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 = 4πA/λ 2. As part of this derivation, it is shown that the integral of G Ω over 4π 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.
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Notes
- 1.
Setting the input SNR to unity makes the output SNR equal to the gain of the ESA.
- 2.
For a transmitting (Tx) antenna, this expression also can be considered to be the input power (i.e., the power radiated by each element of the aperture) giving an identical expression for the power out versus power in Tx gain as derived for a receiving antenna.
- 3.
We are assuming an ideal case where the effective area of the aperture is equal to the actual area. Equivalently, we assume that the array directivity and gain are the same and are allocating any losses and imperfections to the loss terms in the link margin equation.
- 4.
In Appendix 5 we show what happens to our example ESA gain if the noise exhibits some element-to-element correlation or if the signal some decorrelation.
- 5.
The actual number of k-space samples used in the plots in this book is 1024 × 1024, whereas the number of elements is 32 × 32. This is accomplished by zero padding the spatial array outside the boundary of elements, as illustrated in Appendix A for another application. The reason 1024 is used is esthetics and performance accuracy: seeing the k-space grid in gain plots is displeasing, but more annoying and relevant to ESA performance is when the measurement accuracies of pointing errors or beamwidths, for example, are limited by the grid rather than by the true values.
- 6.
Bracewell, R. N., The Fourier Transform and Its Applications, New York: McGraw-Hill Book Company, 1978.
- 7.
Here F and G (in this subsection only) represent voltage quantities with amplitude and phase.
- 8.
Integrating the resulting power gain over all k-space, a change of integration variables will eliminate α from the expressions.
- 9.
Usually element spacing is based on the maximum frequency of ESA bandwidth, so f 0 is the maximum frequency.
- 10.
For a direct sum implementation, the frequency dependence of the grid is explicitly included, so the metrics do not need frequency scaling. However, the foreshortening effect is still missing and must be included in the metrics.
- 11.
Indicating that even though Taylor weights reduce the sidelobes and peak gain, the broadening of the main beamwidth preserves the integral over all visible space, as it should.
- 12.
Normally, when contour plots are made in MATLAB with this “jet colormap” scheme, red colors are at the top of the color bar, and blue colors are at the bottom. We refer to this as “plumber colors” as red is hot and blue is cold. This convention seems to be common in engineering literature. A physicist, however, might consider this backward. So we plot with “physicist colors” where blue is hot and red is cold.
- 13.
This author jumped to the conclusion that the 2-D ISL would be the product of the two 1-D ISLs. Much head scratching took place when the ESA performance code he was developing produced 2-D values that looked like the sum of the two 1-D ISLs. After setting aside preconceived notions, actually writing out the integral expressions, and noting that the 2-D ISL involves the integral over all k-space minus the integral over a rectangular main beam region in the center, it was finally clear that the 2-D ISL involved both the product and sum of the two 1-D ISLs.
References
Abramowitz, M., & Stegun, I. A. (1964). Handbook of mathematical functions (second printing). Washington, DC: National Bureau of Standards.
Bracewell, R. N., Chang, K.-Y., Jha, A. K., & Wang, Y.-H. (1993, February). Affine theorem for two-dimensional Fourier transform. Electronics Letters, 29(3).
Mailloux, R. J. (2005). Phased Array antenna handbook (2nd ed.). Boston: Artech House.
Skolnik, M. I. (2001). Introduction to radar systems (3rd ed.). Boston: McGraw-Hill.
Taylor, T. T. (1955, January). Design of line source antenna for narrow beam width and low side lobes. IRE Transactions on Antennas and Propagation, 3, 16–28.
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Dana, R.A. (2019). K-Space Gain and Antenna Metrics. In: Electronically Scanned Arrays (ESAs) and K-Space Gain Formulation. Springer, Cham. https://doi.org/10.1007/978-3-030-04678-1_3
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