Fig. 1. Uniform Linear Antenna Array diagram
Then, the LLA has the antenna factor (AF) :
\(AF(\theta)=\sum_{n=0}^{N-1}I_{n}e^{-jnkdcos\theta}\) (1)
where \(I_{n}\ \)is the exciting current of the nth element, N is the
total number of the elements of the LLA.
Let \(\varphi=kdcos\theta=\pi cos\theta\), then formular (1)
becomes:
\(\ \ \ \ \ \ AF(\theta)=\sum_{n=0}^{N-1}I_{n}e^{-jn\varphi}\)(2)
On the other hand, the frequency response definition of the FIR filter
is:
\(H(z)=\sum_{n=0}^{N-1}{h(n)z^{-n}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\)(3)
When z is \(e^{\text{jω}}\), formular (2) meets this definition. So, the
LLA can be regarded as the FIR filter of DSP, and the ULAA is just the
FIR filter with RWF.
LLA design with least-square linear-phase method: Let the FIR
filter length be L+1 samples, with L even. Initially it will be centered
about the time origin, i.e., zero phase. Then the frequency response can
be calculated at the specified frequency \(\omega_{k}\ \)by
\({H(\omega}_{k})=\sum_{n=-L/2}^{L/2}{h(n)e^{-j\omega_{k}n},\ \ \ k=0\text{\ to}\ }N-1.\)where N>>L (4)
Regarding even symmetry in the impulse response, i.e., h(n)=h(-n), gives
a zero-phase FIR filter that can be later right-shifted L/2 samples to
make a causal, linear phase filter. then, the frequency response reduces
to a sum of cosines:
\({H(\omega}_{k})=h(0)+2\sum_{n=1}^{L/2}{h(n)\cos(\omega_{k}}n),\)where
k=0,1, 2, …, N-1, N>>L. (5)
Let the desired FIR filter frequency response be \(D(\omega_{k})\), then
the design target is to minimize the following square error:
\(\min{||H(\omega_{k})-D(\omega_{k})||}^{2}\) (6)
When the least square error is reached, the related h(n) can be
obtained, and h(n) is real number since the FIR filter is enforced to be
even symmetry. In our case, this h(n) can be regarded as the exciting
current of each element of the LAA.
So, the LAA design procedure of using method can be:
- Specify the total element number of LLA.
- Specify the normalized cut-off phase of the main lobe.
- Specify the normalized start phase of the sidelobe.
- Use the firls function in signal processing toolbox of MATLAB [3],
Here is a design example with the MATLAB source codes for an 8-element
LAA. For the normalized phase, i.e., φ is between 0 and 1, set the
cut-off phase of the main lobe to 0.1, the start phase of the sidelobe
from 0.2 to 1. By the exciting current of each element for this LAA is
as the follows:
\begin{equation}
I_{0}=I_{7}=0.0804,\nonumber \\
\end{equation}\begin{equation}
I_{1}=I_{6}=0.1147,\ \nonumber \\
\end{equation}\begin{equation}
I_{2}=I_{5}=0.1422\nonumber \\
\end{equation}\begin{equation}
I_{3}=I_{4}=0.1574\nonumber \\
\end{equation}The pattern diagram of this LAA vs that of ULAA is as the follows: