Consider a normal array model with $N$ element. The output of the array is given by,
$$\mathbf{y}=\mathbf{w}^H\mathbf{x}=\mathbf{w}^H\mathbf{as}$$
where
$\mathbf{a}=[a_1,...,a_N]^T=\in\mathbb{C}^{N\times 1}$ is the manifold of the desired signal (I ignore the noise for simplicity).
- $T$ denotes the transpose.
$\mathbf{s}\in\mathbb{C}^{1\times M}$ is the desired signal
- $M$ is the number of snapshots and
- the power of $\mathbf s$ is $\sigma_s^2=\operatorname E(|\mathbf{s}(t)|^2)=\operatorname E(\mathbf{ss}^H)=\frac{1}{M}(|\mathbf{s}(1)|^2+...+|\mathbf{s}(M)|^2), \mathbf{ss}^H=|\mathbf{s}(1)|^2+...+|\mathbf{s}(M)|^2$
- $t$ is the time index,
$\mathbf{x}\in\mathbb{C}^{N\times M}$ is the desired signal incident at each antenna, $\mathbf{x}(t)=[\mathbf{x_1}(t),...,\mathbf{x_N}(t)]^T=[a_1\mathbf{s}(t),...,a_N\mathbf{s}(t)]^T$
$\mathbf{w}\in\mathbb{C}^{N\times 1}$ is the receiving weight (where $\mathbf w = \mathbf a$),
$^H $ denotes the conjugate transpose and
$\operatorname E$ denotes the expectation.
Therefore the power of the output $y$ is given by, $$P_y=\operatorname E (|\mathbf{y}(t)|^2)=\frac{1}{M}(|\mathbf{y}(1)|^2+...+|\mathbf{y}(M)|^2)=\frac{1}{T}(\mathbf{y} \mathbf{y}^H)=\frac{1}{T}(\mathbf{a}^H\mathbf{ass}^H\mathbf{a}^H\mathbf{a})=\sigma_s^2(\mathbf{a}^H\mathbf{a}\mathbf{a}^H\mathbf{a})=\sigma_s^2(N\times N)=N^2\sigma_s^2$$
I think the power of the desired signal that incident at the receiving array is, $$P_{inc}=P_{x_1}+...+P_{x_N}=\operatorname E(|\mathbf{x_1}(t)|^2)+...+\operatorname E(|\mathbf{x_N}(t)|^2)=\frac{1}{M}(|\mathbf{x_1}(1)|^2+...+|\mathbf{x_1}(M)|^2)+...+\frac{1}{M}(|\mathbf{x_N}(1)|^2+...+|\mathbf{x_N}(M)|^2)=\frac{1}{M}a_1\mathbf{s}(a_1\mathbf{s})^H+...+\frac{1}{M}a_N\mathbf{s}(a_N\mathbf{s})^H=\sigma_s^2a_1a_1^H+...+\sigma_s^2a_Na_N^H=N\sigma_s^2$$
($\operatorname{tr}(\mathbf A)$ denotes the trace of $\mathbf A$)
then the beamforming brings gain of $N$ to the power of the received desired signal.
What confuse me is why the total power of desired signal is changed. According to my understanding, the power should remain unchanged as the signal power incident at the array is certain and beamforming won't add power to it.
I think I was wrong but I don't know where. Is it related to the concept of effective array and directivity gain? Am I calculating the desired signal power that incident at the array before beamforming right?
I add the figure below to help me explain my problem. My question can be summarized as why the received signal power after beamforming (which is power of $y(t)$) is $N$ times of signal power of incident signal power or signal power before adding up? Is there a physical explanation for this?
