In this paper, a generalized scenario for SNR maximization is derived employing a beamforming technique on the transmitter side, where "n" number of signal components are transmitted by "N" antenna elements of a linearly phased antenna array. The transmitted signal. components differing in phase and amplitude are spatially multiplexed into a common signal x(t). We consider a receiver with multiple antenna elements as shown in Fig. 1. The antenna elements are arranged in such a way that the spacing \("d"\) between them is uniform and placed linearly. A signal\({y}_{i}\left(t\right)\) is considered to be arriving at an angle of \("\theta "\) with the vertical axis at the uniform linear array on the receiver side, where \("i = \text{1,2},3,\dots N"\) represents the number of linearly spaced antenna elements on the receiver’s side.

If \("{y}_{i}\left(t\right)"\) is the signal received at the \({i}^{th}\) antenna element the we have

$${y}_{i}\left(t\right)= x\left(t\right).{e}^{-j(i-1)\phi }+{\sigma }_{i}$$

1

where \({\sigma }_{i}\)is the Gaussian noise, appearing in the received signal due to inherited disturbances in a wireless channel of the communication system and \(j\) is a complex number. Angle \("\phi "\) is related to \("\theta "\) as

$$\phi =2\pi {F}_{c}\frac{dcos\theta }{c}$$

or\(\)\(\phi =2\pi \frac{dcos\theta }{\lambda }\)

Where \(\theta\)is the angle of arrival for this uniform linear phased array.

The signals received at each antenna element is given as under:

$$\begin{array}{cc}{y}_{1}\left(t\right)= x\left(t\right)+{\sigma }_{1}& Signal received at first antenna element of ULA\\ {y}_{2}\left(t\right)= x\left(t\right).{e}^{-j\phi }+{\sigma }_{2}& Signal received at second antenna element of ULA\\ {y}_{3}\left(t\right)=x x\left(t\right).{e}^{-j2\phi }+{\sigma }_{3}& Signal received at third antenna element of ULA\\ ⋮ ⋮ + ⋮& ⋮\\ {y}_{N}\left(t\right)= x\left(t\right).{e}^{-j(N-1)\phi }+{\sigma }_{N}& Signal received at {N}^{th} antenna element of ULA\end{array}$$

The signal components received at each successive receiver antenna element is delayed by an additional phase factor of \({ e}^{-j\phi }\)corresponding to the signal components received at the previous receiver antenna element. Thus, the received signal vector becomes a phased antenna array vector.

## 3.1 Signal Processing in Uniform Linear Arrays (ULA)

The vector system model for the Uniform linear array is given as:

$$\left[\begin{array}{c}{y}_{1}\\ {y}_{2}\\ {y}_{3}\\ ⋮\\ {y}_{N}\end{array}\right] =\left[\begin{array}{c}1\\ {e}^{-j\phi }\\ {e}^{-2j\phi }\\ ⋮\\ {e}^{-j\left(N-1\right)\phi }\end{array}\right] x+ \left[\begin{array}{c}{\sigma }_{1}\\ {\sigma }_{2}\\ {\sigma }_{3}\\ ⋮\\ {\sigma }_{N}\end{array}\right]$$

2

$$\begin{array}{ccc}\stackrel{-}{y} & \stackrel{-}{h\left(\phi \right)} & \stackrel{-}{\sigma }\end{array}$$

The \(\stackrel{-}{h\left(\phi \right)}\)function is known as the array steering vector. Therefore, the system model for Signal processing in ULA is expressed in terms of the array steering vector\(\stackrel{-}{ h}\left(\phi \right)\). Hence, we have

$$\begin{array}{ccc}\stackrel{-}{y} = & \stackrel{-}{ h}\left(\phi \right). x +& \stackrel{-}{\sigma }\end{array}$$

3

Consider the noise powers \({\sigma }_{1}, {\sigma }_{2}, {\sigma }_{3}\dots . {\sigma }_{N}\) as Independent Identically Distributed (IID) Gaussian with zero mean i.e. each Gaussian noise sample value \({\sigma }_{i}\) has mean \(E\left({\sigma }_{i} \right)=0\) and power\(E\left\{{\left|{\sigma }_{i}\right|}^{2}\right\}={\rho }^{2}\), where \({\rho }^{2}\) is the variance or noise power. Further, it is assumed that the covariance between the noise samples at adjacent antennas is also zero. i.e.\(E\left\{\left|{\sigma }_{i}{\sigma }_{j}^{*}\right|\right\}=0\)

Now the received signal samples \({y}_{1}, {y}_{2}, {y}_{3}\dots . {y}_{N}\) are combined at the ULA employing a combiner or a Beamformer \(\stackrel{-}{a}\) such that \(\stackrel{-}{a}=\left[\begin{array}{c}{a}_{1}\\ {a}_{2}\\ {a}_{3}\\ ⋮\\ {a}_{N}\end{array}\right]\) and\(\tilde{ y}={a}_{1}^{*}{y}_{1}+{a}_{2}^{*}{y}_{2}+{a}_{3}^{*}{y}_{3}+\dots +{a}_{N}^{*}{y}_{N}\).

Where \(\tilde{y}\) is the combined received signal vector in a ULA and can be expressed as the vector product of the beamforming weights \({a}_{1}^{*}+{a}_{2}^{*}+{a}_{3}^{*}+\dots +{a}_{N}^{*}\) and the received signal samples \({y}_{1}, {y}_{2}, {y}_{3}\dots . {y}_{N}\) at different antenna elements as under

\(\tilde{y}={a}_{1}^{*}+{a}_{2}^{*}+{a}_{3}^{*}+\dots +{a}_{N}^{*}\left[\begin{array}{c}{y}_{1}\\ {y}_{2}\\ {y}_{2}\\ ⋮\\ {y}_{N}\end{array}\right]\) Which implies\(\tilde{ y}={ \stackrel{-}{a}}^{H}\stackrel{-}{y}\), where \(H\)represents the Hermitian operator. Substituting the value of \(\stackrel{-}{y}\)from (3), we get

$$\tilde{y}={ \stackrel{-}{a}}^{H}\left(\stackrel{-}{ h}\right(\phi ). x+\stackrel{-}{\sigma })$$

$$\tilde{y}={ \stackrel{-}{a}}^{H}\left(\stackrel{-}{ h}\right(\phi ). x+{ \stackrel{-}{a}}^{H}\stackrel{-}{\sigma })$$

The factor \({ \stackrel{-}{a}}^{H}\stackrel{-}{ h}\left(\phi \right). x\)represents the signal part and \({ \stackrel{-}{a}}^{H}\stackrel{-}{\sigma })\) represents the noise part of the combined receiver signal vector \(\tilde{y}\) in the ULA system.

## 3.2 Signal to Noise Ratio (SNR)

Assuming the transmitted signal power with variance \(E\left\{{\left|x\right|}^{2}\right\}=P\). The ratio of transmitted signal power to noise power is termed as the Signal to Noise Ratio (SNR). By carrying out the linear combination of the Gaussian IID noise samples \({\sigma }_{1},{\sigma }_{2},{\sigma }_{3},\dots .{\sigma }_{N}\) with zero mean and variance \(E\left\{{\left|{\stackrel{-}{a}}^{H}\stackrel{-}{\sigma }\right|}^{2}\right\}\) we have mean \(E\left\{{\stackrel{-}{a}}^{H}\stackrel{-}{\sigma }\right\}=0\)and variance\(E\left\{{\left|{\stackrel{-}{a}}^{H}\stackrel{-}{\sigma }\right|}^{2}\right\}= {\sigma }^{2}{‖\stackrel{-}{a}‖}^{2}\). Therefore

SNR =\(\frac{{\left|{\stackrel{-}{a}}^{H}\stackrel{-}{h}\left(\phi \right)\right|}^{2}E\left\{{\left|x\right|}^{2}\right\}}{E\left\{{\left|{a}^{-H}\stackrel{-}{\sigma }\right|}^{2}\right\}}\)

SNR = \(\frac{{\left|{\stackrel{-}{a}}^{H}\stackrel{-}{h}\left(\phi \right)\right|}^{2}P}{{\sigma }^{2}{‖\stackrel{-}{a}‖}^{2}}\) (4)

Equation (4) represents the general expression of the SNR at the output of the Beamformer.

## 3.3 Maximization of Signal to Noise Ratio

The maximum SNR at the output of the Beamformer is obtained by applying Cauchy-Schwartz Inequality on the general expression of the SNR. This yields the optimum value of the beamforming vector\(\stackrel{-}{a}\), which maximizes the SNR at the output.

According to Cauchy-Schwartz Inequality, we have\({\left|{\stackrel{-}{a}}^{H}\stackrel{-}{h}\left(\phi \right)\right|}^{2} \le {‖\stackrel{-}{a}‖}^{2}{‖\stackrel{-}{h}\left(\phi \right)‖}^{2}\)

Therefore, from (4) we have

SNRmaximum ≤\(\frac{{‖\stackrel{-}{a}‖}^{2}{‖\stackrel{-}{h}\left(\phi \right)‖}^{2}P}{{\sigma }^{2}{‖\stackrel{-}{a}‖}^{2}}\)

SNRmaximum ≤ \(\frac{{‖\stackrel{-}{h}\left(\phi \right)‖}^{2}P}{{\sigma }^{2}}\) (5)

From equation X we have \(\stackrel{-}{h}\left(\phi \right)= \left[\begin{array}{c}1\\ {e}^{-j\phi }\\ {e}^{-2j\phi }\\ ⋮\\ {e}^{-j\left(N-1\right)\phi }\end{array}\right]\) which implies \({‖\stackrel{-}{\text{h}}\left({\phi }\right)‖}^{2}=1+1+1+\dots +1;\text{N}\) times

Hence,

SNRmaximum =\(\frac{N.P}{{\sigma }^{2}}\)

Thus, the SNR at the output of the Beamformer is N times conventional SNR, where N is the number of antenna elements in an array also known as the array Gain of the Uniform Linear Phased Array (ULPA). The ULPA is a novel technology in pursuit of wireless communication systems as it results in a gain of a factor N in the SNR at the output of the receiver. For maximum SNR at the receiver choose \(\stackrel{-}{a}= \stackrel{-}{h}\left(\phi \right)\) i.e., the optimal Beamformer that maximizes the SNR at the receiver output is known as the Maximal Ration Combiner (MRC) for the phased array system. It acts as a matched filter since it matches the receive signal vector \(\tilde{y}\) with the array steering vector\(\stackrel{-}{h}\left(\phi \right)\).