# K-Space Gain and Antenna Metrics

## Abstract

This chapter starts with a derivation, in a straightforward manner, of the relationship between k-space gain *G*_{K} and the more familiar angular-space gain *G*_{Ω}. The two are not the same, for example, the latter is dimensionless but the former has units of area. From this relationship, the well-known formula is derived that the peak gain of an aperture antenna is related to its area *A* and the center frequency wavelength *λ* as *G*_{Ω,max} = 4*πA*/*λ*^{2}. As part of this derivation, it is shown that the integral of *G*_{Ω} over 4*π* steradians and the integral of *G*_{K} over all K-space must both be equal to one, as this is required for conservation of energy. Next the discrete Fourier transform (DFT) implementation of ESA K-space gain is discussed, and important antenna effects that are not included in this formulation are pointed out. In particular, the affine transformation is used to incorporate the effects of aperture foreshortening in the DFT-based gain function. Other effects discussed in this chapter include the cosine taper of element gain, the frequency dependence of antenna gain, and the relationship between the number of elements of a transmit ESA and its effective isotropic radiated power (EIRP). This chapter also includes discussions on phase-comparison monopulse and on computing ESA directivity in angular space directly from power measurements recorded in k-space. It concludes with a derivation of the integrated sidelobe level (ISL) for 1-D and 2-D arrays and for uniformly weighted ESAs.

## Keywords

Electronically scanned array (ESA) Fourier transform Discrete Fourier transform Affine transformation Antenna foreshortening Cosine taper Antenna performance metrics ESA gain K-space gain Directivity Angular-space gain ESA directivity ESA integrated sidelobe level ESA beamwidth ESA maximum sidelobe Effective isotropic radiated power Phase-comparison monopulse## References

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