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One of the goals of the next generation true broadband wireless wave, Fourth Generation (4G), is high speed and high capacity IP based services at a low cost per bit. To achieve this, it is not appropriate to simply increase power or bandwidth of classical communication systems. Increasing the power allows higher-order modulation to be employed for a particular bit error rate. An increase in the bandwidth of systems corresponds to an increase in the bit rate, thus achieving higher capacity. The increase in bandwidth and power to enhance capacity for delivering 4G services is impractical because of the cost. Not only in terms of the financial and environmental angles, but also in terms of the interference threat they pose to the smooth operation of other communication systems in adjacent spectral channels within the same geographic area. Hence, bandwidth and power are strictly regulated by governing agencies.
Another hurdle to achieving the 4G rates is the multipath fading effect due to multipath propagation of electromagnetic waves. The destructive effect of this phenomenon on communication signals puts an upper limit on the channel capacity and transmission rate achievable in these systems. In making significant improvements to the performance of communication systems, overcoming the damaging effects of fading channels is of paramount importance . Techniques must therefore be devised to achieve greatly increased channel capacity without influencing the power and bandwidth which overcomes the multipath fading effect.
Newer communication systems like the IEEE 802.16e-2005 standard for Wireless Metropolitan Area Network (mobile WiMAX) and mobile broadband wireless access scheme tend to put multipath propagation to good use. Techniques such as Orthogonal Frequency Division Multiplexing (OFDM), Multiple-Input-Multiple-Output (MIMO) together with appropriate coding techniques such as Space-Time Block Coding (STBC) that make constructive use of multipath signal components are recently favoured to other ones. The diversities offered by these techniques are essentially required to meet the ever increasing consumer demand for faster data transmission together with good link quality.
Broadband wireless communications is needed to meet the goals of 4G. Transmitted wideband signals are very susceptible to frequency-selective fading; therefore these communication systems will need complex equalization techniques to combat the distortive and detrimental effect of the frequency selective channel.
Space-time coding when employed in MIMO systems in a transmit diversity mode could be used to achieve more reliable high data rates transmissions in wideband systems. Due to the orthogonal nature of the STBCs, decoding can be achieved in such a simple way (based solely on linear processing at the receivers) that complex equalizers are unnecessary  Vahid Tarokh.
Therefore, it is desirable to investigate the effect of frequency-selective fading on space-time code performance. I will be using WiMax as the base technology to perform this study. The aim of the dissertation, "Diversity Exploitation in MIMO-OFDM", is to investigate via MATLAB simulations, the impact of STBC on MIMO- OFDM in the improvement of the link quality of IEEE 802.16 in a frequency selective channel and to compare the resulting performances for various numbers of antennas at the transmitter and the receiver sides.
The bulleted objectives below are to enable the accomplishment of the dissertation's aim:
Extensive literature review of the topic and other related subjects
Modelling in MATLAB of a SISO MIMO-OFDM mobile WiMAX system in a frequency selective channel
Modelling in MATLAB of a 2*1 STBC based MIMO-OFDM WiMAX system with the IEEE 802.16 approved Alamouti's STBC implemented
Extension of the model to the 2*2 MIMO-OFDM
Interpretation and evaluation of obtained results and its extension by intuitive discussions
1.3 Thesis Scope
Beamforming and data rate adaptation.
Space diversity has been a popular technique in wireless microwave communications. Space diversity is also called antenna diversity. It is typically implemented using multiple antennas or antenna arrays arranged together in space for transmission and/or reception. The multiple antennas are separated physically by a proper distance so that the individual signals are uncorrelated. The separation requirements vary with antenna height, propagation environment and frequency. Typically a separation of a few wavelengths is enough to obtain uncorrelated signals. In space diversity, the replicas of the transmitted signals are usually provided to the receiver in the form of redundancy in the space domain. Unlike time and frequency diversity, space diversity does not induce any loss in bandwidth efficiency. This property is very attractive for future high data rate wireless communications.
The multiple antennas are separated physically by a proper distance so that the individual signals are uncorrelated. The separation requirements vary with antenna height, propagation environment and frequency. Typically a separation of a few wavelengths is enough to obtain uncorrelated signals. In space diversity, the replicas of the transmitted signals are usually provided to the receiver in the form of redundancy in the space domain.
1.4 Thesis Organization
The rest of this dissertation is organized as follows:
Chapter 2 presents MIMO-OFDM. MIMO, on its own, is considered together with its benefits so also is OFDM and its benefits. The chapter also presents the allure of MIMO-OFDM and a brief overview of WiMAX, the underlying technology for the study, and some of its important functional blocks.
Chapter 3 presents the concept of Space-Time coding. The wireless channel concept is discussed in this chapter; wireless channel characteristics, fading, and various channel models. Thereafter, STBC is presented with a particular focus on mathematical support for the Alamouti's 2*1 and 2*2 systems.
Chapter 4 presents the model design while Chapter 5 discusses the results. Finally, in Chapter 6, an overall conclusion is given with suggestions for further work.
2.1 Multiple Antenna Configurations
To carry out meaningful Space-Time (ST) wireless processing, there is a need for the communications system to have multiple antennas. Listed below are the different antenna configurations for wireless links. The nomenclature of each configuration depends on whether multiple antennas are present on either ends of the communications link i.e. transmit (input) and receive (output) sides. If there is only one antenna at the transmitter, the first term of each antenna configuration name would be Single Input. If the single antenna is at the receiver side of the link the second term of the antenna configuration name would be Single Output. Otherwise (for multiple antenna at the transmitter and receiver respectively), the antenna configuration name will be termed Multiple Input or Multiple Output.
The multiple antenna configurations are:
2.1.1 Single Input - Single Output (SISO) Channel
The Single Input - Single Output (SISO) is the simplest of all the configurations with only one transmit-antenna and one receive-antenna at both ends of the communication entities. It represents the conventional and classical way of achieving communication links. The situation of having only one antenna at both ends of the link necessarily means that spatial diversity cannot be exploited. SISO is illustrated in the Fig 2.1 below
Fig 2.1: SISO Antenna Configuration
The model of the SISO channel as defined by D. Gore & co  is given below:
"Let h(Ï„,t) be the time-varying channel response from the input of the pulse-shaping filter at the transmitter, through the propagation channel, to the output of the receiver match-filter. We define h(Ï„,t) as the response at time t to an impulse at time t - Ï„. For convenience we normally refer to h(Ï„,t) as the channel from the transmit antenna to the receive antenna.
If a signal s(t) is transmitted, the received signal y(t) is given by
Where â‹† denotes the convolution operator and where we have assumed a casual channel impulse response of duration Ï„total. The signals s(t) and y(t) are also complex envelopes of a narrowband signal".
2.1.2 Single Input - Multiple Output (SIMO) Channel
Single Input - Multiple Output antenna configuration, similarly, as the name implies means that we have only a single transmit antenna but more than one receive antenna. Multiple antennas at the receiver is an indication that some sort of diversity can be exploited (receive diversity in this case). The receive antennas or try to capture the singular transmitted message at each point in time. Since the information will transverse different paths to the receive antennas, thereby suffering different attenuations, the several received messages via different receipt antenna can be discriminated to pick the best or combined to each better reliability as in techniques such as the Maximal - Ratio Receive Combining (MRRC) scheme. Figure 2.2 below illustrates the SIMO antenna configuration.
Fig 2.2: SIMO Antenna Configuration.
The SIMO channel model is shown below :
"Consider a SIMO channel with MR receive antennas. The SIMO channel can be decomposed into MR SISO channels. Denoting the impulse response between the transmit antenna and the ith (i = 1, 2, ..., MR) receive antenna by hi(Ï„,t), we see that the SIMO channel may be represented as an MR * 1 vector, h(Ï„,t) , given by
Further, when a signal s(t) is launched from the transmit antenna, the signal received at the ith receive antenna, yi(t), is given by
, i = 1, 2, ..., MR (2.3)
Denoting that the signals received at the MR receive antennas by the MR x 1 vector, , we see that the the relation in Eq. 2.3 may be concisely expressed as
2.1.3 Multiple Input - Single Output (MISO) Channel
With the Multiple Input - Single Output (MISO) antenna configuration, we have multiple antennas at the transmit end of the communication link while having just one antenna at the receipt end. It therefore offers a transmit diversity. The spatial diversity here unlike the SIMO channel can be exploited not only to improve the reliability of communications but also to increase the capacity and speed. It is shown in Figure 2.3 below.
Fig 2.3: MISO Antenna Configuration
The MISO channel model is described below :
"Consider a MISO system with MT transmit antennas. Analogous to the SIMO channel discussed earlier, the MISO channel comprises of MT SISO links. Denoting the impulse response between the jth (j = 1, 2, ..., MT) transmit antenna and the receive antenna by hj(Ï„,t), the MISO channel may be represented by a 1 x MT vector h(Ï„,t) , given by
Assuming sj(t) is the signal transmitted from the jth transmit antenna and y(t) is the received signal, the input-output relation for the MISO channel is given by
which may alternatively be expressed in vector notation as
where is an MT x 1 vector".
2.1.4 Multiple Input - Multiple Output (MIMO) Channel
The Multiple Input - Multiple Output (MIMO) configurations offers diversity in both direction of the communicating parties. This configuration offers transmit diversity as well as receive diversity as there are multiple antennas of both ends of the communication link. This makes it more advantageous than the SIMO or MISO configurations when practicable. It is illustrated in Figure 2.4, as shown below:
Fig 2.4: MIMO Antenna Configuration
Simlarly, the channel model as discussed in  is given:
"Consider a MIMO system with MT transmit antennas and MR receive antennas. Denoting the impulse response between the jth (j = 1, 2, ..., MT) transmit antenna and the ith (i = 1, 2, ..., MR) receive antenna by hi,j(Ï„,t), the MIMO channel is given by the MR x MT matrix H(Ï„,t) with
The vector is the spatio-temporal signature or channel induced by the jth transmit antenna across the receive antenna array. Further, given that the signal sj(t) is launched from the jth antenna, the signal received at the ith receive antenna, yi(t), is given by
, i = 1, 2, ..., MR (2.9)
The input-output relation for the MIMO channel may be expressed in matrix notation as
where is an MT x 1 vector and is a vector of dimension MR x 1".
2.1.5 Benefits of MIMO
MIMO is one of the techniques that make reliable high data rate wireless communications practicable. The practicability aspect of it means within appropriate emitted power and with significant bandwidth saving. It constitutes a cost effective approach to high-throughput wireless communications. It is therefore a key component of the 4G wireless communication systems.
Since information is transmitted through different paths, from multiple transmit antennas to multiple receive antennas, a MIMO system is capable of exploiting both transmitter diversity and receiver diversity, hence maintaining reliable communications. Furthermore, with the advent of multiple antennas it becomes possible to process/combine jointly the multi-antenna signals and thus improve the system's integrity and/or throughput.
The two most significant advantages of the MIMO system in comparison to the SISO channel are :
"A significant increase of both the system's capacity and spectral efficiency. The capacity of a wireless link increases with the minimum number of transmitter or receiver antennas. The data rate can be increased by spatial multiplexing without consuming more frequency resources and without increasing the total transmit power".
"Dramatic reductions of the effects of fading due to the increased diversity. This is particularly beneficial when the different channels fade independently".
It can be seen therefore that, MIMO techniques based on using multiple antennas at the transmitter and receiver provides gains such as spatial diversity, interference suppression, multiplexing gain of which there is also the capability to make different tradeoffs between them.
2.2 Orthogonal Frequency Division Multiplexing (OFDM)
Over the years, as wireless communications continues to advance, the popularity of the Orthogonal Frequency Division Multiplexing (OFDM) over other multiplexing techniques such as the Frequency Division Multiplexing (FDM) and the Time Division Multiplexing continues. This is due to increase due to the comparative advantages of OFDM over other techniques. It has therefore been widely accepted and has become a successful air-interface technique.
OFDM is thus employed in many wireless standards such as Digital Audio Broadcasting (DAB), Digital Video Broadcasting for Terrestrial television (DVB-T), Digital Video Broadcasting for Handheld terminals (DVB-H), Wireless Local Area Networks (WLANs) and Broadband Radio Access Networks (BRANs).
Furthermore, OFDM is also the baseline technology for both the Fourth Generation (4G) mobile communications candidate: Long Term Evolution (LTE) and Wireless Interoperability for Microwave Access (WiMAX). It has also been ratified in standards by a number of standardization groups of the Institute of Electrical and Electronics Engineers (IEEE), such as the IEEE802.11 and the IEEE 802.16 standard families  L. Wang, & M. Jiang.
OFDM is a parallel transmission scheme, it divides a high rate serial data stream is into a set of lower rate sub-streams, each of which is modulated on a separate single carrier (SC) as in FDM. It is therefore widely referred to as a multicarrier technique. The effect of the serial to parallel conversion ensures that the bandwidth of the SCs becomes small compared with the coherence bandwidth of the channel and each individual SC will now experience flat fading. This is good as it greatly simplifies the equalization technique.
This also implies that the symbol period of the sub-streams is made long compared to the delay spread of the time-dispersive radio channel. R. Prasad showed that mutual interference as a result of this is mitigated by introducing a cyclic prefix (GI), thereby maintaining the orthogonality of SCs in a dispersive channel.
Fig 2.5: Cyclic extension of the OFDM symbol  Prasad
High spectral efficiency (spectrum saving) is obtained using OFDM because the spectra of the SCs are orthogonal and can safely overlap. This is illustrated in the Figure 2.6 below.
Fig 2.6: OFDM Frequency Span showing the Spectrum Overlap 
The three key concepts vital in modern OFDM systems listed by F. Ohrtman  are:
Cyclix prefix insertion
Forward Error Correction (FEC) & interleaving
These three important concepts are directly responsible for OFDM's reliable performance even in bad channel conditions. The Fourier Transform, either Discrete Fourier Transform (DFT) or the Fast Fourier Transform (FFT), enables channels to overlap without losing their individual characteristics (orthogonality). This is a more efficient use of the spectrum and enables the channels to be processed at the receiver more efficiently. The cyclix prefix insertion together with the FFT helps mitigate the effects of multipath fading and intersymbol interference and ensures it exceeds other competing wireless multiplexing techniques in performance.
When FEC with or without interleaving is used together with OFDM technique, the resulting system are called coded OFDM (COFDM) systems. R. Prasad , Investigated the Bit Error Rate (BER) performance of the coded and uncoded OFDM systems. Attenuation of our message signals in frequency selective channels can be severe, leading to a large BER. Further error correcting techniques are usually needed to keep the BER within acceptable limits.
Figure 2.7 below illustrates the block diagram of a simplex point-to-point transmission system using OFDM. The three main principles are shown to be incorporated.
Fig 2.7: Simplex point-to-point transmission using OFDM  Prasad
2.3 MIMO Assisted OFDM (MIMO-OFDM)
OFDM in its own stead is very attractive, and so also is MIMO. Both can be combined to attain the MIMO assisted OFDM (MIMO-OFDM) systems. The aim of this combination is to derive collectively the distinct merits of the two distinct techniques. This combination has proven to be so attractive that several communication systems, including the MIMO option in WiMAX, and wireless Local Area Networks are based on it.
The already discussed benefits of MIMO and OFDM are tabularized here for emphasis in Table 2.1 below.
Table 2.1: Merits of MIMO and OFDM
Increases the system's spectral efficiency by exploiting the diversity.
Increases the system's spectral efficiency by the overlapping SCs.
Increases the system's capacity via spatial multiplexing.
Reductions of the effects of Intersymbol Interference (ISI) by inserting the GI.
Reductions of the effects of fading due to the increased diversity.
Reductions of the effects of fading due to the serial to parallel transformation.
From the Table 2.1 above, one can easily tell that the advantages of both the MIMO and OFDM techniques are very similar but are achieved through different methods. For example, while the MIMO technique increases spectral efficiency by taking advantage of the diverse paths between transmitter and the receiver the OFDM technique saves bandwidth simply because of its overlapping spectrum.
The quality of a wireless link can be described by three basic performance metrics , namely:
The transmission rate
The transmission range
The transmission reliability
Classically, to increase one of the above parameters, it must be to the detriment of another one. For instance, the transmission rate may be increased by reducing the transmission range and reliability. Also, the transmission range may be extended at the cost of a lower transmission rate and reliability, while the transmission reliability may be improved by reducing the transmission rate and range. However, the beauty of the MIMO-OFDM systems is that the above mentioned three parameters may all be improved simultaneously.
L. Wang, & M. Jiang , claims that "Initial field tests of broadband wireless MIMO-OFDM communication system have shows that an increased capacity, coverage and reliability is achievable with the aid of MIMO techniques. Furthermore, although MIMOs can potentially be combined with any modulation or multiple-access technique, recent research suggests that the implementation of MIMO-aided OFDM is more efficient as a benefit of the straight forward matrix algebra invoked for processing the MIMO-OFDM signals ".
2.4 WiMax Physical Layer
â€¢ Space time coding: is an optional feature of 802.16 that can be used in the downlink communication to provide for space transmit diversity. Space time coding assumes that the base station is using two transmit antennas and the subscriber station uses one transmit antenna  Ilyas Standard.
Fig 2.8 Conceptual Diagram of the WiMAX's Uplink Physical Layer  Standards
Randomization is performed for spectral shaping and to ensure bit transitions for clock recovery.
Randomization shall be employed to minimize the possibility of transmission of an unmodulated carrier and to ensure adequate numbers of bit transitions to support clock recovery. The stream of DL packets shall be randomized by modulo-2 addition of the data with the output of the pseudo-random binary sequence (PRBS) generator
Forward Error Correction (FEC)
WiMAX utilizes FEC, a technique that doesn't require the transmitter to retransmit any information that a receiver uses for correcting errors incurred in transmission over a communication channel. The transmitter usually uses a common algorithm and embeds sufficient redundant information in the data block to allow the receiver to correct. Without FEC, error correction would require the retransmission of whole blocks or frames of data, resulting in added latency and a subsequent decline in QoS  Ohrtman.
To maximize utilization of the airlink, the PHY uses a multilevel modulation scheme. The modulation constellation can be selected per subscriber based on the quality of the RF channel. If link conditions permit, then a more complex modulation scheme can be utilized to maximize airlink throughput while still allowing reliable data transfer. If the airlink degrades over time, possibly due to environmental factors, the system can revert to the less complex constellations to allow more reliable data transfer. In the DL, the BS shall support QPSK and 16-QAM modulation and, optionally, 64-QAM  Standards.
3.0 Space - Time Coding
The importance of diversity techniques to wireless communication systems cannot be overemphasized. Previously, frequency diversity and time diversity took the forefront but more recently, space diversity with multiple antennas at the transmitter and/or sufficiently spaced multiple antennas at the receiver, encoding the signals across both the transmit antennas and time intervals is generating greater interest  Saulnier. For systems with several transmit antennas, STBCs offers the simplest type of spatial temporal codes  Tsoulos. STBC is thus widely employed practically today because of its surprising simplicity given its good performance. This means that great improvements can be made to the communication quality cheaply, a noteworthy fact considering its commercial significance.
This chapter, before delving into STBC, looks at the reasons while STBC is important. Space-Time coding came about because we want to overcome the problem of fading in wireless channels. A brief discussion of two important characteristics of wireless channel that leads to fading and various fading channels are discussed before treating STBC.
3.1 Wireless Channel Characteristics
The two major phenomena that occur in the wireless channel which has a relation to the fading of electromagnetic signals that propagates the channel are multipath propagation and Doppler shift. They are responsible for frequency-selective and time-selective fading respectively.
The frequency- and time-selective nature of the mobile wireless channels is one of the most critical elements from the overall wireless link quality point of view. Various transmitter and/or receiver signal processing techniques are in practice utilized to overcome the time- and frequency-selective fading effects in practical communications systems. This includes various channel equalization, coding, and diversity transmission schemes.
3.1.1 Multipath Propagation
In reality, very rarely can radio waves propagate through a practical wireless channel over a reasonable distance without being scattered or reflected. Objects in our environment such as buildings, trees, etc act as reflectors. This means that transmitted signals often arrive at its destination most probably after being multiple reflected from several reflectors. This multipath propagation of the same transmitted signals via different paths causes the replicas received to vary widely in attenuation and phase to the original signal. The amplitude fluctuation of the received signal is called signal fading and it is caused by the time-variant multipath characteristics of the channel  Yuan. Figure 3.1 below shows a multipath propagation in a typical cellular mobile radio environment.
The receiver therefore receives several replicas of the same message which can either add up constructively or destructively. This posed a great problem to earlier communication system. In the design of these systems, the inclination was to mitigate the effect of multipath propagation. The philosophy has changed in the design of newer communication systems and the tendency now is to put the multipath effect to good use via techniques such as MIMO and Space-Time coding.
Fig 3.1: Multipath Propagation 
3.1.2 Doppler Shift
Due to the relative motion between the transmitter and the receiver, each multipath wave is subject to a shift in frequency. The frequency shift of the received signal caused by the relative motion is called the Doppler shift. It is proportional to the speed of the mobile unit. The Doppler shift of the received signal, denoted by fd, is given by :
where v is the mobile speed, c is the speed of light, and fc . The Doppler shift in a multipath propagation environment spreads the bandwidth of the multipath waves within the range of fc Â± Bd, where Bd is the maximum Doppler shift, given by :
The Doppler spectrum is shown in Figure 3.3 below:
Fig 3.2: Doppler Spectrum
As a result, a single tone transmitted gives rise to a received signal with a spectrum of nonzero width, a phenomenon called frequency dispersion of the channel. The Figure 3.2 below is an illustration of Doppler Effect. While the mobile or subscriber station (SS) is stationary (and all other objects in the influence the propagation of the signal within the channel), Figure 3.2(a), no frequency shift is experienced shown by the concentric circles. As soon as objects are in motion within the channel (not necessarily the SS alone), Figure 3.2(b), there is a frequency shift as indicated by the eccentric circles.
Fig 3.3: Doppler Effect
3.1.3 Frequency Selective Fading
If the bandwidth of the used transmission waveforms is much greater than the coherence bandwidth of the channel, the channel is said to be frequency-selective. This means that some parts of the signal spectrum are attenuated more than the others (different amplitude and phase changes for different parts of the spectrum). This means that the channel delay spread is clearly greater than the used symbol duration or signalling interval. In wideband wireless communications, the symbol period becomes smaller relative to the channel delay spread, and consequently, the transmitted signals experience frequency-selective fading.
As mentioned earlier, in wireless communications systems, the transmitted signal typically propagates via several different paths from the transmitter to the receiver. This can be caused, e.g., by reflections of the radio waves from the surrounding buildings or other obstacles. Each of the multipath components have generally different relative propagation delays and attenuations which, when summing up in the receiver, results in filtering type of effect on the received signal where different frequencies of the modulated waveform are experiencing different attenuations and/or phase changes. This is typically termed frequency-selective fading.
Frequency-selective fading channels can be modelled by a tapped-delay line as shown in  Yuan. The other obvious case is that the signal bandwidth is much less than the channel coherence bandwidth. In this case, the channel is frequency-non-selective or frequency-flat, meaning that all the frequency components of the used waveform experience similar behaviour. Essentially, the channel at any time instant can then be modelled with a single complex coefficient (one tap).
3.1.4 Statistical Fading Channels
Because of the multiplicity of factors involved in propagation in a cellular mobile environment, it is convenient to apply statistical techniques to describe signal variations. In a narrowband system, the transmitted signals usually occupy a bandwidth smaller than the channel's coherence bandwidth, which is defined as the frequency range over which the channel fading process is correlated. That is, all spectral components of the transmitted signal are subject to the same fading attenuation. This type of fading is referred to as frequency non-selective or frequency flat. Widely employed statistical channel models are the Rayleigh and Rician fading models, and are mostly used to describe signal variations in a narrowband multipath environment.
Rayleigh fading models assume that the magnitude of a signal that has passed through aÂ communications channel will vary randomly, orÂ fade, according to a Rayleigh distribution. Rayleigh fading is viewed as a reasonable model forÂ heavily built-upÂ urbanÂ environments on radio signals.Â Rayleigh fading is most applicable when there is no dominant propagation along aÂ line of sightÂ between the transmitter and receiver. If there is a dominant line of sight,Â Rician fadingÂ may be more applicable.
3.1.5 Additive White Gaussian Noise (AWGN) Channel
It is a norm in communications to attribute every unwanted distortion of propagated signals from the transmitter to the receiver to the channel. Though most of channel noise and interference is indeed picked up while the signal traverses the channel, some noise also can be added to the communication equipment too. This kind of noise is known as thermal noise and it is due to the vibration of atoms in conductors is modelled by an AWGN channel. The only impairment to communication here is a linear addition ofÂ widebandÂ orÂ white noiseÂ with a constantÂ spectral density.Â
3.2 Multiple Antenna Options in WIMAX
In wireless mobile communications systems such as WiMAX, the use of multiple antennas make possible three vital techniques particularly important to meet the user's demand for better Quality of Service (QoS) and faster speeds. The three concepts are:
Smart antenna techniques (beamforming and data rate adaptation)
Some of the improvement benefits from employing multiple antennas in wireless systems include: An adaptive array with N antennas can provide the following performance benefits: better link capacity and throughput, reliability improvement and interference suppression. Also, various gain improvement such as antenna gain, multiplexing gain, coding gain e.t.c. can also be quantified and measured.  Marca stated more specifically that "an antenna array with Nt transmit antennas and Nr receiver antennas provides an array gain (average SNR increase) of Nt + Nr and a diversity gain (BER slope reduction) of Nt Nr. Alternatively, in rich scattering it provides a min(Nt, Nr) multiplexing gain (data rate increase) or it can nullify Nr interferers on the receive end"  Marca.
It is however important to note that, the standards for WiMAX approves MIMO technique only as an optional feature to a maximum of four antennas. But to achieve great speeds and good communications link quality expected of the WiMAX standard, most probably, MIMO would have to be implemented.
The smart antenna technique such as beamforming is possible due to the presence of more than one in the communication system. The scope of the thesis is more concerned how data is coded and transmitted using multiple antennas. Therefore a closer look into the transmit diversity and the spatial multiplexing is given in the following sub-sections.
3.2.1 Transmit Diversity
The diversity technique requires multiple replicas of the transmitted signals at the receiver, all carrying the same information but with small correlation in fading statistics. The basic idea of diversity is that, if two or more independent samples of a signal are taken, these samples will fade in an uncorrelated manner, e.g., some samples are severely faded while others are less attenuated. This means that the probability of all the samples being simultaneously below a given level is much lower than the probability of any individual sample being below that level. Thus, a proper combination of the various samples results in greatly reduced severity of fading, and correspondingly, improved reliability of transmission. In most wireless communication systems a number of diversity methods are used in order to get the required performance.
One of the WiMAX system profiles is the simple STC scheme proposed by Alamouti  for transmit diversity on the downlink. In the IEEE 802.16e-2005 specifications, this scheme is referred to as Matrix A. Originally, Alamouti's transmit diversity was proposed to avoid the use of receive diversity and keep the subscriber stations simple. This technique is applied subcarrier by subcarrier and can be described as shown in Figure 2.5.
Fig 3.4: STBC in WiMAX
Suppose that (s1, s2) represent a group of two consecutive symbols in the input data stream to be transmitted. During a first symbol period t1, transmit (Tx) antenna 1 transmits symbol s1 and Tx antenna 2 transmits symbol s2. Next, during the second symbol period t2, Tx antenna 1 transmits symbol sâˆ-2 and Tx antenna 2 transmits symbolâˆ’sâˆ-1 . Denoting the channel response (at the subcarrier frequency at hand) from Tx1 to the receiver (Rx) by h1 and the channel response from Tx2 to the receiver by h2, the received signal samples corresponding to the symbol periods t1 and t2 can be written as:
r1 = h1s1 + h2s2 + n1 (2.8)
r2 = h1sâˆ-
+ n2 (2.9)
where n1 and n2 are additive noise terms.
3.2.2 Spatial Multiplexing
One strategy for constructing codes is to maximize rate and achieve the greatest spectral efficiency. Spatial multiplexing achieves this goal by transmitting uncorrelated data over space and time, and relies on a sophisticated decoding mechanism to separate the data streams at the receiver. Strictly speaking, spatial multiplexing is not a method of space-time coding; it does not provide transmit diversity. However, it can be considered an extreme case in the diversity-multiplexing trade-off curve; it trades off diversity for maximal rate. The performance of these codes is highly dependent on the decoding algorithm at the receiver and the instantaneous channel characteristics. Codes exploiting spatial multiplexing occur in the standard for two transmit antennas (Section 220.127.116.11.3, Code B), three transmit antennas (Section 18.104.22.168.4, Code C), and four transmit antennas (Section 22.214.171.124.5, Code C). The symbols may support different code rates  Ilyas QoS.
The second multiple antenna profile included in WiMAX systems is the 2x2 MIMO technique based on the so-called matrix B = (s1, s2)T . This system performs spatial multiplexing and does not offer any diversity gain from the Tx side. But it does offer a diversity gain of 2 on the receiver side when detected using maximum-likelihood (ML) detection. To describe the 2x2 spatial multiplexing, we omit the time and frequency dimensions, leaving
only the space dimension. The symbols transmitted by Tx1 and Tx2 in parallel are denoted as s1 ands2, respectively. Denoting by h ji the channel response from Tx i to Rx j (i , j = 1, 2), the signals received by the two Rx antennas are given by
r1 = h11s1 + h12s2 + n1 (2.18)
r2 = h21s1 + h22s2 + n2 (2.19)
which can be written in a matrix form as
The ML detector makes an exhaustive search of all possible values of the transmitted symbols
and decides in favor of (s1, s2)which minimizes the Euclidean distance:
D (s1, s2) = |r1 âˆ’ h11s1 âˆ’ h12s2|2 + |r2 âˆ’ h21s1 âˆ’ h22s2|2
The complexity of theMLdetector grows exponentially with the size of the signal constellation,
and this motivates the use of simpler suboptimum detectors in practical applications  Marca .
Fig 3.5: SM in WiMAX
3.3 Space - Time coding
In general, Space Time Code (STC) is the technique to exploit spatial diversity. It consists of a set of coding algorithms which adjust and optimize the joint encoding across the transmit antennas aiming at increasing the reliability of a wireless link. Alamouti  proposed Space Time Block Code (STBC) which can be implemented in the DL of the WiMAX systems with 2*1 or 2*2 transmit-receive antennas at rate 1. STBC can be used in low delay spread environment and it enhances PER performance. STBC was later generalized to support arbitrary number of transmit antennas. Research has been conducting in the MIMO coding algorithms over the years.
3.3.1 Historical Background of STBC
Early work carried out in antenna diversity by Telatar  and Foschini  showed that significant gains can be obtained by increasing the number of transmit antennas while maintaining a fixed number of receive antenna(s). Tarokh, Seshadri, & Calderbank  delved only into the actual transmit diversity. They came up with the concept of Space-Time Trellis Coding (STTC): trellis coding used together with space-time diversity. Although the performance of their method was good, it suffered the disadvantage that the system complexity grows exponentially with increase in the number of antennas.
Alamouti  work is the foundation of STBC. He proposed the space-time diversity together with orthogonal block codes for two transmit antennas. The result he obtained for two transmit antenna and a single receive antenna was very similar to the receive diversity scheme, maximum ratio receiver combining, with a single transmit antenna and two received antenna. This is important as it provides an alternative method instead of having to live with the problem of how to appropriately space multiple antennas at the receiver, especially in small mobile terminals.
Tarokh, Jafarkhani, & Calderbank  further worked on the Alamouti's two antenna scheme, taking advantage of the inherent orthogonal structure of STBC, to develop an algorithm for any number of transmit antennas. The orthogonal nature of STBC makes linear Maximum Likelihood Detection (MLD) possible.
3.3.2 Alamouti's Scheme
Alamouti scheme is a simple transmit diversity scheme suitable for two transmit antennas. Before coding the base band signal would be modulated at each antenna using an appropriate modulation scheme as stipulated in the WiMAX standard. Then the modulated signal is encoded using STBC technique in sets of two modulated symbols. We assume here that the transmitter does not have channel knowledge but the receiver has full Channel State Information (CSI) and that it does not change over the transmission of a symbol i.e. the channel be a slow fading frequency selective channel. We will first see, two transmit antenna and one receive scheme then two transmit and two receive scheme.
126.96.36.199 Two transmitter and one receiver scheme
Two symbols are considered at a time, say x1 and x2 they are transmitted in two consecutive time slots. In first time slot, x1 is transmitted from antenna one and x2 is transmitted from antenna two. In the second time slot - x2* is transmitted from antenna one while x1* is transmitted from antenna two as shown in Fig 3.6 below.
Figure 3.6: The new two-branch transmit diversity scheme with one receiver .
Also the equivalent space time block coding matrix shown in Table 3.1
Table 3.1: Encoding in space and time.
t + T
The fading coefficient from antenna one and two are denoted by h1(t) and h2(t) respectively at time t .By assuming slow fading Rayleigh channels, we can take the equations below to be true:
The received signal at time t and t +T in matrix form expressed as (3.2)
Where y1 and y2 are the received signals at time t and t +T and n1 and n2 are independent zero mean Additive White Gaussian Noise (AWGN).
The combiner combines the received signal as:
By substituting equations (3.1) and (3.2) in equation (3.3) we get:
188.8.131.52 Two transmitters and two receivers scheme
Figure 3.7: The new two-branch transmit diversity scheme with two receivers .
The figure above shows the Alamouti scheme with two transmit and two receive antennas. There are therefore two distinct channels use at time t and time t + T. We obtain:
Where H is a 2*2 channel transfer matrix, c(t) and c(t+T) are Alamouti code words and n(t) and n(t+T) are zero mean uncorrelated additive noise.
By substituting the terms as expressed in equation (3.6) into equation (3.5) we obtain:
The two signals built by the combiner and sent to the maximum likelihood decoder for decision are:
Further computation resolves to:
Equations (3.4) and (3.9) are mathematically indications that to make decisions on the received symbols only linear operations are required.
The combination with OFDM system can extend the advantages of STBC to frequency-selective fading channel since the wide-band multipath fading channel is split up into many narrow-band flat fading sub-channels in OFDM system. Therefore, STBC MIMO-OFDM system is adopted as many wireless standards, such as IEEE 802.11n Wireless Local Area Network (WLAN) and IEEE 802.16 Wireless Metropolitan Area Network (WMAN).
are widely used to reduce the effects of multipath fading and improve the reliability of transmission without increasing the transmitted power or sacrificing the bandwidth  .
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