NonCoherent Detection in MultiAntenna Fading Channels

1
Non-Coherent Detection in Multi-Antenna Fading
Channels

• Lizhong Zheng
• Department of EECS
• University of California
• Berkeley

2
Channel Model

• Independent gain between each pair of transmit
  and receive antennas.
• All the parameters evolving with time. And could
  be modeled as invariant in a coherence time Tc,
  which decides the speed that the channel is
  evolving.

N Tran. antennas
N Recv. antennas

• All the transmit antennas can cooperate with each
  other. So we can do coding over different
  antennas. The receiver decode with the output
  from all receive antennas.
• Typical channel Rayleigh Fading.

3
Training and Feedback
Training Sequence
data
Tc

• In slow fading channels (indoor systems), we can
  send a training sequence at the beginning of each
  coherence time, so the receiver can estimate the
  channel parameters, and then communicate with
  these estimations.
• If coherence time Tc N, we can send long
  enough training such that receiver know the
  channel accurately.
• With the assumption of perfect knowledge of the
  channel, the capacity is known to increase
  linearly with N, for fixed total power.

4
Fast fading channels

• The time needed to estimate channel parameter is
  proportional to N, so we can not increase the
  number of antennas arbitrarily. When N is close
  to Tc, the perfect knowledge assumption fails.
• For outdoor system (mobile) or high carrier
  frequency , the channel is changing fast, Tc is
  small, there is not enough time to do a good
  estimation of the channel.
• We use information theory to study the optimal
  way to code over coherence times, add in
  structure in the code words so we can estimate
  the channel and communicate simultaneously.

5
Single Antenna Channel Capacity

• In narrow band applications, we can assume high
  SNR.
• For the case with 1 transmit antenna and 1
  receive antenna, with high SNR, we can compute
  the capacity, which is achieved by sending the
  signal vectors of dimension Tc to be uniform in
  all directions, but with constant norm.
• Information theory suggest to encode all
  information in the direction of code words, which
  is not affected by fading.

• No information is conveyed by the norm of code
  word, so we need not to estimate the channel gain
  at all. And only do a non-coherent detection is
  optimal.
• All constellation points are in the surface of
  sphere of dimension Tc.

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Non-Coherent Detection

• With information theory, we maximize the mutual
  information without any assumption of knowledge
  of the channel parameters, or training structure.
  Instead, we optimize over all input
  distributions, so that is a more general result
  than channel estimating and tracking.
• In fading channel, the signal is only affected by
  the fadings in some directions, in other
  directions, there is only additive noise. So to
  receive information in these directions, we do
  not need to estimate the channel parameters. On
  the other directions, we can also compute the
  optimal input distribution and design optimal
  receiver.
• The decomposition of the signal space decompose
  the communication in fading channel into two
  different channels, and to use each channel, we
  can achieve the capacity with no prior knowledge
  of the channel parameters.

7
Multi-Antenna Capacity

• For N transmit antenna, N receive antenna system,
  the total number of degree of freedom is N Tc,
  and the number of degree of freedom with no
  fading is , so the capacity increases
  with SNR as function .
• If Tc is close to N, the capacity no longer
  increase linearly with N. For any given Tc, we
  can always compute the optimal number of antennas
  used to maximize capacity in high SNR, which is
  approximately Tc/2.
• In real system, where the SNR is finite, the
  information conveyed by norms is not negligible,
  so we want to compute the optimal input norm
  distribution.

8
Achieve the Capacity

• To achieve the capacity we can modulate the two
  parts of information (subspace and norm)
  independently.

• The subspace detection is given by

9
Conclusions and Open Issues

• For multiple antenna fading system with high SNR,
  we have found the way to compute channel
  capacity, without any assumption of knowledge on
  the channel parameters, or any training
  structure.
• The limit of increasing the number of antenna
  with given coherence time Tc is around Tc/2.
• The optimal input distribution of norms, or a
  suboptimal distribution that achieves the
  coherent part of capacity with small difference.
• The connection between existing channel
  estimation and tracking algorithms and the
  optimal input distribution in Shannon capacity
  sense.
• Need simulation to see how well is the high SNR
  approximation works in real system, and
  comparison of performance to the training scheme.