Multiple Input Multiple Output
MIMO
• MIMO is introduced fairly recently.
• Will be used in 802.11n. 802.11n will be
OFDM over MIMO.
• The basic idea is simple – having multiple
transmitting antennas and multiple receiving
antennas.
MIMO
• First, having multiple receiving antennas means that you
can pick up more energy.
• Also, when one antenna is having trouble receiving signal,
others are unlikely to be having the same problem. That is
why commercial APs sometimes have multiple antennas
also. It compares the received signal strength from different
antennas and use the strongest one to decode the data.
Called ``antenna diversity.’’
• As long as the antennas are sufficiently apart from each
other, the signals are likely experiencing different fading.
The space needs to be half of the carrier wavelength. If we
are using 2.4GHz, the wavelength is about 10cm.
MIMO
• Having multiple transmitting antennas does not necessarily mean
that you can send more energy, because the transmitting energy is
determined by other issues, such as your battery.
• However, it does mean that you can have multiple paths between
the sender and the receiver. With nt transmitting and nr receiving
antennas, you have nt times nr paths that can be assumed to be
independent. If one path is in trouble, i.e., there is someone in the
blocking position right now, other paths are unlikely to be in the
same situation at the same time. Much better than depending
everything on only one path!
• Also, what makes MIMO possible is that the receiver antennas can
operate in the linear range such that the received signal is the
ADDITON of signals from multiple transmitting antennas.
• So, based on these high-level intuitions, MIMO is likely able to
improve the performance. But how exactly?
SIMO
• Single Input Multiple Output.
• Consider one transmitting antenna and two
receiving antennas.
• Assume flat-fading, meaning that there is no
multi-path, i.e., the received sample is
relevant only to the current data symbol. We
write it as y[n]=x[n] + w[n].
• We can make this assumption because of
OFDM.
SIMO
• With two receiving antennas, we will receive
that is, from the waveform received at each antenna,
we can take a sample, and call it y1 and y2, respectively.
Both samples are excited by x, but they are from
different paths, therefore their channel coefficients
(i.e., h1, h2) are different. One important thing to
remember is that the noise from both antennas are
usually assumed to be following the same distribution
and have the same power and are independent from
each other.
SIMO Receiver
• For the simplest receiver, let’s just add y1 with y2
and make a decision.
• Is this the optimal one?
– What if h1=10 while h2=1 (yes, this is possible!)?
Remember that the noises are the same (random but
following the same distribution) at both channels).
Assume the data is 1 (BPSK), and this moment, the
noises at both channels are -6. So, we will get (10+(6)) + (1+(-6)) = -1, and we will think the sender sent 0!
– What is the problem? If we only use the strong
channel we won’t make the wrong decision!
SIMO receiver
• The problem is that we are treating the
information from a good channel and a weak
channel in the same way.
• The information from the strong channel is
more valuable than the weak channel.
• The optimal -- Maximum Ratio Combining
(Section 3.2.1 in the Tse book). We should
weight the samples from the antennas
according to the channel strength:
MISO
• Now consider the case when the sender has
multiple antennas and the receiver has only one
antenna.
• The sender has a power budget – the total
transmitting power cannot exceed a threshold.
• Assume that all antennas are sending the same
data symbol at any time, so the receiver will
receive
where a1 and a2 represent the power allocated for
antenna 1 and antenna 2, respectively.
MISO
• The problem is to maximize the magnitude of
the received signal x(h1a1+h2a2) subject to the
constraint that
• Any ideas?
MISO
• Still maximum ratio combining. Define Lagrange
• Take the partial derivative of L over a1 and a2 :
• Means that a1 and a2 should be proportional to h1
and h2.
• But this requires the sender knows the
channel – not always the case.
The Altamonte Scheme
• The key is that the transmitting antennas are
NOT restricted to sending the same data
symbols at the same time.
• The Altamonte Scheme (Tse book Section
3.3.2). Consider two data symbols to be sent
in two consecutive symbol times, u1 and u2 . At
time 1, ant1 transmits u1 and ant2 transmits u2.
At time 2, ant1 transmits –u*2 and ant2
transmits u*1.
The Altamonte Scheme
• (These two formulas are from the Tse book.)
So,
• Rearrange it, we have
The Altamonte Scheme
• So, we have
• Note that
that is, the two vectors are orthogonal to each
other. So, to recover u1 and u2, we can multiply
with the conjugate of either of the vectors.
The Altamonte Scheme
• So, the magnitude of the received signal is
proportional to
, even when the
transmitter is not aware of the channel
coefficient at all.
• If the transmitter simply sends the same
symbol over two antennas at the same power,
the received signal is proportional to h1 + h2 ,
and depends on the phase, they may cancel
each other out!
2 by 2 MIMO
• Now consider we have two transmitting
antennas and two receiving antennas.
• A simple scheme called ``V-BLAST:’’ Send
independent data symbols over the
transmitting antennas as well as over time.
MIMO
• MIMO receiver. Will receive two samples per
time slot. hij: the channel coefficient from Tx
ant j to Rx ant i.
• How to decode the data?
MIMO receiver
• The simplest receiver just do a matrix
inversion:
• This is NOT the optimal decoder! The
maximum likelihood decoder is better.