Multiplex communications – Communication over free space – Combining or distributing information via code word channels...
Reexamination Certificate
1999-09-17
2004-01-13
Chin, Vivian (Department: 2682)
Multiplex communications
Communication over free space
Combining or distributing information via code word channels...
C370S335000, C375S299000, C375S267000
Reexamination Certificate
active
06678263
ABSTRACT:
FIELD OF THE INVENTION
The invention relates generally to PSK-modulated space-time codes and more specifically to using fundamental code constructions for quasi-static and time-varying channels to provide full spatial diversity for an arbitrary number of transmit antennas.
BACKGROUND OF THE INVENTION
Recent advances in coding theory include space-time codes which provide diversity in multi-antenna systems over fading channels with channel coding across a small number of transmit antennas. For wireless communication systems, a number of challenges arise from the harsh RF propagation environment characterized by channel fading and co-channel interference (CCI). Channel fading can be attributed to diffuse and specular multipath, while CCI arises from reuse of radio resources. Interleaved coded modulation on the transmit side of the system and multiple antennas on the receive side are standard methods used in wireless communication systems to combat time-varying fading and to mitigate interference. Both are examples of diversity techniques.
Simple transmit diversity schemes (in which, for example, a delayed replica of the transmitted signal is retransmitted through a second, spatially-independent antenna and the two signals are coherently combined at the receiver by a channel equalizer) have also been considered within the wireless communications industry as a method to combat multipath fading. From a coding perspective, such transmit diversity schemes amount to repetition codes and encourage consideration of more sophisticated code designs. Information-theoretic studies have demonstrated that the capacity of multi-antenna systems significantly exceeds that of conventional single-antenna systems for fading channels. The challenge of designing channel codes for high capacity multi-antenna systems has led to the development of “space-time codes,” in which coding is performed across the spatial dimension (e.g, antenna channels) as well as time. The existing body of work on space-time codes relates to trellis codes and a block coded modulation scheme based on orthogonal designs. Example code designs that achieve full diversity for systems with only a small number of antennas (L=2 and 3) are known for both structures, with only a relatively small number of space-time codes being known. Thus, a need exists for a methodology of generating and using code constructions which allow systematic development of powerful space-time codes such as general constructions that provide full diversity in wireless systems with a large number of antennas.
The main concepts of space-time coding for quasi-static, flat Rayleigh fading channels and the prior knowledge as to how to design them will now be discussed. For the purpose of discussion, a source generates k information symbols from the discrete alphabet X, which are encoded by the error control code C to produce code words of length N=nL
t
over the symbol alphabet Y. The encoded symbols are parsed among L
t
transmit antennas and then mapped by the modulator into constellation points from the discrete complex-valued signaling constellation &OHgr; for transmission across a channel. The modulated streams for all antennas are transmitted simultaneously. At the receiver, there are L
r
receive antennas to collect the incoming transmissions. The received baseband signals are subsequently decoded by the space-time decoder. Each spatial channel (the link between one transmit antenna and one receive antenna) is assumed to experience statistically independent flat Rayleigh fading. Receiver noise is assumed to be additive white Gaussian noise (AWGN). A space-time code consists as discussed herein preferably of an underlying error control code together with the spatial parsing format.
Definition 1 An L×n space-time code
of size M consists of an (Ln,M) error control code C and a spatial parser a that maps each code word vector {overscore (c)} &egr;C to an L×n matrix c whose entries are a rearrangement of those of {overscore (c)}. The space-time code
is said to be linear if both C and &sgr; are linear.
Except as noted to the contrary, a standard parser is assumed which maps
{overscore (c)}
=(
c
1
1
, c
1
2
, . . . ,c
1
Lt
, c
2
1
,c
2
2
, . . . ,c
2
Lt
, . . . ,c
n
1
,c
n
2
, . . . ,c
n
Lt
)&egr;
C
to the matrix
c
=
[
c
1
1
c
2
1
…
c
n
1
c
1
2
c
2
2
…
c
n
2
⋮
⋮
⋰
⋮
c
1
L
t
c
2
L
t
…
c
n
L
t
]
.
In this notation, it is understood that c
t
i
is the code symbol assigned to transmit antenna i at time t.
Let f:y→&OHgr; be the modulator mapping function. Then s=f(c) is the baseband version of the code word as transmitted across the channel. For this system, the following baseband model of the received signal is presented:
y
t
j
=
∑
i
=
1
L
t
α
ij
s
t
i
E
s
+
n
t
j
,
(
1
)
where y
t
j
is the signal received at antenna j at time t; &agr;
ij
is the complex path gain from transmit antenna i to receive antenna j; s
t
i
=f(c
t
i
) is the transmitted constellation point corresponding to c
t
i
; and n
t
j
is the AWGN noise sample for receive antenna j at time t. The noise samples are independent samples of a zero-mean complex Gaussian random variable with variance N
0
/2 per dimension. The fading channel is quasi-static in the sense that, during the transmission of n code word symbols across any one of the links, the complex path gains do not change with time t, but are independent from one code word transmission to the next. In matrix notation,
{overscore (Y)}={square root over (E)}
s
{overscore (A)}D
c
+{overscore (N)},
(2)
where
Y
_
=
[
y
1
1
y
2
1
…
y
n
1
y
1
2
y
2
2
…
y
n
2
…
y
1
L
r
y
2
L
r
…
y
n
L
r
]
,
N
_
=
[
n
1
1
n
2
1
…
n
n
1
n
1
2
n
2
2
…
n
n
2
…
n
1
L
r
n
2
L
r
…
n
n
L
r
]
,
A
_
=
[
α
11
α
21
…
α
L
t
1
α
12
α
22
…
α
L
t
2
…
α
1
L
r
α
2
L
r
…
α
L
t
L
r
]
,
D
c
=
[
f
(
c
)
0
…
0
0
f
(
c
)
…
0
⋮
⋰
⋰
0
0
0
…
f
(
c
)
]
L
r
L
t
×
L
r
n
.
Let code word c be transmitted. Then the pairwise error probability that the decoder prefers the alternate code word e to c is given by
P
(
c→e|{&agr;
ij
})=
P
(
V
<0|{&agr;
ij
}),
where V=||{overscore (A)}(D
c
−D
c
)+{overscore (N)}||
2
−||{overscore (N)}||
2
is a Gaussian random variable with mean E{V}=||
{overscore (A)}(D
c
−D
e
)||
2
and variance Var{V}=
2
N
0
E{V}. Thus,
P
(
V
<
0
|
{
α
ij
}
)
=
Q
(
&LeftDoubleBracketingBar;
A
_
(
D
c
-
D
e
)
&RightDoubleBracketingBar;
2
N
0
)
(
3
)
≤
1
2
exp
{
-
1
4
N
0
&LeftDoubleBracketingBar;
A
_
(
D
c
-
D
e
)
&RightDoubleBracketingBar;
2
}
.
(
4
)
For the quasi-static, flat Rayleigh fading channel, equation (4) can be manipulated to yield the fundamental bound:
P
(
c
→
e
|
{
α
ij
}
)
≤
(
1
∏
i
=
1
r
(
1
+
λ
i
E
s
/
4
N
0
)
)
L
r
(
5
)
≤
(
η
E
s
4
N
0
)
-
rL
r
,
(
6
)
where r=rank(f(c)−f(e)) and &eegr;=(&lgr;
1
&lgr;
2
. . . &lgr;
r
)
1/r
is the geometric mean of the nonzero eigenvalues of A=(f(c)−f(e))(f(c)−f(e))
H
.
This leads to the rank and equivalent product distance criteria for space-time codes.
(1) Rank Criterion: Maximize the diversity advantage r=rank(f(c)−f(e)) over all pairs of distinct code words c,
El Gamal Hesham
Hammons, Jr. A. Roger
Chin Vivian
Craver Charles
Hughes Electronics Corporation
Sales Michael
Whelan John T.
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