Pulse or digital communications – Receivers – Angle modulation
Reexamination Certificate
2001-03-12
2004-12-21
Deppe, Betsy L. (Department: 2637)
Pulse or digital communications
Receivers
Angle modulation
C375S261000, C375S340000
Reexamination Certificate
active
06834088
ABSTRACT:
FIELD OF THE INVENTION
The present invention relates generally to communications systems, and more particularly to methods for demodulating and decoding encoded data bits transmitting in a code division multiple access (CDMA) system.
BACKGROUND OF THE INVENTION
Various communication systems are known in the art. In many systems an information signal is modulated onto a carrier signal and transmitted from a first location to a second location. At the second location, the information signal is demodulated and recovered.
Typically, the communication path used by such systems has various limitations, such as bandwidth. As a result, there are upper practical limitations that restrict the quantity of information that can be supported by the communication path over a given period of time. Various modulation techniques have been proposed that effectively increase the information handling capacity of the communication path as measured against other modulation techniques.
One modulation technique is M-ary quadrature amplitude modulation (QAM). QAM provides a constellation of an M number of modulation values (distinguished from one another by each having a different combination of phase and amplitude) wherein each constellation point represents a plurality of information bits. In order to achieve greater spectral efficiency in CDMA systems, for example, higher order modulations are used. In general, the number of bits that are represented by each symbol in an M-ary QAM system is equal to log M. For example, 16 QAM uses 16 distinct constellation points (four points in each quadrant of a complex Cartesian plane having I and Q axes) that each represent log
2
16 or four information bits. Another example is 64-QAM wherein 64 constellation points each represent a combination of log
264
or six information bits.
FIG. 1
illustrates a constellation for a 64-QAM communication system that is a map of 64 points each representing a combination of six bits on the complex plane defined by a horizontal axis representing the real portions (i.e., I) and a vertical axis representing imaginary portions (i.e., Q) of a complex number. Transmitted QAM information symbols on a communication channel (and the pilot and sync symbols as well) are discrete packets of a carrier signal modulated to convey information using both the amplitude and phase-angle displacement of the carrier from some reference. QAM information symbols are represented on the constellation of
FIG. 1
as complex quantities represented as vectors having both magnitude (represented as length or distance from origin) and phase angles &phgr; (angles measured with respect to one of the axes). In a 64-QAM system, having 64 different magnitude and phase angle combinations that correspond to 64 different possible bit patterns of six binary digits (which bits are from a serial stream of bits from an information source), each of the 64 points on the constellation is identified as representing one combination of six bits.
A vector
100
(expressed in rectangular coordinates as 5I+5Q and having a length=(5
2
+5
2
)/2 and a phase angle
102
equal to the arc tangent of 5/5 or forty-five degrees with respect to the real axis), points to the point
104
in the constellation. This point is shown in
FIG. 1
as representing the series of six binary digits (i.e., 000000). Another QAM symbol represented by point
106
(−1I−1Q) is illustrated in this constellation and represents another series of six binary digits (i.e., 110011).
For any modulation scheme, the demodulator preferably should provide the decoder with the log-likelihood ratio (LLR) of the received encoded bits. As shown in
FIG. 2
, a receiver
200
in a CDMA system includes a log-likelihood ratio calculator
202
that receives demodulated symbols from a demodulator
204
and calculates a LLR for each bit of the demodulated symbols. The decoder
206
receives the LLR's calculated by the log-likelihood ratio calculator
202
and decodes the individual bits based on the received LLR's.
When two-dimensional modulations (i.e., having I and Q components) yield spectral efficiencies greater than 2 bps/Hz, each encoded bit cannot be mapped to an orthogonal symbol component. As a result, bit LLR's must be calculated from Euclidean distances between the demodulated symbol and the constellation symbols. The LLR of each actual j
th
bit associated with a symbol y received at time k (referred to as y
i
) is determined by the following relationship:
LLR
(
u
k
,
j
)
=
log
{
P
(
u
k
,
j
=
0
&RightBracketingBar;
y
k
)
P
(
u
k
,
j
=
1
&RightBracketingBar;
y
k
)
}
(
1
)
where u
k,j
is the hypothesis j
th
bit value of the symbol u at time k and P represents the probability that the j
th
bit of u
k
is a binary value of 0 or 1 given that y
k
is the received symbol. This equation reduces to the expression:
LLR
(
u
k
,
j
)
=
log
{
∑
u
k
u
k
,
j
=
0
P
(
y
k
&RightBracketingBar;
u
k
)
∑
u
k
u
k
,
j
=
1
P
(
y
k
&RightBracketingBar;
u
k
)
}
(
2
)
assuming that all transmitted symbols are equiprobable. The probability P is represented by the following relationship assuming a perfect channel correction, such that the real and imaginary components are independent:
P
(
y
k
|u
k
)=
p
(
y
k
i
,y
k
i
|u
k
i
,u
k
i
)=
p
(
y
k
i
|u
k
i
)
p
(
y
k
i
|u
k
i
) (
3
)
Assuming additive Gaussian noise, the above equation reduces to the following expression:
P
(
y
k
&RightBracketingBar;
u
k
)
=
1
2
πσ
2
exp
-
D
k
2
2
σ
2
(
4
)
where D
k
2
is the squared Euclidean distance between the received symbol and the hypothesis symbol and &sgr;
2
is the variance of the Gaussian noise. Accordingly, the squared Euclidean distance D
k
2
is calculated by summing the squares of the differences between the received and hypothesis symbol of both the real (r) and imaginary (i) components as represented by the following relationship:
D
k
2
=|y
k
−u
k
|
2
=(
y
k
i
−u
k
i
)
2
+(
y
k
i
−u
k
i
)
2
(5)
Substituting equation (4) into equation (2) allows calculation of the LLR based on the squared Euclidean distances and yields the following expression:
LLR
(
u
k
,
j
)
=
log
∑
u
k
u
k
,
j
=
0
ⅇ
-
D
k
2
2
σ
2
-
log
∑
u
k
u
k
,
j
=
1
ⅇ
-
D
2
2
σ
2
(
6
)
This expression may be approximated by simply taking the difference between the minimum squared distances between the received symbol and the hypothesis symbol having respective bit values of 1 and 0. Thus, equation (6) may be reduced to the following expression to approximate the LLR:
LLR
(
u
k
,
j
)
=
min
u
k
u
k
,
j
=
1
[
D
k
2
]
-
min
u
k
·
u
k
,
j
=
0
[
D
k
2
]
(
7
)
where the LLR of a bit u
k . . . j
in a symbol is found by computing the difference of the minimum distance between the received symbol and a QAM constellation point where that particular bit equals one and the minimum distance between the received symbol and a QAM constellation point where that particular bit equals zero. In M-ary QAM systems the M number of symbols is equal to 2
m
with m being the number of bits per symbol. Hence, a m number of LLR calculations are required per modulation symbol (i.e., finding the LLR for each of the m number of bits in the symbol). In order to determine the m number of LLR calculations, a M number of squared Euclidian distances (D
2
) must first be determined between the received QAM symbol and each of the QAM symbols in the QAM constellation of symbols. Given the above mathematical operations, the calculation of LLR values for each received modulation symbol require a M number of squared Euclidean distance calculations (i.e., D
2
calculations, each of which involve 2 multiplication operations and an addition operation), a M number of comparison/selections per bit to determine the minimum squared Euclidean to the par
Agami Gregory
Corke Robert John
Rotstein Ron
Deppe Betsy L.
Jacobs Jeffrey K.
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