

$IBW=\frac{kc}{L\sin {\text{θ}}_o}$

(B.1)

where k is the beamwidth factor, L is the length of the ESA, and θ_o is the maximum required scan angle. An alternative expression can be shown that ultimately reduces to Equation (B.1). We begin with the expression for the AF shown in Chapter 1:

$AF={\displaystyle \sum _{m=1}^M{a}_m{e}^{j\left(\frac{2\text{π}}{\text{λ}}{x}_m\sin \text{θ}-\frac{2\text{π}}{{\text{λ}}_o}{x}_m\sin {\text{θ}}_o\right)}}$

(B.2)

Rewriting Equation (B.2) in terms of f, the AF expression becomes

$AF={\displaystyle \sum _{m=1}^M{a}_m{e}^{j\frac{2\text{π}}{c}{x}_m(f\sin \text{θ}-f_o\sin {\text{θ}}_o)}}={\displaystyle \sum _{m=1}^M{a}_m{e}^{j\frac{2\text{π}}{c}{x}_m(\text{Ψ})}}$

(B.3)

When operating at the tune frequency, f = f_o, and the ESA is scanned to θ = θ_o, Ψ = 0. However, for operation away from the tune frequency, f = f_o + Δ f, Ψ = (f_o + Δ f) sin θ – f_o sin θ_o and Ψ is no ...