Implement angular-domain model of MIMO channel, and evaluate the Radiation and Reception patterns of Uniform Linear Arrays (ULA). This project is divided to 2 parts, which are SIMO(Single input multiple output) and MISO(Multiple input single output). Besides, we have to set some input parameters as following.
- The number of antennas N
- The normalized antenna separation $ \delta $ (normalized to the wavelength
$\lambda_c $ ) - The radiation or reception directions of the desired signal
- The radiation or reception direction of the interference signal
- The correlation between two signal with different radiation or reception direction
- Polar beamforming pattern of the ULA
- The SINR of multiple input signals (multiple reception directions) with diversity combining (considering fading for the signals and interference)
The time-invariant channel is described by
$$ y = Hx + w $$
x: transmitted signal; y: received signal; w: white Gaussian noise;
The following two figure shows the LOS(Line-of-sight) channel with SIMO and MISO.
Uniform linear antenna arrays: the antenna are evenly spaced on a straight line.
The following picture shows SIMO model:
Channel gain:
$$ h_i = 𝑎∙𝑒𝑥𝑝(−\frac{𝑗2\pi 𝑓_𝑐 𝑑_𝑖}{c}) = 𝑎∙𝑒𝑥𝑝 (−\frac{𝑗2\pi 𝑑_𝑖}{\lambda_𝑐}) $$
a: attenuation of the path, which we assume to be the same for all antenna pairs
Because antenna space is much smaller than distance between transmitter and receiver, we can write distance between each antenna pair as:
$$ di\approx 𝑑+(𝑖 − 1)\Delta _𝑟 \lambda _𝑐 \cos \phi , i = 1, ... , n_r $$
Define directional cosine: $ \Omega = cos\phi $. Channel gain will be:
The following picture shows MISO model:
Similar to SIMO model, the channel gain is:
Every linear transformation can be represented as a composition of three operations: a rotation operation, a scaling operation, and another rotation operation. H has a singular value decomposition (SVD): $$ H=U\Lambda V^* $$
U & V: (rotation) unitary matrices ; $ \Lambda $: a rectangular matrix
Define:
$$ \tilde{x}=V^* x ; \tilde{y}=U^* y; \tilde{w}=U^* w $$
We can rewrite the channel to angular domain as:
$$ \tilde{y}=\Lambda \tilde{x}+\tilde{w} $$
Orthonormal basis for received signal space will be: $$ U_r=[e_r (0),e_r (\frac{1}{L_r} ),… e_r (\frac{N_r -1}{L_r }]=U $$
Orthonormal basis for transmitted signal space will be: $$ U_t=[e_t (0),e_t (\frac{1}{L_t }),… e_t (\frac{N_t -1}{L_t })]=V $$
We now represent the MIMO fading channel $ y=Hx+w $ in the angular domain. $ U_t $ and $ U_r $ are respectively the $ n_t \times n_t $ and $ n_r \times n_r $ unitary matrices. Transformations: $$ x^a = U_t^* x, y^a = U_r ^* y $$ are the changes of coordinates of the transmitted and received signals into angular domain. Substitute this into y=Hx+w, we have channel in angular domain: $$ y^a =U_r ^* HU_t x^a +U_r^* w=H^a x^a +w^a $$ where $$ H^a = U_r ^* HU_t $$
Two transmit antenna are placed very far apart, so the two channel are independent. Channel matrix $ H=[h_1,h_2] $
Angle $ \theta $ between the two spatial signatures is
$$ |cosθ|=|e_r (\Omega _{r1} )^* e_r (\Omega _{r2})| $$
It only dependes on difference $ \Omega _r =\Omega _{r2} -\Omega _{r1} $ . Define
$$ f_r (\Omega _{r2}- \Omega _{r1} ) = e_r (\Omega _{r1} )^* e_r (\Omega _{r2} ) =\frac{1}{n_r }\sum _{i=1} ^{n_r } e^{-j2\pi (i-1) \Delta _r \Omega _r } =\frac{1}{n_r } \frac{1-e^{-j2\pi \Delta _r n_r \Omega _r }}{1-e^{-j2\pi \Delta _r \Omega _r }}$$
Since $ |1-e^{-j2\theta } |=|2sin\theta | $
Normalized length of the receive antenna array: $ L_r =n_r \Delta_r $
$$ |cos\theta |=\frac{1}{n_r }\frac{|1-e^{-j2π∆_r n_r Ω_r } |}{|1-e^{-j2π∆_r Ω_r } |} =|\frac{sin(πn_r ∆_r Ω_r)}{n_r sin(π∆_r Ω_r)}|=|\frac{sin(πL_r Ω_r)}{n_r sin(\frac{πL_r Ω_r }{n_r } )}| $$
Beamforming pattern : If the signal arrives from a single direction $ \phi _0 $, then the optimal receiver projects the received signal onto the vector $ e_r (cos \phi _0)
For geographically separated received antennas, the concepts is similar to separated transmitted antennas.
Input parameters:
- Number of received antenna: $ N_r $=5
- The normalized antenna separation: 1/2
- The reception directions of the desired signal: π/4
- The reception direction of the interference signal: π/2
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Correlation between two signal with different radiation or reception direction $ \theta $
We use $ |f_r (Ω_r )|=|f_r (Ω_{r2}-Ω_{r1} )|=|cos\theta | $ to represent the correlation between two different signal, which is shown as following image.
Besides, here will show some correlation comparison with different $ n_r $ and $ ∆_r $
For the last figure $ (∆_r =1/1000) $, we can see that, as $ n_r \rightarrow \infty $ and $ ∆_r \rightarrow 0$. $ $ f_r (Ω)\rightarrow e^{jπL_r Ω_r } sinc(L_r Ω_r) $$ -
Polar beamforming pattern of ULA
Polar plot of $ |f_r (Ω_{r2}-Ω_{r1} )|=|f_r (cosφ-cosφ_0 )| $. Main lobe is around reception direction of signal $ \phi _0 $ and also any $ \phi $ for: $ cos\phi =cos\phi _0 $ mod $ 1/∆_r $. Moreover, Beamwidth is determined by $ 2/L_r $. The following figure is our received gain pattern with $ \phi _0=π/4 $, and $ ∆_r=1/2 $, $ L_r=5/2 $. We can see that main lobe is around +45° and-45°.
Here, I will show other beamforming patterns with different $ phi_0 $, $ L_r $, $ ∆_r $
- Different $ \phi _0 $
- Different $ L_r $
- Different $ ∆_r $
- The SINR of multiple input signals (multiple reception directions) with diversity combining
I implemented this part by MRC (Maximal ratio combining), and the SINR = 16.9243 dB
Input parameters:
- Number of received antenna: $ N_t =7 $
- The normalized antenna separation: 1/2
- The reception directions of the desired signal: π/6
- The reception direction of the interference signal: π/3
Basically, the result of MISO is similar to SIMO.
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Correlation between two signal with different radiation or reception direction $ \theta $
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Polar beamforming pattern of ULA
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The SINR of multiple input signals (multiple reception directions) with diversity combining
I view different transmitted signals as different received signals and combined them together. I use MRC strategy and get the SINR = 35.75 dB
