Alamouti code

To introduce Space Time Block Codes, we present The Alamouti code, an early space time code and still one of the most commonly used.

In this page, we discuss Alamouti code for 2transmitters-1receiver system(2x1) and 2transmitters-2receivers system(2x2).

                                                      ALAMOUTI 2X1

Encoding is done in the following manner:

            Y=HX+N ;                    Note: H is an Orthogonal Matrix.

            It is readily apparent that this is a rate-1 code. It takes two time-slots to transmit two symbols. The bit error rate (BER) of this STBC is equivalent to 2n_R-branch maximal ratio combinig (MRC). This is a result of the perfect orthogonality between the symbols after receive processing — there are two copies of each symbol transmitted and n_R copies received.

           This is a very special STBC. It is the only orthogonal STBC that achieves rate-1. That is to say that it is the only STBC that can achieve its full diversity gain without needing to sacrifice its data rate. Strictly, this is only true for complex modulation symbols. Since almost all constellation diagrams rely on complex numbers however, this property usually gives Alamouti's code a significant advantage over the higher-order STBCs even though they achieve a better error-rate performance.

Matlab Simulation

Alamouti Code for 2 transmitters and 1 receiver(2X1) in 16-QAM

          In this case, 2 channels are involved in every time slot whose channel impulse responses follow Rayleigh distribution.i.e h=x+iy; where x and y are Gaussian random variables.

First generate H matrix(Channel matrix) and then generate X matrix(16-QAM modulated input symbols) .

         Y=HX+N is the received matrix.

                                           DECODING OF ALAMOUTI 2X1 CODE
          For decoding Alamouti code, there are several methods like Zero Forcing(ZF), Minimum Mean Square Estimation(MMSE) ,Brute Force ML decoding etc..

          In the above mentioned decoding techniques ,ML decoding gives the best performance at the cost of maximum complexity when compared to other two.

As mentioned above Alamouti is an Orthogonal Code . So,the ML decoding simplifies to ZF decoding.

                                                              ZERO FORCING

     

         

Click the link below to view matlab code for Alamouti 16-QAM  2 Transmitters and 1 Receiver

Alamouti Code 2X1 16-Qam.m

Now, we discuss the Alamouti code for two transmitters and two receivers.

                                                      ALAMOUTI 2X2

 We assume that the channel parameters remain constant during the two time slots.Encoding is done in the same manner as discussed in Alamouti 2X1 code.

Matlab Simulation
Alamouti Code for 2 transmitters and 2 receivers(2X2) in 16-QAM

In this case, 4 channels are involved in every time slot whose channel impulse responses follow Rayleigh distribution.i.e h=x+iy; where x and y are Gaussian random variables.First generate H matrix(Channel matrix) and then generate X matrix(16-QAM modulated input symbols) .

         Y=HX+N is the received matrix. 

                                              DECODING OF ALAMOUTI 2X2 CODE
         This is similar to the decoding problem of Alamouti 2X1 Code.
As discussed earlier, the best performance is given by the brute force ML decoder which searches for the matrix X which minimizes the overall noise power.

          But, the orthogonality of H matrix ensures that the Brute force ML decoding simplifies to Zero forcing.

                                                         ZERO FORCING
          As discussed earlier 

 

Click the link below to view matlab code for Alamouti 16-QAM  2 Transmitters and 2 Receivers
Alamouti Code 16-QAM 2X2.m

Comparision Graph for 2X1 and 2X2 Alamouti Code for 16-QAM



Reference
A Simple Diversity Technique for Wireless Communication Siavash M Alamouti, IEEE Journal on selected areas in Communication, Vol 16, No, 8, October 1998

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