Unlike conventional phased array (PA), frequency diversity array (FDA) can perform the beampattern synthesis not only in an angle dimension but also in a range dimension by introducing an additional frequency offset (FO) across the array aperture, thus greatly enhancing the beamforming flexibility of an array antenna. Nevertheless, an FDA with uniform inter-element spacing that consists of a huge number of elements is required when a high resolution is needed, which results in a high cost. To substantially reduce the cost while almost maintaining the antenna resolution, it is important to conduct a sparse synthesis of FDA. Under these circumstances, this paper investigated the transmit–receive beamforming of a sparse-fda in range and angle dimensions. In particular, the joint transmit–receive signal formula was first derived and analyzed to resolve the inherent time-varying characteristics of FDA based on a cost-effective signal processing diagram. In the sequel, the GA-based low sidelobe level (SLL) transmit–receive beamforming of the sparse-fda was proposed to generate a focused main lobe in a range-angle space, where the array element positions were incorporated into the optimization problem. Numerical results showed that 50% of the elements can be saved for the two linear FDAs with sinusoidally and logarithmically varying frequency offsets, respectively termed as sin-FO linear-FDA and log-FO linear-FDA, with only a less than 1 dB increment in SLL. The resultant SLLs are below −9.6 dB, and −12.9 dB for these two linear FDAs, respectively.

The thinned and sparse beamforming for semicircular FDAs were investigated, where the excitation amplitudes were also considered in thinned semicircular FDAs, and only the elements’ positions were incorporated into the sparse semicircular FDA. Firstly, the transmit–receive model was introduced to handle the inherent time-varying issue of FDA, followed by the thinned and sparse implementations successively. Note that three types of non-linearly varying frequency offsets (FO), i.e., log-FO, sin-FO, and tanh-FO, were adopted during the investigations. Under the same assumption that 50% of the elements should be saved, the sidelobe levels (SLLs) of the thinned semicircular -FDAs were reduced by 5.8 dB, 4.4 dB, and 4.4 dB, and the widths of the mainlobes were all widened by 3° in their angle dimension. Compared with the thinned semicircular FDAs, the phenomenon of mainlobe widening was alleviated in the sparse semicircular FDAs where the SLLs were reduced by 2.2 dB, 3.7 dB and 3.5 dB, and the mainlobes’ widths in the angle dimension were widened by 1°, 0° and 1°, respectively. It should be highlighted that the sparse semicircular FDA with sin-FO did not broaden the mainlobe in the angle dimension. Therefore, it can be concluded that a sparse semicircular FDA is superior over a thinned semicircular FDA, since it can reduce the same cost with a higher array resolution.