MC-CDMA 53
Pseudo noise (PN) spreading codes: The property of a PN sequence is that the sequence
appears to be noise-like if the construction is not known at the receiver. They are typically
generated by using shift registers. Often used PN sequences are maximum-length shift
register sequences, known as m-sequences. A sequence has a length of
n = 2
m
− 1 (2.17)
bits and is generated by a shift register of length m with linear feedback [40]. The sequence
has a period length of n and each period contains 2
m−1
ones and 2
m−1
− 1 zeros, i.e., it
is a balanced sequence.
Gold codes: PN sequences with better cross-correlation properties than m-sequences are
the so-called Gold sequences [40]. A set of n Gold sequences is derived from a preferred
pair of m-sequences of length L = 2
n
− 1 by taking the modulo-2 sum of the first preferred
m-sequence with the n cyclically shifted versions of the second preferred m-sequence. By
including the two preferred m-sequences, a family of n + 2 Gold codes is obtained. Gold
codes have a three-valued cross correlation function with values {−1, −t(m),t(m) −2}
where
t(m) =
2
(m+1)/2
+ 1form odd .
2
(m+2)/2
+ 1form even
(2.18)
Golay codes: Orthogonal Golay complementary codes can recursively be obtained by
C
L
=
C
L/2
C
L/2
C
L/2
−C
L/2
, ∀L = 2
m
,m 1, C
1
= 1,(2.19)
where the complementary matrix
C
L
is defined by reverting the original matrix C
L
.If
C
L
=
A
L
B
L
,(2.20)
and A
L
and B
L
are L ×L/2 matrices, then
C
L
= [
A
L
−B
L
].(2.21)
Zadoff-Chu codes: The Zadoff–Chu codes have optimum correlation properties and are
a special case of generalized chirp-like sequences. They are defined as
c
(k)
l
=
e
j 2πk(ql+l
2
/2)/L
for L even
e
j 2πk(ql+l(l+1)/2)/L
for L odd
, (2.22)
where q is any integer, and k is an integer, prime with L.IfL is a prime number,
a set of Zadoff–Chu codes is composed of L −1 sequences. Zadoff–Chu codes have
an optimum periodic autocorrelation function and a low constant magnitude periodic
cross-correlation function.
Low-rate convolutional codes: Low-rate convolutional codes can be applied in CDMA
systems as spreading codes with inherent coding gain [50]. These codes have been applied
as alternative to the use of a spreading code followed by a convolutional code. In MC-
CDMA systems, low-rate convolutional codes can achieve good performance results for
54 MC-CDMA and MC-DS-CDMA
moderate numbers of users in the uplink [30][32][46]. The application of low-rate con-
volutional codes is limited to very moderate numbers of users since, especially in the
downlink, signals are not orthogonal between the users, resulting in possibly severe mul-
tiple access interference. Therefore, they cannot reach the high spectral efficiency of
MC-CDMA systems with separate coding and spreading.
2.1.4.2 Peak-to-Average Power Ratio (PAPR)
The variation of the envelope of a multi-carrier signal can be defined by the peak-to-
average power ratio (PAPR) which is given by
PAPR =
max |x
v
|
2
1
N
c
N
c
−1
v=0
|x
v
|
2
.(2.23)
The values x
v
, v = 0, ,N
c
− 1, are the time samples of an OFDM symbol. An addi-
tional measure to determine the envelope variation is the crest factor (CF) which is
CF =
√
PAPR.(2.24)
By appropriately selecting the spreading code, it is possible to reduce the PAPR of the
multi-carrier signal [4][36][39]. This PAPR reduction can be of advantage in the uplink
where low power consumption is required in the terminal station.
Uplink PAPR
The uplink signal assigned to user k results in
x
v
= x
(k)
v
.(2.25)
The PAPR for different spreading codes can be upper-bounded for the uplink by [35]
PAPR
2max
L−1
l=0
c
(k)
l
e
j2πlt/T
s
2
L
,(2.26)
assuming that N
c
= L. Table 2-1 summarizes the PAPR bounds for MC-CDMA uplink
signals with different spreading codes.
The PAPR bound for Golay codes and Zadoff–Chu codes is independent of the spread-
ing code length. When N
c
is a multiple of L, the PAPR of the Walsh-Hadamard code is
upper-bounded by 2N
c
.
Downlink PAPR
The time samples of a downlink multi-carrier symbol assuming synchronous transmission
are given as
x
v
=
K−1
k=0
x
(k)
v
.(2.27)
MC-CDMA 55
Table 2-1 PAPR bounds of MC-CDMA uplink signals;
N
c
= L
Spreading code PAPR
Walsh –Hadamard 2L
Golay 4
Zadoff–Chu 2
Gold 2
t(m)− 1 −
t(m) + 2
L
The PAPR of an MC-CDMA downlink signal with K users and N
c
= L can be upper-
bounded by [35]
PAPR
2max
K−1
k=0
L−1
l=0
c
(k)
l
e
j2πlt/T
s
2
L
.(2.28)
2.1.4.3 One- and Two-Dimensional Spreading
Spreading in MC-CDMA systems can be carried out in frequency direction, time direc-
tion or two-dimensional in time and frequency direction. An MC-CDMA system w ith
spreading only in the time direction is equal to an MC-DS-CDMA system. Spreading in
two dimensions exploits time and frequency diversity and is an alternative to the conven-
tional approach with spreading in frequency or time direction only. A two-dimensional
spreading code is a spreading code of length L where the chips are distributed in the
time and frequency direction. Two-dimensional spreading can be performed by a two-
dimensional spreading code or by two cascaded one-dimensional spreading codes. An
efficient realization of two-dimensional spreading is to use a one-dimensional spreading
code followed by a two-dimensional interleaver as illustrated in Figure 2-3 [23]. With two
cascaded one-dimensional spreading codes, spreading is first carried out in one dimension
with the first spreading code of length L
1
. In the next step, the data-modulated chips of
the first spreading code are again spread with the second spreading code in the second
dimension. The length of the second spreading code is L
2
. The total spreading length
with two cascaded one-dimensional spreading codes results in
L = L
1
L
2
.(2.29)
If the two cascaded one-dimensional spreading codes are Walsh–Hadamard codes, the
resulting two-dimensional code is again a Walsh–Hadamard code with total length L.
For large L, two-dimensional spreading can outperform one-dimensional in an uncoded
MC-CDMA system [13][42].
Two-dimensional spreading for maximum diversity gain is efficiently realized by using
a sufficiently long spreading code with L
D
O
,whereD
O
is the maximum achievable
two-dimensional diversity (see Section 1.1.7). The spread sequence of length L has to be
appropriately interleaved in time and frequency, such that all chips of this sequence are
faded independently as far as possible.
56 MC-CDMA and MC-DS-CDMA
1D spreading 2D spreading
1st direction
2nd direction
interleaved
Figure 2-3 1D and 2D spreading schemes
Another approach with two-dimensional spreading is to locate the chips of the two-
dimensional spreading code as close together as possible in order to get all chips similarly
faded and, thus, preserve orthogonality of the spreading codes a t the receiver as far as
possible [3][38]. Due to reduced multiple access interference, low complex receivers can
be applied. However, the diversity gain due to spreading is reduced such that powerful
channel coding is required. If the fading over all chips of a spreading code is flat, the
performance of conventional OFDM without spreading is the lower bound for this spread-
ing approach; i.e., the BER performance of an MC-CDMA system with two-dimensional
spreading and Rayleigh fading which is flat over the whole spreading sequence results
in the performance of OFDM with L = 1 shown in Figure 1-3. O ne- or two-dimensional
spreading concepts with interleaving of the chips in time and/or frequency are lower-
bounded by the diversity performance curves in Figure 1-3 which are assigned to the
chosen spreading code length L.
2.1.4.4 Rotated Constellations
With spreading codes like Walsh–Hadamard codes, the achievable diversity gain degrades,
if the signal constellation points of the resulting spread sequence s in the downlink con-
centrate their energy in less than L sub-channels, which in the worst case is only in one
sub-channel while the signal on all other sub-channels is zero. Here we consider a full
loaded scenario with K = L. The idea of rotated constellations [8] is to guarantee the
existence of M
L
distinct points at each sub-carrier for a transmitted alphabet size of M
and a spreading code length of L and that all points are nonzero. Thus, if all except one
sub-channel are faded out, detection of all data symbols is still possible.
With rotated constellations, the L data symbols are rotated before spreading such that
the data symbol constellations are different for each of the L data symbols of the transmit
symbol vector s. This can be achieved by rotating the phase of the transmit symbol
alphabet of each of the L spread data symbols by a fraction proportional to 1/L.The
rotation factor for user k is
r
(k)
= e
j 2πk/(M
rot
L)
,(2.30)
where M
rot
is a constant whose c hoice depends on the symbol alphabet. For example,
M
rot
= 2 for BPSK and M
rot
= 4 for QPSK. For M-PSK modulation, the constant
MC-CDMA 57
(a) (b)
I
Q
Q
I
Figure 2-4 Constellation points after Hadamard spreading a) nonrotated, b) rotated, both for
BPSK and L = 4
M
rot
= M. The constellation points of the Walsh-Hadamard spread sequence s with BPSK
modulation w ith and without rotation is illustrated in Figure 2-4 for a spreading code
length of L = 4.
Spreading with rotated constellations can achieve better performance than the use of
nonrotated spreading sequences. The performance improvements strongly depend on the
chosen symbol mapping scheme. Large symbol alphabets reduce the degree of freedom
for placing the points in a rotated signal constellation and decrease the gains. Moreover,
the performance improvements with rotated constellations strongly depend on the chosen
detection techniques. For higher-order symbol mapping schemes, relevant performance
improvements require the application of powerful multiuser detection techniques. The
achievable performance improvements in SNR with rotated constellations can be in the
order of several dB at a BER of 10
−3
for an uncoded MC-CDMA system with QPSK in
fading channels.
2.1.5 Detection Techniques
Data detection techniques can be classified as either single-user detection (SD) or mul-
tiuser detection (MD). The approach using SD detects the user signal of interest by not
taking into account any information about multiple access interference. In MC-CDMA
mobile radio systems, SD is realized by one tap equalization to compensate for the distor-
tion due to flat fading on each sub-channel, followed by user-specific despreading. As in
OFDM, the one tap equalizer is simply one complex-valued multiplication per sub-carrier.
If the spreading code structure of the interfering signals is known, the multiple access
interference could not be considered in advance as noise-like, yielding SD to be subopti-
mal. The suboptimality of SD can be overcome with MD where the apriori knowledge
about the spreading codes of the interfering users is exploited in the detection process.
The performance improvements with MD compared to SD are achieved at the expense
of higher receiver complexity. The methods of MD can be divided into interference
cancellation (IC) and joint detection. The principle of IC is to detect the information of
the interfering users with SD and to reconstruct the interfering contribution in the received
signal before subtracting the interfering contribution from the received signal and detecting
the information of the desired user. The optimal detector applies joint detection with
maximum likelihood detection. Since the complexity of maximum likelihood detection
grows exponentially with the number of users, its use is limited in practice to applications
58 MC-CDMA and MC-DS-CDMA
y
. . .
r
parallel-to-serial
converter
d
^
(k)
inverse OFDM
single-user
or
multi-user
detector
d
^
R
0
R
L−1
Figure 2-5 MC-CDMA receiver in the terminal station
with a small number of users. Simpler joint detection techniques can be realized by using
block linear equalizers.
An MC-CDMA receiver in the terminal station of user k is depicted in Figure 2-5.
2.1.5.1 Single-User Detection
The principle of single-user detection is to detect the user signal of interest by not tak-
ing into account any information about the multiple access interference. A receiver with
single-user detection of the data symbols of user k is shown in Figure 2-6.
After inverse OFDM the received sequence r is equalized by employing a bank of
adaptive one-tap equalizers to combat the phase and amplitude distortions caused by the
mobile radio channel on the sub-channels. The one tap equalizer is simply realized by
one complex-valued multiplication per sub-carrier. The received sequence at the output
of the equalizer has the form
u = Gr= (U
0
,U
1
, ,U
L−1
)
T
.(2.31)
The diagonal equalizer matrix
G =
G
0,0
0 ··· 0
0 G
1,1
0
.
.
.
.
.
.
.
.
.
00··· G
L−1,L−1
(2.32)
of dimension L ×L represents the L complex-valued equalizer coefficients of the sub-
carriers assigned to s. The complex-valued output u of the equalizer is despread by
correlating it with the conjugate complex user-specific spreading code c
(k)∗
. The complex-
valued soft decided value at the output of the despreader is
v
(k)
= c
(k)∗
u
T
.(2.33)
r
d
^
(k)
equalizer
G
despreader
c
(k)*
quantizer
u
n
(k)
Figure 2-6 MC-CDMA single-user detection
MC-CDMA 59
The hard decided value of a detected data symbol is given by
ˆ
d
(k)
= Q{v
(k)
},(2.34)
where Q{·} is the quantization operation according to the chosen data symbol alphabet.
The term equalizer is generalized in the following, since the processing of the received
vector r according to typical diversity combining techniques is also investigated using the
SD scheme shown in Figure 2-6.
Maximum Ratio Combining (MRC): MRC weights each sub-channel with its respective
conjugate complex channel coefficient, leading to
G
l,l
= H
∗
l,l
,(2.35)
where H
l,l
,l = 0, ,L− 1, are the diagonal components of H. The drawback of MRC
in MC-CDMA systems in the downlink is that it destroys the orthogonality between the
spreading codes and, thus, additionally enhances the multiple access interference. In the
uplink, MRC is the most promising single-user detection technique since the spreading
codes do not superpose in an orthogonal fashion at the receiver and maximization of the
signal-to-interference ratio is optimized.
Equal Gain Combining (EGC): EGC compensates only for the phase rotation caused by
the channel by choosing the equalization coefficients as
G
l,l
=
H
∗
l,l
|H
l,l
|
.(2.36)
EGC is the simplest single-user detection technique, since it only needs information about
the phase of the channel.
Zero Forcing (ZF): ZF applies channel inversion and can eliminate multiple access
interference by restoring the orthogonality between the spread data in the downlink with
an equalization coefficient chosen as
G
l,l
=
H
∗
l,l
|H
l,l
|
2
.(2.37)
The drawback of ZF is that for small amplitudes of H
l,l
the equalizer enhances noise.
Minimum Mean Square Error (MMSE) Equalization: Equalization according to the
MMSE criterion minimizes the mean square value of the error
ε
l
= S
l
− G
l,l
R
l
(2.38)
between the transmitted signal and the output of the equalizer. The mean square error
J
l
= E{|ε
l
|
2
} (2.39)
can be minimized by applying the orthogonality principle, stating that the mean square
error J
l
is minimum if the equalizer coefficient G
l,l
is chosen such that the error ε
l
is
orthogonal to the received signal R
∗
l
, i.e.,
E{ε
l
R
∗
l
}=0.(2.40)
60 MC-CDMA and MC-DS-CDMA
The equalization coefficient based on the MMSE criterion for MC-CDMA systems re-
sults in
G
l,l
=
H
∗
l,l
|H
l,l
|
2
+ σ
2
.(2.41)
The computation of the MMSE equalization coefficients requires knowledge about the
actual variance of the noise σ
2
. For very high SNRs, the MMSE equalizer becomes iden-
tical to the ZF equalizer. To overcome the additional complexity for the estimation of σ
2
,
a low-complex suboptimum MMSE equalization can be realized [21].
With suboptimum MMSE equalization, the equalization coefficients are designed such
that they perform optimally only in the most critical cases for which successful transmis-
sion should be guaranteed. The variance σ
2
is set equal to a threshold λ at which the
optimal MMSE equalization guarantees the maximum acceptable BER. The equalization
coefficient with suboptimal MMSE equalization results in
G
l,l
=
H
∗
l,l
|H
l,l
|
2
+ λ
(2.42)
and requires only information about H
l,l
. The value λ has to be determined during the
system design.
A controlled equalization can be applied in the receiver, which performs slightly worse
than suboptimum MMSE equalization [23]. Controlled equalization applies zero forcing
on sub-carriers where the amplitude of the channel coefficients exceeds a predefined
threshold a
th
. All other sub-carriers apply equal gain combining in order to avoid noise
amplification.
In the uplink G and H are user-specific.
2.1.5.2 Multiuser Detection
Maximum Likelihood Detection
The optimum multiuser detection technique exploits the maximum a posteriori (MAP)
criterion or the maximum likelihood criterion, respectively. In this section, two optimum
maximum likelihood detection algorithms are shown, namely the maximum likelihood
sequence estimation (MLSE), which optimally estimates the transmitted data sequence
d = (d
(0)
,d
(1)
, ,d
(K−1)
)
T
and the maximum likelihood symbol-by-symbol estimation
(MLSSE), which optimally estimates the transmitted data symbol d
(k)
. It is straightforward
that both algorithms can be extended to a MAP sequence estimator and to a MAP symbol-
by-symbol estimator by taking into account the apriori probability of the transmitted
sequence and symbol, respectively. When all possible transmitted sequences and symbols,
respectively, are equally probable apriori, the estimator based on the MAP criterion and
the one based on the maximum likelihood criterion are identical. The possible transmitted
data symbol vectors are d
µ
, µ = 0, ,M
K
− 1, where M
K
is the number of possible
transmitted data symbol vectors a nd M is the number of possible realizations of d
(k)
.
Maximum Likelihood Sequence Estimation (MLSE): MLSE minimizes the sequence
error probability, i.e., the data symbol vector error probability, which is equivalent to
MC-CDMA 61
maximizing the conditional probability P{d
µ
|r} that d
µ
was transmitted given the received
vector r. The estimate of d obtained with MLSE is
ˆ
d = arg max
d
µ
P {d
µ
|r},(2.43)
with arg denoting the argument of the function. If the noise N
l
is additive white Gaussian,
(2.43) is equivalent to finding the data symbol vector d
µ
that minimizes the squared
Euclidean distance
2
(d
µ
, r) =||r − Ad
µ
||
2
(2.44)
between the received and all possible transmitted sequences. The most likely transmitted
data vector is
ˆ
d = arg min
d
µ
2
(d
µ
, r). (2.45)
MLSE requires the evaluation of M
K
squared Euclidean distances for the estimation of
the data symbol vector
ˆ
d.
Maximum Likelihood Symbol-by-Symbol Estimation (MLSSE): MLSSE minimizes the
symbol error probability, which is equivalent to maximizing the conditional probability
P {d
(k)
µ
|r} that d
(k)
µ
was transmitted given the received sequence r. T he estimate of d
(k)
obtained by MLSSE is
ˆ
d
(k)
= arg max
d
(k)
µ
P {d
(k)
µ
|r}.(2.46)
If the noise N
l
is additive white Gaussian the most likely transmitted data symbol
is
ˆ
d
(k)
= arg max
d
(k)
µ
∀d
µ
with same
realization of d
(k)
µ
exp
−
1
σ
2
2
(d
µ
, r)
.(2.47)
The increased complexity with MLSSE compared to MLSE can be observed in the
comparison of (2.47) with (2.45). An advantage of MLSSE compared to MLSE is that
MLSSE inherently generates reliability information f or detected data symbols which can
be exploited in a subsequent soft decision channel decoder.
Block Linear Equalizer
The block linear equalizer is a suboptimum, low-complex multiuser detector which requires
knowledge about the system matrix A in the receiver. Two criteria can be applied to use
this knowledge in the receiver for data detection.
Zero Forcing Block Linear Equalizer: Joint detection applying a zero forcing block
linear equalizer delivers at the output of the detector the soft decided data vector
v = (A
H
A)
−1
A
H
r = (v
(0)
,v
(1)
, ,v
(K−1)
)
T
,(2.48)
where (·)
H
is the Hermitian transposition.
MMSE Block Linear Equalizer: An MMSE block linear equalizer delivers at the output
of the detector the soft decided data vector
v = (A
H
A + σ
2
I)
−1
A
H
r = (v
(0)
,v
(1)
, ,v
(K−1)
)
T
.(2.49)
62 MC-CDMA and MC-DS-CDMA
Hybrid combinations of block linear equalizers and interference cancellation schemes (see
the next section) are possible, resulting in block linear equalizers with decision feedback.
Interference Cancellation
The principle of interference cancellation is to detect and subtract interfering signals from
the received signal before detection of the wanted signal. It can be applied to reduce intra-
cell and inter-cell interference. Most detection schemes focus on intra-cell interference,
which will be further discussed in this section. Interference cancellation schemes can use
signals for reconstruction of the interference either obtained at the detector output (see
Figure 2-7), or at the decoder output (see Figure 2-8).
Both schemes can be applied in several iterations. Values and functions related to the
iteration j aremarkedbyanindex
[j]
,wherej maytakeonthevaluesj = 1, ,J
it
,and
J
it
is the total number of iterations. The initial detection stage is indicated by the index
[0]
.
Since the interference is detected more relia bly at the output of the channel decoder than
at the output of the detector, the scheme with channel decoding included in the iterative
process outperforms the other scheme. Interference cancellation distinguishes between
parallel and successive cancellation techniques. Combinations of parallel and successive
interference cancellation are also possible.
Parallel Interference Cancellation (PIC): The principle of PIC is to detect and subtract
all interfering signals in parallel before detection of the wanted signal. PIC is suitable for
equalizer
despreader
k
channel
decoder
Π
−1
equalizer
despreader
g ≠ k
distortion
spreader
g ≠ k
symbol
demapper
symbol
mapper
symbol
demapper
hard interference evaluation without channel decoding
Figure 2-7 Hard interference cancellation scheme
equalizer
despreader
k
channel
decoder
Π
−1
equalizer
despreader
g ≠ k
distortion
spreader
g ≠ k
symbol
demapper
soft symbol
mapper
symbol
demapper
soft interference evaluation exploiting channel decoding
soft out
chan. dec.
Π
−1
Π tanh(.)
Figure 2-8 Soft interference cancellation scheme
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