Summary by Defne Aktas and Siwaruk Siwamogsatham

@article{Hug99,
   author = "Brian L.~Hughes",
   title = "Differential Space-Time Modulation",
   journal = {IEEE Transactions on Information Theory},
   year = 1999,
   note = "submitted"
}

1. Summary:

A general differential modulation scheme based on group codes is proposed. This scheme can be applied to any number of transmit
antennas and any signal constellation. A low complexity differential demodulator is derived and design criteria for differential modulation
is given. The optimal differential space-time codes for two transmit antennas are presented.

2. Key Points and Main Contributions: 

2.1. The derivation of the optimal receiver, bound on pairwise error probability (PWEP) and code design criteria for a multi-antenna system
equipped with t transmit and r receive antennas using space-time block coding is summarized for two cases:

a) The receiver have perfect channel state information (CSI).

b) The receiver has no CSI.

For case a), the design criteria are well-known in space-time coding theory; the objective is to maximize the diversity gain which is the
rank of the txn codeword difference matrix (C_0-C_1) and to maximize the product distance which is related to the product of nonzero
eigenvalues of the matrix (C_0-C_1)(C_0-C_1)^{H}.

For case b), with unitary space time codes , the objective is to maximize the diversity gain which is the rank of the matrix in (10)
and to maximize the coding advantage which is given in (10). For t=1, it is shown that the coding advantage is a function of the angle
between two codeword matrices and therefore this quantity is referred to as "angular distance" as opposed to product distance in perfect CSI
case.

2.2. A unitary group code of length n is defined for n >= t as the collection of the matrices of form {DG} where G is from a group of nxn
unitary matrices and D is a txn matrix such that DG is a matrix whose elements are drawn from a particular constellation. This definition is
a generalization of Slepian group codes to multiple antennas and complex valued constellations.

This group code formulation can be used as space-time block codes when perfect CSI is available and also, it can be used to generate a
differential space-time modulation (DSTM) scheme. In the proposed unitary differential modulation scheme for t=n, the transmitter
initially sends X_0=D and then to send the message G_k at the block k, the transmitter sends X_k=X_{k-1}G_k.

A simple suboptimal differential demodulator that estimates the message matrix using only the last two received blocks is derived. It
is shown that the receiver has an estimator/correlator form where the receiver uses the previous received signal block to obtain channel and
previous transmitted signal information. The PWEP bound for this differential scheme is derived and it is observed that DSTM achieves
full diversity in the absence of CSI if and only if the unitary group code achieves full diversity for perfect CSI. Coding advantage of DSTM
for no CSI case is half of that of the group code with perfect CSI. Furthermore, choice of D matrix does not affect the performance.

It is stated that there is a 3dB performance loss relative to that of the group code with perfect CSI since the receiver is
suboptimal. However, the question is how much gain we get in terms of performance when we use more complex receiver. 

2.3. In Appendices B-E, it is shown that every full diversity unitary group code with cardinality M and n=t=2 is equivalent to an (M,k)
cyclic code or a dicyclic code. The dicyclic codes only exists for M>=8. The optimal cyclic and dicyclic codes (with the highest coding
gain) with t=n=2 for different M are listed.

The optimal unitary group codes with different rates and constellations for two transmit antennas are listed and compared to
some of Alamouti's and Tarokh-Jafarkhani's codes.

Tarokh-Jafarkhani's differential modulation scheme is based on orthogonal designs and thus the authors state that the orthogonal
block code design is limited to t<=8 for real constellations and t<=2 for complex constellations. However, Hughes's differential scheme is
not limited by the number of transmit antennas.

3. Questions:

3.1. How complicated would the differential receiver be if n>t?

3.2. How does the code design change when t=n >2 or n >t ? The
extension does not seem obvious.

3.3. It is not surprising that it is possible to find the optimal codes for t=n =2 case since codeword matrices are just 2x2. However,
intuitively codes with n>t might have better coding gain due to added time diversity. How difficult will it be to search for optimal codes
for n > t case?

3.4. The authors state that 3dB loss in performance in DSTM compared to perfect CSI is due to the fact that the receiver uses two most recent
blocks to decode the message. How much gain we can get in performance if the receiver is more complex? Will Viterbi decoding help?

3.5. How does the performance of unitary group codes compare to other space-time codes in literature?

3.6. It is stated that the channel is constant over n symbols. However, in Section III-B, in the derivation of the
differential demodulator, it is assumed that the channel is constant for all the observations.