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:

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:

The pattern diagram of this LAA vs that of ULAA is as the follows: