Digital Beamforming - Channel Matrix Singular Value Decomposition

Consider a point-to-point MIMO beamforming system, assuming that the transmitter has $N_t$Antennas, $N_r$The root receiving antenna, the transmitting end sends the modulated complex number information vector as $x$ , after the transmit beamforming matrix$After\; the\; function\; of\; F$ , it is loaded on each array element of the transmitting antenna, and passes through the MIMO channelHHThe received signal vector after$H$
$$y=HFx+$$nwhere
,$$nn$$$n$ is complex Gaussian white noise, the mean is$\theta$ , the variance is$p^2$ .
At the receiver using the receive beamforming matrix$W$ performs a weighted combination operation on the received signals:
$$W^{H}y=W^{H}HFx+W^{H}nFor$$
the convenience of analysis, the beamforming matrix$W$ and$F$ normalization, ie$W^{H}W=F^{H}F=I$ , using the criterion of maximizing the received signal-to-noise ratio to obtain the beamforming matrix. In order to maximize the receiving SNR, the transmitting beamforming matrix should be obtained according to the optimization criterion of the following formula:
$$F_t=\underset{F}{\mathrm{argmax}}\left[(\mathbf{H}\mathbf{F}{)}^{H}HF\right]$$
According to the KKT condition in convex optimization theory,$F_t$The columns are represented by $H^{H}H$的$N_s$The eigenvectors corresponding to the largest eigenvalues ​​are formed, where $N_s$is the number of data streams. Similarly, at the receiver, the receive beamforming matrix $W_r$The columns are represented by $H^{H}H$的$N_s$The eigenvectors corresponding to the largest eigenvalues ​​are formed. According to matrix theory, $F_t$Give $W_r$They are the largest N s N_s in the right singular matrix and the left singular matrix obtained after Singular Value Decomposition (SVD) of the channel matrix$N_s$Singular vectors corresponding to singular values.
For channel matrix $H$ is decomposed by SVD to get:
$$H=U\Lambda V^H$$
使用，$U=[u_1,u_2,...,u_{N_r}]$ is$N_r$Order unitary matrix, $V=[v_1,v_2,...,v_{N_t}]$ is$N_t$Order unitary matrix, $\Lambda$是$N_r\times N_t$Order diagonal matrix, the elements on the diagonal are H's $p=min(N_t,N_r)$ singular values$p_1,p_2,...,p_p$, arrange them in descending order, that is, $p_1>p_2>...>p_p$. Then, the known channel matrix $H$ , the optimal beamforming matrix$W_r$and $F_t$are the right singular vector and left singular vector corresponding to the largest singular value of H respectively: $W_r=U[1:N_s]$ and$F_t=V[1:N_s]$ , if the transmitted signal power is normalized, the maximum SNR is$p_i^2/p^2,i=1,...,N_s$

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References:
[1] Fu Zigang. Research on Beamforming in Millimeter Wave MIMO System[D]. 2016.