Blind Estimation of Direct Sequence Spread Spectrum Signals in Multipath

Abstract—Self-recovering receivers for direct-sequence spreadspectrum signals with unknown spreading codes are discussed in this paper. Applications include signal interception, jamming, and low probability of intercept (LPI) communications. A multirate/m

IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.45,NO.5,MAY19971241 Blind Estimation of Direct Sequence

Spread Spectrum Signals in Multipath

Michail K.Tsatsanis,Member,IEEE,and Georgios B.Giannakis,Fellow,IEEE

Abstract—Self-recovering receivers for direct-sequence spread-spectrum signals with unknown spreading codes are discussed in this paper.Applications include signal interception,jamming, and low probability of intercept(LPI)communications.A mul-tirate/multichannel,discrete-time model of the spread spectrum signal is introduced,which establishes links with array processing techniques.Borrowing blind channel estimation ideas,which were originally developed in the context of fractionally spaced equalizers or receivers with multiple antennas,linear solutions are obtained that are independent of the input distribution.The signal interception problem is further studied,and a zero-forcing (ZF)receiver/equalizer is proposed to recover the transmitted data.Its performance is analyzed,and some illustrative simula-tions are presented.

I.I NTRODUCTION

S PREAD-SPECTRUM(SS)signals have been used for secure communications,command,and control for sev-eral decades[17].For certain applications,their antijamming capabilities and low probability of intercept justify the price to be paid in increased bandwidth.In direct-sequence spread-spectrum(DS-SS)systems,the information signal is modu-lated by a pseudo-noise(PN)sequence prior to transmission resulting in a wideband signal resistant to narrowband jam-ming or multipath.

In multiuser CDMA systems,the PN spreading sequence is typically known to the receiver,where it is used to perform the matched?ltering(or“despreading”)operation and recover the transmitted data(e.g.,[14]).In single-user systems,however, there are cases where the receiver may have no knowledge of the transmitter’s PN sequence(e.g.,when intercepting an unfriendly transmission in LPI communications).Then,all the related issues of synchronization,multipath equalization,and data detection become more challenging.

There is an abundance of references in the literature on the problem of synchronization or“code-delay acquisition”under various interference and multipath environments,when the PN sequence is known to the receiver[1]–[3].However,

Manuscript received July5,1995;revised December27,1996.This work was supported by the National Science Foundation under Grant NSF-MIP 9424305.The associate editor coordinating the review of this paper and approving it for publication was Prof.Michael D.Zoltowski.

M.K.Tsatsanis is with the Electrical and Computer Engineering Depart-ment,Stevens Institute of Technology,Hoboken,NJ07030USA(e-mail: mtsatsan@stevens-tech.edu).

G.B.Giannakis is with the Department of Electrical Engineering,Uni-versity of Virginia,Charlottesville,V A22903-2442USA(e-mail:geor-gios@virginia.edu).

Publisher Item Identi?er S1053-587X(97)03343-6.the literature is not equally rich for the case where the PN sequence is unknown.

In[11]and[26],the problem of identifying the code-generating mechanism(i.e.,the primitive generating poly-nomial)from a subset of noiseless or noisy chip sequence observations has been studied.However,these methods as-sume perfect chip synchronization and absence of multipath and data modulation,thereby limiting their applicability. Contrary to these approaches,the combined effects of the unknown spreading code and multipath channel are estimated in this work without assuming a speci?c structure for the code generating mechanism.In particular,novel blind algo-rithms are developed for estimating the time-delay,multipath parameters,and spreading code of the received signal.The multirate nature of the spreading and despreading operations is exploited,and a multirate/multichannel discrete-time model of the DS-SS system is introduced.This model links the spread-spectrum signal estimation problem to recently developed methods for the estimation of multiple FIR channels[13],[18], [21],[28].Multichannel methods obviate the use of higher order statistics and provide subspace-based algorithms with considerably improved performance.

The introduction of multirate/multichannel models for frac-tionally spaced equalizers has facilitated the development of fast blind algorithms for channel equalization applications [21].Multirate models of CDMA systems have also been studied in order to improve interference suppression schemes [23],[24].In this paper,the multirate spread-spectrum model is exploited to derive second-order-statistics-based,linear blind signal estimation methods.The signal parameters may be useful for purposes of intercepting or jamming the received spread-spectrum signal.

The focus of the present paper is on the interception/user acquisition problem,that is,on the recovery of the trans-mitted information data stream.Zero-forcing(ZF)linear re-ceivers/equalizers are proposed based on the estimated signal parameters.Their performance is analyzed and compared with matched?lter-based solutions.Optimal mean-square-error ZF receivers are also derived,and their performance is studied. The rest of the paper is organized as follows.In the next section,the spread-spectrum signal estimation problem is stated in a discrete-time framework,whereas in Section III,the proposed algorithms are delineated,and identi?abil-ity issues are addressed.In Section IV,the receiver design problem is studied,and some linear solutions are proposed. The performance of these methods is evaluated in Section V,

1053–587X/97$10.00©1997IEEE

Abstract—Self-recovering receivers for direct-sequence spreadspectrum signals with unknown spreading codes are discussed in this paper. Applications include signal interception, jamming, and low probability of intercept (LPI) communications. A multirate/m

1242IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.45,NO.5,MAY

1997

Fig.1.Direct-sequence spread-spectrum

modulation.

Fig.2.Direct-sequence spreading operation.

whereas issues related to timing and bit rate estimation are discussed in Section VI.Finally,some illustrative simulation results and conclusions are presented in Sections VII and VIII,respectively.

II.DS/SS S IGNAL M ODEL

In most studies of DS/SS systems,the continuous-time transmitted signal

1is modeled as a product of two waveforms (see Fig.

1)

(1)

where is the information bearing signal before spreading,whereas is the (periodically extended)spreading code waveform that is typically of much larger bandwidth

than

.

Despite its simplicity,this formulation does not reveal the multirate nature of the spreading operation and does not lend itself to the derivation of convenient discrete-time models.Since the spreading procedure is typically performed in discrete time,it will be useful in the sequel to describe it as a multirate convolution.A.Multirate Framework

Let

be a spreading code

chip sequence of

length

chips per

information symbol.Then,the transmitted discrete-time signal at the chip rate

is

and ?ltered

by

.

In some DS/SS systems,a spreading code that is longer than the bit period is used.In this case,consecutive information

1Subscript

c is use

d to denot

e continuous time signals,and index t denotes

continuous time.

symbols modulate different consecutive portions of the spread-ing code in a circular fashion.Equation (2)can be extended to cover this case,as we explain next.

Let be a code sequence of

length .

Let

be a family of sequences de?ned

as

and,

if

mod

,.Then,the transmitted

discrete-time signal can be expressed

as

is transmitted through a possibly dispersive

channel at a chip

rate denote the complex baseband representation of the overall impulse response of the channel,including multipath effects and the transmitter and receiver spectral shaping ?lters.Then,the received signal at the receiver

is

(4)

where

is additive Gauss-ian noise.

Substituting

from (2)to (4),we

obtain

is the combined impulse response of the channel

and the spreading code.The summation limits in (6)come from the nonzero support

of

over

(7)

where

is the sampled impulse response,

and .In addition,from (6),the overall impulse

response

is

Abstract—Self-recovering receivers for direct-sequence spreadspectrum signals with unknown spreading codes are discussed in this paper. Applications include signal interception, jamming, and low probability of intercept (LPI) communications. A multirate/m

TSATSANIS AND GIANNAKIS:BLIND ESTIMATION OF DIRECT SEQUENCE SPREAD SPECTRUM SIGNALS IN MULTIPATH

1243

Fig.3.Multirate model of DS/SS system.

order

(9)

,and the code

length

is a circular complex Gaussian white noise

process,uncorrelated

with

.

Of the above assumptions,AS1)may be dif?cult to meet

in some applications.

If

,which is a tool that is

commonly used in multirate theory(e.g.,[27]).

B.Multichannel Framework

Let vector representation

of

(11)

with

vector.Similarly,let us

de?ne

.It can be shown that the operation of

oversampling

by(the interpolating?lter

in Fig.3)is described in the polyphase domain by[27,ch.

4]

(12)

or in

the

is a scalar quantity

multiplying

is an

FIR vector system since the

original

(14)

where

is the polyphase representation

of

be the polyphase representation of the signal

received at array

element.Then,by stacking

all the polyphase vectors in a supervector,we obtain the

description

..

.

..

.

..

.

Abstract—Self-recovering receivers for direct-sequence spreadspectrum signals with unknown spreading codes are discussed in this paper. Applications include signal interception, jamming, and low probability of intercept (LPI) communications. A multirate/m

1244IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.45,NO.5,MAY1997 advantage is that it does not require the

input

,have no

common roots.

ii)for at least

one

for at least

one

Of the above three conditions,only i)is indeed restrictive;

ii)and iii)are always satis?ed,provided that the vector

channel

order

has

no

Proposition2is easier to interpret in the context of DS/SS

systems.The identi?ability requirement amounts to not

having

,

i.e.,

,equispaced on a circle.

The

sequence has

only zeros,and they are

typically not distributed around a circle(or can be designed

that way).Moreover,since,typically,the order

of is such

that has less

than

,and the condition is

valid for any

code of

length

.Then,using

(12),can be written

as

(16)

where

rows.

Finally,

.From(16),the data correlation

matrix

(provided that the number of

columns is less than the number of

rows is also

full rank,then

range.

Let us de?ne the noise subspace

of

be the matrix

containing those eigenvectors.

Then,

(21)

where

and

(22)

The

matrix

is

.

Some remarks on the application of this method to the

DS/SS setup are now in order:

1)The subspace identi?cation method requires an SVD to

be performed on

a

can be fairly small

since

,

and

,the

order increases only

to

(and,hence,the number of equivalent FIR

channels)is large,typically in the order of hundreds

of chips.To ease the computational burden and memory

requirements,one could divide the problem into two(or

more)

with(or a fraction

of

Abstract—Self-recovering receivers for direct-sequence spreadspectrum signals with unknown spreading codes are discussed in this paper. Applications include signal interception, jamming, and low probability of intercept (LPI) communications. A multirate/m

TSATSANIS AND GIANNAKIS:BLIND ESTIMATION OF DIRECT SEQUENCE SPREAD SPECTRUM SIGNALS IN MULTIPATH 1245

set the sampling rate equal

to

in this way resulting in a problem of smaller

dimensionality.

2)The proposed method can estimate the convolution of the spreading code with the multipath channel and is useful when the code is unknown.It is still applicable,however,if the

code is known,and we are interested in the estimation of the multipath

parameters .In

this case,the

relationship

can be written in the time domain

as

after a permutation of its elements.Hence,after substi-

tuting

],the former can be solved with

respect to the

parameters

(i.e.,

is large.

Due to the inherent dif?culty of the problem,it is common practice when dealing with long spread-spectrum codes to focus on identifying the parameters of the code-generating mechanism (e.g.,code-generating polynomial)rather than in estimating the entire code sequence [26].In this approach,the parameter space is systematically searched,and each generating polynomial candidate is tested against the given data.Although in the presence of multipath and bit modulation the method of [26]is not applicable,the idea of testing each candidate code can still be applied.In this framework,(24)

2We

wish to thank the anonymous reviewer who pointed out this direction.

can be extended to

describe

,

contains the elements

of

.If the candidate se-quence coincides with the true one,there exists a non-trivial solution of (29).Hence,the smallest eigenvalue

of

should be zero,and the

solution is given by the corresponding eigenvector.Thus,a test on the smallest singular value (against zero)provides a means of determining the validity of a candidate code sequence.This test should be used in connection with a systematic search of the parameter space of the generating polynomial.Uniqueness questions regarding the solution of (29),as well as identi?ability conditions for this case,however,will not pursued here any further.

IV.R ECEIVER D ESIGN

The estimated signal

parameters

from the received

data .

It is well known that in a multipath environment,the maximum likelihood receiver can be implemented using the Viterbi algorithm (e.g.,[15]).Due to its computational com-plexity,however,considerable interest exists for simpler,linear solutions.In this section,we are interested in developing linear

receivers

in some

sense.Although simpler solutions may be possible if more than one sample per chip is available (e.g.,RAKE receivers),we will not pursue this direction here.

In the special case where no multipath is present,the matched ?lter followed by a hard limiter is known to be the optimal way to process and decode the received data (e.g.,[15]).Hence,in this

case,

Abstract—Self-recovering receivers for direct-sequence spreadspectrum signals with unknown spreading codes are discussed in this paper. Applications include signal interception, jamming, and low probability of intercept (LPI) communications. A multirate/m

1246IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.45,NO.5,MAY 1997

This approach may provide satisfactory performance even when mild ISI is present,but different techniques should be applied when the ISI is severe.A.Zero-Forcing Receivers

A simple linear design that can completely eliminate ISI is the ZF receiver [9],[18].In this approach,the

receiver

in (30)is

identical

to ,i.e.,the transmitted signal is perfectly recovered within a possible delay

of

],the ZF receiver must

satisfy

).

Let

be an FIR vector receiver of

order

.Then,the ZF

constraint of (32)can be written in the time domain

as

with one in

the

,(33)represents an exact or

underdetermined system of equations and,therefore,admits an exact solution.

As an example,consider a system

where

,i.e.,a constant

vector is equivalent

to

(34)

An exact solution of (34)can be found

if

and if

is chosen greater than 1or

if (as is usually the

case),(34)represents an underdetermined system and admits an in?nity of different solutions.Since all of these solutions are equivalent in the absence of noise,it is natural to select the one that,on top of the ZF property,minimizes the noise interference.This problem is discussed next.B.Optimal ZF Receivers

Let us consider the optimal

receiver

is given by (30),subject to the ZF constraint of

(32).This problem was addressed in [23]in the more general context of multiuser systems.The solution (which has been

adapted to the current scenario)for an FIR receiver of

order

such that (33)has at least one

solution,the parameter

vector

(36)

where

.

is white,

hence,

,(36)simpli?es

to

and can tolerate colored inputs.Moreover,

contrary to Wiener solutions,they do not depend on the power of the additive noise.For this reason,an optimal ZF solution may be preferable to the MMSE one in a blind scenario,where the noise power may not be available.

Optimal ZF equalizers in a cyclostationary setup were studied in [9],whereas extensions to multiuser systems were presented in [23]and [24].The contribution of the current work lies in their adoption for the recovery of distorted DS-SS signals.In addition,similar results for the special case of white noise and under a different communications setup were independently obtained in [7].These results were brought to our attention by the reviewers of the current paper.

V.P ERFORMANCE A NALYSIS

In Section IV,we presented two alternatives for the receiver structure,namely,the matched ?lter and the ZF receiver.Depending on the severity of multipath effects and ISI,either of the two could be best suited for a particular setup.It would be informative,therefore,to study the performance of the two schemes as a guide in selecting the appropriate structure.In the sequel,we consider the output SNR and the probability of error for each structure.We focus on BPSK modulation for reasons of simplicity and clarity of presentation.

Let be an i.i.d.BPSK signal with

variance

be white Gaussian noise of

variance

(39)

where

Abstract—Self-recovering receivers for direct-sequence spreadspectrum signals with unknown spreading codes are discussed in this paper. Applications include signal interception, jamming, and low probability of intercept (LPI) communications. A multirate/m

TSATSANIS AND GIANNAKIS:BLIND ESTIMATION OF DIRECT SEQUENCE SPREAD SPECTRUM SIGNALS IN MULTIPATH

1247

(a)

(b)

(c)

(d)

Fig.4.Spreading code and multipath effects.If no multipath is present,

then (41)

where

ISI 1x e 0

x o te th o o p por t

of

v of t

t

th

wh th o o p on t

ou t pu t

o powe

th opti wh v th

Abstract—Self-recovering receivers for direct-sequence spreadspectrum signals with unknown spreading codes are discussed in this paper. Applications include signal interception, jamming, and low probability of intercept (LPI) communications. A multirate/m

1248IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.45,NO.5,MAY

1997

Fig.5.Monte Carlo results on estimated impulse response. which for white noise reduces

to

,and the probability of error is given

by

.Hence,the

factor

depends explicitly on time is

nonstationary.It is clear from(49)

that is a periodic

function of

time,it

admits a Fourier series representation with

coef?cients

,using delta functions

as

,i.e.,is not

synchronized with the transmitting

rate

Abstract—Self-recovering receivers for direct-sequence spreadspectrum signals with unknown spreading codes are discussed in this paper. Applications include signal interception, jamming, and low probability of intercept (LPI) communications. A multirate/m

TSATSANIS AND GIANNAKIS:BLIND ESTIMATION OF DIRECT SEQUENCE SPREAD SPECTRUM SIGNALS IN MULTIPATH

1249

(a)

(b)

(c)

(d)

Fig.6.Received and equalized symbols.will be close to 1,but we assume it is not exactly equal to

one.The correlation of the sampled system

is is a periodic function

of ,assuming that the aliasing effects

are not severe (note that

if

).Hence,an estimate of the spectral peaks

can be used to provide information

about .The Fourier

transform in (53)should be computed in the neighborhood of the “assumed”bit

rate for

some

can be estimated

as

(54)

Once

using (2),where the spreading

code was a Gold sequence of length 31[see Fig.4(a)].The chip rate signal was then modulated using a rectangular spectral

pulse

Abstract—Self-recovering receivers for direct-sequence spreadspectrum signals with unknown spreading codes are discussed in this paper. Applications include signal interception, jamming, and low probability of intercept (LPI) communications. A multirate/m

1250IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.45,NO.5,MAY

1997

Fig.7.Performance of matched ?lter and ZF

receiver.

Fig.8.Bit period estimation.

The path delays

are (57)

where

(58)By sampling (57)

every

Abstract—Self-recovering receivers for direct-sequence spreadspectrum signals with unknown spreading codes are discussed in this paper. Applications include signal interception, jamming, and low probability of intercept (LPI) communications. A multirate/m

TSATSANIS AND GIANNAKIS:BLIND ESTIMATION OF DIRECT SEQUENCE SPREAD SPECTRUM SIGNALS IN MULTIPATH1251

response

).

Test Case1:The subspace method of(19)and(20)was

used to estimate the unknown impulse

response

.Fig.4(d)shows the es-

timated

was used(equivalent to two bit

periods),in this way overestimating the actual channel order

of.The overestimation can be seen in Fig.4(d),

where

bits,

SNR

the standard deviation(dashed-dotted line)of the100

runs.We observe a reasonable estimation accuracy despite the

low SNR.

Test Case2—Receiver Design:Based on the

estimated

)were implemented[using(31)and(33),

respectively].Fig.6(a)shows the transmitted BPSK

sequence

,whereas Fig.6(b)shows the received

data,which

are corrupted by ISI and additive noise.In Fig.6(c)and(d),

the estimated is shown using the matched?lter and the

ZF structures,respectively.Both receivers manage to equalize

the data adequately,as shown in the?gures.The remaining

?uctuations at the output of the matched?lter are due to

residual ISI,whereas those at the output of the ZF receiver

are due to noise effects.

A better picture of the performance of the two schemes

is obtained by the results of Section V.The probability of

error for the two alternatives was evaluated using(43)and

(48)and is plotted in Fig.7versus SNR.The matched?lter

receiver(dashed line)is slightly superior to the ZF receiver

(solid line)in this example,indicating that the ISI removal of

the ZF approach does not outweigh the noise enhancement.In

other cases of more severe ISI,the ZF solution is expected to

outperform the matched?lter receiver.

Test Case3—Bit Period Estimation:In this experiment,we

show how the receiver may be synchronized to the transmitter

bit rate,according to the developments of Section VI.The

same multipath channel of(57)was used,but the receiver

was assumed to sample

every in this

case).This effect was simulated by oversampling the channel’s

continuous-time impulse response by a factor of10and then

undersampling the?ltered signal by a factor of9.

Fig.8shows the estimated cyclic correlation

(for)

using(53)with a record of50bits.

The

is

exactly positioned

at.

VIII.C ONCLUSIONS

A discrete-time multirate framework for spread-spectrum

signals was introduced in this paper,leading to novel algo-

rithms for blind estimation of the signal parameters and for

signal interception.Extensions to the cases where very long

spreading codes are used present an interesting future topic.

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Abstract—Self-recovering receivers for direct-sequence spreadspectrum signals with unknown spreading codes are discussed in this paper. Applications include signal interception, jamming, and low probability of intercept (LPI) communications. A multirate/m

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Michail K.Tsatsanis(M’93)was born in Patras,

Greece,in1964.He received the Diploma degree

in electrical engineering from the National Technical

University of Athens,Greece,in1987and the M.Sc.

and Ph.D degrees in electrical engineering from the

University of Virginia,Charlottesville,in1990and

1993,respectively.

From1986until1988,he was with Binary Logic

Applications,Athens,Greece,where he worked on

the design and development of digital systems for

industrial control.From1994to1995,he worked as a research associate at the Department of Electrical Engineering,University of Virginia.In1995,he joined the Electrical and Computer Enginering Department,Stevens Institute of Technology,Hoboken,NJ,as an assistant professor.His general research interests lie in the areas of statistical signal processing,digital communications,system identi?cation,pattern recognition, higher order statistics,and multiresolution analysis.His current interests focus on equalization and synchronization problems in single-user and multiuser communication systems.

Dr.Tsatsanis has served as a member of the organizing committee for the IEEE1996SSAP Workshop and is a member of the Technical Chamber of

Greece.

Georgios B.Giannakis(F’96)received the

Diploma degree in electrical engineering from the

National Technical University of Athens,Greece,

1981.From September1982to July1986,he was

with the University of Southern California(USC),

Los Angeles,where he received the M.Sc.degree

in electrical engineering in1983,the M.Sc.degree

in mathematics in1986,and the Ph.D.degree in

electrical engineering in1986.

After lecturing for one year at USC,he joined the

University of Virginia,Charlottesville,in September 1987,where he is now a professor with the Department of Electrical Engineering.His general interests lie in the areas of signal processing, estimation and detection theory,and system identi?cation.Speci?c research areas of current interest include diversity techniques for channel estimation and multiuser communications,nonstationary and cyclostationary signal analysis,wavelets in statistical signal processing,and non-Gaussian signal processing using higher order statistics with applications to sonar,array,and image processing.

Dr.Giannakis received the IEEE Signal Processing Society’s1992Paper Award in the Statistical Signal and Array Processing(SSAP)area.He co-organized the1993IEEE Signal Processing Workshop on Higher Order Statistics,the1996IEEE Workshop on Statistical Signal and Array Processing, and the?rst IEEE Signal Processing Workshop on Wireless Communications in1997.He guest co-edited two special issues on high-order statistics (International Journal of Adaptive Control and Signal Processing and the EURASIP journal Signal Processing)and a special issue on signal processing for advanced communications(IEEE T RANSACTIONS ON S IGNAL P ROCESSING, Jan.1997).He has served as Associate Editor for the IEEE T RANSACTIONS ON S IGNAL P ROCESSING and the IEEE S IGNAL P ROCESSING L ETTERS,as secretary of the Signal Processing Conference Board,and as member of the SP Publications board and the SSAP Technical Committee.He is also a member of the IMS and the European Association for Signal Processing.

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