Set-up No Scattering Channel

### Channel model

$$\bf{y}=\bf{H}\cdot \bf{x}$$ $\bf{y}$: received vector
$\bf{x}$: transmitted vector
$\bf{H}$: channel matrix with elements $h_{m,n}$

free-space path loss is considered
$$h_{m,n}=\frac{c}{f\cdot 4\cdot \pi \cdot d}\cdot e^{-j\frac{2 \cdot \pi \cdot f \cdot d}{c}}$$ $f$: frequency
$c$: speed of light
$d$: distance to receiver

### Precoding:

$$\bf{x}=\bf{P}\cdot \bf{s}$$ $\bf{P}$: precoding matrix
TxMF: Transmitter Matched Filter
$$\bf{P}=\beta_{\text{TxMF}}\cdot \bf{H}^H$$ constant TX power $E_{tx}=\text{E}||\bf{x}||^2_2$: $$\beta_{\text{TxMF}}=\sqrt\frac{E_{tx}}{trace\left(\bf{H}^H\cdot \bf{s}\cdot \bf{s}^H\cdot \bf{H}\right)}$$ TxMF: Transmitter Matched Filter (norm)
$$\bf{P}=\beta_{\text{TxMF,norm}}\cdot \bf{H}^H$$ constant RX power $E_{rx}=\text{E}||\bf{y}||^2_2$: $$\bf{y}=\bf{H}\cdot \bf{x} =\beta_{TxMF,norm}\cdot \bf{H} \cdot \bf{H}^H \cdot \bf{s}$$ $$\beta_{\text{TxMF,norm}}=\sqrt\frac{E_{rx}}{trace\left(\bf{H}\cdot \bf{H}^H\cdot \bf{s}\cdot \bf{s}^H\cdot \bf{H} \cdot \bf{H}^H\right)}$$ TxZF: Transmitter Zero Forcing
$$\bf{P}=\beta_{\text{TxZF}}\cdot\bf{H}^H \cdot \left( \bf{H}\cdot \bf{H}^H\right)^{-1}$$ constant TX power $E_{tx}=\text{E}||\bf{x}||^2_2$: $$\beta_{\text{TxZF}}=\sqrt{\frac{E_{tx}}{trace\left(\bf{H}^H\cdot \left(\bf{H}\cdot\bf{H}^H\right)^{-1}\bf{s}\cdot \bf{s}^H \left(\left(\bf{H}\cdot\bf{H}^H\right)^{-1}\right)^H\cdot \bf{H}\right)}}$$ TxZF: Transmitter Zero Forcing (norm)
$$\bf{P}=\beta_{\text{TxZF,norm}}\cdot\bf{H}^H \cdot \left( \bf{H}\cdot \bf{H}^H\right)^{-1}$$ constant RX power $E_{rx}=\text{E}||\bf{y}||^2_2$: $$\bf{y}=\bf{H}\cdot \bf{x} =\beta_{TxZF,norm}\cdot \bf{H} \cdot\bf{H}^H \cdot \left( \bf{H}\cdot \bf{H}^H\right)^{-1} \cdot \bf{s}=\beta_{TxZF,norm}\cdot\bf{s}$$ $$\beta_{\text{TxZF,norm}}=\sqrt{\frac{E_{rx}}{trace\left(\bf{s}\cdot \bf{s}^H\right)}}$$ • Different transmission (TX) antennas geometries useable
- linear horizontal array
- linear vertical array
- cyclic array
• Simulation of a two-dimensional coverage area
• Precoding results in beamforming patterns
• Dipols used as TX antennas