Bit error probability of Rake receiver in the presence of code synchronization errors and correlated Rayleigh fading channels

Abstract

We derive here a simple and accurate theoretical method for computing the bit-error-probability (BEP) of a direct-sequence spread spectrum (DS-SS) receiver using maximum ratio com-biner (MRC), in the presence of code synchronization errors, root raised cosine (RRC) pulse shaping, and Rayleigh fading channels with correlated paths. Theoretical BEP curves with QPSK modulation are validated by link level simulations for different downlink Rayleigh fading channels. The impact of the delay estimation errors on the performance of Rake receiver is analyzed for both closely-spaced and distant channel taps, and for different path correlation coefficients.

Full text

Bit error probability of Rake receiver in the presence
of code synchronization errors and correlated Rayleigh
fading channels
Elena-Simona Lohan and Markku Renfors
Institute of Communications Engineering, Tampere University of Technology
P.O.Box 553, FIN-33101, Tampere, Finland; e-mails: {elena-simona.lohan, markku.renfors}@tut.fi
ABSTRACT
We derive here a simple and accurate theoretical method for
computing the bit-error-probability (BEP) of a direct-sequence
spread spectrum (DS-SS) receiver using maximum ratio com-
biner (MRC), in the presence of code synchronization errors,
root raised cosine (RRC) pulse shaping, and Rayleigh fading
channels with correlated paths. Theoretical BEP curves with
QPSK modulation are validated by link level simulations for
different downlink Rayleigh fading channels. The impact of the
delay estimation errors on the performance of Rake receiver is
analyzed for both closely-spaced and distant channel taps, and
for different path correlation coefficients.
I. I NT ROD UC TI ON
Rake receivers are one of the most common receiver types
used nowadays in direct sequence spread spectrum systems.
Errors in the code synchronization (or delay estimation) process
affect significantly the performance of the Rake receiver [1]-[4].
The focus of this paper is on the theoretical modeling of
the bit-error-probability (BEP) degradation due to the code
synchronization errors in the presence of correlated Rayleigh
fading channels, root raised cosine (RRC) pulse shaping, and
BPSK or QPSK modulation. Basic understanding over the
impact of code synchronization errors on the performance of
Rake receivers has been formerly developed in [1]-[6]. However,
fast algorithms to compute BEP of Rake receivers in the
presence of delay estimation errors and Rayleigh multipath
channels, with arbitrarily spaced channel taps and correlated
fading are still hard to be found in the existing literature. BEP
of a Rake receiver can be derived based on the probability
distribution function (pdf) of the signal-to-noise ratio (SNR)
at the output of the maximum ratio combiner (MRC) [2], [3],
[7], or, equivalently, based on the characteristic function (CF)
of SNR at the output of the combiner [8] or on the CF of the
quadratic receiver [9], [10], [11]. For Rayleigh fading channels
and in the absence of code synchronization errors, the SNR
at the output of MRC can be modeled as a quadratic form in
complex Gaussian variables [7], [13], [14]. If both fading and
code synchronization errors are present, the statistics of SNR are
changed compared to the case with no delay errors, and closed
form formulas of its pdf are not easy to be derived. The solution
usually adopted to deal with multipath fading channels and
code synchronization errors is to assume flat fading channels,
whose pdf over the observation interval is approximatively
constant [2], [3]. Alternatively, characteristic function-based
methods can be employed, but they have a large computational
complexity, especially when channel imperfections, such as
code synchronization errors, are taken into account [11].
Here, we introduce a simple and accurate method of com-
puting BEP, based on the extended pdf of SNR at the output
of the combiner in the presence of frequency selective Rayleigh
fading channels and code synchronization errors. The focus here
is on the correlated channels. The detailed theoretical framework
for Rayleigh and Rician uncorrelated fading channels has been
developed by the authors in [12]. Although the starting point
is based on the traditional expression of the instantaneous SNR
[7], [14], the consequent derivations are structured into a novel
framework, taking into consideration the errors in the delay
estimates, the pulse shaping function and the possible correla-
tions between MRC branches. The SNR with delay estimation
errors can no longer be seen as a quadratic form in complex
Gaussian variables as in [7], [13], [14], but rather as a ratio
of quadratic forms, which are dependent on the pulse shaping
waveform and the multipath delay spacings. Each quadratic
form characterizes the desired user signal, the additive white
Gaussian noise interference and the multiple access interference,
respectively. Based on this new SNR model, BEP is derived in a
semi-analytical fashion. Our model is generic in the sense that
it characterizes the BEP deterioration due to delay errors for
arbitrary channel profiles, it can take into account the correlation
between MRC branches, and it can be applied to RRC band-
limited pulse shapes as well as to any other general pulse shape.
For conformity with the WCDMA standard, the derivation is
given for BPSK and QPSK modulations, but the extension to
higher order PSK modulations is quite straightforward.
Section II deals with the theoretical modeling of SNR and
BEP. Section III shows the comparison between the theoretical
BEP and the bit error rates obtained via simulation, using
COSSAP environment. In section IV, the theoretical analysis is
applied in the context of two practical scenarios. Two important
questions related to the MRC diversity in the presence of
code synchronization errors are answered here. The first one
is how much diversity gain we can achieve with Rake receiver
when closely-spaced paths and code synchronization errors are

present. The second one is about the performance degradation
of the Rake receiver, when the Rake branches are affected by
the correlations between the fading paths and imperfect delay
estimation. Section V presents the conclusions.
II . SE MI -A NALYTI CA L AP PROAC H FO R BEP
CO MP UTATIO N
If a BPSK or QPSK signal is spread with a DS-SS code and
transmitted over a multipath fading channel, the signal seen at
the receiver can be modeled as
r(t) =
Nu
X
v=1 pEbv
∞
X
m=−∞
b(m)
v
SFv
X
k=1
c(m)
k,v
L
X
l=1
α(m)
l,v
g(t−mTv−kTc−τ(m)
l,v ) + w(t).(1)
Here, Ebvis the bit energy of v-th user (we assume that all
the bits of a certain user have the same energy), b(m)
vis the m-th
transmitted data symbol of the v-th user. α(m)
l,v is the complex
fading coefficient of the l-th path of v-th user, corresponding
to the m-th symbol and τ(m)
l,v is the delay introduced by the
l-th multipath component of the v-th user, corresponding to the
m-th symbol, c(m)
k,v is the complex code value for k-th chip of
v-th user, during the m-th symbol, Tvis the symbol interval of
v-th user, Tcis the chip interval, g(·)is the chip pulse shape
after the matched filtering, SFvis the spreading factor of v-th
user, Lis the number of channel paths, and w(·)is a complex
additive white Gaussian noise of double-sided power spectral
density N0. The code chips are normalized in such a way that
PSFv
k=1 |c(m)
k,v |2= 1,∀v, m.
The despread signal of u-th user, at the output of l1-th finger
during the n-th symbol can be written as:
z(n)
l1,u =ZnTu
(n−1)Tu
r(t)
SFu
X
k1=1 c(n)
k1,u∗g1(t−nTu−k1Tc−ˆτ(n)
l1,u)dt,
(2)
where ˆτ(n)
l1,u is the estimated delay of l1-th path of u-th user,
during n-th symbol, and g1(·)is the pulse shape of the replica
code at the receiver (for shaped replica codes, g1≡g, and for
unshaped replica codes, g1(·)is the rectangular pulse shape).
The output of the MRC for n-th symbol is
y(n)
u=
ˆ
L
X
l1=1 ˆα(n)
l1,u∗z(n)
l1,u,(3)
where ˆα(n)∗
l1,u is the conjugate of the estimated complex coeffi-
cient of tap l1during the n-th symbol and L0is the number
of Rake fingers used in the MRC. In what follows we assume
perfect channel coefficient estimates (ˆαn
l1,u=αn
l1,u), and we are
interested only in the impact of synchronization errors. We also
assume that the number of Rake fingers is equal to the number
of paths L=L0.
It follows from (1), (2) and (3) that
y(n)
u=pEbub(n)
u
SFu
X
k=1
S(n,n)
k,k,u,u +pEbub(n)
u
SFu
X
k=1
SFu
X
k1=1
k16=k
S(n,n)
k,k1,u,u
+pEbu
∞
X
m=−∞
m6=n
b(m)
u
SFu
X
k=1
SFu
X
k1=1
S(m,n)
k,k1,u,u
+
Nu
X
v=1
v6=u
pEbv
∞
X
m=−∞
b(m)
v
SFv
X
k=1
SFu
X
k1=1
S(m,n)
k,k1,v,u +
SFu
X
k=1
N(n)
k,u ,
(4)
where the notations S(m,n)
k,k1,v,u and N(n)
k,u stand for:
S(m,n)
k,k1,v,u =c(m)
k,v c(n)
k1,u∗L
X
l=1
L
X
l1=1
α(m)
l,v α(n)
l1,u∗
RnTu−mTv+ (k1−k)Tc+ ˆτ(n)
l1,u −τ(m)
l,v ,(5)
N(n)
k,u =c(n)
k,u
L
X
l=1 α(n)
l,u ∗ZnTu
(n−1)Tu
w(t)
g1(t−nTu−kTc−ˆτ(n)
l,u )dt, (6)
and R(·)is the normalized correlation function between the
received signal pulse shape and the reference code pulse shape:
R(τ) = RnTu
(n−1)Tug(t−nTu)g1(t−nTu−τ)dt (normalization
means that R(0) = 1).
The first term in the summation of (4) represents the signal
component, the second one is due to inter-chip interference
(ICI), the third one is due to inter-symbol interference (IS I),
the fourth one is due to multi-access interference (MAI) and
the last one is due to the additive noise.
The SNR at the output of the MRC, conditional on the fading
coefficient vector ξ(n)= (α(n)
1,u,...,α(n)
1,L)Tis defined as [7]
γ(n)=

Ey(n)
u|ξ(n)


2
var(y(n)
u|ξ(n)).(7)
Here, E(·)and var(·)are the mean and the variance operators,
respectively, conditioned by the fading coefficients, and Tstands
for transpose operator.
From equations (4) to (7), and assuming ideal code properties
(i.e., the ICI and ISI components can be neglected), and using
the fact that RnTu
(n−1)Tu|w(t+τ)|2dt =N0δ(τ), it follows that


E(y(n)
u|ξ(n))


2≈Eb

ξ(n)HRξ(n)


2,(8)
and
var(y(n)
u|ξ(n))≈N0(ξ(n))HRcξ(n)
+(Nu−1)Eb
SFu

ξ(n)HRξ(n)


2,(9)
where Rand Rcare L×Lpulse shaping matrices of u-th user
(we dropped the user subscript for convenience), including the

imperfect code synchronization errors, with elements Rl,l1=
R(ˆτl1−τl), and Rcl,l1=Rc(ˆτl1−ˆτl),l= 1,...,L,l1=
1,...,L . Here, Rc(·)is the autocorrelation function (ACF) of
the received signal pulse shape g(·).
Using eq. (7), (8) and (9), we obtain the SNR in the presence
of delay errors as follows:
γ(n)=Eb
N0


ξ(n)HRξ(n)

2
ξ(n)HRcξ(n)+Eb
N0
Nu−1
SFu
ξ(n)HRξ(n)

2
,
(10)
We notice that eq. (10) is valid for any type of pulse shaping
functions, and for any spacing between paths. We also notice
that γ(n)of (10) is well defined, since matrix Rcis always
distinct from an all-zeros matrix (diagonal elements are always
1). The only restriction that we imposed was related to the code
properties: we assumed ideal code correlation properties, an
assumption which is usually approached when the spreading
factor is sufficiently high.
Eq. (10) is fundamentally different from the traditional SNR
without code synchronization errors [7], [13], [14]. The imper-
fections in the delay estimation process are reflected in the terms
at the denominator and in the particular structure of the matrix
Rat the nominator.
Until now, we have not imposed any conditions on the fading
path coefficients ξ(n). If we assume Rayleigh fading channels,
the vector ξ(n)is complex Gaussian-distributed vector of covari-
ance matrix Σξξ , with elements Σξξ (l, l) = Pl, l = 1,...,L,
where Plis the average tap power of path l, and Σξξ(l, l1) =
ρg(l, l1),if l6=l1, where ρg(l, l1)is the correlation coefficient
between the complex Gaussian coefficients corresponding to
paths land l1.
The BEP of BPSK and QPSK modulations, conditional to
fading process is [7] Pe(γ(n)|ξ(n)) = Q(p2γ(n)),Q(·)being
the Gaussian Q-function. Hence, the overall BEP is
Pe=Z∞
0
Q(p2γ(n))∗pγ(γ(n))dγ(n),(11)
where pγ(γ(n))is the pdf of γ(n). With the above considerations
in mind, we can use the following semi-analytical approach for
computing BEP:
Step 1. Generate Nobs vectors of Lcomplex corre-
lated Gaussian distributed variables of zero means and
covariance matrix Σξξ. These vectors are denoted by
ξ(n),n= 1,...,Nobs, where Nobs is the number of
realizations.
Step 2. Build the decision variable γ(n), given by eq.
(10).
Step 3. Using the histogram method, compute an
estimate of pdf of γ(n):∆pγ(γ(n)).
Step 4. Compute BEP as
Pe=
Nobs
X
n=1
Qp2γ(n)∆pγ(γ(n)).(12)
The procedure used here to generate correlated com-
plex Gaussian variables is that one proposed in [15].
The relationship between the Gaussian correlation
coefficients ρg(l, l1)and the envelope correlation co-
efficients ρe(l, l1)is also given in [15] (usually, the
envelope correlation coefficients are employed as mea-
sures of correlation between paths).
0 2 4 6 8 10 12 14 16 18 20
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
SNR at the output of MRC [linear scale]
Estimated PDF of instantaneous SNR
Estimated PDF of instant. SNR, with delay errors, Eb/N0=10 dB, 1 user, ρe=0.1
Path delays τ=[1.25, 2.25] Tc
Average path powers P=[0, 0] dB
No delay errors
Delay errors [0.25, −0.25] Tc
Delay errors [0.5, −0.5] Tc
Delay errors [1, −1] Tc
Fig. 1. Estimated pdf of γ(n)for a two path Rayleigh fading channel and
different delay estimation errors, envelope correlation coefficient ρe= 0.1.
As an example, Fig. 1 shows the estimated pdf of SNR at
the output of MRC, in the absence and in the presence of code
synchronization errors, for a 2path correlated fading channel
and RRC pulse shaping (both for received signal and for the
replica code at the receiver).
Clearly, as the synchronization errors approach to 1chip, the
SNR decreases towards 0, and its pdf tends to a Dirac pulse
δ(γ). At the limit, for code synchronization errors greater or
equal to Tc, BEP tends to R∞
0Q(√2γ)δ(γ)dγ =Q(0) = 0.5,
as expected.
II I. B EP I N TH E PR ES EN CE O F UN CO RR EL ATED
FADI NG
The theoretical BEP was validated via the simulation re-
sults obtained with COSSAP simulator. The COSSAP model
was based on WCDMA specifications [16] for synchronous
downlink multiuser systems. The COSSAP simulations were
carried out for a chip rate of 3.84 Mcps and raw bit rates
of 60 kbps (same bit rates for all users). The data symbols
were spread with the same Walsh code of SF= 128 and
scrambled with a 10 ms segment of a Gold code [16]. The
pulse shaping filter was RRC filter with rolloff factor 0.22 and
the oversampling factor was Ns= 8. We used shaped replica
codes at the receiver. The mobile velocity was 120 km/h. The
complex channel coefficients were assumed to be known at the
receiver, but we incorporated the code synchronization errors
in our simulations. The COSSAP channel was a frequency
selective Rayleigh fading channel with uncorrelated paths. The
bit-error-ratio (BER) curves from COSSAP model are computed
for 200 000 data bits, while the theoretical approach needs
only Nobs = 10 000 realizations, which makes it very fast
from the computational point of view (increasing Nobs showed
no difference in theoretical BEP). The theoretical BEP was
computed with eq. (12) and (10). The results are shown in

Figures 2 and 3, for 1and 16 users, respectively, in the presence
of different code synchronization errors. Because the pseudo-
−5 0 5 10 15
10−5
10−4
10−3
10−2
10−1
100 4 distant−spaced Rayleigh paths, 1 user
Eb/N0
BER
del. errors=[−0.125 0.5 0.75 1]Tc, sim
del. errors=[−0.125 0.5 0.75 1]Tc, th
del. errors=[−0.125 0.5 0.5 0.5]Tc, sim
del. errors=[−0.125 0.5 0.5 0.5]Tc, th
del. errors=[−0.125 0.125 0.125 0.125]Tc, sim
del. errors=[−0.125 0.125 0.125 0.125]Tc, th
Fig. 2. Comparison between the theoretical BEP (th) and COSSAP simulation
results (sim) for 4 path-fading channel and delay estimation errors, channel
paths delays τ= [1.25,2.25,4.5,6.625] Tc, average tap powers (before
normalization) P= [0,−3,−6,−9] dB, 1user, uncorrelated fading.
random code properties for SF= 128 are quite close to
ideal, we notice a very good match between the simulated and
theoretical results. At high bit-energy to noise ratios (Eb/N0),
the theoretical results tend to be slightly optimistic, because
ICI and ISI was neglected in our analysis. Of course, decreasing
the spreading factor, the gap between theoretical and simulation
results will increase, especially at high Eb/N0, when ICI and
ISI become more significant. However, the theoretical results
will still give a close lower bound on the system performance
also for low spreading factors.
−5 0 5 10 15
10−1
100 3 closely−spaced Rayleigh paths, 16 users
Eb/N0 [dB]
BER
delay errors eτ=[−0.5, −0.5, 0.25] Tc, th
[delay errors eτ=−0.5, −0.5, 0.25] Tc, sim
delay errors eτ=[−0.5, 0, 0] Tc, th
delay errors eτ=[−0.5, 0, 0] Tc, sim
no delay errors, sim
no delay errors, th
Fig. 3. Comparison between the theoretical BEP (th) and COSSAP simulation
results (sim) for 3 path-fading channel and delay estimation errors, channel paths
delays τ= [1.25,2.00,2.5] Tc, average tap powers (before normalization)
P= [0,−1,−2] dB, 16 users, uncorrelated fading.
IV. BEP I N TH E PR ES EN CE O F CO RR EL ATED
FADI NG
The comparison between semi-analytical and link-level sim-
ulations showed a good agreement between the two. Therefore,
in this section, we will rely on the theoretical results. One
important question when dealing with Rake receivers is how
much diversity can be gained if closely-spaced paths are used
in the combiner. Usually, it is assumed that the paths are at least
one chip apart [2], [3], [7]. However, some diversity might be
achieved even if the paths are closely-spaced, and our theoretical
model allows us to study this effect. For paths spaced less
than one chip apart, there is an inherent correlation due to the
pulse shaping ACF. Also, closely-spaced paths are more likely
to exhibit correlated fading. The BEP curves as functions of
the spacing between taps is illustrated in Fig. 4 for a 2path
Rayleigh fading channel. The upper plot of Fig. 4 shows BEP
in the presence of 0.25 Tccode synchronization error on the
first path, for different envelope correlation coefficients. The
lower plot of Fig. 4 compares BEP in the absence and in the
presence of a delay error on the first path. When the correlation
between paths comes only from the ACF of the pulse shaping
(i.e., envelope correlation coefficient ρe= 0), the paths spaced
at more than half chip achieve almost the same performance as
the paths which are more than one chip apart. This means, that
the correlation due to RRC pulse shaping is significant only up
to 0.5Tcspacing between taps. However, the degradation due to
envelope correlation is more significant: even a factor ρeof 0.05
can deteriorate BEP quite much. When the envelope correlation
factor tends to 1, the BEP tends to single path performance (no
diversity).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
10−3
10−2
10−1
2 Rayleigh paths, 0.25 Tc delay error on the first path, Eb/N0=10 dB, 1 user
Spacing between taps [chips]
BEP
Single path BEP (no diversity)
Envelope Correlation =0.8
Envelope Correlation =0.5
Envelope Correlation =0.2
Envelope Correlation =0.1
Envelope Correlation =0.05
Envelope Correlation =0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
10−3
10−2
10−1
Spacing between taps [chips]
BEP
Env. Corr. = 0.05, Del. err. = 0.5
Env. Corr. = 0, Del. err. = 0.5
Env. Corr. = 0.05, Del. err. =0.25
Env. Corr.=0.05, Del. err. = 0
Env. Corr. = 0, Del. err. = 0.25
Env. Corr. = 0, Del. err. = 0
Fig. 4. BEP for 2-path Rayleigh fading channel with correlated fading, different
(constant) envelope correlation coefficients, and different code synchronization
errors on the first path, equal average tap powers, single user.
Another question related to MRC performance is how fast
BEP deteriorates in the presence of code synchronization errors

and correlated MRC branches. Fig. 5 illustrates BEP curves for
single and multi-user cases, for different delay estimation errors,
fading channels with 2and 6taps, and exponential correlation
model [13] (i.e., the envelope correlation coefficient between
the l-th and l1-th tap is ρ|l−l1|
e).
We notice that for more than 0.25 Tcdelay error, the BEP de-
teriorates drastically. The slope of BEP degradation at low delay
errors (less than 0.25 Tc) is smaller for high correlation factors
ρethan for low ρe, but the overall performance for strongly
correlated paths is very poor, approaching the single-path case.
For single user case, the diversity is almost completely lost at
high ρe(there is no much diversity gain when we increase the
number of paths from 2to 6when ρe≥0.4). For 16 user-case
and strongly correlated fading (ρe≥0.4), the performance with
6correlated paths becomes poorer than the performance with
2correlated paths. This apparent contradictory behaviour might
be explained by the fact that the Rake receiver cannot cope well
with multiple users and correlated fading, and its output SNR
decreases when the number of strongly correlated paths of the
interfering users increases.
V. CO NC LU SI ON S
In this paper, we presented a simple and fast quasi-analytical
method for computing BEP for Rake receivers in the presence
of code synchronization errors and Rayleigh fading channels.
The verification of BEP formulas was done by comparison with
COSSAP simulation results for uncorrelated fading Rayleigh
channels. The analysis was then extended to the correlated
fading channels, and two practical scenarios with code syn-
chronization errors, RRC pulse shaping and correlated fading
were discussed. Further research will focus on the modeling
of imperfect code correlation properties on the SNR at the
output of the Rake combiner and on the impact of BER of both
imperfect delay estimation and imperfect channel coefficient
estimation.
ACK NOW LE DG ME NT S
This work was carried out in the project ”Digital and Analog
Techniques in Flexible Receivers” funded by the National Tech-
nology Agency of Finland. This work is also partly supported
by Nokia Foundation and by the Graduate School in Electron-
ics, Telecommunications, and Automation. The authors express
special thanks to Professor M. Juntti for fruitful discussions and
helpful suggestions related to WCDMA receivers and channel
estimation issues.
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[4] J. Baltersee, G. Fock, V. Simon, and H. Meyr, “Performance bounds for a
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10−4
10−3
10−2
10−1
100
Exponential correlation model [ρe ρe
2 ... ρe
L−1], Eb/N0=10 dB, 1 user
Delay error in chips (all paths have the same delay error)
BEP
ρe=0.8, L=2
ρe=0.8, L=6
ρe=0.4, L=2
ρe=0.4, L=6
ρe=0.02, L=2
ρe=0, L=2
ρe=0.02, L=6
ρe=0, L=6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Exponential correlation model [ρe ρe
2 ... ρe
L−1], Eb/N0=10 dB, 16 users
Delay error in chips (all paths have the same delay error)
BEP
ρe=0.8, L=6
ρe=0.4, L=6
ρe=0.8, L=2
ρe=0.4, L=2
ρe=0.02, L=2
ρe=0.02, L=6
ρe=0, L=2
ρe=0, L=6
Fig. 5. BEP for L-path Rayleigh fading channel with correlated fading
(L= 2 and L= 6), exponential correlation model, equal average tap powers,
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