SYSTEM MODEL 523
where
˜
kl
is the estimate of
kl
and
y
ikl
(k
l
) = y
iikl
(k
l
) + y
iqkl
(k
l
)
= A
k
l
[d
ik
ρ
ik
l
,ikl
cos ε
k
l
,kl
+ d
qk
ρ
qk
l
,ikl
sin ε
k
l
,kl
] (15.9)
and
y
(n)
qkl
=
(n+1)T +τ
k
+τ
kl
nT +τ
k
+τ
kl
r(t)c
qk
(t − nT − τ
k
+ τ
kl
) sin(ω
0
t +
˜
kl
) dt
=
k
l
y
qkl
(k
l
) (15.10)
y
qkl
(k
l
) = y
qqkl
(k
l
) + y
qikl
(k
l
)
= A
k
l
[d
qk
ρ
qk
l
,qkl
cos ε
k
l
,kl
− d
ik
ρ
ik
l
,qkl
sin ε
k
l
,kl
] (15.11)
with ρ
x,y
being the cross-correlation functions between the corresponding code compo-
nents x and y. A scaling factor 1/2 is dropped in the above equations for simplicity.
Basically by dropping this coefficient for both signal and noise, the signal-to-noise ratio
(SNR) that determines the system performance will not change. Each of these components
is defined with three indices (i or q, user and path). Parameter ε
a,b
=
a
−
˜
b
where a
and b are defined with two indices (user and path). Let the L-element vectors
L
(·) of
matched filter output samples for the nth symbol interval be defined as
y
(n)
ik
=
L
(y
(n)
ikl
) = (y
(n)
ik1
,y
(n)
ik2
, ,y
(n)
ikL
)
T
∈ C
L
(15.12)
y
(n)
qk
=
L
(y
(n)
qkl
) = (y
(n)
qk1
,y
(n)
qk2
, ,y
(n)
qkL
)
T
∈ C
L
(15.13)
y
(n)
k
= y
(n)
ik
+ jy
(n)
qk
(15.14)
y
(n)
=
K
(y
T(n)
k
) ∈ C
KL
(15.15)
y =
N
b
(y
T(n)
) ∈ C
N
b
KL
(15.16)
Let in general R
(n)
(i) ∈ (−1, 1]
KL×KL
be a cross-correlation matrix with the follow-
ing partition:
R
(n)
(i) =
R
(n)
1,1
(i) R
(n)
1,2
(i) ··· R
(n)
1,K
(i)
R
(n)
2,1
(i) R
(n)
2,2
(i) ··· R
(n)
2,K
(i)
.
.
.
.
.
.
.
.
.
.
.
.
R
(n)
K,1
(i) R
(n)
K,2
(i) ··· R
(n)
K,K
(i)
∈ R
KL×KL
=
K
(R
(n)
k,k
(i)) (15.17)
524 WIDEBAND CDMA NETWORK SENSITIVITY
For the final representation of the complex matched filter output signal, given by
equations (15.49) and (15.50), we now define four specific matrices of the form given by
equation (15.17) with the following notation:
R
ii(n)
(i) =
K
(R
ii(n)
k,k
(i)) (15.18)
R
qi(n)
(i) =
K
(R
qi(n)
k,k
(i)) (15.19)
R
iq(n)
(i) =
K
(R
iq(n)
k,k
(i)) (15.20)
R
qq(n)
(i) =
K
(R
qq(n)
k,k
(i)) (15.21)
where matrices R
ab(n)
k,k
(i) ∈ R
L×L
, ∀k, k
∈{1, 2, ,K} in equations (15.18) to (15.21)
have elements
(R
ii(n)
k,k
(i))
l,l
= cos ε
k
l
,kl
×
∞
−∞
c
(n)
ik
(t − τ
k
− τ
kl
)c
(n−i)
ik
(t + iT − τ
k
− τ
k
l
) dt
(15.22)
(R
qi(n)
k,k
(i))
l,l
= sin ε
k
l
,kl
×
∞
−∞
c
(n)
qk
(t −τ
k
− τ
k,l
)c
(n−i)
ik
(t +iT − τ
k
− τ
k
l
) dt
(15.23)
(R
iq(n)
k,k
(i))
l,l
=−sin ε
k
l
,kl
×
∞
−∞
c
(n)
ik
(t −τ
k
− τ
k,l
)c
(n−i)
qk
(t +iT − τ
k
− τ
k
l
) dt
(15.24)
(R
qq(n)
k,k
(i))
l,l
= cos ε
k
l
,kl
×
∞
−∞
c
(n)
qk
(t − τ
k
− τ
k,l
)c
(n−i)
qk
(t + iT − τ
k
− τ
k
l
) dt
∀l, l
∈{1, 2, ,L} (15.25)
In order to simplify the notation, we present equation (15.22) as R = ρ cos ε and its
estimation as
ˆ
R =ˆρ cos ˆε(15.26)
In general, the estimated phase difference ˆε between the two users (e.g. users with index
k = 1andk = 2) can be represented as
ˆε = φ
1
− φ
1
− φ
2
− φ
2
= ε + ε (15.27)
where ε = φ
1
− φ
2
and ε =−(φ
1
+ φ
2
).
The noise samples at the output matched filters for different users are uncorrelated.
So, if φ is a process with zero mean and variance σ
2
φ
,thenε is a zero mean process
with variance 2σ
2
φ
. The estimated correlation function can be represented as
ˆρ = ρ + ρ
∼
=
ρ +ρ
ε
τ
= ρ
1 +
ρ
ε
τ
ρ
= ρ(1 + s
ρ
)(15.28)
SYSTEM MODEL 525
where ρ
is the slope of the ρ function at the point of zero delay estimation error and
ε
τ
= τ
1
− τ
2
(15.29)
is the difference between the two delay estimation errors. For a given class and code
length, ρ
is a parameter [30]. If τ is a zero mean variable with variance σ
2
τ
,thenε
τ
is
a zero mean variable with variance 2σ
2
τ
. The second component of equation (15.26) can
be represented as
cos ˆε = cos(ε +ε) = cos ε cos ε − sin ε sin ε
= (1 −ε
2
/2) cos ε − ε sin ε
= (1 +s
ε
) cos ε (15.30)
where
s
ε
=−(ε
2
/2 + ε · tan ε) (15.31)
Now, equation (15.26) becomes
ˆ
R = R +R (15.32)
where
R = ρ cos ε
R = ρ cos ε(s
ε
+ s
ρ
+ s
ε
s
ρ
) (15.33)
Whenever k
= k, the average value of the cross-correlation ρ = 0 and parameter R
can be considered as an additional noise component with a zero mean and variance
σ
2
R
= ρ
2
[(1 + 2σ
2
ρ
σ
2
τ
/ρ
2
)(3σ
4
φ
+ 2σ
2
φ
) + 2σ
2
ρ
σ
2
τ
/ρ
2
] (15.34)
Similar expressions can be derived for the estimation of equations (15.23) to (15.25).
In the case when multipath delay spread produces severe intersymbol interference
(ISI), the overall received signal should be further modified. When the delay spread is
limited to less than one symbol interval, then for an asynchronous network the vector
equation (15.15) can be expressed as [4]
y
(n)
(R, H, A, d) = R
(n)
(2)H
(n−2)
Ad
(n−2)
+ R
(n)
(1)H
(n−1)
Ad
(n−1)
+ R
(n)
(0)H
(n)
Ad
(n)
+ R
(n)
(−1)H
(n+1)
Ad
(n+1)
+ R
(n)
(−2)H
(n+2)
Ad
(n+2)
+ n
(n)
(15.35)
where
A = diag(A
1
,A
2
, ,A
K
) ∈ R
K×K
(15.36)
526 WIDEBAND CDMA NETWORK SENSITIVITY
is a diagonal matrix of transmitted amplitudes,
H
(n)
= diag(H
(n)
1
, H
(n)
2
, ,H
(n)
K
) ∈ C
KL×K
(15.37)
is the matrix of channel coefficient vectors
H
(n)
k
= (H
(n)
k,1
,H
(n)
k,2
, ,H
(n)
k,L
)
T
∈ C
L
(15.38)
d
(n)
= (d
(n)
1
,d
(n)
2
, ,d
(n)
K
)
T
∈
K
(15.39)
is the vector of the transmitted data and n
(n)
∈ C
KL
is the output vector due to noise. This
component is due to processing the second term of equation (15.7) that was dropped for
simplicity in derivation of equations (15.8) to (15.34). It is easy to show that R
(n)
(i) =
0
KL
, ∀|i| > 2andR
(n)
(−i) = R
T(n+1)
(i),where0
KL
is an all-zero matrix of size KL ×
KL. Thus, the concatenation vector of the matched filter outputs (15.16) has the expression
y(R, H,A, d ) = RHAd + n = RHh + n(15.40)
where
R =
R
(0)
(0) R
T(1)
(1) R
T(2)
(2) ··· 0
KL
R
(1)
(1) R
(1)
(0) R
T(2)
(1) ··· 0
KL
R
(2)
(2) R
(2)
(1) R
(2)
(0) ··· 0
KL
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
KL
0
KL
0
KL
··· R
(N
b
−1)
(0)
∈ R
N
b
KL×N
b
KL
(15.41)
H = diag(H
(0)
, H
(1)
, ,H
(N
b
−1)
) ∈ C
N
b
KL×N
b
K
(15.42)
A = diag(A, A, ,A) ∈ R
N
st
K×N
b
K
(15.43)
d = (d
T(0)
, d
T(1)
, ,d
T(N
b
−1)
)
T
∈
N
b
K
(15.44)
h = Ad is the data-amplitude product vector and n is the Gaussian noise output vector
with zero mean and covariance matrix σ
2
R.Ifwedefine
y
ii
= y(R
ii
,H,A,d
i
) (15.45)
y
qi
= y(R
qi
,H,A,d
q
) (15.46)
y
iq
= y(R
iq
,H,A,d
i
) (15.47)
y
qq
= y(R
qq
,H,A,d
q
) (15.48)
then we have
y
i
= y
ii
+ y
qi
(15.49)
y
q
= y
iq
+ y
qq
(15.50)
CAPACITY LOSSES 527
On the basis of these equations in the sequel, a complex decorrelator receiver structure
is derived.
15.2.1 Complex decorrelator
As a starting point, we represent equations (15.49) and (15.50) as
y
i
= y
ii
+ y
qi
=
ii
d
i
+
qi
d
q
+ n
i
(15.51)
y
q
= y
iq
+ y
qq
=
iq
d
i
+
qq
d
q
+ n
q
(15.52)
where
=RHA(15.53)
and we use an additional step to produce
ii−1
y
i
= d
i
+
ii−1
qi
d
q
+
ii−1
n
i
iq−1
y
q
= d
i
+
iq−1
qq
d
q
+
iq−1
n
q
qi−1
y
i
=
qi−1
ii
d
i
+ d
q
+
qi−1
n
i
qq−1
y
q
=
qq−1
iq
d
i
+ d
q
+
qq−1
n
q
(15.54)
From the last set of equations one can show that the data estimates should be obtained as
ˆ
d =
ˆ
d
i
+ j
ˆ
d
q
ˆ
d
q
= sgn{D
qi
y
i
+ D
qq
y
q
}
D
qi
={
qi−1
ii
−
qq−1
iq
}
−1
qi−1
D
qq
=−{
qi−1
ii
−
qq−1
iq
}
−1
qq−1
(15.55)
and similarly
ˆ
d
i
= sgn{D
ii
y
i
+ D
iq
y
q
}
D
ii
={
ii−1
qi
−
iq−1
qq
}
−1
ii−1
D
ii
={
ii−1
qi
−
iq−1
qq
}
−1
ii−1
(15.56)
Bearing in mind that all current wideband code division multiple access (WCDMA)
standards are based on using complex signal formats, future research in the field of
multiuser detectors should be focused on the structures defined above.
15.3 CAPACITY LOSSES
The starting point in the evaluation of CDMA system capacity is parameter Y
bm
=
E
bm
/N
0
, the received signal energy per symbol per overall noise density in a given
528 WIDEBAND CDMA NETWORK SENSITIVITY
reference receiver with index m. For the purpose of this analysis, we can represent this
parameter in the general case as
Y
bm
=
E
bm
N
0
=
ST
I
oc
+ I
oic
+ I
oin
+ η
th
(15.57)
where I
oc
,I
oic
and I
oin
are the power densities of intracell, intercell and overlay type inter-
network interference, respectively, and η
th
is the thermal noise power density. Parameter
S is the overall received power of the useful signal and T = 1/R
b
is the information bit
interval. Contributions of I
oic
and I
oin
to N
0
have been discussed in Chapter 8 and in a
number of papers, for example, in Reference [28]. In order to minimize repetition in our
analysis, we will parameterize this contribution by introducing
η
0
= I
oic
+ I
oin
+ η
th
(15.58)
and concentrate on the analysis of the intracell interference, I
oc
, in a CDMA network
based on advanced receivers using imperfect RAKE, and MAI cancellation based on
an imperfect decorrelator. An extension of the analysis to include both intercell and
internetwork interference is straightforward. A general block diagram of the receiver is
shown in Figure 15.1.
If for user m an L
0
-finger RAKE receiver (L
0
≤ L) with combiner coefficients w
mr
(r = 1, 2, ,L
0
) and an imperfect decorrelator is used, the SNR will become
Y
bm
=
r
(L
0
)
m
ς
0
ηR
b
/S
(15.59)
where
ς
0
=
L
0
r=1
w
2
mr
= w
m
w
T
m
; w
m
= (w
m1
,w
m2
, ,w
mL
0
)(15.60)
Bank of
matched
filters
Other users
Rake
receiver
of user
m
Demodulation
&
decoding
Information
output
Imperfect parameter estimation
Complex
decorrelator
Figure 15.1 General receiver block diagram.
CAPACITY LOSSES 529
is due to Gaussian noise processing in the RAKE receiver, and the noise density η
0
,after
decorrelation, becomes η. The relation between these two parameters is elaborated later in
equations (15.74) to (15.77). The parameter r
(L
0
)
m
in equation (15.59) is called the RAKE
receiver efficiency and is given by
r
(L
0
)
m
=
L
0
r=1
w
mr
cos ε
θmmr
√
α
mr
2
= (w
m
· α
mm
√
)
2
(15.61)
with α
mm
√
= (cos ε
θmm1
√
α
m1
, cos ε
θmm2
√
α
m2
, )
T
. Parameter ε
θmmr
= θ
mmr
−
ˆ
θ
mmr
is
the carrier phase synchronization error in receiver m for signal of user m in path r.We
will drop index mkl whenever it does not result in any ambiguity. In the sequel we will
use the following notation: α
kl
= A
2
kl
/2,
ˆ
A
mkl
is the estimation of A
kl
by the receiver
m, ε
a
= A
mkl
/A
kl
= (A
kl
−
ˆ
A
mkl
)/A
kl
is the relative amplitude estimation error and
ε
θ
is the carrier phase estimation error.
For the equal gain combiner (EGC), the combiner coefficients are given as w
mr
=
1. Having in mind the notation used so far, in the sequel we will drop index m for
simplicity. For the maximal ratio combiner (MRC), the combiner coefficients are based
on estimates as
ˆw
r
=
cos ε
θr
cos ε
θ1
·
ˆ
A
r
ˆ
A
1
∼
=
(1 −ε
2
θr
/2)
(1 −ε
2
θ1
/2)
·
A
r
(1 − ε
ar
)
A
1
(1 − ε
a1
)
(15.62)
E{ˆw
r
}=w
r
(1 − σ
2
θr
)(1 +σ
2
θ1
)(1 −ε
ar
)(1 +ε
a1
) (15.63)
E{ˆw
2
r
}=w
2
r
(1 − 2σ
2
θr
+ 3σ
4
θr
)(1 + 2σ
2
θ1
− 3σ
4
θ1
)(1 −ε
ar
)
2
(1 +ε
a1
)
2
(15.64)
By using notation A
r
/A
1
=
√
α
r
/α
1
, averaging equation (15.61) gives for EGC
E{r
(L
0
)
}=E
L
0
r=1
cos ε
θr
√
α
r
2
∼
=
E
L
0
r=1
(1 − ε
2
θr
/2)
√
α
r
2
=
r
l
l = r
(1 −σ
2
θr
)(1 −σ
2
θl
)
√
α
r
α
l
+
r
α
r
(1 − 2σ
2
θr
+ 3σ
4
θr
) (15.65)
For MRC the same relation becomes
E{r
(L
0
)
}=E
L
0
r=1
ˆw
r
cos ε
θr
√
α
r
2
∼
=
E
L
0
r=1
α
r
√
α
1
(1 − ε
2
θr
/2)
2
(1 −ε
2
θ1
/2)
(1 −ε
ar
)
(1 −ε
a1
)
2
= E
L
0
r=1
α
2
r
α
1
(1 − ε
2
θr
/2)
4
(1 − ε
2
θ1
/2)
2
(1 − ε
ar
)
2
(1 −ε
a1
)
2
+
r
l
l = r
α
r
α
l
α
1
(1 − 2σ
2
θr
+ 3σ
4
θr
)
× (1 −2σ
2
θl
+ 3σ
4
θl
)(1 +2σ
2
θ1
− 3σ
4
θ1
)(1 −ε
ar
)(1 −ε
al
)(1 +ε
a1
)
2
(15.66)
530 WIDEBAND CDMA NETWORK SENSITIVITY
One should note that even when ˆw
1
= 1, the value of the first term in the above sum is
cos ε
θ1
√
α
1
, which takes into account the error in the estimation of the phase for the first
finger. In order to avoid dealing with the fourth power terms of the type (1 − ε
2
θr
/2)
4
in the evaluation of the first term in equation (15.66), we use limits. For the upper limit
we have
ε
2
θr
⇒ ε
2
θ1
(15.67)
By using this we have
(1 −ε
2
θr
/2)
4
(1 −ε
2
θ1
/2)
2
⇒ (1 −ε
2
θ1
/2)
2
(15.68)
and the first term becomes [31]
L
0
r=1
α
2
r
α
1
(1 − 2σ
2
θ1
+ 3σ
4
θ1
)
(1 −ε
ar
)
2
(1 −ε
a1
)
2
(15.69)
For the lower limit we use
ε
2
θ1
⇒ ε
2
θr
(15.70)
and the first term becomes
L
0
r=1
α
2
r
α
1
(1 − 2σ
2
θr
+ 3σ
4
θr
)
(1 − ε
ar
)
2
(1 − ε
a1
)
2
(15.71)
For a signal with I and Q components, the parameter cos ε
θr
should be replaced by
cos ε
θr
⇒ cos ε
θr
+ bρ sin ε
θr
(15.72)
where b is the information in the interfering channel (I or Q) and ρ is the cross-correlation
between the codes used in the I and Q channels. For small tracking errors, this term can
be replaced as
cos ε
θr
+ bρ sin ε
θr
≈ 1 +bρε − ε
2
/2 (15.73)
where the notation is further simplified by dropping the subscript θr.Byusing
equation (15.73) in equations (15.64) to (15.72), similar expressions can be derived for
the complex signal format.
We will assume that a linear decorrelator is used for IC in the system. The detector will
operate with the inverse of the estimated correlation matrix
ˆ
R
−1
where the real correlation
matrix of the system is R =
ˆ
R + R. The elements of R have zero mean and variance
given by equation (15.34). So, after decorrelation by using
ˆ
R
−1
the residual noise in the
receiver will have variance
Var
ˆ
R
−1
(n
r
+ n)
(15.74)
CAPACITY LOSSES 531
where Gaussian noise components of vector n have variance σ
2
n
and components of
residual vector n
r
have variance σ
2
r
that can be approximated as
σ
2
r
=
k,l
α
k,l
σ
2
k,l
(15.75)
where σ
2
k,l
is given by equation (15.34) for specific indices k,l. The residual noise is
composed of a large number of components with the same distribution that suggests using
the central limit theorem to approximate the overall distribution as Gaussian with average
variance represented as
σ
2
r
∼
=
αKLσ
2
(15.76)
where σ
2
is given by equation (15.34). One should keep in mind that the residual noise
n
r
is created in front of the decorrelator. After the decorrelation and the RAKE combiner,
the components of the overall noise variance given by equation (15.74) become
σ
2
∼
=
ς
0
R
+
mm
(σ
2
r
+ σ
2
n
) =
2
r
+
2
n
(15.77)
where R
+
mm
is the mmth component of
ˆ
R
−1
. In this relation,
2
r
is the contribution of
system imperfections due to the overall noise variance σ
2
and
2
n
is the contribution of
Gaussian noise after decorrelation. So, the equivalent noise variance is expressed in terms
of phase and code delay estimation errors (see equation 15.34). One should notice that
the same arguments about using the central limit theorem apply in the case of the noise
after decorrelator too because decorrelation is a linear operation. These results should
be now used for analysis of the impact of large scale of channel estimators on overall
CDMA network sensitivity. A performance measure of any estimator is the parameter
estimation error variance that should be directly used in equation (15.77) for equivalent
noise variance and equations (15.62) to (15.73) for the RAKE receiver. If joint parameter
estimation is used, on the basis of the maximum likelihood (ML) criterion, then the
Cramer–Rao bound could be used for these purposes. For Kalman type estimators, the
error covariance matrix is available for each iteration of estimation. If each parameter is
estimated independently, then for carrier phase and code delay estimation error a simple
relation σ
2
θ,τ
= 1/SNR
L
canbeusedwhereSNR
L
is the SNR in the tracking loop. For the
evaluation of this SNR
L
, the noise power is in general given as N = B
L
N
0
. In this case,
the noise density N
0
is approximated as a ratio of the overall interference plus noise power
divided by the signal bandwidth. The loop bandwidth will be proportional to f
D
where f
D
is the fading rate (Doppler). If decorrelation is performed prior to parameter estimation,
N
0
is obtained from the equivalent noise having the variance defined by equation (15.77).
If parameter estimation is used without decorrelation, then the overall noise consists of
MAI and Gaussian noise.
For the numerical analysis, further assumptions and specifications are necessary. First of
all we need the channel model. The exponential multipath intensity profile (MIP) channel
model is a widely used analytical model and is realized as a tapped delay line [32]. It is
532 WIDEBAND CDMA NETWORK SENSITIVITY
very flexible in modeling different propagation scenarios. The decay of the profile and
the number of taps in the model can vary. Averaged power coefficients in the MIP are
α
l
= α
0
e
−λl
l, λ ≥ 0 (15.78)
where λ is the decay parameter of the profile. Power coefficients should be normalized as
L−1
l=0
α
0
e
−λl
= 1 (15.79)
For λ = 0 the profile will be flat. The number of resolvable paths depends on the channel
chip rate and this must be taken into account. We will start with the average SNR that
can be expressed as
Y
b
=
Sr
(L
0
)
G
σ
2
=
Sr
(L
0
)
(K)G
σ
2
(K)
=
Y
b
(K) (15.80)
where r
(L
0
)
is given in Section 3B, σ
2
is given by equation (15.77) and G = 1/ρ
2
is
the processing gain. RAKE receiver efficiency and overall noise variance depend on
the number of users K. If we accept some quality of transmission, bit error rate (BER)
= 10
−e
, that can be achieved with given SNR = Y
0
= Y
b
(K = C),whereC is the system
capacity, then in the case of perfect channel estimation we have C
max
= K, which is the
solution to the equation
Y
0
=
Sr
(L
0
)
(K)G
2
n
(K)
(15.81)
In the case of imperfect channel estimation we have C = K, which is the solution to
the equation
Y
0
=
Sr
(L
0
)
(K)G
2
r
(K) +
2
n
(K)
(15.82)
The system sensitivity function is defined as
=
C
max
− C
C
max
(15.83)
For illustration purposes, we use system parameters based on UTRA FDD wideband
CDMA concept [1] where the chip rate is 4.096 Mchips s
−1
.
We will see in Chapter 17 that in the latest version of the standard this value is
modified to 3.84 Mchips s
−1
. The chosen coded data rate is 16 kbit s
−1
, which means
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