Pencil to Flat-Top Beampattern Transform and Its Application in Gaussian Array Design

Synthesis of flat‐top beampatterns based on simple polynomial transforms of Gaussian excitations

“…For N ≥ 3, we define quadratic transform in ref. [9] as $$\begin{align*}{h}_{n}=\frac{{w}_{n}^{2}-a{w}_{n}}{1-a}& ;& n=0,1,...,N-1\end{align*}$$ where a ≠ 1 is the transform coefficient. Note that since w is symmetrical, h = [ h 0 , h 1 , …, h N −1 ] is symmetrical as well.…”

“…By substituting Equation () into (), and Equation () into (), we obtain the coefficient a as in ref. [9], and it equals $$$a=\left\{\begin{array}{@{}lll@{}}\frac{\sum\limits _{k=1}^{(N-1)/2}{k}^{2}{w}_{k+(N-1)/2}^{2}}{\sum\limits _{k=1}^{(N-1)/2}{k}^{2}{w}_{k+(N-1)/2}}& ,& \text{odd}\ N\\ \frac{\sum\limits _{k=0}^{N/2-1}{\left(k+\frac{1}{2}\right)}^{2}{w}_{k+N/2}^{2}}{\sum\limits _{k=0}^{N/2-1}{\left(k+\frac{1}{2}\right)}^{2}{w}_{k+N/2}}& ,& \text{even}\ N\end{array}\right.$$$ …”

“…For the quadratic transform, the limit is given in ref. [9] and it has the form $$${h}_{n}=c\,\left[1-b\,{\left(n-\frac{N-1}{2}\right)}^{2}\right]$$$ where c = 1 for an odd N , c = 4/(4 − b ) for an even N , b = S 1 / S 2 , and $$${S}_{i}=\left\{\begin{array}{@{}lll@{}}\sum\limits _{k=1}^{(N-1)/2}{k}^{2i}& ,& \text{odd}\,N\\ \sum\limits _{k=0}^{N/2-1}{\left(k+\frac{1}{2}\right)}^{2i}& ,& \text{even}\,N\end{array}\right.$$$ …”

“…For a given sidelobe level from the interval −32 dB ≤ R ≤ −13 dB and N ≥ 10, α can be estimated as $$$\alpha \approx {p}_{0}+{p}_{1}R+{p}_{2}{R}^{2}+{p}_{3}{R}^{3}$$$ where coefficients p = [ p 0 , p 1 , p 2 , p 3 ] are given by the expression () in ref. [9] for the quadratic transform and by $$$\mathrm{p}=\left\{\begin{array}{@{}lll@{}}[-3.20,-0.4263,-0.0156,-2.20\mathrm{e}-4]& ,& 10\le N< 20\\ {[-3.35,-0.4285,-0.0149,-2.00\mathrm{e}-4]} & ,& 20\le N< 40\\ {[-3.34,-0.4179,-0.0140,-1.82\mathrm{e}-4]}& ,& N\ge 40\end{array}\right.$$$ for the cubic transform. For α given in Equation (), the 3dB‐beamwidth expressed in degrees can be estimated as $$${\text{BW}}_{-3\text{dB}}\approx \frac{30\alpha }{N-2}+\frac{250}{N}$$$ for the quadratic [9] and as $$${\text{BW}}_{-3\text{dB}}\approx \frac{174...$$…”

“…However, as a consequence of integer orders of flatness, this method provides a limited number of beam widths. This drawback can be overcome by compensation of pencil beams [8] or by using pencil to flat‐top beam transform [9]. The former approach flattens the pencil beam by a second‐order broadening polynomial, whereas the latter introduces flatness by applying quadratic function to an arbitrary symmetrical excitation.…”