## mmWave Channel Model

The mmW channel matrix is randomly distributed following a random geometry with a small number of propagation paths (order of tens) grouped in very few clusters of similar paths (average $1.9$)[4,27] $$\textbf{H} = \sqrt{\dfrac{N_{t}N_{r}}{\rho N_cN_p}}\sum_{k=1}^{N_c}\sum_{\ell=1}^{N_p}g_{k,\ell}\textbf{a}_{r}(\phi_{k}+\Delta\phi_{k,\ell}) \textbf{a}_{t}^H(\theta_k+\Delta\theta_{k,\ell})$$ where $\rho$ is the distance dependent path-loss, $N_c$ is the number of independent clusters, $N_p$ represents the number of paths per cluster, $g_{k,\ell}\sim \mathcal{CN}(0,1)$ is the small scale fading associated with the $\ell^{th}$ path of the $k^{th}$ cluster, $\phi_k$ and $\theta_k$ $\in [0,2\pi)$ represent the mean angle of arrival (AoA) and angle of departure (AoD) of the $k^{th}$ cluster at the receiver and at the transmitter, respectively. The AoA and AoD of each path within each cluster vary around the mean direction of that cluster, with a standard deviation $\theta_{RMS}$. We represent by $\Delta\phi_{k,\ell}$ and $\Delta\theta_{k,\ell}$ $\sim \mathcal{N}(0,\theta_{RMS}^2)$ the differential AoA and AoD of the $\ell^{th}$ path of the $k^{th}$ cluster. Here, we model the antenna arrays at both the transmitter and the receiver as uniform linear arrays (ULA) with adjacent antenna spacing of half the wavelength of the transmitted signal ($\lambda/2$). Under this model, a spatial signature vector $\textbf{a}_{t}$ for the transmit array can be expressed as a function of the AoD as follows $$\textbf{a}_{t} = \dfrac{1}{\sqrt{N_{t}}}[1, e^{j{\pi}\sin (\theta)}, ... , e^{j(N_{t} - 1){\pi}\sin (\theta)}]^T$$ where $T$ represents the transpose. Likewise, an analogous expression characterizes the spatial signature vector for the receiver, $\textbf{a}_{r}$. Finally, for a 28 GHz channel, the path-loss is computed with [28] $$\rho_{LOS} (dB)=61.5+20\log_{10} (d)+\xi,\; \xi\sim\mathcal{N}(0,5.8),$$ $$\rho_{NLOS} (dB)=72+29.2\log_{10} (d)+\xi,\; \xi\sim\mathcal{N}(0,8.7),$$ where $d$ represents the distance between the transmitter and the receiver on a straight line while the variation of distances traversed by different paths is captured in $g_{k,\ell}$. The parameters suggested to model mmW 28 GHz channels in the literature [28] are $N_p=20$, $N_c\sim\max\{Poisson(1.8),1\}$ and $\theta_{RMS}\sim 10^o$. In our simulations, however, we set specific non-random values for $N_c$ to study the effect of the rank of the channel matrix $\mathbf{H}$ in the performance of receiver architectures. We consider a rank-1 channel with $N_c=N_p=1$; a low-rank channel with $N_c=1$ and $N_p=10$, where $\mathbf{H}$ typically has $4-5$ dominant eigenvalues; and a not-so-low rank channel $N_c=2$, $N_p=10$ where $\mathbf{H}$ typically has $8-10$ dominant eigenvalues.