# On the transmit weight vector in MIMO-MRC systems

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

• 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) – Marcus Müller Feb 24 '19 at 17:09
• Could you write down "maximize the signal component" in term of mathematical expression? – AlexTP Feb 24 '19 at 22:42
• @Wei-ChengLiu exactly! So, what's the value of $\|\mathbf H_0^H \mathbf w_R\|$, or: how does one compute that? – Marcus Müller Feb 25 '19 at 16:51
• 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 :) – Marcus Müller Mar 1 '19 at 7:33
• @Wei-ChengLiu did you get the answer for this question? I think that's logic. . – Zeyad_Zeyad Apr 24 '19 at 12:38