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Introduction
In the previous sections, it has been shown that transmit diversity techniques allow us to transmit at a much higher transmission rate over the same channel, retaining the channel SNR qualities. The multiplier is given by the number of independent transmit receive paths achieved. We have not discussed any active participation of the transmitter in the entire operation. It is assumed that the properties required for optimal performance are naturally present in the transmitted sequence i.e. the assumption of non-correlation between transmitted symbols on different antennae.
To look at what the transmitter can do, it is important to look at what the objectives of the transmitter are. In the most simple case the transmitter can simply increase the information transmission rate - this is the preferred approach when the channel quality is acceptable and the emphasis is on increasing capacity. However, the transmitter should also be able to utilize spatial diversity to increase reliability. Thus there are three answers to the question of utilization.
- Transmit completely different data streams in parallel, one per independent transmit-receive path, to increase capacity.
- Transmit the identical same data on all transmit-receive paths to achieve maximum reliability, with no capacity benefits.
- Use a technique in-between, which gives some improvement in both.
Space Time Diversity coding techniques are techniques which actually seek to introduce desirable properties in transmitted signals so as to facilitate reception using spatial diversity techniques and achieve the reliability-rate trade-off that the transmitting system needs. We shall discuss two flavours of this coding: STC coding as popularized by Alamouti and others and the more complex STBC
Space Time Coding
We will first discuss STC by Alamouti, because of the simplicity of the idea and the widespread use. STC is a technique by which a single receiver can emulate a dual transmit dual receive system and achieve a significant improvement in the reliability of the signal. Thus, if we have a remote with a single receiver and a base-station with multiple transmitters (a frequent occurence in real-life), the remote can emulate a dual-receive diversity mechanisms. This works as follows: In the first time instant, the base-station transmits two symbols s_0 and s_1 out of its two antennae. In the second time instant, the base-station transmits {-s_1}^* and {s_0}^*, where a^* is the complex conjugate of a. Assuming that the path characteristics from antenna zero to the receiver is h_0 and from antenna one to the receiver is h_1, the receiver gets the following received symbols in the two time instances
| Receiver from antenna 0 | Receiver from antenna 1 |
| r_0 = h_0 s_0 + n | r_1 = h_1 s_1 + n |
| r_0 = -h_0 {s_1}^* + n | r_1 = h_1 {s_0}^* + n |
We can write this as a matrix product
\left[ \begin{array}{c} r_0 \\ r_1 \end{array}\right] = \left[ \begin{array}{c c} h_0 & h_1 \\ {h_1}^* & {-h_0}^* \\ \end{array} \right]\left[ \begin{array}{c} s_0 \\ s_1 \end{array}\right]
The reader will notice that by transmitting the modified symbols in the second instance, the transform matrix is now invertible and is in fact a Wishart matrix as described in the previous sections, with the added advantage that it is orthogonal.
This is what gives the STC technique its unique advantages. There are another issue to be considered at this point: to retain the symbol rate, the transmission rate has to be doubled, since each symbol is effectively being transmitted twice. This is both to maintain the symbol rate and also to maintain the basic assumption that the channel characteristics remain unchanged over the first and second transmissions. A doubling of the symbol rate (while keeping the bandwidth, power etc unchanged) will have a logarithmic degradation on SNR, whereas by using two transmitters, we can achieve a linear improvement on SNR. In other words, the overall improvement is of the order of 2/ln(2), which can be significant. The STC technique can be extended to 4, 8 and 16 antennae.
Alamouti shows that the standard techniques used for diversity combining i.e. maximum likelihood estimation can be used in the receiver. This is an added advantage, since there is no significant additional complexity in the receiver, other than standard RAKE functionality. In fact, by using the feature that the channel matrix H is orthogonal, even a simple linear combiner can extract the transmitted symbols:
\begin{eqnarray}
H = \left[ \begin{array}{c c} h_0 & h_1 \\ {h_1}^* & {-h_0}^* \\ \end{array} \right] \\
c = \arg\min_{c \in C} \vert \tilde{r} - \alpha \tilde{c} \\
\tilde{r} = H^* r \\
\tilde{c} = cH \\
\alpha = HH^*
\end{eqnarray}
Ofcourse, this assumes perfect knowledge of the channel matrix H, using pilot transmission or other techniques as mentioned in previous sections.
Note that \alpha = HH^* is the real part of the channel transform multiplier {h_0}^2 + {h_1}^2
Space Time Block Codes (STBC)
The idea of STC can be extended to a more general form in the study of space time block codes. The idea of STBC derives from the idea of block coding for reliability. For more information about Coding Theory the reader may refer to the page on CodingTheory?).
Space Time Block codes allow us to choose the optimal trade-off between rate increase and signal reliability. It is obvious that STC actually uses (2) i.e. the same information is transmitted on both paths. In other words, STC is a specialized form of STBC. STBC acts on blocks of data and employes spatial and temporal diversity techniques(ie: Multiple streams are transmitted (optionally also received) through multiple antennas and at multiple times) to mitigate the effects of scattering,fading etc. - each stream can be somewhat redundant with one or more other streams. This technique increases the throughput of wireless channels which otherwise have low through put capacities on account of the varying nature of the channel.
A Space Time Block Code is represented as follows
\left[ \begin{array}{c c c c} a_{11} & a_{12} & \ldots & a_{1 N_T} \\ a_{21} & a_{22} & \ldots & a_{2 N_T} \\ \ldots & \ldots & \ldots & \ldots \\ a_{N_R 1} & a_{N_R 2} & \ldots & a_{N_R N_T} \end{array} \right]
Where
- a_{ij} is the signal transmitted by the
- N_T is the number of transmitting antennae
- N_R is the number of receiving antennae
- T is a time-slot
The columns in this matrix denote the symbols transmitted (received) by all the antennas during a particular time slot. The rows in this matrix indicate the symbols transmitted (received) by a particular antenna during a time span.
The code rate for a given Space Time Block Code is defined as follows: r = k/T
Space Time Block codes can be divided into two parts
- Fully Orthogonal Block Codes
- Quasi Orthogonal Block Codes
Fully Orthogonal Block Codes
Full orthogonality implies that all the columns in the Space Time Block code matrix are mutually orthogonal (ie: The signals transmitted by the antennas in the system are mutually orthogonal in nature).
This strict criteria for orthogonality, implies that we have a scheme which requires linear decoding at the receiver.However, this orthogonality criteria also implies that the codes which do not satisfy this criteria are unable to to achieve the optimum data rate.
There exist only a finite number of matrices which have real coefficients and are completely orthogonal in nature (these matrices only exist for n=2,4,8 symbols). This result is know as the Hurwitz-Radon criteria for real orthogonal matrices. {5},{9}
The only known code which has rate =1, uses complex symbols and is fully orthogonal is the code developed by Sivash Alamouti. However this code is capable of using only two transmit antennas simultaneously {2}.
Quasi Orthogonal Block Codes
These codes are not fully orthogonal in nature (ie: not all the columns in the code matrix are pair wise mutually orthogonal).These codes are constructed by dividing the set of transmitted symbols into separate sub-groups. The division of symbols into sub-groups is done in a manner to ensure that all the symbols in any particular sub-group are orthogonal to the symbols in all the other sub-groups of the code. However the symbols within a particular sub-group may not be necessarily mutually orthogonal.
At the receiver's end the received symbols can be separated into sub-groups by using simple linear processing. Then we can run the Maximum Likelihood Estimate algorithm on all the separate sub-groups in parallel thus speeding up the decoding process.
The fully orthogonal Space Time Block coding scheme described in the last section has some limitations, due to which it is not possible to construct orthogonal codes for systems having the following properties
- Number of transmit antennas in the system is >2.
- Transmission rate = 1.
- All the columns in the transmission matrix are mutually orthogonal in nature.
- The modulation scheme used by the system uses complex values also.
For detailed proofs of the above mentioned results the reader may kindly refer to the following papers {4},{5}.
However, for systems with more than two transmit antennas we can construct codes which are quasi orthogonal in nature and have the following properties
- The system does not have full diversity (ie: The channel matrix is rank deficient)
- The system uses modulation schemes which have only real valued variables.
Studies have indicated that for systems with more than two antennas and the above mentioned conditions holding true, Quasi Orthogonal Space Time Block Codes give higher data rate than Fully Orthogonal Space Time Block Codes with the same number of antennas. Also studies {8} have proved that for degraded channel conditions (ie. low SNR), it is better to use coding schemes which have partial diversity but high transmission rate.
References
[1] Space Time Block Codes [http://en.wikipedia.org/wiki/Spacetime_block_code]
[2] Sivash Alamouti "A Simple Transmit Diversity Technique for Wireless Communications", IEEE JSAC, VOL. 16, NO. 8, OCTOBER 1998
[3] Li Liu, and Hamid Jafarkhani, "Combining Beamforming And Quasi-Orthogonal Space-Time Block Coding Using Channel Mean Feedback ", Globecom 2003
[4] Vahid Tarokh, Nambi Seshadri,and A. R. Calderbank, "SpaceTime Codes for High Data Rate Wireless Communication: Performance Criterion and Code Construction", IEEE Transactions on Information Theory, Vol. 44, No. 4, March 1998
[5] Vahid Tarokh, Hamid Jafarkhani and A. R. Calderbank "SpaceTime Block Codes from Orthogonal Designs" IEEE Transactions on Information Theory, Vol. 45, No. 5, July 1999
[6] Vahid Tarokh, Hamid Jafarkhani A. R. Calderbank and Ayman Naguib, "SpaceTime Codes for High Data Rate Wireless Communication: Performance Criteria in the Presence of Channel Estimation Errors,Mobility, and Multiple Paths", IEEE Transactions on Information Theory, Vol. 47, No. 2, February 1999
[7] Sushanta Das, Naofal Al-Dhahir, and Robert Calderbank, Novel Full-Diversity High-Rate STBC for 2 and 4 Transmit Antennas, IEEE Communications Letters, Vol. 10, NO. 3, March 2006
[8] Hamid Jafarkhani, A Quasi-Orthogonal SpaceTime Block Code, IEEE Transactions on Communications, Vol. 49, No. 1, January 2001
[9] A. V. Geramita and J. Seberry, Orthogonal Designs, Quadratic Forms and Hardamard
Matrices, Lecture Notes in Pure and Applied Mathematics, Vol. 43, New York and Basel:
Marcel Dekker, 1979.
Category: Wireless
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