# 2x1 TX beamforming vs 2x1 Alamouti

Started by June 10, 2008
```Dear all,

I have the following doubt. Consider a 2x1 MIMO system, 2 antennas in
the Tx and 1 antenna in the receiver. Which scheme should work better
2x1 TX beamforming (with perfect knwoledge in the transmitter) or 2x1
Alamouti ?

Where I understand by beamforming the following:

The MIMO channel matrix is in this case H = [h_0 h_1]. Then the
transmitter would perform the SVD of the channel matrix obtaining a
2x2 transmit processing matrix V that would represent the transmit
beamforming. What the transmitter would then transmit is:

x = V*s, where s = [a 0]', a vertical vector containing the modulation
symbol "a" and a zero.

In my understanding what the transmitter is doing in this case is
analyzing the two spatial channels between himself and the receiver
and putting all the energy through the best spatial channel. Instead
Alamouti is doing MRC (maximal ratio combining) among the two spatial
channels. Is it then correct to say that in this case Alamouti should
be better than beamforming ?

Regards

Daniel

```
```> The MIMO channel matrix is in this case H = [h_0 h_1]. Then the
> transmitter would perform the SVD of the channel matrix obtaining a
> 2x2 transmit processing matrix V that would represent the transmit
> beamforming. What the transmitter would then transmit is:
>
> x = V*s, where s = [a 0]', a vertical vector containing the modulation
> symbol "a" and a zero.

for MISO case, the optimal beamforming vector is simply the conjugate
of H, as the eigenspace of the channel is only 1 and this is spanned
by the channel vector.

> In my understanding what the transmitter is doing in this case is
> analyzing the two spatial channels between himself and the receiver
> and putting all the energy through the best spatial channel. Instead
> Alamouti is doing MRC (maximal ratio combining) among the two spatial
> channels. Is it then correct to say that in this case Alamouti should
> be better than beamforming ?

Depends on your definition of doing better. Diversity-wise, both have
diversity order of 2. Performance-wise, beamforming is probably
better. Beamforming is actually doing MRC at the transmitter instead
of the receiver, while Alamouti is equivalent to doing an averaging of
the two channels. The advantages of  Alamouti is that it is open loop
(no need to information at transmitter), and ML is decoupled at the
```
```On 12 jun, 11:20, Kenneth Chin <whc...@gmail.com> wrote:
> > The MIMO channel matrix is in this case H = [h_0 h_1]. Then the
> > transmitter would perform the SVD of the channel matrix obtaining a
> > 2x2 transmit processing matrix V that would represent the transmit
> > beamforming. What the transmitter would then transmit is:
>
> > x = V*s, where s = [a 0]', a vertical vector containing the modulation
> > symbol "a" and a zero.
>
> for MISO case, the optimal beamforming vector is simply the conjugate
> of H, as the eigenspace of the channel is only 1 and this is spanned
> by the channel vector.
>
> > In my understanding what the transmitter is doing in this case is
> > analyzing the two spatial channels between himself and the receiver
> > and putting all the energy through the best spatial channel. Instead
> > Alamouti is doing MRC (maximal ratio combining) among the two spatial
> > channels. Is it then correct to say that in this case Alamouti should
> > be better than beamforming ?
>
> Depends on your definition of doing better. Diversity-wise, both have
> diversity order of 2. Performance-wise, beamforming is probably
> better. Beamforming is actually doing MRC at the transmitter instead
> of the receiver, while Alamouti is equivalent to doing an averaging of
> the two channels. The advantages of  Alamouti is that it is open loop
> (no need to information at transmitter), and ML is decoupled at the

Dear Kenneth,

sentence?

> for MISO case, the optimal beamforming vector is simply the conjugate
> of H, as the eigenspace of the channel is only 1 and this is spanned
> by the channel vector.

I understand that what you propose will result in MRC in the reciver
but is not that the same than Alamouti in a 2x1 system then ?
And another question is that what I was doing is instead of directly
applying the conjugate of the spatial channel, I was applying the SVD
to the MISO channel vector and obtaining a 2x2 precoding matrix to be
applied in the TX, then the beamforming gain is the eigenvalue of the
SVD decomposition, that as I obeserve is exactly the same than the MRC
gain (coherently add all the components of the spatial channel), is
that correct ?

Best Regards

Daniel
```
```On Jun 15, 10:42 pm, Dani Camps <danicamp...@gmail.com> wrote:

> I understand that what you propose will result in MRC in the reciver
> since the two spatial channels are coherently added in the receiver,
> but is not that the same than Alamouti in a 2x1 system then ?

No. Alamouti is an open loop system, which means that the transmitter
has no channel information. Hence, it is not able to ensure that the
information signals are sent through the two antennas, so theres
definitely no way the signals can add constructively.

> And another question is that what I was doing is instead of directly
> applying the conjugate of the spatial channel, I was applying the SVD
> to the MISO channel vector and obtaining a 2x2 precoding matrix to be
> applied in the TX, then the beamforming gain is the eigenvalue of the
> SVD decomposition, that as I obeserve is exactly the same than the MRC
> gain (coherently add all the components of the spatial channel), is
> that correct ?

when you apply SVD to your channel vector, you will get a single
eigenmode, which means that only 1 eigenvector is useful, the other is
just meaningless. and if you look carefully at your useful
eigenvector, it is simply a scaled version of your channel vector
conjugated. Essentially, what you are doing is the same as applying
the conjugated channel vector as the beamforming weights. Just
remember to take care of the transmit power.

-K
```
```On Jun 17, 12:17&#2013266080;pm, Kenneth Chin <whc...@gmail.com> wrote:
> On Jun 15, 10:42 pm, Dani Camps <danicamp...@gmail.com> wrote:
>
> > I understand that what you propose will result in MRC in the reciver
> > since the two spatial channels are coherently added in the receiver,
> > but is not that the same than Alamouti in a 2x1 system then ?
>
> No. Alamouti is an open loop system, which means that the transmitter
> has no channel information. Hence, it is not able to ensure that the
> information signals are sent through the two antennas, so theres
> definitely no way the signals can add constructively.
>
> > And another question is that what I was doing is instead of directly
> > applying the conjugate of the spatial channel, I was applying the SVD
> > to the MISO channel vector and obtaining a 2x2 precoding matrix to be
> > applied in the TX, then the beamforming gain is the eigenvalue of the
> > SVD decomposition, that as I obeserve is exactly the same than the MRC
> > gain (coherently add all the components of the spatial channel), is
> > that correct ?
>
> when you apply SVD to your channel vector, you will get a single
> eigenmode, which means that only 1 eigenvector is useful, the other is
> just meaningless. and if you look carefully at your useful
> eigenvector, it is simply a scaled version of your channel vector
> conjugated. Essentially, what you are doing is the same as applying
> the conjugated channel vector as the beamforming weights. Just
> remember to take care of the transmit power.
>
> -K

Hi,

that in Alamouti the two signals transmitted at one symbol time do not
i.e. the channel stays constant over two consecutive symbol times,
then after the two symbol times and the processing done in the
receiver what I have is the SNR boosted by something equivalent to the
coherent addition of the two channel components, i.e. (|h_0|^2+|h_1|
^2) for each transmitted constellation symbol. Should not in this
sense, post detection SNR, give Alamouti the same performance than
transmit beamforming ?

Regards

Dani
```
```> Thanks again for your clear answer. Just one addition. I understand
> that in Alamouti the two signals transmitted at one symbol time do not
> add coherently in the receiver, but under the assumptions of Alamouti.
> i.e. the channel stays constant over two consecutive symbol times,
> then after the two symbol times and the processing done in the
> receiver what I have is the SNR boosted by something equivalent to the
> coherent addition of the two channel components, i.e. (|h_0|^2+|h_1|
> ^2) for each transmitted constellation symbol. Should not in this
> sense, post detection SNR, give Alamouti the same performance than
> transmit beamforming ?

Not really. Work out the SNR for Alamouti and work out the SNR for
beamforming and you'll find that they are different. Remember, that
noise doubles as well when you observe over two time slots.

-K

```