Abstract
In this paper we analysis the channel capacity of wireless communication systems and to define the Shanon capacity is limitation and this capacity can be improved by using the number of transmitter and receiver antennas and it exploit the advantages and also increased throughoutput, broad range in multipath fading environment and is capable to provide highest data capacity and also established a reliable wireless systems over the multipath fading channel like Rayleigh or additive white Gaussian (AWGN). In our observation , we have to implementation of different capacity i.e. outage and ergotic for different number of multi antenna systems in the terms of channel is known and unknown for transmitter as well as receiver. Furthermore, it takes the advantage of space time coding (STC) and provides coding and diversity gain and also support to MIMO log det formula.
Keywords:MIMO channel model, diversity, spatial multiplexing, information theory, channel capacity and spacetime codes (STCs).
Introduction
Multiple input multiple output (MIMO) take numerous benefits over conventional wireless systems in either data rate or reliable link. A seminal work demonstrated [3], the wireless channel capacity namely Shannon capacity is limitation and the bandwidth of wireless systems is very scarce. Thus, the applicable approach [1] of this phenomenon technology is implementation of various techniques and algorithm exploit to wireless systems. The performance of MIMO systems depend on some term i.e. array gain, spatial multiplexing and diversity and so on. Channel characteristic play a significant role and consider as deterministic as well as random in wireless systems.
In this paper we have to explore wireless systems capacity is limitation and capacity can be obtain by using number of transceiver. The capacity is explored when the channel is known and unknown for transmitter and receiver. The MIMO channel is also random channel for different capacity i.e. 10% outage ,Ergodic and theier number of transmitter and receiver. However, the signal attitude of real wireless systems is abnormal so it’s distributed as Rayleigh in Line of Sight (LOS) case are well result. Moreover, we have to define the channel model as SISO, SIMO, MISO and MIMO systems and their input output relations and also mention as frequency selective channel. Thus MIMO is the best candidate for next generation wireless standard and guarantee achieve to best capacity in wireless communication systems.
MIMO is an abstract mathematical model of general matrix systems more specifically it produce array of antenna at both sides respectively transmitter and receiver.
Before starting MIMO technology, to take flavor about some others systems like SISO, SIMO, MISO and MIMO. Conventionally SISO (single input single output) provide single antenna at transmitter and receiver respectively. On the other hand SIMO referred single transmitter and multiple receiver is called SIMO (Single Input Multiple Output) systems. To do this trend the use of multiple antennas at transmitter and single receiver in wireless link MISO (Multiple Input Single Output) systems. MIMO (Multiple Input Multiple Output) provide same fashion in this scenario. Lastly, in this technology included MU (multi user)MIMO whether provide a system, user can also communicate with base station by using multiple antennas.
Array gain
Array gain is employed [11] at the both side receiver and transmitter for increased average signal to noise ratio (SNR) at the receiver those signal comes from coherent combining effect in the multiple antennas. Channel knowledge is required for transmitter/receiver to obtain array gain and depends on number of transmitter/ receiver antenna. If the transmitters know the channel then transmitter will weight the transmission with weights, depending on the channel coefficients, so that there is coherent combining at the single antenna receiver. The array gain in this system is called transmitter array gain. Alternatively, if we have only one antenna at the transmitter and no knowledge of the channel and a multiple antenna receiver, which has perfectly knowledge of the channel, the receiver can suitably weight the incoming signals so that they coherently add up at the output (combining), thereby enhancing the signal and is known receiver array gain. So in MIMO systems provide both side array gains is available.
Diversity
In wireless channel, signal is always fluctuate and create fading if the signal fluctuate very fast then it’s create fast fading, however diversity is one kind of technique that is capable to combat fading in wireless links. Multipath fading is common scenario in wireless channel causing by Receian or Rayleigh fading. If the signal strength is very low normally it given fade and increased high bit error rate (BER). Diversity techniques involve with time, frequency and space.

 Temporal diversity:
It provides the replica of the transmitted signal across the time by using channel coding and time interleaving. In this situation for diversity needs channel sufficient variations in time. We can achieve diversity when the channel coherence time smaller than desired interleaving symbol so it is assumed that interleaved symbol is independent of the previous symbol, thus makes a completely the new replica of the signal [11][12].

 Frequency Diversity:
Signal is always fluctuate into the channel. It transmitted by using different types of frequency and reached at the receiver by using multipath, if the coherence bandwidth of the channel is less than compared with signal bandwidth then we can apply this technique to get the replicas of the accurate signal and thus established a reliable link in wireless channel.
 Spatial (Antenna) Diversity:
It can mitigate fading in wireless channel and associated with time/frequency diversity. This diversity can be applied when the antenna spacing is larger than the coherence space. If the MIMO channel fade is independently and transmitted signal suitably constructed, the receiver can also received signal coherently and reduce the signal amplitude then we can get MTxMRx(The number of transmitter and Receiver) order diversity. This diversity depend design of the transmitted signal and SpaceTime Coding (STC) can be done. Spatial diversity can be categorized receive and transmit diversity
 Receive Diversity:
At the receiver end using maximum ratio combining (MRC) to improve signal quality but it’s very costly in wireless communication systems that’s why transmit diversity is becoming a popular and it’s less complexity to implement at the transmitter side and also exciting topics in MIMO systems. Receive Diversity improve capacity and range capability at the base station, except cost it’s very efficient technique to mitigate fading within a signal.
 Transmit Diversity:
Earlier we have to mention why it is very popular for researchers and wireless companies. Transmit diversity is applicable when multiple antennas are used at the transmitter. It’s a suitable signal construction. A significant effort has been devoted in 3GPP to develop efficient transmit diversity solutions to enhance downlink capacity. Transmit diversity methods also provide space diversity for terminals with only one receive antenna, and in that sense retain the complexity at the base station. Typically, in 3G base stations, the transmitting antenna elements are relatively close to each other. [13] In later section we will discuss more about diversity with spacetime coding.
 Spatial Multiplexing:
Spatial multiplexing offers a linear (in minimum number of transmit and receive antenna) increase capacity without additional power expenditure and bandwidth. It is only provide MIMO channels [5, 6]. This is commonly known spatial multiplexing gain and is considered for two transmit and receive antennas. It can be extended in MIMO channel. Let us consider 2×2 MIMO systems, in this case, we want to send bit stream, at first bit stream will split and modulated then transmitted simultaneously from both antennas. Channel knowledge is available at the receiver so it can completely decoded data thus provide receiver diversity whether transmitter has no knowledge about channel. In such event transmitter cannot provide diversity and data stream is completely different from each other so they carry totally different data. Thus, spatial multiplexing increases data capacity in MIMO systems.
 Multi Antenna System Model
We consider the number of transmitte antenna (i=1,2…………….MT) and the number of reciver antenna (j =1,2……….MR) respectively. Hence the create MIMO channel denoted Hij .
It gives us MT×MR complex matrix is called MIMO channel . However, if consider signal s is transmitted from ith transmit antenna. At the receive end, will get a complex weighted version of the transmitted signal. As we know jth receiver antenna by hji, where hij is the path gain or channel response between receive antenna jth and transmit antenna ith. The vector [h1i, h2i……..hMRi] Tis the signature induced by the ithtransmit antenna across the receive antenna array. Using this assumption, MIMO channel H for MT transmitter antenna and MR receive antenna can be represent as
The channel defines the inputoutput relation of the MIMO system and is also known as the channel transfer function. We assume that channel is Gaussian distributed (i.i.d.) means Gaussian variables. Hence the systems consider channel is unknown at the transmitter and assumed that the signals transmitted from each antenna have same power . So the covariance matrix of this transmitted signal is given by [4]
Where is the power across the transmitter irrespective of the number of antennas and is an identity matrix.
Hence we can ignore the signal attenuation, scatterings, and so on. In this scenario the channel matrix as deterministic as
If the channel is random, so this result can be apply for normalization.
The channel realization in real wireless communication systems is very difficult.
In the receiver, the channel estimation can be found at the receiver to send training sequence from the transmitter. On the other hand, the transmitter can get the channel information via feedback information. Hence the channel matrix is known for receiver but unknown for transmitter.
The covariance matrix of the receiver is given by [4]
If there is no correlation of components n. the covariance matrix is can be obtain as
Where is the identical noise power for each receiver.
For Simplicity, if we send signal vector from ithtransmitter antenna array (xi) then the signal received at the receiver antenna array is . At the receiver end is applied maximum likelihood (ML) algorithm over receiver antennas. We assumed that each received power level denoted by and the total power of receive antenna is equal to the total transmitted power. So the average SNR at each receive antenna is defined as
So linear model of the received vector can be written as
And the received covariance matrix can be define as and can be written as
While the total signal power can be represent as tr( ).
Where n is the additive white noise random variable with MR×1 column matrix distributed elements with zero mean complex Gaussian random variables with variance 0.5 per real dimension.
MIMO Capacity
MIMO channel H affected by large number of scatters like the superposition of delayed, reflected, scattered (buildings, vehicle and other terrain objects) in the wireless spectrum. So any receive antenna received transmitted signal with several multipath component. In such an event the replica of transmitted signal at each antenna will be complex random variable. The element of channel matrix H can be assumed to be independent, zero mean, complex Gaussian random variables that are distributed by Rayleigh (Raleigh fading). When signal introduce rich multipath with large delay spread then H varies as a function of time, the channel delay spread, which is a measure of the difference in the time of arrival of various multipath components at the receiver antenna, is less than the symbol rate. This assumption guarantees flat fading.
The capacity of MIMO channel is explain [3,5].
To control radio frequency spectrum in time varying channel with multipath propagation environment is really difficult for both case forward (base station to mobile) and reverse (mobile to base station).Actually,
receiver signal is generally weaker than transmitted signal due to the propagation phenomena like slow fading, propagation loss and fast fading. The mean propagation comes from angles of spreading by water and foliage and effect of ground reflections, slow fading arise by building and natural features and fast fading caused by multipath scattering. All fades expressed by Rayleigh fading [15]. So needless to say that channel is always unpredictable normally its behavior is random. On the other hand bandwidth is limited. In this event, a very essential systems designed was required in wireless communication that will done fill up all of requirement within a systems. MIMO is phenomenon’s that fill up all necessity in Wireless industry. According to MIMO definition we can get highest capacity in wireless channel. How we can get highest capacity in multi antenna system and several types of channel behaviors detailed can be found [5] within an Additive Gaussian channel with fading and without fading. This seminal paper also provides computational procedure for these dump antenna systems. Now we have to discuss MIMO capacity within an information theory. Before then, how we can achieve a sufficient data transmission within a MIMO systems possibly 1 Gb/s [2]. Let us consider a system to achieve this rate. When spectral efficiency 4 b/s/Hz over 250 MHz. Bandwidth then we can achieve 1 Gb/s. In real systems to get 250 MHz bandwidth available in 40Ghz frequency, normally frequency bands below should be 6 GHz. A potential paper proposed [2], where MIMO wireless constitutes technological breakthroughs that will allow1 Gb/s within NLOS environment. To do this, need 10×10 array of antenna at the both sides. In SISO systems to get 1 Gb/s need 220 MHz bandwidth whether in MIMO systems require only 20 MHz bandwidth and also does not need additional transmit power or receive SNR to deliver such performance gains. Thus MIMO provide a very strong and high data capacity rate in wireless systems.
However, consider [1] [5] [6] [10] provide rich capacity in several system that is exploit a MIMO channel and apply with signal scheme STC in practical wireless systems.
If channel is Rayleigh fading, in SISO systems provide capacity
Where h is channel with additive white Gaussian and complex value, is the SNR for any MR antenna, in such case if we add more antenna at the receiver side to get more capacity is given (SIMO case)
Where hiis the channel gain with number of MR receive antenna. It is also provide receiver diversity. In contrast of this system we can say MISO case whether add more antennas at the transmitter, whether transmitter has no knowledge about channel. In such event MISO is given capacity
Where hi is AWGN channel with number of MT antenna. It can worked as a transmit diversity.
Lastly at the both side multi antenna (MIMO) systems is given tremendous capacity
Where (*) means transposeconjugate and H is the MT×MR channel matrix. H* is the conjugate transpose of H. Till now this capacity is best capacity for MIMO systems.
Generally receiver has perfect knowledge for the channel but it can be implementation in different channel situation when channel is unknown and known to the transmitter.
Conclusion
The increasing demand for the development of wireless communication systems for high data rate transmission and high quality information exchange leads to the new challenging subject in communication research area. MIMO principles are able to provide future wireless communication systems with significant increased capacity or higher link reliability using the same bandwidth and transmit power as today. From the literature review, significant performance improvement possible over traditional wireless communication systems by using several kind of STC technique, that will drive in MIMO systems. This technique guaranteed maximum code rate, excellent diversity, rich coding gain and lastly not least reliable wireless communications. A good tutorial can be found [3] for MIMO STC. However, Spacetime coding is poised to play an important role in MIMO systems. Furthermore, MIMO technology is a strong candidate for 4G and beyond. Numerous vendors, such as Airgo, Lucent, are promoting MIMO as the IEEE802.11 standard, 802.11n, which the activities will complete by 2006.
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Author:Almas Uddin Ahmed
©Almas Uddin Ahmed, 2006
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