Wednesday, 6 July 2011


A space time code is a method employed to improve the reliability of data transmission in wireless communication systems using multiple transmit antennas.
            Essentially, two different space-time coding methods, namely space-time trellis codes (STTCs) and Space time block codes (STBCs) have been proposed. STTC has been introduced as a coding technique that promises full diversity and substantial coding gain at the price of a quite high decoding complexity. To avoid this disadvantage, STBCs have been proposed by the pioneering work of Alamouti.
             Space–time block coding is a technique used in wireless communications to transmit multiple copies of a data stream across a number of antennas and to exploit the various received versions of the data to improve the reliability of data-transfer. The fact that the transmitted signal must traverse a potentially difficult environment with scattering, reflection, refraction and so on and may then be further corrupted by thermal noise in the receiver means that some of the received copies of the data will be 'better' than others. This redundancy results in a higher chance of being able to use one or more of the received copies to correctly decode the received signal. In fact space time coding combines all the copies of the received signal in an optimal way to extract as much information from each of them as possible.
             In our posts, we consider only coherent STBCs; i.e. the receiver knows the channel impairments through training or some other form of estimation.Matlab simulations and the relevant results are given for each of the STBCs discussed.
             We start off with the description of the simplest STBC which is the Alamouti STBC for 2 transmit and 1 receive antenna.Then, we discuss the Alamouti code for 2 transmit and 2 receive antennasThe decoding is done by  Zero Forcing( ZF ) technique. The modulation used is 16-QAM. 
             We describe the Golden code, a code for 2 transmit and 2 receive antennas which achieves full diversity and full rate. In the subsequent pages, we describe  two decoding algorithms for Golden code. They are Brute force ML and Sphere decoding.
             We give a comparision of Alamouti code and the Golden code. For the comparision to be fair, we maintain the  bit rates same  for both the codes. So we use 4-QAM modulation for the Golden code.
             We describe the Silver code, a code for 2 transmit and 2 receive antennas whose coding gain is slightly less than that of the Golden code but easier in implementation owing to its lesser decoding complexity. We decode it using the Sphere decoder.

              We then describe an
8X8 MIMO system that does not use any coding and decode it using ZF and V-BLAST techniques. Our aim here is to compare the performances of these two decoders.

Related Posts
     Alamouti Code
     Golden Code
     Silver Code
     8 X 8 Uncoded MIMO using VBlast and ZF

A short note on Perfect STBCs

Perfect Codes are Space-Time codes for the coherent MIMO channel.

They are algebraic codes, built on non-commutative fields (or division algebras ).
The channel model considered is the following: if M is the number of transmit and receive antennas,
Y = H X + N                  
where H ={hij} is the MxM channel matrix with complex fading coefficients and N the MxM complex Gaussian noise matrix.

A square nt × nt STBC is called a perfect code if and only if:

It is a full rate linear code using n2t information symbols.
The determinant of the difference of any two distinct codewords is different  from 0 which ensures full-rank and in turn full-diversity.
Non-vanishing determinant is the minimum determinant of a perfect code that is lower bounded away from zero by a constant. This constant is the measure of coding gain.( The coding advantage is an approximate measure of the gain over an uncoded system operating with the same diversity advantage).

The energy required to send the linear combination of the information symbols on each layer is similar to the energy used for sending the symbols themselves (we do not increase the transmitted energy in encoding the information symbols).

It induces uniform average transmitted energy per antenna in all T time slots, i.e., all the coded symbols in the code matrix have the same average energy.

Perfect codes only exist in dimension 2, 3, 4, and 6.

  • For M=2 antennas, QAM symbols are sent. There are infinitely many of them, but the most famous is the Golden code. The minimum determinant is 1/5. 
  • For M=3 antennas, HEX symbols are sent. The minimum determinant is 1/49.
  • For M=4 antennas, QAM symbols are sent. The minimum determinant is 1/1125.
For M=6 antennas, HEX symbols are sent. The minimum determinant is between 1/(26 74) and 1/(26 75) .

Related Posts
     Alamouti Code
     Golden Code