Radioelektronika 20051 Spread Spectrum Signals in Modern Communications Jan Šimša Institute of...

Radioelektronika 2005 1

Spread Spectrum Signals in Modern Communications

Jan ŠimšaInstitute of Radio Engineering and Electronics AS

[email protected]


What signal has spread spread spectrumspectrum?

Any digitally modulated signalAny digitally modulated signal

whose ratio of bandwidth to itswhose ratio of bandwidth to its

data (modulation) symbol ratedata (modulation) symbol rate

is substantially greater than is substantially greater than

one.one.

Spreading of spectrum Spreading of spectrum – – generation of signal using shaping or modulationgeneration of signal using shaping or modulation


Contents• Motivation Classification of signals and methods of

spreading spectrum Features of SS signals Optimum reception (detection) Synchronization Code division multiplex (multiple access) Conclusions


Motivation Why is spreading utilized?

Transmission of an additional information? Decrease of error rate in AWGN channel? ?????????? Decrease of error rate in any more disturbing channel? Preserving of unauthorized reception? Advantageous multiple utilization of a channel?

What is the price we pay for these advantageous features? Complexity of a system realization

No (user)

No

Yes

Yes

Yes


Heavily distorting Heavily distorting communication channelscommunication channels

Time variant channelsMultipath channelsFrequency selective channelsChannels with interferences


Frequency diversityFrequency selective / nonselective

channelsChannel coherence bandwidth is given by

parameters of the channelNarrowband signals can fade completelyFrequency nonselective selective

f f

H(f), S(f)H(f), S(f)


Frequency diversityDistortion by frequency selective

channelOne deep fade model – rectangular fade

f

H(j2πf)

1

0 0

f +B/2f -B/2 0

1 1 0 0( ) 1 ( ) ( ) 1 ( , )

0

H j H j where H j for B B

elswhere


Frequency diversityImpulse response of the rectangular fade

1 1 0

1 sin( ) ( ) 2 cos

2j t Bt

h t H j e d B tBt

Response to one chip of the rectangular shape

1 0 0 0

0 0 0 0

20 0 0

( 1)

0 0

sin( ) 2 cos ( ) sin[ ( ) ]

sin2 sin[ ] cos ( ) cos

sin2 sin[ ] cos

sinsin[ ]

c

c

c

c

i i T c

i T c

t iT

i

t i T

i

By t c B p t iT t d

B

Bc B t p t iT d

B

Bc B t d

B

ct

( )

( 1)

c

c

B t iT

Bt i T

d


Frequency diversityResponse y1i(t) to positive chip

for B.Tc = 0.01 (solid line )

B.Tc = 0.05 (dotted line)


Classification of signals and methods of spreading spectrum

Any spread spectrum signal belongs to one of the three categories:

signals without a carriersignals with a single carrier signals with multiple carriers


Signals without carrierare created by a sequence of very

narrow pulses (< 1ns) of a proper shape (Gaussian, wavelets)

their spectrum bandwidth is by some orders of magnitude wider than the modulation rate. Such signals are often referred to as Ultra Wide-Band (UWB) signals (and related UWB systems). Their bandwidth B≥500 MHz, B/f0 ≥ 0.2

it can reach bandwidth > 5 GHz


Single carrier signals

Harmonic carrier spread spectrum (SS) signals three basic subgroups

Direct Sequence (DS) – spreading by BPSK, QPSK

keying

Frequency Hopping (FH)

Time Hopping (TH)

- slow (SFH)- fast (FFH)


Direct Sequence (DS) Spreading

Transmitter Receiver

BP filter

DEM ReplicaSpreading modulation

data

carrier

channel

timing

Spectra - PSD

f0 f0f0 f0

ffff

B ~ 1/T




BP filter


data

carrier

channel

timing

Spectra - PSD

f0 f0 f0f0 f0

fffff

B ~ 1/T B ~ 1/T = L/Tc c




BP filter


data

carrier

channel

timing

Spectra - PSD

f0 f0 f0f0 f0

fffff

B ~ 1/T B ~ 1/T = L/Tc c

Spreading (system) gain L = T / T = B / B = R / Rc cc


Direct Sequence Spreading

Replica of spreading signal

I ntegration interval

Product

Output value of integrator

1

-1

1

-1

1

-1

00

t

t

t

t

Spreading modulation of received signal

Symbol interval = spreading signal period






Product


1

-1

1

-1

1

-1

00

t

t

t

t






Product


1

-1

1

-1

1

-1

0 0

t

t

t

t

1

-1/L



Direct Sequence SpreadingBPSK data modulation b(t) and BPSK

spreading modulation c(t)

0 0( ) ( ) ( ) cos( )s t A c t b t t

Spreading modulation c(t)( ) ( ) ( ) 1( ) 1( )

ci T ci

c t c p t iT where p t t t Data modulation b(t)

( ) ( )k Tk

b t b p t kT where T- symbol interval, Tc- chip intervalThe period of spreading modulation is T=L.Tc


Optimization of linear receiver

Matched filter (MF) – correlatorIts response

MF impulse response is defined by the signal to which MF is matched - MF[s(t)]

MF response at t=t0=T to finite signal s(t)

whose interval of nonzero values is (0,T),is

0 0( ) ( ) ( ) ( )h t s t t h t s

2

0 0

( ) ( ) ( ) ( )T T

y T s t dt n t s t dt

( ) [ ( ) ( )] ( )y t s n h t d


Optimization of linear receiver

MF

correlators(t) + n(t)

s(t)

0

T

y(T)

h(t)s(t) + n(t) y(t)


InterferencesResponse of MF[s(t)] to interfering signal s1

Correlation coef.Orthogonal signals do not cause any

interferenceCorrelation in time domain – frequency

domainParseval’s formula

1 1

0 0

( ) ( ) ( ) ( ) ( )T T

y T s t s t dt n t s t dt

1 2 1 2

1( ) ( ) ( ) ( )

2s t s t dt S j S j d

( ) { ( )} , 1,2i iS j s t i F{

Orthogonal signals have orthogonal spectra


Code synchronization

Code synchronization is an alignment of the spreading modulation of received signal and the replica at the multiplier producing despreaded signal

It consists of two steps Acquisition – coarse alignment of the

modulation and the replica Tracking – accurate alignment and time-

variations tracking


DS Code acquisitionwith the aid of a system of correlators - Matched filters Uncertainty region – the interval of prospective alignments. The

position is nonsensitive to shifts by an integer multiple of the spreading signal period (modulo L.Tc )

The goal (penalty) function of code acquisition process optimization

• Mean acquisition time – minimization • Probability of acquisition within given time interval – max

Code acquisition process is defined

• by acquisition detector rule • by search strategy - a sequence of points defining the replica

positions within an uncertainty region Classification according to changes of replica position

Stepping correlator Sliding correlator


Classification of DS code acquisition detectors

as to the number of channels Single channel detector – for serial search on

uncertainty region Multiple channel detector – for parallel / serio-

parallel search

as to the realization of each channel of the detector

Passive correlator - multiplier/integrator Active correlator - Matched Filter (DSP, SAW)

as to the rule of the final decision making Single dwell detector Multiple dwell detector


Spreading signal - optimization

Sharp and narrow main lobe of autocorrelationDC component-freeLow- level side lobes of autocorrelationLong linear spanLow croscorrelation between signal and interferencesGenerating constant- envelope signal

Noise-like signals – pseudonoise (PN) signals

M-sequences, Gold codes, Kasami codes, Walsh Hadamard sequences, bent sequences,


Slow Frequency Hopping

cT T

0 Tc Tc Tc2 3 Tc4 Tc6 Tc8 Tc Tc10 13f1

f

f

f

f

f

f

3

5

7

9

11

13

t

Carrier frequency


Slow Frequency Hopping

f

Spectrum of subchannels

f f f f1 2 3 7

1i if f B


Fast Frequency Hopping

0 Tc Tc Tc2 3 Tc4 Tc6 Tc8 Tc Tc10 13f1

f

f

f

f

f

f

3

5

7

9

11

13

t

Carrier frequency

T

cT T


Time hopping

Packet transmissionFixed length of packetsPseudorandom position of packets

within framePosition control by code sequence

TFTFT

FTF0 2 3 4 t


Two-path channelSpreading modulation of received signal (fi rst path)



1

-1

1

-1

-0,5

t

t


0,5

t

Spreading modulation of the second path signal (Delayed in the channel)

Spreading modulation of the fi rst path signal delayed in the receiver

t


Multipath channel – Rake receiver

Replica Gen.

Delay T c Delay T c Delay T cDelay T c

Combiner - Weighted summation

• •

• •••

• •

I nput signal

OUT

•• • •

• • ••


InterferencesOrthogonal signals do not cause interferences

(orthogonality is not generally invariant to mutual shift of signals)

Nonorthogonal (correlated) signals cause interference

Interference is proportional to amplitude of interfering signal

This amplitude can be greater than the amplitude

of useful (target) signal = near-far effect

Efekt nestejných vzdáleností (efekt nestejných amplitud)


Multicarrier CDMATransmitter of the k-th user

Data inOutput

MC CDMA∑

C k1

C kN

cos 1t

cos Nt

•••

••••

••T

TN

TN

TN = T, = fi+1 – fi =(i+1-i)/2 = T-1

Components are orthogonal on the interval T


Multicarrier DS CDMA

TN = T/N , = fi+1 – fi =(i+1-i)/2 =Tc-1

Components are orthogonal on the interval Tc

Transmitter of the k-th user

Data inOutput

MC DS CDMA∑

Ck(t)

Ck(t)

cos 1t

cos Nt

•••

•••

T

TN

TN

S/P


Multitone CDMATransmitter of the k-th user

Data inOutput

MT CDMA∑

cos 1t

cos Nt

•••

T

TN

TN

S/P

ck(t)

TN = N.T, = fi+1 – fi =(i+1-i)/2 =(N• T)-1

Components are orthogonal on the interval TN


Comparison in frequency domain

Spectra

OFDM

MC CDMA

MC DS CDMA

MT CDMA

f

f

f

f

1/TN1/NT

1/T

=1/Tc=L/TN=L/NT

1/TN


Shared communication Shared communication channelchannel Multiplex (Multiple Access)

• Frequency division• Time division• Code division

Timing of symbols – identical = synchronous CDMA - nonidentical = symbol asynchronous CDMA

> identical chip timing = chip synchronous CDMA > nonidentical chip timing = chip asynchronous


Code Division Multiple Access

t

f

B

T

FDM

TDM

CDM

Signal space - dimensionality 2BT = number of mutually orthogonal signals


Synchronous and Asynchronous CDMA

Signature

Synchronous CDMA (chip synchronous – symbol synchronous)

Asynchronous CDMA

t

t

t

s1(t)


CDMA signalSignal at detector input

K

i i i i ii 1

y(t) A b (t ) s (t ) n(t)

where bi(t) is data signal of the i-th user, is its delay and si(t) is its spreading modulation, which in CDMA is labeled to as a signaturesignature.Usually, signature has unit energy ( a system of orthogonal signatures is in the same time orthonormal).As ci

2(t)=1, the unit energy signature si(t) is preserved if

i

1( ) ( )i is t c t

T


Synchronous CDMACDMA is synchronous, iff

1,2, ...i konst for i K

Then K

i i ii 1

y(t) A b s (t) n(t) t (0,T)

Without loss of generality let it be 0i

This signal causes response ym of MF matched to the signature of m-th userT K

m m i i im mi 10

K

m m i i im mi 1i m

y y(t) s (t)dt A b n

A b A b n

MUI


The second component of the right side of the equation represents Multiuser Interference (MUI / MAI)This component is zero if the signaturesat the detector input are ortogonal.If system designer is not able ensure validity of this condition (it is the usual case as parameters of channel are unknown and time variable)Detector minimizing MUI is not MF any more; it is more complex, usually nonlinear.

To keep the complexity of receiver acceptably low, suboptimum linear detectors are used.

Multiuser interference (MUI)Multiuser interference (MUI)


Multiuser detectionVector notation

Matrix of amplitudes and correlation matrix

1 2 K

ij

= diag{A ,A ,...,A }

= { }A

R

T1 2 K

T1 2 K

T1 2 K

= [y , y ,..., y ]

= [b ,b ,..., b ]

= [n ,n ,..., n ]

y

b

n

b̂

T K2

i i ii 10

ˆ = { [y(t) - A b s (t)] dt}arg minb

b

Detector estimates vector of dataOptimum detector is described by the equation

Vector y can now be expressed as y ARb n


It can be expressed using likelihood

where

and

Averaging over random variable bk is not performed . For a priori equiprobable symbols

the above average is

Optimum receiver is nonlinear.

kk (k)b 1,1

b̂ arg max E [P( )]

y b

K1 2P( ) P( b ,b ,...,b )y b y

K1 2 k 1 k 1K1 2(k) b ,b ,...,b ,b ,...,b

E [P( )] E [P( b ,b ,...b )]

y b y

k k1P(b 1) P(b 1) , k 1, 2, ... , K2

K1 k 1 k 1K1 2(k) K 1

b 1,1 b 1,1 b 1,1 b 1,1

1E P( ) ... ... P( b ,b ,...,b )2

yb y


Using vector notation it isˆ = {2 T T

bb b Ay - b Hb }arg min

where=H ARA

Approximation of the optimum detector - linear detectors

m mb̂ Mysign

Linear detector consists of the bank of filters matched to the signatures of individual users and weighted sum of their outputs is created. Weights in summation are chosen in a way minimizing MUI or noise plus MUI, respectively.

Linear detectors


Block diagram of linear multiuser detector

MF 1

y(t)

1b̂

2b̂

3b̂

Kb̂

Sync 1

Sync 2

Sync 3

Sync K

M

MF 2

MF 3

MF K


Conventional detector

M I

This detector is optimum in the case of uncorrelated signatures, i.e. if

k kb̂ sign(y )

If signatures are correlated, MAI is nonzero.

Conventional detector – bank of MFs


Decorrelating detector Matrix M is the inverse to correlation matrix R

1M R

Then1 1 1 My R RAb R n Ab R n

and k-th component of the detector output isK

k k ik ik i 1A b n

My

MUI component is totally MUI component is totally compensatedcompensated

but noise component has increased.but noise component has increased.


MMSE detectorIt minimizes men square error of the vector b estimate. We search for the matrix M minimizing this error

0

2

0min E

MM b M y

K*KRarg

where the norm of vector is defined asK2 2 T

ii 1

x trace{ }

x xx

After some manipulation

0

0 0

T0 0

0M

2

0M Mtrace ( )( )

min trace [cov( )]

min E min E

b M y b M y

b M y

b M y

K*K

K*K K*K

R

R R


We expand T0 0

( )( )E

b M y b M y

T T T T T T0 0 0 0 0 0

( )( ) E[ ] E[ ] E[ ] E[ ]E

b M y b M y bb by M M yb M yy M

After substitution y it can be rearranged to the shape 2 1 2 2 T

0 0 0cov[ ] [ ] ( )[ ]( ) b M y I ARA M M RA R R M M

where 1 2 2 1[ ] M A R A

Minimum MSE is reached forM M

Decision rule of MMSE detector is

2 2 1 2 2 1k k k

k

1b̂ sign {( ) } sign {( ) }A

R A y R A y

MMSEMMSE detectordetector


Asymptotic cases

AAdaptive methods2 20 R A R

2 2 2 2 R A A MF detectorMF detector

Decorrelating detectorDecorrelating detector

- Trained methods- Blind methods

To avoid periodical repetition of measurements of values of matrices A and R in the case of time variant channeladaptive MMSE methods can be used.


Spread Spectrum Signals AdvantagesAdvantages

» Interference resistant»Multipath resistant»Selective fades resistant»Sharing communication channel by multiple

users»Unauthorized reception resistant» Interference into other systems reduced

CostsCosts»Code synchronization»More complex system realization


Spread Spectrum Signals

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