Mimo Design
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We began the work on our project by studying the concept of Fading and Diversity in transmission channels. For our particular case, we shall be considering a Rayleigh-fading model, in which the complex gain is actually a random variable with circularly-symmetrical zero-mean Gaussian distribution, so that the amplitude has a Rayleigh distribution whereas its phase has a uniform distribution.
We also understood the basic concept behind diversity which is that transmission ought to be done simultaneously across multiple channels, because it is highly unlikely that all the channels in consideration will fade simultaneously. The diversity can be achieved through three ways: in time, in frequency or in space.
Receiver Diversity
Consider the case of a SIMO model (Single Input Multiple Output). The model follows the following equation –
r = ah + n
where
a the transmitted symbol,
h = [h1, h2, h3 hn]T the channel gain vector,
n the noise matrix.
For the current case of study of receiver diversity, we consider n=1, i.e. single transmitter and m receivers.
There can be multiple ways in which we can combine the outputs from various antennae 1. Switched combining.
2. Selection combining.
3. Equal-gain combining.
4. Maximal-ratio combining.
Assuming that the transmitted code was modulated by 4-QAM, i.e. the transmitted symbol a can be equal to either of the four complex values {±1± j}(Eb)1/2. Also, the alphabet has been scaled so that the energy per symbol is 2Eb .Since 4-QAM transmits 2 bits of information, the energy per bit is Eb. Furthermore, AWGN is assumed with zero mean and variance equaling No/2.
Maximal-ratio combining:-
The analysis in [1] conveys that the Average BER (bit error rate) without diversity,
p (4 Eb /No) -1
whereas for the case of m receiver antennae, i.e. receiver diversity equaling m, the Average BER changes to
p Kmpm,
= Km (4 Eb /No)-m.
The comparative plots between Average BER and Average SNR per bit per antenna thereby show[1] that on a log-log scale, the curve approaches a straight line at high SNR values with an asymptotic slope proportional to m. This essentially implies that a higher diversity leads to a steeper slope and thus implying that for a given BER, a more diverse system requires a much lesser SNR than a receiver with no diversity.
Selection combination technique:-
There is again a marked improvement in the Average SNR per bit per antenna for a given value of Average BER, which furthers the motivation behind the usage of diversity in a transmitter receiver model.
(WRITE EQUATION HERE)
However for selection combining technique, the results are not as good as in the case of maximal-ratio combining technique.
Qualitatively studies showed that there is a trade-off between diversity and interference, or in other words, there exists a fundamental trade-off between the capacity of a system to mitigate interference or mitigate fading. This implies that the realizable diversity order in actuality is dependent not only upon the diversity, but also upon the number of interfering independent signal users/receivers.
Transmit diversity.
Unlike receiver diversity scheme wherein an antenna array at the receiver end was utilized, transmit diversity scheme uses linear or non-linear processing to spread information across multiple antennae at the transmitter. It can be classified into direct and indirect schemes. Indirect schemes are further include delay diversity and frequency offset diversity whereas direct schemes include closed-loop pre-compensation, space-time block codes and space-time trellis codes. For the purpose of our project, we have studied the case of Alamouti Space-Time Block Code which is an example of direct and open-loop method for transmit diversity.
Alamoutis scheme is a very simple an efficient way of achieving spatial diversity when the transmitter has an array with two antennae. Consider that we wish to send a pair of complex symbols {x1, x2} that have been chosen from some QAM or PSK constellation. To do the same, we have a transmitter with two antennae. Now for using this technique, we require two signaling intervals. Essentially, during the first time interval, x1 and x2 are transmitted whereas in the second time interval, -x2* and x1* are transmitted from the first and the second transmitter antennae respectively. Correspondingly, a space-time codeword or codematrix is obtained as follows –
Thus as we can see, the rows in the above matrix are showing the time axis whereas the columns are showing the spatial axis.
Applying equation (i), we get:
r1 = [h1 h2] [ x1 x2 ] T + n1
r2 = [h1 h2] [-x2* x1*] T + n2
combining which, we essentially obtain:
r = Heff . x + n
where
r = [r1 r2*] T
x = [x1 x2] T