Space-Time Block Codes for Wireless Systems - The ...

hence the worst case PEP is still a valid design criterion. Furthermore, due to our consideration
of the maximum likelihood decoder, we focus on a single receive antenna. Note
that for other decoding rules like Minimum Mean Squared Error (MMSE), the number
of receive antennae does matter in code design [GM01].
Due to the large dimensionality of our problem, a variable X might have up to 4
indices. To assist the reader in deciphering the notation X l(k)
i
(t), we generally conform
to the following rules: the subscript i is the transmit antenna index, the first superscript
l is the multipath index, the second superscript (k) is the user index, and finally t is
the time index.
Sometimes a matrix or vector is independent of the time index t or
transmit antenna index i, but we still keep that index to help track the dimension of the
corresponding matrix or vector. In this case, we will point it out explicitly. Two kinds of
codeword matrices are constantly used throughout the chapter, the calligraphic D, which
represents the block code in its original block form, and the regular D, which represents
a reshaped version of D. A prefix ∆ in front of D or D denotes the difference between
two codewords.
We summarize the constants used in this chapter for quick reference:
K: the # of active users
L t : the # of transmit antennae
L r : the # of receive antennae
L c : the # of maximum resolvable multipaths for a single transmit antenna
N c : the time dimension of block code in units of the symbol period
L u : the # of samples per symbol period, or sampling rate, for non-spread systems,
or spreading gain for chip-sampled DS-CDMA systems
τ max : the largest path delay
L s : the # of samples between two consecutive transmissions
Note that L s is used for non-spread systems only, for spread systems, L s = L u .
In this chapter, a unique integer, the code index, is used to represent a codeword. If
nPSK is employed, u is the n th root of unity u = e j 2π n and d i (t) = u ci(t) , a block code
matrix of size N c × L t can be represented by the index obtained from a base n number:
(c Lt (N c )c Lt−1(N c )c Lt−2(N c ) · · · · · · + c 2 (1)c 1 (1)) n . (2.1)
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