知识储备-信道模型整理

整理不同的信道模型以及它们的使用场景

Rician Fading

介绍

MISO System: Transmitter with M antennas and the receiver has only one antenna.

It is a very general assumption that allows modeling a wide variety of channels by tuning the Rician factor \(\kappa\geq0\)

• \(\kappa=0\), the channel envelope is Rayleigh distributed
• \(\kappa\to\infty\), the channel is a fully deterministic LOS channel

莱斯衰落信道指除了经反射折射散射等来的信号外， 还有从发射机直接到达接收机 （如从卫星直接到达地面接收机）的信号，那么总信号的强度服从分布莱斯分布。有一条主路径，其余多径传输过来的信号仍如瑞利衰落所述。

瑞利衰落只适用于从发射机到接收机不存在直射信号（LOS，Line of Sight）的情况，即信道模型能够描述由电离层和对流层反射的短波信道，以及建筑物密集的城市环境。否则应使用莱斯衰落信道作为信道模型。

建模

\[ \textbf{h}=\sqrt{\frac{\kappa}{1+\kappa}}e^{\text{i}\varphi_0}\textbf{h}_{\text{los}}+\sqrt{\frac{1}{1+\kappa}}\textbf{h}_{\text{nlos}} \]

其中\(\varphi_0\)是初始相位。有LOS部分： \[ \textbf{h}_{\text{los}}=[1,e^{\text{i}\Phi_1},\cdots,e^{\text{i}\Phi_{M-1}}]^\mathrm{T} \] 且\(\Phi_t=-t\pi\sin\phi\)，\(\phi\)是azimuth angle relative to the boresight of the transmitting antenna array。

有Rayleigh部分： \[ \textbf{h}_{\text{nlos}}\sim\mathcal{CN}(\textbf{0},\textbf{R}) \]

适用范围

CSI-Free中，对Wireless Energy Transfer的信道进行建模。

• On CSI-Free Multiantenna Schemes for Massive RF Wireless Energy Transfer. Onel L. A. López et.al. IEEE Internet of Things Journal, Jan.1, 1 2021 (pdf) (Citations 8)

实现

function h = RicianFadingChannel(kappa,M,x,y, choice) 
    phi = atan(y/x);
    Phi = -(0:1:M-1)*pi*sin(phi);
    h_los = PreventiveAdjustmentofMeanShift(exp(1i.*Phi.'), M, choice);
    hnlos = 1/sqrt(2)*(randn(M,1)+1i*randn(M,1));
    h = sqrt(kappa/(1+kappa))*h_los*exp(1i*pi/4)+sqrt(1/(1+kappa))*hnlos;
end

?? Wideband geometric channel model

介绍

Consider a transmitter-IRS channel, \(\textbf{h}_{T,k}\), (and similarly for the IRS-receiver channel) consisting of L clusters. Each cluster contributes with one ray from the transmitter to the IRS. The ray parameters are: azimuth/elevation angles of arrival, \(\theta_l,\phi_l\in[0,2\pi)\); complex coefficient ; \(\alpha_l\in\mathbb{C}\); time delay \(\tau_l\in\mathbb{R}\). The transmitter-IRS path loss is denoted by \(\rho_T\). The pulse shaping function, with \(T_s\)-spaced signaling, is defined as \(p(\tau)\) at \(\tau\) seconds. The frequency-domain channel vector, \(\textbf{h}_{T,k}\), can then be defined as

###　建模

\[ \mathbf{h}_{\mathrm{T}, k}=\sqrt{\frac{M}{\rho_{\mathrm{T}}}} \sum_{d=0}^{D-1} \sum_{\ell=1}^{L} \alpha_{\ell} \mathbf{a}\left(\theta_{\ell}, \phi_{\ell}\right) p\left(d T_{S}-\tau_{\ell}\right) e^{-j \frac{2 \pi k}{K} d} \]

where \(\mathbf{a}\left(\theta_{\ell}, \phi_{\ell}\right)\in\mathbb{C}^{M\times 1}\) is the IRS array response vector. Assume a block-fading channel model, where \(\textbf{h}_{T,k}\) and \(\textbf{h}_{T,k}\) are assumed to stay constant over the channel coherence time.

适用范围

Both transmitter and Receiver have only one antenna.

• Deep Learning Coordinated Beamforming for Highly-Mobile Millimeter Wave Systems. Ahmed Alkhateeb et.al. IEEE Access, 2018 (pdf) (Citations 186)
• Deep Reinforcement Learning for Intelligent Reflecting Surfaces: Towards Standalone Operation. Abdelrahman Taha et.al. 2020 IEEE 21st International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), 26-29 May 2020 (pdf) (Citations 24)

Saleh-Valenzuela (SV) model (Narrowband geometric channel model)

介绍

Saleh-Valenzuela (SV) model where a geometric channel model is adopted with limited scattering.

eg: BS->IRS

建模

\[ \textbf{H}=\sqrt{\frac{NM}{\rho}}\sum\limits_{l=1}^L\varrho_l\textbf{a}_r(\vartheta_l,\gamma_l)\textbf{a}_l^\mathrm{H}(\phi_l) \]

其中，\(\rho\) average path-loss between BS and IRS; \(L\) is the number of paths; \(\varrho\) denotes the complex gain associated with the \(l\)-th path; \(\vartheta_l\) and \(\gamma_l\) denote the azimuth angle and elevation angle of arrival (AoA), respectively. \(\phi_l\) is the angle of departure (AoD), \(\textbf{a}_r\) and \(\textbf{a}_t\) represent the receive and transmit array response vectors, respectively.

假设IRS为一个有\(M_x\times M_y\)个elements的UPA (Uniform Planner Array)，则有： \[ \textbf{a}_r(\vartheta_l,\gamma_l)=\textbf{a}_x(u)\otimes\textbf{a}_y(v) \] 其中，\(\otimes\)表示Kronecker product（克罗内克积）。\(\textbf{a}_x(u)\)和\(\textbf{a}_y(v)\)表示导向矢量 or 相应向量 (steering vector) or 阵列流形 (array manifold)。 且\(u\triangleq2\pi d\cos(\gamma_l)/\lambda\)（即空间相位）,\(v\triangleq2\pi d\sin(\gamma_l)\cos(\vartheta_l)/\lambda\)，\(d\)表示天线位置：

在更多的文章中，\(v\triangleq2\pi d\sin(\gamma_l)\sin(\vartheta_l)/\lambda\)，这和选取的参考系相关。我个人认为选\(\sin\sin\)比较好

\[ \begin{aligned} &\boldsymbol{a}_{x}(u) \triangleq \frac{1}{\sqrt{M_{x}}}\left[\begin{array}{llll} 1 & e^{j u} & \ldots & e^{j\left(M_{x}-1\right) u} \end{array}\right]^{T} \\ &\boldsymbol{a}_{y}(v) \triangleq \frac{1}{\sqrt{M_{y}}}\left[\begin{array}{llll} 1 & e^{j v} & \ldots & e^{j\left(M_{y}-1\right) v} \end{array}\right]^{T} \end{aligned} \] 由于毫米波信道的稀疏散射特性，路径L的数目相对于G的维度很小。

理解

毫米波信道可以简化为单径信道模型（求和）。主要是LoS场景。毫米波绕射能力差，路径稀疏，信道模型具有丰富的几何特征。

上面的\(\mathbf{H}\)其实就是接收为面阵，发射为线阵的导向矢量乘积，也就是==直射信道==。

适用范围

考虑毫米波稀疏散射特性，有\(L\)个路径。

• Compressed Channel Estimation for Intelligent Reflecting Surface-Assisted Millimeter Wave Systems. Peilan Wang et.al. IEEE Signal Processing Letters, 2020 (pdf) (Citations 72)
• Deep Channel Learning for Large Intelligent Surfaces Aided mm-Wave Massive MIMO Systems. Ahmet M. Elbir et.al. IEEE Wireless Communications Letters, Sept. 2020 (pdf) (Citations 51)

实现

function H = generate_channel(Nt, Nr, L)
    AOD = pi*rand(L, 1) - pi/2;  %-2/pi~2/pi
    AOA = pi*rand(L, 2) - pi/2;  %-2/pi~2/pi

    alpha(1) = 1; % gain of the LoS
    if L >1
        alpha(2:L) = 10^(-0.7)*(randn(1,L-1)+1i*randn(1,L-1))/sqrt(2);
    end
%     alpha(2:L) = 0;
    H = zeros(Nr, Nt);
    Nx = sqrt(Nr);
    Ny = sqrt(Nr);
    for ll=1:1:L
        ax = sqrt(1/Nx)*exp((0:Nx-1)*1j*pi*cos(AOA(ll, 2))).';
        ay = sqrt(1/Ny)*exp((0:Ny-1)*1j*pi*sin(AOA(ll, 1))*sin(AOA(ll, 2))).';
        ar =kron(ax, ay);
        at =  sqrt(1/Nt)*exp((0:Nt-1)*1j*pi*sin(AOD(ll, 1))).';
        H = H + sqrt(Nr * Nt)*alpha(ll)*ar*at';
    end
end

还有简化版本，只有一个主径：

function H = generate_channel_simple(Nt, Nr, AOA1, AOA2, AOD)
    alpha = 1; % gain of the LoS
    Nx = sqrt(Nr); % 因为我这里是有Nr个element的IRS
    Ny = sqrt(Nr); % 方阵排布
    ax = sqrt(1/Nx)*exp((0:Nx-1)*1j*pi*cos(AOA2)).';
    ay = sqrt(1/Ny)*exp((0:Ny-1)*1j*pi*sin(AOA1)*sin(AOA2)).';
    ar =kron(ax, ay);
    at =  sqrt(1/Nt)*exp((0:Nt-1)*1j*pi*sin(AOD)).';
    H = sqrt(Nr * Nt)*alpha*ar*at';
end