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# Transmit beamforming

by on April 13, 2009

In this post lets discuss a closed-loop transmit diversity scheme, where the transmitter has the knowledge of the channel. As there is a feedback path required from the receiver, to communicate the channel seen by the receiver to the transmitter, the scheme is called closed-loop transmit diversity scheme. Recall that the transmit diversity using Space Time Coding (Alamouti STBC) does not require the knowledge of the channel. In this post, we will restrict our discussion to a 2 transmit, 1 receive case. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

## Channel Model

1. We have 1 receive antennas and two transmit antenna.

2. The channel is flat fading – In simple terms, it means that the multipath channel has only one tap. So, the convolution operation reduces to a simple multiplication. For a more rigorous discussion on flat fading and frequency selective fading, may I urge you to review Chapter 15.3 Signal Time-Spreading from [DIGITAL COMMUNICATIONS: SKLAR]

3. The channel experienced by each receive antenna is randomly varying in time. For the $i^{th}$ receive antenna, each transmitted symbol gets multiplied by a randomly varying complex number $h_i$. As the channel under consideration is a Rayleigh channel, the real and imaginary parts of $h_i$ are Gaussian distributed having mean $\mu_{h_i}=0$ and variance $\sigma^2_{h_i}=\frac{1}{2}$.

4. The channel experience by each transmit antenna to receive antenna is independent from the channel experienced by other transmit antennas.

5. On each receive antenna, the noise$n$ has the Gaussian probability density function with

$p(n) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(n-\mu)^2}{2\sigma^2}$ with $\mu=0$ and $\sigma^2 = \frac{N_0}{2}$.

6. At each transmit antenna, the channel $h_i$ is known.

Figure: 2 transmit 1 receive beam steering

## Transmit Beamforming

$y = \begin{eqnarray}[h_1 & h_2]\end{eqnarray} \left[ \begin{eqnarray}x \\ \\ x \end{eqnarray}\right]+n = \underbrace{(h_1+h_2)}x + n$ where,

$y$ is the received symbol,
$h_i$ is the channel on the $i^{th}$ transmit antenna,
$x$ is the transmitted symbol and
$n$ is the noise on the receive antenna.

When transmit beamforming is applied, we multiply the symbol from each transmit antenna with a complex number corresponding to the inverse of the phase of the channel so as to ensure that the signals add constructively at the receiver. In this scenario, the received signal is,

$y = \begin{eqnarray}[h_1 & h_2]\end{eqnarray} \left[ \begin{eqnarray}e^{-j\theta_1} \\ \\ e^{-j\theta_2} \end{eqnarray}\right]x+n$,

where,

$h_1 = |h_1|e^{j\theta_1}$ and

$h_2 = |h_2|e^{j\theta_2}$.

In this case, the signal at the receiver is,

$y = \underbrace{\left(|h_1| + |h_2|\right)}x + n$.

For equalization, we need to divide the received symbol $y$with the new effective channel, i.e,

$\hat{y} = \frac{y}{\left(|h_1| + |h_2|\right)}=x + \frac{n}{\left(|h_1| + |h_2|\right)}$.

## BER Simulation Model

The Matlab/Octave script performs the following

(a) Generate random binary sequence of +1′s and -1′s.

(b) Multiply the symbols with the beam steering matrics – corresponding to the phase of the channel

(c) Perform equalization at the receiver

(d) Perform hard decision decoding and count the bit errors

(e) Repeat for multiple values of $\frac{E_b}{N_0}$ and plot the simulation and theoretical results.

Figure: BER plot for 2 transmit 1 receive beamforming for BPSK in Rayleigh channel

## Observations

1. Sending the same information on multiple transmit antenna does not provide diversity gain. Intuituvely, this is due to the fact that the effective channel $h_1 + h_2$ in a 2 transmit antenna case is again a Rayleigh channel; hence the BER performance is identical to 1 transmit 1 receive Rayleigh channel case.

2. If the transmit symbols are multiplied by a complex phase to ensure that the phases align at the receiver, there is diversity gain. However, the BER performance seems to be slighly poorer than the 1 transmit 2 receive MRC case. I guess, the noise is scaled by $|h_1| + |h_2|$ in the case of transmit beamforming, whereas the noise scaling is different in the case of Maximal Ratio Combining. I need to study bit more for a precise answer.

## Reference

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