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Recently, I am reading paper [1]. In this paper, the author wrote:

In MIMO-MRC systems in the absence of interferers, the signal vector at the received $n_R$ antennas is given by $$\mathbf{r} = \sqrt{P_0} \mathbf{H}_0 \mathbf{w}_T s_0 + \mathbf{n}, \tag{1}$$ and the decision variable for detecting the transmitted symbol $s_0$ is obtained by combining with a weight vector $\mathbf{w}_R$. The combined signal at the receiver is given by $$ z = \mathbf{w}^H_R \mathbf{r} = \sqrt{P_0} \mathbf{w}^H_R \mathbf{H}_0 \mathbf{w}_T s_0 + \mathbf{w}^H_R \mathbf{n}. \tag{2}$$ To maximize the signal component, the principle of maximum ratio transmission (MRT) can be used, then the transmit weight vector $\mathbf{w}_T$ can be chosen as $$ \mathbf{w}_T = \frac{\mathbf{H}_0^H \mathbf{w}_R}{\| \mathbf{H}_0^H \mathbf{w}_R \|}. \tag{3} $$

My question is: Why the transmit weight vector can be chosen as $$ \mathbf{w}_T = \frac{\mathbf{H}_0^H \mathbf{w}_R}{\| \mathbf{H}_0^H \mathbf{w}_R \|}? \tag{4}$$ Thank you very much.

Note

  • Vectors are expressed in boldface lower case letters, e.g., $\mathbf{a}$, and matrices are boldface capital letters, e.g., $\mathbf{A}$.
  • $P_0$ is the average received power.
  • $\mathbf{H}_0$ is the $n_R \times n_T$ channel matrix.
  • $n_T$ is the number of transmit antennas.
  • $\mathbf{w}_T$ is the beamforming vector at the transmitter with unit norm, i.e., $\| \mathbf{w}_T \|^2 = 1$.
  • The additive thermal noise $\mathbf{n}$ is a complex Gaussian vector with mean vector $\mathbf{0}$ and covariance matrix $\sigma_n^2 \mathbf{I}$.
  • We assume that the noise power per antenna is identical.
  • The matrix Hermitian is denoted by $\mathbf{A}^H$.
  • The Euclidean norm of vector $\mathbf{a}$ is denoted by $ \| \mathbf{a} \| $.

Reference

[1] K. S. Ahn, "Performance analysis of MIMO-MRC system in the presence of multiple interferers and noise over Rayleigh fading channels," in IEEE Transactions on Wireless Communications, vol. 8, no. 7, pp. 3727-3735, July 2009. doi: 10.1109/TWC.2009.080940

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    $\begingroup$ Good question! So, what happens when you insert $\mathbf w_T$ into equation $(2)$? Hint: what does $\mathbf H_0 \mathbf H_0^H$ w.r.t symmetry? ($\mathbf H$ has symmetry properties) $\endgroup$ – Marcus Müller Feb 24 at 17:09
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    $\begingroup$ Could you write down "maximize the signal component" in term of mathematical expression? $\endgroup$ – AlexTP Feb 24 at 22:42
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    $\begingroup$ @Wei-ChengLiu exactly! So, what's the value of $\|\mathbf H_0^H \mathbf w_R\|$, or: how does one compute that? $\endgroup$ – Marcus Müller Feb 25 at 16:51
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    $\begingroup$ right :) @Wei-ChengLiu since we ² afterwards, we can just drop the squared root, and then we end with "summing up the squared absolute values", and that can be represented by a vector operation: $\|a\|^2 = a^H a$ . So, when considering $$\|\mathbf H_0^H \mathbf w_r\|^2=\mathbf w_r^H\mathbf H_0\mathbf H_0^H \mathbf w_r\text,$$ you'll recognize the term from the enumerator and denominator :) $\endgroup$ – Marcus Müller Mar 1 at 7:33
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    $\begingroup$ @Wei-ChengLiu did you get the answer for this question? I think that's logic. . $\endgroup$ – Zeyad_Zeyad Apr 24 at 12:38

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