Chapter 2 MIMO-OFDM Technique Overview
2.3 Diversity Based on MIMO Techniques
log ( ) bits/sec/Hz
k
where “+＂ denotes taking only those terms which are positive and u is a scalar, representing the portion of the available transmit power going into the ith subchannel which is chosen to satisfy
1 channel capacity appears intractable, even in the Wishart case when the joint distribution of λ λ1, 2,…,λk is known. Nevertheless, the channel capacity can be simulated using Equations (2.26) and (2.27) for any given HHH so that the optimal capacity can be computed numerically for any channel [29].
2.3 Diversity Based on MIMO Techniques
Antenna diversity, or spatial diversity, can be obtained by placing multiple antennas at the transmitter and/or the receiver. If the antennas are placed sufficiently far apart, the channel gains between different antennas pairs fade more independently, and independent signal paths are created. The required antenna separation depends on the local scattering environment as well as on the carrier frequency. For a mobile which is near the round with many scatterers around, the channel decorrelates over shorter spatial distances, and typical antenna separation of half to one carrier wavelength is sufficient. For base stations on high towers, larger antenna separation of several to 10’s of wavelengths may be required.
We will look at both receive diversity, using multiple receive antennas (single-input, multi-output SIMO channels), and transmit diversity, using multiple transmit antennas (multi-input, single-output MISO channels). Interesting coding problems arise in the latter and have led to recent excitement in space-time codes.
(I) (II) Figure 2.2: (I) Receive diversity (II) Transmit diversity
2.3.1 Receive Diversity
In a flat fading channel with one transmit antenna and L receive antennas (Figure 2.2 (I)), the channel model is as follows:
[ ] [ ] [ ] [ ] 1,...,
l l l
y m =h m x m +w m l = L (2.28)
where the noise w ml[ ] ~CN(0,N0) and independent across the antennas. We would like to detect x[1] based on y1[1],...,yL[1]. This is exactly the same detection problem as in the use of a repetition cod over time, with L diversity branches now over space instead of over time. If the antennas are spaced sufficiently far apart, then we can assume that the gains hl[1]are independent Rayleigh, and we get a diversity gain of L.
With receive diversity, there are actually two types of gain as we increase L. this can be seen for the error probability of BPSK conditioned on the channel gains:
( 2 2SNR )
Q h . (2.29)
We can break up the total received SNR conditioned on the channel gains into a product of two terms:
2 1 2
SNR= SNRL
⋅L
h h . (2.30)
The first term corresponds to a power gain (also called array gain): by having multiple receive antennas and coherent combining at the receiver, the effective total received signal power increases linearly with L: doubling L yields a 3 dB power gain. The second term reflects the diversity gain: by averaging over multiple independent signal paths, the probability that the overall gain is small is decreased. The diversity gain L is reflected in the SNR exponent; the power gain affects the constant before the 1/ SNRL. Note that if the channel gains hl[1] are fully correlated across all branches, then we only get a power gain but no diversity gain as we increase L. on the other hand, even when all the h are independent there is a diminishing marginal return as L increases: l due to the law of large numbers, the second term in (2.30),
2 2
1
1 1
[1]
L l l
L h = L
∑
= h , (2.31)Converges to 1 with increasing L (assuming each of the channel gains is normalized to have unit variance). The power gain, on the other hand, suffers from no such limitation: a 3 dB gain is obtained for every doubling of the number of antennas.
2.3.2 Transmit Diversity: Space-Time Codes
Now consider the case when there are L transmit antennas and one receive antenna, the MISO channel (Figure 2.2 (II)). This is common in the downlink of a cellular system since it is often cheaper to have multiple antennas at the base station than to having multiple antennas at every handset. It is easy to get a diversity gain of L: simply
one time, only one antenna is turned on and the rest are silent. This is simply a repetition code, and , as we have seen in the previous section, repetition codes are quite wasteful of degrees of freedom.
More generally, any time diversity code of block length L can be used on this transmit diversity system: simply use one antenna at a time and transmit the coded symbols of the time diversity code successively over the different antennas. This provides a coding gain over the repetition code. One can also design a code specifically for the transmit diversity system. There have been a lot of research activities in this area under the rubric of space-time coding and here we discuss the simplest, and yet one of the most elegant, space-time code which is called Alamouti scheme. This is the transmit diversity scheme proposed in several third-generation cellular standards. Alamouti scheme is designed for two transmit antennas;
generalization to more than two antennas is possible, to some extent.
2.3.2.1 Alamouti Scheme
With flat fading, the two transmit, single receive channel is written as
1 1 2 2
[ ] [ ] [ ] [ ] [ ] [ ]
y m =h m x m +h m x m +w m (2.32)
where hi is the channel gain from transmit antennas i. the Alamouti scheme transmits two complex symbols u1 and u2 over two symbol times: at time 1, x1[1]=u1, x2[1]=u2; at time 2, x1[2]= -u2*, x2[1]= -u1. If we assume that the channel remains constant over the two symbol times and set h1=h1[1]= h1[2], h2=h2[1]= h2[2], then we can write the matrix form:
[ ] [
1 2]
1 *2[ ]
2 1
[1] [2] u u [1] [2]
y y h h w w
u u
⎡ − ⎤
= ⎢⎣ − ⎥⎦+ . (2.33)
We are interested in detecting u1, u2, so we rewrite this equation as
1 2 1
* * *
2 1 2
[1] [1]
[2]* [2]
h h u
y w
h h u
y w
⎡ ⎤ ⎡ ⎤
⎡ ⎤ ⎡ ⎤
=⎢ ⎥ ⎢ ⎥+
⎢ ⎥ − ⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (2.34)
We observe that the columns of the square matrix are orthogonal. Hence, the detection problem for u1,u2 decomposes into two separate, orthogonal, scalar problems. We project y onto each of the two columns to obtain the sufficient statistics
1, 2
i i i
r = h u +w i= (2.35)
where h=[ ,h h1 2]t and wi ~CN(0,N0) and w1, w2 are independent. Thus, the diversity gain is 2 for the detection of each symbol. Compared to the repetition code, 2 symbols are now transmitted over two symbol times instead of 1 symbol, but with half the power in each symbol (assuming that the total transmit power is the same in both cases).
The Alamouti scheme works for any constellation for the symbols u1, u2, but suppose now they are BPSK symbols, thus conveying a total of two bits over two symbol times. In the repetition scheme, we need to use 4-PAM symbols to achieve the same data rate. To achieve the same minimum distance as the BPSK symbols in the Alamouti scheme, we need 5 times the energy per symbol. Take into account the factor of 2 energy saving since we are only transmitting one symbol at a time in the repetition scheme, we see that the repetition scheme requires a factor of 2.5 (4dB) more power than the Alamouti scheme. Again, the repetition scheme suffers from an inefficient utilization of the available degrees of freedom in the channel: over the two symbol times, bits are packed into only one dimension of the received signal space, namely along the direction [ ,h h1 2]t. In contrast, the Alamouti scheme spreads the information onto two dimensions- along the orthogonal directions [ ,h h1 *2]t and [ ,h2 −h1*]t.