## 2 Array Design
### Analytical Techniques  
#### 2.1.1 Sidelobe Level  
We first examine the abovementioned specifications to find the most suitable technique for the realization of an efficient digital beamformer. The requirement of an invariant SLL considerably restricts the available beamforming techniques. As a matter of fact, apart from optimization algorithms (see, e.g., the excellent reviews in [14,15] and refs. therein), only the Tseng-Cheng technique [13] ensures a constant SLL at arbitrary azimuthal cuts and for any scan angle. Moreover, the Tseng-Cheng technique has the following advantages compared to other techniques: (i) it is fully analytical; (ii) it is computed once, without requiring the amplitude coefficients to be refreshed as the scan angle changes; and (iii) being based on the Dolph-Chebyshev technique, it provides the minimum beamwidthfor a given SLL [13]. All these aspects make the Tseng-Cheng technique particularly attractive for its implementation in an SDR-driven beamforming network.  
In order to implement the Tseng-Cheng technique, we assume an even number of elements along the two main directions, i.e., \(N_{x}=N_{y}=M=2N\), and quadrantal symmetry of the amplitude coefficients, i.e., \(a_{mn}=a_{-m,n}=a_{m,-n}=a_{-m,-n}\) (the extension to rectangular lattices has been proposed in [16]). Under this hypothesis, the AF can conveniently be recast as [13; 17]  
\[\text{AF}(\theta,\phi)=4\sum_{m=1}^{N}\sum_{n=1}^{N}a_{mm}\cos[(2m-1)u]\cos[(2 n-1)v] \tag{1}\]  
where \(u=0.5 kd_{x}(\sin\theta\cos\phi-\sin\theta_{0}\cos\phi_{0})\) and \(v=0.5 kd_{y}(\sin\theta\sin\phi-\sin\theta_{0}\sin\phi_{0})\); \(d_{x}(d_{y})\) is the spacing along the \(x(y)\)-axis, \(\theta\) and \(\phi\) are the elevation and azimuthal angles of the spherical coordinate system, and \(k=2\pi/\lambda\) is the free-space wavenumber, with \(\lambda\) being the free-space wavelength. As shown in [13], an invariant SLL over the entire angular range can be achieved if the amplitude coefficients are set equal to  
\[a_{mn}=\frac{1}{N^{2}}\sum_{p=1}^{N}\sum_{q=1}^{N}T_{M-1}\left[w _{0}\cos\frac{\pi(p-\frac{1}{2})}{2N}\cos\frac{\pi(q-\frac{1}{2})}{2N}\right]\] \[\cos\left(\frac{\pi(m-\frac{1}{2})(p-\frac{1}{2})}{N}\right)\cos \left(\frac{\pi(n-\frac{1}{2})(q-\frac{1}{2})}{N}\right) \tag{2}\]  
where \(T_{M-1}[\cdot]\) is the Chebyshev polynomial of the first kind and order \(M-1\), \(w_{0}\) is related to the SLL through the expression \(w_{0}=\cosh[1/(M-1)\text{arccosh}(\text{PSR})]\), where PSR is the peak-to-sidelobe ratio, i.e., \(\text{PSR}=\sqrt{1/\text{SLL}}\) (assuming all quantities in linear scale). In order to avoid grating lobes, the usual condition for untapered distributions is modified as follows:  
\[\frac{d_{x(y)}}{\lambda}<\frac{1-[\arccos(1/w_{0})]/\pi}{1+\sin\theta_{\text{ max}}} \tag{3}\]  
where \(\theta_{\text{max}}\) is the maximum scan angle. As is clear from expression (2), its straight implementation would require four nested loops to evaluate all the coefficients (viz., two for the \(m\) and \(n\) elements indices, and two for the \(p\) and \(q\) summation indices), thus resulting in time-consuming computations. To take advantage of the MATLAB potentialities in handling matrix calculations, it is therefore important to recast (2) to avoid as much as possible the use of nested loops. Indeed, the two summations in (2) can conveniently be expressed in a quadratic form:  
\[a_{mn}=N^{-2}t_{p}(m)\underline{T}_{pq}t_{q}(n) \tag{4}\]  
where \(t_{p}(m)=\cos[\pi(m-0.5)(p-0.5)]/N\), \(t_{q}(n)=\cos[\pi(n-0.5)(q-0.5)]/N\), and \(\underline{T}_{pq}=T_{2N-1}\{w_{0}\cos[\pi((p-0.5)/2N]\cos[\pi((q-0.5)/2N]\}\), with \(\underline{p}=1,2\ldots N\) and \(\underline{q}=1,2\ldots N\) being row and column vectors of \(N\) elements, respectively.  
This calculation for the Tseng-Cheng tapering coefficients is very efficient, considering that the MATLAB implementation of (4) considerably reduces the execution time from 18 min to fractions of a second (on a laptop with 32 GB RAM and a 3.5 GHz quad-core processor) for the previous case study.