On the Performance of Blind Chip Rate Estimation in Multi-Rate CDMA Transmissions Using Multi-Rate Sampling in Slow Flat Fading Channels

doi:10.4236/wsn.2009.12011

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Wireless Sensor Network, 2009, 2, 61-121

doi:10.4236/wsn.2009.12011 lished Online July 2009 (http://www.SciRP.org/journal/wsn/).

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On the Performance of Blind Chip Rate Estimation in

Multi-Rate CDMA Transmissions Using Multi-Rate

Sampling in Slow Flat Fading Channels

Siavash GHAVAMI, Bahman ABOLHASSANI

School of Electrical Engineerin g, Iran University of Science and Technology, Tehran, Iran

Email: sghavami@ee.iust.ac.ir, abolhassani@iust.ac.ir

Received March 7, 2009; revised March 19, 2009; accepted March 20, 2009

Abstract

This paper considers blind chip rate estimation of DS-SS signals in multi-rate and multi-user DS-CDMA

systems over channels having slow flat Rayleigh fading plus additive white Gaussian noise. Channel impulse

response is estimated by a subspace method, and then the chip rate of each signal is estimated using zero

crossing of estim ated differential channel i mpulse response. For chip rate estim ation of each user, an algorithm

which uses weighted zero-crossing ratio is propos ed. Maximum value of the weighted zero cros sing ratio takes

place in the Nyquis t rate sam pling frequen cy, which equ als to the twice of the chip rate. Furth ermor e, bit time

of each user is estimated using fluctuations of autocorrelation estimators. Since code length of each user can

be obtained using bit time and chip time ratio. Fading channels reduce reliability factor of the proposed algo-

rithm. To overcome this problem, a receiver with multiple antennas is proposed, and the reliability factor of

the proposed algorithm is analyzed over both spatially correlated and independent fading channels.

Keywords: Multi-Rate Sampling, Chip Time, Multi-Rate CDMA, Blind Estimation

1. Introduction

Direct-sequence code division multiple accesses (DS-

CDMA) systems are nowadays of increasing importance

in wireless cellular communications because of their in-

clusion in most of the proposals about both terrestrial and

satellite based standards for third-generation (3G) wire-

less networks [1,2]. On the other hand, direct sequence

spread spectrum (DS-SS) signals are well-known and are

used in secure communication for their low probability

of interception, their statistics are similar to those of

noise; furthermore, they are usually transmitted below

the noise level.

One of the salient features of 3G cellular systems is

the capability of supporting transmission data as diverse

as voice, packet data, low-resolution video, and com-

pressed audio. Since these heterogeneous services pro-

duce digital information streams with differen t da ta rates,

their implementation requires the use of multi-rate

CDMA systems where each user may transmit his data at

one among a set of available data rates. An easy way to

view the multi-rate CDMA transmission is to consider

the variable spreading length (VSL) technique where all

users employ sequences with the same chip period;

moreover, the data rate is tied to the length of the

spreading code of each user. Another way to view a

multi-rate CDMA transmission is to consider a constant

spreading length where users employ sequences with

different chip periods. In general, chip time and spread-

ing sequence l e n gth of users can be sel ec t ed variable.

Two recent systems are often applied in military sys-

tems based on spread spectrum. In the literature, differ-

ent methods have been presented for chip time estima-

tion, many number of those methods are based on cyclic

cumulant method, which have been presented in litera-

S. GHAVAMI ET AL.69



tures [3-7]. Cyclic cumulant method in multi-rate and

multi-user system doesn’t exhibit good performance be-

cause cyclic frequencies of different users overlap and

chip time estimation of different users is difficult. In [8]

we proposed a blind chip time estimation algorithm

based on multi-rate signal processing.

In this paper, blind chip time estimation technique is

considered, which is based on channel impulse response

estimation using singular value decomposition of esti-

mated received signal covariance matrix. For spreading

sequence estimation in slow flat Rayleigh fading channel

plus additive white Gaussian noise (AWGN), length of

code must be determined. Bit time and chip time must be

estimated, for length of code estimation. In the proposed

method in this paper, chip time is determined using

multi-rate sampling of channel impulse response. Maxi-

mum number of zero crossing in differential of channel

impulse response take places in sampling frequency

twice of the chip rate. Therefore, number of zero cross-

ing is obtained as a function of sampling rate. Then bit

time is determined using fluctuations of correlation esti-

mator. Code length of each user is determined using their

corresponding chip times and bit times. Finally, per-

formance of the proposed method is analyzed in fading

channels and a receiver with multiple antennas is pro-

posed for performance improvement in fading channels.

Estimated parameter using this method is useful and ap-

plicable in noted systems in pervious paragraph in very

low signal to noise ratio (SNR) (negative SNR in dB).

This needs no prior knowledge about transmitter in the

receiver side; it is typically the case in blind signals in-

terception in the military field or in spectrum surveil-

lance.

The remaining of this paper is organized as follows. In

Section 2, System model are introduced. In Section 3,

subspace methods for channel impulse response estima-

tion are reviewed. Section 4 proposed blind chip time

estimation based on multi-rate sampling. Blind bit time

estimation is considered in Section 5. In Section 6, the

performance of the proposed chip rate estimation algo-

rithm will be analyzed over spatially correlated and non

correlated flat fading channels. Simulation results are

expressed in Section 7. Finally conclusions are per-

formed in Section 8.

2. System Model

We consider the down link scenario of a multi-rate

DS-CDMA network. The base station transmits signals

and a mobile station receives a combined signal of K

active users, which is given by

()[ ]()(),

kk kis ki

ikj

rtAd jhtjTnt









 (1)

where k

, , , []

dj ()

ht i

T and ,ki



are the receiv-

ed amplitude, jth data symbol, convolution of channel

impulse response and spreading sequence waveform,

symbol time and delay of kth user in ith rate respectively.

Also, Ki is number of active users in ith rate, S is the

number of available rates, and is additive white

Gaussian noise. In the down link scenario,

(nt)

,ki



s are

equal, and the is expressed by

,ki

h()t

1/2

()( )

kiiki ic

ht LcptjT





 (2)

where is the spreading sequence length,



is the chip time of ith rate, is explained in

the next paragraph, is the value of the mth chip

and ith rate with

()

,[]

,ki

cm[] 1.



Data symbols





[]

different users have independent identically distributions

(i.i.d.). In this paper, channel model is considered slow

flat Rayleigh fading plus AWGN. We assume that the

channel in the duration of one processing window re-

mains approximately constant.

In (2), is the total channel impulse response of

the kth user in ith rate, it is convolution of the transmitter

filter , channel filter

()

)(et ()

tand receiver filter()

i.e.

()()()* ()

ptetstgt



 (3)

where these filters have unit energy. ()

()()k

st te







()t



, k



s have Rayleigh distribution and k



s have

uniform distribution, * operation denotes convolution

operation. We assume that

1) Channel model is slow flat Rayleigh fading plus

AWGN.

2) Signal power is lower than noise power (SNR < 0 in

dB).

3) A symbol time is equal to the spreading sequence

length, i.e. ii

TTL



(i.e. short spreading code).

4) Symbols have zero mean and are uncorrelated.

3. Estimation of Channel Impulse Response

Covariance matrix of the received signal can be written as









HHH

EEE RrrxxnnR

R (4)

where r is vector of received signal, x is vector of re-

ceived signal when noise is absent , H denotes hermitian

operation, and n is vector of additive white Gaussian

S. GHAVAMI ET AL.

noise. Furthermore,

R andn show covariance ma-

trix of the received signal when no ise is absent and noise

covariance matrix, respectively. Covariance matrix of the

received signal can be written as [9]





200*11

0(1)()( )

nkkkkkk k

 







vvv vI



 (5)

where, 2



is noise variance, kk





,



ksig











, in which is power of signal of kth

user in ith rate. and are normalized eigenvector

of the estimated covariance matrix of received signal,

they correspond to kth asynchronous user in ith rate,

is the sampling period, and is the identity matrix.



sig





In the downlink scenario, when signal of one user are

synchronized all user are synchronized (k



s =0) and

covariance matrix of the received signal is given by



{

nkkk







vv I}

(6)

4. Blind Chip Rate Estimation

In singular value decomposition of the received signal

covariance matrix, if the chip rate is less than half of the

sampling frequency, estimated eigenvector of the re-

ceived signal covariance matrix (corresponds to maxi-

mum eigenvalue) is the convolution of ,

()et ()

()

t and spreading sequence of synchronized user. If

the impulse response of transmitter and receiver filters

have been assumed ideal, and channel is constant in the

duration of one processing window, estimated eigenvec-

tor will be the multiplication of spreading sequence of

synchronized user in constant value. Hence for estimat-

ing spreading sequence from estimated impulse response

of channel by subspace method i.e., ch ip rate must be

known exactly. k

Sampling ratio is defined as

 (7)

The total impulse response of the channel related to

each user, (2), can be written as follows

()( ).

mL m

ki ki

ht cptj







 (8)

If the sampling frequency is known, by estimation

using eigenvector of synchronized signal and

relation between sampling time and chip time (i.e.

,()

TTm



), we can obtain chip rate of the signal. Hence

we focus on the estimation of sampling ratio (SR).

First, we calculate sign of to obtain the nor-

malized spreading sequence, it is performed due to bi-

nary phase shift keying (BPSK) spreading assumption of

transmit signal. Sign of is given by

,()

sgn(( ))((1))()

mL m

ki ki

htcutjutj



















 (9)

where is the unit step function. Then, the differen-

tial of is calculated with respect to t to es-

timate the number of zero crossings of the spreading se-

quence, which is given by

()ut

sgn( ,( ))

,1/2 c

sgn(( ))[]()

mL m

ki ki

dht T

Ncmtj

dt m









 (10)

sgn(())

dhtdt

includes impulse sequences by zero

crossings at times c

tjTm



. It is obvious that in sam-

pling frequencies greater than the Nyquist rate, by in-

creasing the sampling frequency, number of zero cross-

ings on differential of spreading sequence remains con-

stant. Figure 1(a) shows a sequence with length of 63,

and Figure 1(b) shows the same sequence with the SR of

4. Figure 1(c) shows the differential of the sequence

shown in Figure 1(a) and Figure 1(d) is similar to Figu re

1(c) for differential of sequence which has been shown in

Figure 1(b). It can be seen in this figure, the number of

zero crossings remains constant in differential of each

sequence by over sampling greater than the Nyquist rate.

Figure 2(a) shows the number of zero crossings of the

differential of sequence in terms of different sampling

frequencies. Bandwidth of signal is considered 10 MHz

and it can be seen the number of zero crossings remains

constant for sampling frequency greater than 20 MHz.

Zero crossing ratio is defined as the following

Number of zero crossings

Length of diffre ntiate d Sequence

ZCR (11)

Figure 2(b) shows the number of zero crossings ratio

of the differential sequence in terms of sampling fre-

quency. It is obvious that for sampling frequencies

greater than the Nyquist rate,

CR reduces by increas-

ing the sampling frequency and the maximum value of

ZCR occurs at the sampling frequency equal to the chip

rate of the sequence (i.e. length of 63). But for sampling

frequencies less than the Nyquist rate, for loss of some

data samples and decreasing length of differential

S. GHAVAMI ET AL.71

Figure 1. (a) Spreading sequence (length of 63). (b) Over

sampled spreading sequence (length of 63) and SR = 4. (c)

Differential of Spreading sequence (length of 63). (d) Dif-

ferential of Over sampled spreading sequence (length of 63)

and SR = 4.

sequence, ZCR may be greater than ZCR corresponding

to the Nyquist rate. In Figure 2(b) an example of this

situation can be seen. Sampling frequencies less than

Nyquist rate losses some of data samples and decrease

length of differential of sequence, it causes ZCR in-

creases for sampling frequency lower than the Nyquist

rate. Since, by weighting ZCR, it is possible to reduce

ZCR for sampling frequencies less than the Nyquist rate.

Therefore, we propose weighted zero crossing ratio,

which is defined as

The Number of zero crossingsWZCR ZCR (12)

Figure 2(c) shows weighted zero crossing ratio in

terms of frequency sampling, it can be seen the maxi-

mum of WZCR corresponds to the sampling frequency is

equal to the chip rate.

Figure 2. (a) Number of Zero crossing in terms of sampling

frequency in MHz. (b) Zero crossing ratio in terms of sam-

pling frequency. (c) Weighted zero crossing ratio in terms

of sampling frequency.

Increasing the code length increases the computational

complexity of subspace decomposition of the received

signal covariance matrix due to increasing the dimension

of covariance matrix, instead, output SNR in output of

covariance matrix estimator increases by the factor of

NN, as,

out

SNR N

NSNR  (13)

where and are spreading sequence lengths

corresponds to and , respectively.

Therefore by increasing code length it is possible to es-

timate spreading sequence of active users in lower SNR

values.

SNR 2

out 1

out

SNR

5. Blind Bit Rate Estimation

For blind bit rate estimation, we use fluctuation of corre-

lation estimator [10]. To compute the fluctuations, we

divide the received signal into M temporal windows with

duration of TF for each window. Then, a correlation es-

timator is applied to each window prior to the computa-

tion. For the mth window, an estimation of the autocorre-

lation for any signal r(t) is given by

ˆ()() ()

rmm

Rrtrt

Tdt





 (14)

where is the signal sample over the mth window.

Replacing in (14) with the mth window of ,

and taking expectation of

()

r()t()rt

ˆ()



for M windows, lead

to the following equation, which can be used for both the

uplink (asynchronous user) and downlink (synchronous)

transmission,

ˆˆˆ

()()()

rsn

RRR







(15)

where ˆ()



is autocorrelation of noise and

() ().









 (16)

In (16), ,

ˆ()



is the noise-unaffected estimation of

the autocorrelation for the (k, i)th signal. Indeed, since

the fluctuations are computed from many ran-

domly-selected windows, th ey do not depend on th e rela-

tive delays of signals. The use of M windows allows us

to estimate the second-order moment of the estimated

correlation ˆ()







ˆˆ ˆ

() ()()

ER R









  (17)

S. GHAVAMI ET AL.

where is the estimated expectation of



E()



. Hence,

()



 is a measure of the fluctuations of ˆ(



. The

difference between the successive and equal amplitude

peaks in (16) determines symbol time for each rate [10].

6. Performance of the Proposed Chip

Rate Estimation Algorithm

In the proposed chip rate estimation algorithm, if the

eavesdropping receiver fallen in the deep fading; the

estimated chip rate is not valid. In this section, effect of

fading channel is analyzed on the performance of the

proposed chip rate estimation algorithm. This analysis is

first done for a receiver using a single antenna. Second,

the analysis is done for a receiver using multiple anten-

nas to overcome fading.

The threshold value of SNR for chip rate estimation

algorithm is defined as , which is the minimum

SNR for achieving a reliability factor (RF) of chip rate

estimation greater than 0.99. is obtained using

computer simulation in non-fading channels and its value

depends di r ec t ly on the sprea d ing factor.

min

SNR

min

SNR

By assuming Rayleigh distribution for the channel, the

probability that the received signal voltage r is less than a

given the threshold R, which is corresponding value of

, found from the following equation

min

SNR



Pr( )1exp()

rR prdr



 

 (18)

where is the probability density function (PDF) of

the Rayleigh distribution , and

()pr

rms



is the value of

the specified level R, normalized to the local rms ampli-

tude of the fading duration envelope, hence

2min

SNR



. The chip rate estimation algorithm is

failed, when in M processing blocks of signal is

less than .

SNR

min

So, the for AWGN channels is ch anged to



1Prexp().

fading

RFRFr RRF



 

(19)

From the above equation, it is obvious that the reli-

ability factor of chip rate estimation algorithm reduces

by the factor of 2

exp( )



 when all M processing win-

dows have be en faded.

Table 1 shows the for achieveing a

greater than 0.99 in terms of spreading factor (SF) over

AWGN channels. These results are extracted using

computer simulations. It can be seen, by 3 dB increase in

SF, the increases by 3 dB.

min

SNR RF

min

SNR

Table 1. min

NR for achieving a

F greater than 0.99

in terms of SF over AWGN channels.

SF [dB] 15 18 21 24 27 30

min

SNR [dB] -5 -8 -11 -14 -17-20

Figure 3. in terms of received SNR for different

values of SF. fading

Figure 3 shows in terms of SNR in flat fading

channels with Rayleigh distribution for different values

of SF. It can be seen that the is greater than 0.95

for SNR = 8 dB in SF = 15 dB. It is obvious that the

performance of the proposed algorithm degraded 13 dB

with respect to that of AWGN channels.

Now, we propose a solution to overcome effect of

fading channels on the performance of the proposed al-

gorithm. If the distance between two neighboring anten-

nas in eavesdropping receiver is large enough and com-

munication environment is rich scattering, fading chan-

nels experienced by these antennas are independent. So,

the reliability factor of the proposed chip rate estimation

algorithm improves. So that







1Pr,

fading

RFRFr R 



(20)

where denotes reliability factor of the proposed

chip rate estimation algorithm with L receiving antennas

in eavesdropping receivers over independent fading

channels. Figure 4 shows in terms of SNR for

spreading factor of 15 dB and different numbers for re-

ceiving antennas in eavesdropping receiver. It can be

seen from this Figure that by increasing the number of

receiving antennas from 1 to 4, increases from

0.53 to 0.95 at SNR = -3 dB.

fading

RF m

fading

In practice, assuming independent channel gains for

receiving antennas is not realistic due to limitation size

S. GHAVAMI ET AL.73

of the eavesdropping receiver. The correlation matrix of

an antenna array depends not only on the array configu-

ration but also on the incident angle of the incoming sig-

nal. Different parameters affect the correlation in differ-

ent ways. Branch correlation depends on the antenna

height and antenna separation through their ratio [11].

For a given antenna array, correlation between signals

received by two neighbouring antennas varies with their

corresponding angle of incidence so that it reaches its

maximum value when the signal comes from the endfire

direction and reduces gradually as the signal direction

moves towards the broadside [12]. The covariance ma-

trix among channel power gains e.g. in

the receiver is denoted by

,1 ,ii









1,1 1,21,

2,1 2,22,

,1 ,2,

LL LL

 



















(21)

corr fading

RF , which is the reliability factor of the pro-

posed chip rate estimation algorithm for correlated fad-

ing channel, is obtained using Bayes’ theorem for corre-

lated fading channels as follows



1Pr

corr fading

RFRFr R















 









(22)

In the above equation, ,ij



s for are less than

one. So



and is less than 1.

Therefore,

ij







,ij





















 (23)

Figure 4. in terms of received SNR for different

number of received antennas and SF = 15 dB .

fading

Figure 5. in terms of received SNR for a 3

antenna array with correlated and independent fading

channels and SF = 15 dB.

corr -fading

Therefore, the power of in equation (22)

is less than L. In comparison to (20), it is obvious that,

is less than .



Pr rR

fading



corr fading

RF RF

Figure 5 shows in terms of received SNR

for a 3 antenna array receiving correlated signals. Similar

to that of [12], we assume triangular array at the eaves-

dropping receiver, located at the height of 30.43 m, and

its correlation matrix is given by

corr fading

RF 

10.727 0.913

0.727 1 0.913

0.9130.9131























It can be seen from Figure 5, that for tri-

angular and antenna arrays is 0.62 at SNR = -3 dB. It is

obvious that decreases by 0.27 compared

with of an antenna array consisting of 3 an-

tennas with independent fading channels.

corr fading

RF 

corr fading

RF 

fading

7. Simulation Results

Computer simulations have been performed using two

streams of spread spectrum signals with both BPSK

spreading and data modulation. Gold sequences with

lengths of 63 (user 1) and 31 (user 2) are considered as

spreading sequences. Their chip rates are 20 Mega

chips/sec and 40 Mega chips/sec, respectively. Process-

ing window size is considered to be 160



, and M =

10 is the number of windows, which is used for averag-

ing. Chip rates of each signal have been obtained using

estimated impulse response of the channel. Bit rate of

each signal has been estimated using the time difference

S. GHAVAMI ET AL.

between the successive equal amplitude peaks in the

output of the correlation estimator.

Figure 6 shows fluctuations of the correlation estima-

tor at SNR of -5 dB. The time differences between two

successive equal amplitude peaks have been estimated

0.775



and 3.175



Figure 7(a) shows the estimated channel impulse re-

sponse for the spreading sequence with length of 63 and

sampling frequency of 20

MHz. This signal has

more SNR relative to the other user and produces the

greatest eigenvalue in the estimated received signal co-

variance matrix; therefore, its first corresponding esti-

mated eigenvector is selected for the chip rate estimation.

A window of estimated channel impulse response as long

as bit time = 3.175



is considered for the chip rate

estimation. Figure 7(b) shows the estimated, which is

given by (9), and exact spreading sequence, it is obvious

they are matched together. Figure 7(c) shows

sgnd

(())

htdt

, it can be seen from this figure, zero crossing

occurs when sign of signal changes.

Figure 8(a) shows that th e distribu tion of zero cro ssing

number for the ,

sgn(( ))

dhtdtin terms of sampling

frequency. Figure 8(b) shows the ZCR of the

(())

htdt, which is defined in (11), in terms of sam-

pling frequency. Figure 8(c) shows WZCR for

sgnd

(())

htdt, which is defined in (12), in terms of sam-

pling frequency. It obvious maximum number of WZCR

corresponds to the sampling frequency equal to Nyquist

rate, which is equivalent to a chip rate 20 Mega chips/sec.

Hence,



c12010 50secTn, code length is ob-

tained as 3.15sec/50sec63

TT n



.

Figure 9(a) shows the probability of chip rate detection

related to first user for 10000 times Monte Carlo test. It

can be seen in this Figure, sampling frequency of 40

MHz is estimated by probability of more than 97.5% as a

Figure 6. Fluctuations of correlation estimator output.

Figure 7. (a) Estimated eigenvector corresponding to the

maximum eigenvalue of the received signal covariance ma-

trix. (b) Exact and estimated spreading sequence of user 1.

Figure 8. (a) Number of Zero crossings in terms of sampling

frequency. (b) Zero crossing ratio in terms of sampling

frequency. (c) weighted zero crossing ratio in terms of sam-

pling frequency, all of them are related to user 1.

Figure 9. (a) Probability of chip rate detection related to the

first user for 10000 times Monte Carlo Test. (b) Probability

of chip rate detection related to the second user for 10000

times Monte Carlo Test.

S. GHAVAMI ET AL.75

sampling frequency with maximum WZCR, which is

equivalent to chip rate of 20 Mega chips/sec. Figure 9(b)

is similar to Figure 9(a) except it is corresponds to sec-

ond user. It can be seen sampling frequency of 80 MHz is

estimated by probability greater than 97% as a sampling

frequency with maximum WZCR, which is equivalent to

a chip rate 40 Mega chips/sec.

8. Conclusions

This paper considered the problem of blind chip rate es-

timation of direct sequence spread spectrum (DS-SS)

signals in multi-rate and multi-user direct-sequence code

division multiple accesses (DS-CDMA). The estimation

is based on the multi-rate sampling of the estimated dif-

ferential channel impulse response. Simulation results

showed that the chip rate and bit rate can be determined

exactly in very low SNR (-5 dB) and in multi-rate and

multi-user field. Therefore, it is possible to blindly esti-

mate the spreading factor of each user. In fading chan-

nels, the reliability factor of the proposed chip rate esti-

mation algorithm is analyzed and a receiver with multi-

ple antennas is proposed to improve the reliability factor

of the proposed algorithm.

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