Spread Spectrum Communicationsscholar.fju.edu.tw/課程大綱/upload/049469/scheme/971... · 2008-09-09 · FJU-EE – YUJL - Spread Spectrum Communications– Outline - Page 3 TEXT

FJU-EE – YUJL - Spread Spectrum Communications– Outline - Page 1

Spread Spectrum Communications

Jung–Lang Yu

Department of Electrical EngineeringFu Jen Catholic University

Taipei, TaiwanTel: +866-2-29052102Fax:+866-2-29042638

e-mail : [email protected]://www.course.fju.edu.tw/student/Main.htm


OUTLINE

1. Introduction (p1~p50)2. Introduction to Digital Communications (p51~p152)3. Fundamentals of Spread-Spectrum Techniques (p153~p194)4. Pseudo-Random Code Sequences for Spread-Spectrum

Systems (p195~p252)5. Time Synchronization of Spread-Spectrum Systems

(p253~p336)6. Cellular Code Division Multiple Access (CDMA) Principles

(p337~p400)7. Multi-User Detection in CDMA Cellular Radio (p401~p472)8. CDMA Wireless Communication Standards (p473~p504)


TEXT BOOKS

Textbook: 1. Mosa Ali Abu-Rgheff , “Introduction to CDMA Wireless

Communications,” 2007, Elsevier Science & Technology, 天瓏書局

Reference:1. R.L. Peterson, et. al., “Introduction to spread spectrum

communications,” 1995,2. IS-95 CDMA and CDMA 2000 Cellular/PCS Systems Implementation,

V.K. Grag, 2000, Chap 1~Chap73. S.G. Glisic, “Adaptive WCDMA,” 2003. 全華書局4. S. Verdu, “Multi-user Detection,” 19985. V.P. Ipatov, “Spread spectrum and CDMA, Principles and

Applications,”, 2005 6. A.J. Viterbi,”Principles of spread spectrum communication,” 1995


GRADE

Exercise 40% [重點整理、讀書心得報告 (手寫)]Midterm Exam. : 23rd Nov. 25%Term project and presentation 35% (duplicate its simulation results) : 12th. Jan.

Due dateTopic of term project : on 23rd Nov.


Project topics:

Topics :Multi-user detection RAKE receiversChannel EstimationSynchronization in spread spectrum systemsPower control in CDMA systemsHandover techniquesWireless LANs based on spread spectrum technologyAdvanced wireless techniques: OFDM, UWB, etc.Any topics related CDMA are OK after discussing with Dr. Yu.

References :IEEE Transaction on CommunicationIEEE Transaction on Vehicle TechnologyIEEE Transaction on Signal ProcessingIEEE Transaction on Wireless CommunicationSignal Processing

FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 1

Chapter 1 : Introduction

1. Mobile communications2. Development of CDMA3. 3G development history4. International Telecommunication Union5. 4G possible techniques6. Important Research Topics7. Reference


§1.1 Mobile communications

1st Generation: analog voice service (9.6Kbps)AMPS(USA), Advances Mobile Phone Service, IS-54NMTS(Europe), Nordic Mobile Telephone SystemTACS(England), Total Access Communication SystemNAMTS(Japan), NEC Advances Mobile Telephone System

2nd Generation: voice and lower-rate data service (9.6Kbps)D-AMPS(USA), Digital-AMPS, IS-136GSM(Europe), Global System for Mobile CommunicationDCS(England), Digital Cellular SystemPDC(Japan), Personal Digital CellularCDMA(North American), IS-95


2.5 Generation: enhanced data service for GSMGPRS for packet switching system (9k, 13.4k, 15.6k, 21.4k/slot, 8 slots/channel)HSCSD for high-speed circuit switching data (14.4k/slot, 8 slots/channel)EDGE integration of GPRS and HSCSD (384kbps)

3rd Generation: voice, data and multi-media service (2Mbps)

4th Generation: voice, data and interactive-media service (156Mbps)



Quality of Service in old generations: Voice Quality (improved) , Coverage (world-wide seamless access) & Costs (low) Quality of Service aspects : low BER and low delay time

New Services and Capabilities in new generations Enabling new voice and data service that are not currently available with 1G and 2G technology High bandwidth services (data, image, multimedia)





Spread spectrum communications originate from MIT Lincoln Labs since 1920’s.The theory of spread spectrum communications has been well known since the late 1940's.It has been used somewhat intensively in the field of secure military communications since 1950's, but in commercial applications it is a relatively new technique.The spread spectrum technique has been released from military since 1970’s.The first major commercial application of spread spectrum techniques was the Global Positioning System (GPS).

§1.2 Development of CDMA


§1.3 3G development history

R.G. Cooper and Nettleton proposed the North American DS-CDMA systems in 1977. It is further commercialized by Qualcomm as narrowband CDMA(IS-95)In 1985, ITU (International Telecommunication Union) proposed the 3G specification, which is called FPLMTS (Future Public Land Mobile Telecommunication Systems). In 1996 it is renamed as IMT-2000(International Mobile Telecommunication) and defines the specifications

144K bps in fast moving speed384K bps in walking, slow moving speed2M bps in standstill environment

Proposals for 3G StandardsWideband-CDMA (Europe)CDMA-2000 (North American)TD-SCDMA (China)


Seamless World-wide Access



W-CDMA: It is proposed by Ericsson (Sweden) and NTT DoCoMo (Japan) which is an extension of GSM systems. In 2001, The first W-CDMA 3G service is proposed in Japan by the DoCoMo company.

CDMA-2000It is an extension of narrowband CDMA (IS-95)

CDMA one, integration of IS-95 in 1997, 8 voices, 64K bps/channelCDMA-2000 1X, wideband service, 307K bps in 1.25M Hz BWCDMA-2000 3X, wideband service 2M bps in 5M Hz BWCDMA 2000 1X EV-DO, 2.5M downlink /307K uplink bps in 1.25M Hz BW



TD-SCDMAIt is the combination of TDMA system and synchronization CDMA, which is proposed by the Simens (Germany) and Datang (China) in 1999.



3G Telecommunication licenses Taiwan : A,B,C,D for W-CDMW at 2G Hz and E for CDMA2000 at 800M Hz.Japan: 2 for W-CDMA and 1 for CDMA2000Korean: 3 for W-CDMA and 1 for CDMA2000



http://www.cdg.org & http://www.umtsworld.com/umts/links.htm

Network Operators



§1.4 International Telecommunication Union

IMT-2000 specification : ←→ ITUInternational Mobile Telecommunications 2000 the time schedule for the first trial system : year 2000 the frequency range to be used : around 2000 MHz

The International Telecommunication Union (ITU) is responsible for the IMT-2000 specification.

The requirements for the 3G standardisation have been discussed under the term FPLMTS (Future Public Land Mobile Telecommunications System) since the early 1990s. In the mid 1990s the term FPLMTS was changed to the term IMT-2000.


UMTS (WCDMA) ←→ ETSIUMTS stands for Universal Mobile Telecommunications System UMTS is a member of the ITU‘s IMT-2000 global family of 3G mobile communication systems The European Telecommunication Standards Institute (ETSI) is responsible for the UMTS standardization UMTS is the successor standard to the second generation GSM. UMTS will play a key role in creating the future mass market for high-quality wireless multimedia communications that will approach 2 billion users worldwide by the year 2010



Air Interfaces for 3G : WCDMA

GSM

WCDMA



Air Interfaces for 3G : WCDMA



Air Interfaces for 3G : CDMA2000



Spectrum Allocation for 3G



§1.5 4G possible techniques

W-CDMA with OFDM technique → Multi-Carrier CDMALAS-CDMA (large area synchronization CDMA by China)Position CDMAUWB (ultra wideband) technique4G standards will be proposed in 2010 ( NTT DoCoMoplans to propose the 4G standard in 2007).


§1.6 Important Research Topics

1. PN sequences2. Code acquisition / Code tracking3. Modulation/demodulation4. Power control5. Handover techniques6. RAKE receivers7. Channel Estimation8. Adaptive CDMA networks9. Radio fading channel10. Multiuser detection11. Advanced CDMA systems, MC-CDMA, OFDM


§1.7 Reference

1. CDMA development group, http://www.cdg.org2. 3G Partnership Project 2，http://www.3gpp2.org/3. 3 G Partnership Project，http://www.3gpp.org/4. UMTS World，http://www.umtsworld.com/umts/links.htm5. 3G Today - IMT-2000 Standard，http://www.3gtoday.com/index.html6. CDMA2000，http://www.ericsson.com/7. Cellular Online，http://www.cellular.co.za/main.htm8. Cellular Technologies Of The World，http://www.cellular.co.za/main.htm9. Philips Consumer Communications，

http://www.wca.org/dgibson/index.htm10. TDD White Paper，http://www.tddcoalition.org/11. TD-SCDMA White Paper，http://www.siemens-mobile.com/mobile12. Wireless Web Features - TD-SCDMA and W-CDMA make ideal partners

for 3G，http://wireless.iop.org/13. UMTS World，http://www.umtsworld.com/umts/links.htm


Chapter 3: Fundamentals of Spread-Spectrum Techniques

1. Historical2. Benefits of spread-spectrum3. Principles of spread-spectrum communications (Scholtz,

1977)4. Most common types of spread-spectrum systems5. Processing gain6. Correlation functions (Sarwate and Pursley, 1980)7. Performance of spread-spectrum systems (Pursley, 1977) .


3G Telecommunication licenses Taiwan : A,B,C,D for W-CDMW at 2G Hz and E for CDMA2000 at 800M Hz.Japan: 2 for W-CDMA and 1 for CDMA2000Korean: 3 for W-CDMA and 1 for CDMA2000

§3.1 Historical


http://www.cdg.org & http://www.umtsworld.com/umts/links.htm

Network Operators

§3.1 Historical


§3.2 Benefits of spread-spectrum

Avoiding interceptionThe successful interceptor usually measures the transmitted power in the allocated frequency band. Spreading the transmitted power over a wider band undoubtedly lowers the power spectral density, and thus hides the transmitted information within the background noise.Because of its low power level, the spread spectrum transmitted signal is said to be a Low Probability of Interception (LPI) signal.



Privacy of transmissionThe transmitted information over the spread-spectrum system cannot be recovered without knowledge of the spreading code sequence. Thus, the privacy of individual user communications is protectedin the presence of other users.

Resistance to fadingThe resistance of the spread-spectrum signals to multipath fading is brought about by the fact that multipath components are assumed to be independent. This means that if fading attenuates one component, the other components may not be affected, so that unfaded components can be used to recover the information.



Accurate low power position findingThe distance (range) between two points can be determined by measuring the time in seconds, taken by a signal to move from one point to the other and back. This technique is exploited in the Global Positioning System (GPS). GPS provides two services. The precise positioning service uses very long code sequence at a code rate of 10.23 MHz. The standard positioning service, on the other hand, uses a shorter code (1023 bits) at a rate of 1.023 MHz.

Improved multiple access schemeMultiple access schemes are designed to facilitate the efficient use of a given network resource by a group of users.Frequency Division Multiple Access (FDMA), Time Division Multiple Access (TDMA) and Code Division Multiple Access (CDMA) are commonly used schemes for multiple access systems.The spread spectrum (CDMA) offers a new network access scheme due to the use of unique code sequences. Users transmit and receive signals with access interference that can be controlled or even minimized.



Anti-jamming ability



Anti-jamming ability


Skip補充 Power Spectral Density

Formula for PSD-- Wiener Khinchin theorem, (ref. ZiemerChapter 5, textbook p44, eq(1.115) )If v(t) is a stationary R.P. andwhere g(t) is the specified waveform, then the PSD of v(t) is given by

If , then the PSD of u(t) is given by

§3.3 Principles of spread-spectrum communication

( ) ( ), . .i iiv t a g t iT a RV∞

=−∞= − ∈∑

[ ]

21( ) ( ) ( ) ,

( ) ( )exp( 2 )

( ) , ( ) { ( )}

v a

a am

a i m i

S f S f G fT

S f R m fmT

R m E a a G f FT g t

π∞

=−∞

+

=

= −

= =

∑

0( ) ( )cos(2 )u t v t f tπ=

[ ]1( ) ( ) ( )4u v o v o

S f S f f S f f= − + +


Example : Baseband

Example : Passband

§3.3 Principles of spread-spectrum communication

2 2

( ) ( ), { 1}, ( ) ( )

( ) 1; ( ) sinc( )1{ ( )} ( ) ( ) sinc ( ) ( )

i ii

b

b

td t b p t iT b p tT

S f P f T fT

PSD d t S f P f T fT D fT

∞

=−∞

= − ∈ ± =

= =

= =

∑ ∏

[ ]

2 2

( ) 2 ( )cos( )2( ) ( ) ( ) 41 sinc ( ) sinc ( )2

d o

d o o

o o

s t Pd t w tPS f D f f D f f

PT f f T f f T

=

= − + +

⎡ ⎤= − + +⎣ ⎦


§3.4 Types of spread-spectrum systems

Direct-sequence (DS) spread spectrum Frequency-hoping (FH) spread spectrum Hybrid DS/FH spread spectrum


§3.4.1 Direct-sequence spread spectrum

BPSK : Transmitter (Model I)

1

0

1

0

( ) ( ); ( ) ( ); , { 1}

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )cos( )

i b j c i ii j

N

n i j b ci j

N

n i j b ci j

s

m t m t iT C t c t jT c m

s t m t C t m c t iT jT

y t s t h t m c h t iT jT

s t y t wt

δ δ

δ

∞ ∞

=−∞ =−∞

∞ −

=−∞ =

∞ −

=−∞ =

= − = − ∈ ±

= = − −

= ∗ = − −

=

∑ ∑

∑ ∑

∑ ∑

If periodic spreading code is used, then Tb=NTcand C(t) is periodic with period N

Data stream

Spreading code

Pulse shaping filter

Oversamplingshould be executed



BPSK : Transmitter (Model II)

1

0

( ) ( ); ( ) ( ); , { 1}

( ) ( ) ( ); ( ) ( / ), ( ) ( / )

( ) ( ) ( )

( ) ( )cos( )

i b b j c c i ii j

n b b c cN

n i j c b ci j

s

m t m p t iT C t c p t jT c m

s t m t C t p t t T p t t T

y t s t m c p t iT jT

s t y t wt

∞ ∞

=−∞ =−∞

∞ −

=−∞ =

= − = − ∈ ±

= = Π = Π

= = − −

=

∑ ∑

∑ ∑

If periodic spreading code is used, then Tb=NTc and C(t) is a periodic waveform with period NTc

Data waveform

Spreading waveform

Shaping filter is included in c(t)and m(t)



BPSK : Receiver

2

2 2

( 1) ( 1) ( 1)

( 1)

( ) 2 ( )cos( ) 2 ( )cos( )cos( ) 2 ( ) ( )cos ( )

( ) ( ) ( ) 2 ( ) ( ) ( )cos ( ) 2 ( )cos ( )

( ) ( ) ( )(1 cos(2 )) ( )

, i( )

b b b

b b b

b

b

s

KT KT KT

K T K T K T

KT bKK T

A t s t wt y t wt wt m t C t wtB t A t C t m t C t C t wt m t wt

D T B t dt m t wt dt m t dt

TD T m dt

− − −

−

= = =

= = =

= = + =

= =

∫ ∫ ∫

∫f 1

, if 1K

b K

mT m

=⎧⎨− = −⎩

A(t) B(t) D(T)

( ) ( )j c cj

C t c p t jT= −∑



Waveforms Periodic spreading code aperiodic

spreading code

m(t)

C(t)

y(t)

BW=?

BW=?



Waveforms

Figure (a) Product signal

y(t) = C(t)m(t). (b) Sinusoidal carrier. (c) DS/BPSK signal.

Ss(t)





QPSK :

Figure 3.5 (a) Quadrature spread-spectrum modulator; (b) Quadrature spread-spectrum receiver.



Example3.3: A binary data stream of 4 digits [1011] is spread using an 8-chip code sequence C(t)= [01 10 10 01]. The spread data phase modulates a carrier using binary phase shift keying. The transmitted spread-spectrum signal is exposed to interference from a tone at the carrier frequency but with 30 degrees phase shift. The receiver generates an in-phase copy of the code sequence and a coherent carrier from a local oscillator.

i. Determine the baseband transmitted signal.ii. Express the signal received. Ignore the background noise.iii. Assuming negligible noise, determine the detected signal atthe output of the receiver.

Solution



Ans iLet the data stream be denoted as m(t). The baseband spread-spectrum data mS(t) can be represented as:mS(t)=m(t)C(t)=[01101001, 10010110, 01101001, 01101001]

Ans iithe received signal mr(t)= mt(t)+ I(t)=mt(t)+cos(ωCt+30)= mS(t)cos(ωCt)+ cos(ωCt+30)

Ans iiiThe demodulated signal is mb(t)= mr(t)2(cosωCt). Therefore: mb(t)=(mS(t)cos(ωCt)+ cos(ωCt+30))2(cosωCt) ~= mS(t)+cos30The de-spread signal md(t) ismd(t) = mb(t)C(t) = [mS(t) + cos30]C(t)= m(t)C(t)C(t) + 0.866C(t) = m(t) + 0.866C(t) ~=m(t)


§3.4.2 Frequency-hoping (FH) spread spectrum

Concept:

, ,

h b

h b

for slow FHSS T Tfor fast FHSS T T

>

<



Coherent FHSS : Transmitter

N=2k subbandsin FH systems



Coherent FHSS : Receiver




Noncoherent FHSS: FH/MFSK combines the FH technique with Noncoherent M-ary FSK demodulation


This method is applied in BlueTooth®



k=2, L=2 and 使用MFSK的慢速FHSS

dW

dW

dW

dW

SW頻率

時間

ST

CT

T

0 0 1 0 0 1 0 0 1 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0二進位資料

PN序列1 10 0 1 00 1

2:

:

ktotal s d d

d

total

BW W W NWWBW

= = =BW before spreading

BW after spreading

2LdBW W M f f= = Δ = Δ

There are M=2Lsubcarriers. The bandwidth is equal to

.

4h sT T=



k=2, L=2 and 使用MFSK的快速FHSS

.

4s hT T=


§3.4.3 Hybrid DS/FH spread spectrum

In special applications such as anti-jamming work, there may be a need for a hybrid system using both the DS and FH spread-spectrum schemes. Two code sequences are employed in this system.Transmitter :

.


§3.4.3 Hybrid DS/FH spread spectrum

Receiver :

.


§3.5 Processing gain

The effectiveness of the processor is measured with a factor called the processing gain Gp defined as:

In spread-spectrum systems, the processing gain (Gp) expresses the bandwidth expansion factor.

For a DS-SS system:

For a FH-SS system: Gp=N=2k

.

Modified signal parameter at processor outputSignal parameter at input

(signal spectrum at the input )(signal spectrum at the output )s

pb

BGB

=

(code sequence rate) 1/(data bit rate ) 1/

c c bp

b b c

R T TG NR T T

= = = =


§3.5 Processing gain

Example 3.4: A speech conversation is transmitted by a DS-SS system. The speech is converted to PCM using an anti-aliasing filter with a cut-off frequency of 3.4 kHz and using 256 quantization levels. It is anticipated that the processing gain should not be less that 23 dB.

i. Find the required chip rate.ii. If the speech was transmitted by an FH-SS system, what would be the number of hopping channels?

Solutioni. the PCM bit rate=Rb =n×6.8=54.4 k bits/sec

Processing gain=23dB=199.53=Gp =Rc/RbSubstituting for Rb gives Rc =10854.2 k chip/sec.

ii. the number of FH channels=N≈200.

.


§3.6 Correlation functions

The interaction and the interdependence between two time (or frequency) varying signals are defined by the correlation function derived from the comparison of the two signals. The comparison of a signal with itself is described as the autocorrelation function. On the other hand, the cross-correlation is a measure of similarity between two autonomous signals.Consider two binary sequences {a} and {b} with elements anand bn that can be real or complex such that:

We assume the two sequences to be periodic in Sec. 3.6.1 with long period N and aperiodic in Sec.3.6.2.

.

0 1 2 1

0 1 2 1

{ } { , , , , }{ } { , , , , }

N

N

a a a a ab b b b b

−

−

=

=


§3.6.1 Periodic correlation function

The periodic correlation function Ra,b(τ) of N-element sequences {a} and {b} is defined by:

Periodic Auto-Correlation Function [PACF]: Ra,a(τ) Periodic Cross-Correlation Function [PCCF]: Ra,b(τ) The normalized correlation function: Ra,b(τ) /NThe periodic correlation Ra,b(τ) can be expressed by modulo operation in eq(3.18), ((.))N

Ra,b(τ) can be separated into two parts:

.

1 *, 0

( ) Na b n nnR a b ττ−

+==∑

1 1* *, (( ))0 0

( )N

N Na b n n n nn n

R a b a bτ ττ− −

+ += == =∑ ∑



Ra,b(τ) can be separated into two parts:

.

1 1* *, (( ))0 0

* * * *0 1 1 0 1 1

, ,

1* * *, 0 1 1 0

* * *, 0 1 1 (( ))

( )

( ) ( )

( )

( )

N

N

N Na b n n n nn n

N N N N

a b a b

Na b N N n nn

a b N N n n

R a b a b

a b a b a b a bR R

R a b a b a b

R a b a b a b

τ τ

τ τ τ τ

ττ τ τ

τ τ τ

τ

τ τ

τ

τ

− −+ += =

− − − − − −

− −− − − +=

− − − +

= =

= + + + + +′ ′′= +

′ = + + =

′′ = + + =

∑ ∑

∑1N

n N τ

−

= −∑



Example 3.5: Sequences {a} and {b}, each with period N=15, are given by:{a}={1, 1, 1, −1, 1, 1,−1,−1, 1,−1, 1,−1,−1,−1,−1}{b}={1, −1,−1,−1,−1, 1,−1,−1,−1,−1, 1,−1,−1,−1,−1}Find the periodic autocorrelation and cross-correlation functions of the sequences.

.



Periodic autocorrelation functions of sequence {a}

.



Periodic autocorrelation functions of sequence {b}

.



Periodic cross-correlation function of sequences {a} and {b}

.


§3.6.2 Aperiodic correlation function

The aperiodic correlation function between sequence {a} and {b} is defined by Ca,b(τ)

If {a}={b} , the expression Ca,b (τ) represents the AperiodicAuto-Correlation Function [AACF]. When {a}~= {b} , the expression defines the Aperiodic Cross-Correlation Function [ACCF].

.

1 1* *, 0 0

1 *0

( ) , 0 1

, 1 0

0,

N Na b n n n nn n

Nn nn

C a b a b N

a b N

N

ττ τ

ττ

τ τ

τ

τ

− − −+ += =

− +−=

= = ≤ ≤ −

= − ≤ ≤

= ≥

∑ ∑∑

,1* * *

, 0 1 1 0

1* * *, 0 1 ,1 0

( )

( )

( )

( )

a bN

a b N N n nn

a b N N N n an bn

R a b a b a b

R a b a b a

C

C Nb

ττ τ τ

ττ τ ττ

τ

τ

τ − −− − − +=−

− − − − +=

′ = + + =

′′ = +

≡

≡+ −=

∑∑

Geometric

interpretation


§3.6.5 Interference rejection capability

Interference can be caused by an external transmitter tuned to afrequency within the passband of the intended receiving equipment, possibly with the same modulation and with enough power to override any signal at the intended receiver.Consider a spread-spectrum system transmitting information signal m(t) between two fixed points. Further, assume that the transmission is being exposed to a jamming signal, j(t). The channel noise and the interfering signal are assumed to be uncorrelated. The received signal r(t) can be expressed as:

The reference signal used by the matching filter receiver is given by:

.

( ) 2 ( )cos( )refr t C t wtτ θ= − +

( ) ( ) ( ) ( ) ( ) ( )cos( ) ( ) ( )sr t S t j t n t m t C t wt j t n t= + + = + +



The signal component at the matched filter output is:

.

請多2倍



The output noise component and the interference component at matched filter receiver output are respectively given by:

DS spectra in tone jamming: Assume DS - BPSK transmission, with a single tone jamming (jamming power J [W] ). The received signal is

At the receiver r(t) is multiplied with a reference signal 2C(t)cos(wt) (=despreading+demodulation)

.

( ) ( )ω ω= +0 0( ) 2 ( ) ( )cos 2 cosr t PC t m t t J t

( ) ( )( ) ( )

2 20 0

0 0

( )2 ( )2( ) 2 ( ) ( ) cos 2 cos

2 ( )[1 cos 2 ] 2 [1 cos( ]) 2

y t Pm t C t t J t

Pm t t J

C t

C t

C t

t

ω ω

ω ω

= +

= + + +


DS spectra in tone jamming (cont.)W.W. Ali-Ahmad, “The CDMA Receiver System in an IS-98-A Standard”, Electronics Engineer, July 2000


DS spectra in tone jamming (cont.)

Despreading distributed the jammer power in frequency:


DS spectra in tone jamming (cont.)

Receiver filtering suppresses the jammer power:


Characterizing SS systems

Code gain, for BPSK

where Tb is bit period, Tc is chip period, Bb is bandwidth before spreading and Bs is bandwidth after spreading.

Let the interference power at the input of the matched filter be J, and assume it is uniformly distributed across the spread-spectrum bandwidth Bs. Consequently, we can assume the average interference power spectral density to be J/Bs The noise considered has white spectral density and zero mean value. Let the one-sided noise power density at the input of the receiver be No in W/Hz.

.

/ /p s b b CG B B T T= =



The signal power to noise power ratio at the input of the receiver is:

.

( )

2

ri

os

PSNR N B J=

+

PSD=No/2 for AWGN

PSD=J/Bs for interference

Bs

(SNR)o

(SNR)i

Bb



The MF acts like a low-pass filter with BW=Bb. The power of noise and interference at the MF are given by

The ratio of output signal power to noise power, (SNR)0 is expressed as:

.

( ) ( )

2 2

sr ro p i

o obb b s

s

BP PSNR G SNRN NJ BB B B JB

= = =+ +

, 2

on b j b

s

N JP B P BB

= =



Jamming margin:The interference rejection capability of the spread-spectrum system can be evaluated in terms of the jamming margin, Mj, which is defined as the level of interference (jamming) that the system is able to tolerate and still maintain a specified level of performance such as specified bit error rate even though the signal-to-noise ratio is



Jamming margin:

Let L be system losses between transmitter and receiver. We include the system loss into the jamming margin

.

1 1( ) ( ) ( )

j n pj j

r i i o

P GM M

P SNR SNR SNR+= = ⇒ = =

( ) ( ) ( ) ( ) ( ) ( )j i p oM dB SNR dB G dB SNR dB= − = −

A. J. Viterbi, “Spread spectrum communications - myths and realities,” IEEE Communication Magazine, May 1979.

1 ( ) ( ) ( ) ( )( )

j nj j i

r i

PM M dB SNR dB L dB

P L SNR L+= = ⇒ = − −

( ) ( ) ( ) ( ) ( )j p oM dB G dB SNR dB L dB⇒ = − −

Related to specified BER =Q(sqrt(SNRo))

If SNRo is the minimum bit energy-to-noise density ratio needed to support a given bit error rate, then Mj is the maximum tolerable jamming power-to-signal power ratio, also known as the jamming margin.



Jamming margin: for example

.

30dB,available code gainpG =2dB,margin for system lossesL =

10dB,required SNR after despreading (at the RX)oSNR =

18dB,limit for additional interference and noisejM⇒ =


§3.7 Performance of spread-spectrum systems

The performance of a spread-spectrum system is measured in terms of the Bit Error Rate (BER).The multiple access interference has to be considered when evaluating system performance in an asynchronous system.

M. B. Pursley, “Performance evaluation for phase-coded spread-spectrum multiple-access communication — Part I: System analysis,” IEEE Trans. Commun., vol. COM-25, pp. 795–799, Aug. 1977.



Worst-case probability of error : Consider the Pursley model with K users such that data bk(t) generated by kth user is transmitted using code sequence Ck(t), at a time delay τk and carrier phase offset θk relative to the intended user. Pursleyhas shown that for a large community of users (N>>1), the worst-case probability of error Pmax is given by:

where Φ(.) is cdf for N(0,1) and Cc is the maximum magnitude of the aperiodic cross-correlation given by:

is the magnitude of the cross-correlation between code sequences that belong to users k and i.

max2 21 [1 ( 1)( )]c b

o

C EP KN N

⎛ ⎞≤ −Φ − −⎜ ⎟

⎝ ⎠

,max{ ( )}, 1 1c k iC C N Nτ τ= − ≤ ≤ −

, ( )k iC τ



System analysis for average signal-to-noise ratio: we treat the phase shifts, time delays, and data symbols as mutually independent random variables. The interference terms are random and are treated as additional noise.This signal-to-noise ratio is computed by means of expectations with respect to the phase shifts, time delays, and data symbols.The interference-to-signal ratio from (K−1) other active users is:

The signal-to-noise ratio for the ith channel, SNRi, is given by:

The probability of error Pe is given by:( ) for K userse iP Q SNR=

( 1)3

Ia

s

P KQP N

−= =

1 1( 1)

2 3s n oI

in I s s b

P P NP KSNRP P P P E N

− −⎛ ⎞ ⎛ ⎞−

= = + = +⎜ ⎟ ⎜ ⎟+ ⎝ ⎠ ⎝ ⎠SNR=Ps/Pn=2Eb/No in AGWN


§3.8 HomeWorks

第三章重點整理、讀書心得報告 (手寫)Problems: 3.3, 3.4, 3.8


Chapter 4: Pseudo-Random Code Sequences for Spread-Spectrum Systems

1. Introduction2. Basic Algebra concepts 3. Arithmetic of binary polynomial 4. Computing elements of GF(2m) 5. Binary pseudo-random sequences 6. Complex sequences


§4.1 Introduction

The technique for generating code sequences should be aimed at a large family of sequences in order to accommodate a number of users and, with an impulse-type autocorrelation which enhances the system synchronization and possibly with low cross-correlation functions, to reduce multiple access interference.Some important topics for pseudo-random sequence generators are studied:

Basic binary algebra maximal-length sequences or simply m-sequences.decimation and the preferred pairs of the m-sequenceGold, Kassami and Walsh sequences


§4.2 Basic Algebra concepts

Number theory: number set, group and field. An algebraic set of M elements is defined by an array of M real or complex numbers acted upon by an operator for addition or a multiplication The set is said to be a closed set if the algebraic operations on the original set, yield a new element already existing in the same set. Informal Definitions (math.stanford.edu/~brubaker/152groups.pdf ) : A GROUP is a set in which you can perform one operation (usually addition or multiplication mod n for us) with some nice properties. A RING is a set equipped with two operations, called addition and multiplication. A RING is a GROUP under addition and satisfies some of the properties of a group for multiplication. A FIELD is a GROUP under both addition and multiplication. A group is a set of elements acted upon by an operator for addition (additive group) or multiplication (multiplicative group). A ring is a set of elements operated upon by addition and multiplication. A field is defined as a ring with every element in the ring (except zero) having an inverse.



A field with a finite number of elements, M, is called a Galois (pronounced as ‘Gal-Wah’, http://tw.knowledge.yahoo.com/question/question?qid=1608031004610) field (finite field) and is denoted GF(M). Generally, finite fields only exist when M is prime or M is the power of a prime, i.e. M=Pm when m is integer. Galois field GF(M) has M elements with index 0, 1,2, . . .,M−1. The simplest Galois field uses modulo 2 arithmetic and is denoted GF(2) with elements drawn from {0, 1} which is also called a binary field.The field requires the set to possess the following properties:

Commutative propertyAssociative propertyDistributed propertyInverse property: additive inverse and multiplication inverse Closure property



Example 4.1: Consider a set of binary elements drawn from {0, 1}. Find the basic algebraic operations (addition/subtraction and multiplication/division) acted upon each pair of the set.

Mod-2 addition

Mod-2 multiplication

Mod-2 subtraction

Mod-2 division


§4.3 Arithmetic of binary polynomial

Binary finite field, GF(2)={bi∈0,1} .Extension field : with a binary field GF(2), GF(2m) is an extension Galois field having 2m elements, GF(2m)={b0 b1…bm-1 ,bi∈0,1}, where m is the number of the Galois elements in an extension Galois element (bits/symbol). E.g., m=3, then GF(2m)={000, 001, 010, 011, 100, 101, 110, 111} having 23=8 elements.Polynomials of extension field GF(2m) :

Element b0b1…bm-1∈GF(2m) can be represented by a polynomial of degree m-1, There are 2m possible polynomials corresponds to the elements of the extension field GF(2m) E.g.

1 2 10 1 2 1( )

mmb x b b x b x b x

−−= + + + +

2

3 2

2 (2 ) { | , 0,1} {0,1, ,1 }

3 (2 ) { | , , 0,1}i j i j

i j k i j k

m GF b b x b b x x

m GF b b x b x b b b

= ⇒ = + ∈ = +

= ⇒ = + + ∈



Example 4.2: Consider the polynomials P1(x) and P2(x) such that: P1(x) = 1+x+x3, P2(x) = x+x2+x3. Evaluate the following mathematical expressions: P1(x)(+-*/)P2(x).

P1(x)+P2(x)=1+x+x3 +x+x2+x3=1+x2 since x+x=0 and x3 +x3 =0P1(x)−P2(x)=1+x2

P1(x) P2(x)=(1+x+x3 )(x+x2+x3)=x+x5 +x6

P1(x)/P2(x): using long division, we get: P1(x)/P2(x)=1+(1+x2)/(x+x2+x3)

Example 4.3: Consider polynomials P1(x) and P2(x) with coefficients drawn from Galois field GF(3) such that: P1(x) = x+2x2+x3, P2(x) = 1+2x+x2 . Evaluate the following mathematical expressions: P1(x)(+-*/)P2(x).



P1(x)+P2(x)= x+2x2+x3+1+2x+x2 =1+x3 since x+2x=0 P1(x)−P2(x)= x+2x2+x3-1-2x-x2 =2+2x+x2+x3

P1(x) P2(x)=(x+2x2+x3 )(1+2x+x2)=x+x2+x4 +x5

P1(x)/P2(x): using long division, we get: P1(x)/P2(x)=x



Irreducible polynomial : An irreducible polynomial is a polynomial that cannot be factored into non-trivial polynomials over the same field. Let p(x)=p0 +p1x+p2x2+…+xm be a polynomial of degree m.Primitive polynomial : irreducible polynomial of degree m is primitive if it divides [1+xn] for which the smallest positive integer n=2m −1. (note : p0= pm=1)The addition of any two elements of GF(2m) is defined as mod-2 addition of two binary polynomials. A multiplication of two elements of GF(2m) is referred to as modulo-h(x) multiplication where h(x) is a primitive polynomial of order m.



Example 4.4: Find the elements that arise from the addition and multiplication of each pair of elements of the polynomials in GF(22).

The elements in GF(22) can be expressed as binary digits: 00, 01, 10, 11 and in binary polynomial as: 0, 1, x, 1+x.Addition:

Multiplication: we choose the following primitive polynomial, h(x) of degree 2 : h(x) = 1+x+x2


§4.4 Computing elements of GF (2m)

Representation of elements of extension field GF(2m)={b0 b1…bm-1 , bi∈0,1}: two polynomial schemes

b(x) = b0 +b1x+b2x2+…+bm-1xm-1

Alternative :Select a primitive polynomial p(x) of degree m which is primitive over GF(2m). p(x)=p0 +p1x+p2x2+…+xm

GF(2m)={0, xk , k=0,1,…, 2m -2, with modulo-p(x) }

For example: the element xm corresponds to the polynomial mod(xm , p(x))= p0 +p1x+p2x2+…+pm-1xm-1

( , ( ))kelement Mod x p x=



Example : let p(x)=1 +x+x3 be a primitive polynomial. The polynomial in GF(23) has 8 elements. They can be expressed by {b(x) = b0 +b1x+b2x2 } or {0, xk , k=0,1,…, 6, with modulo-p(x) }.

0

1

2 2

3

4 2

5 2

6 2

7

. (8) . (8)0 0 0 0 0

1 0 0 10 1 01 0 0

1 0 1 11 1 0

1 1 1 11 1 0 1

1 0 0 1

Powers of x Poly over GF Sequence over GF

xx xx xx xx x xx x xx xx

+++ +

+



Example : An example of GF(23) , generated from p(D)=1+D2+D3 with D3=1 +D2.

0

1

2 2

3 2

4 2

5

6 2

7

. (8) . (8)0 0 0 0 0

1 0 0 10 1 01 0 0

1 1 0 11 1 1 1

1 0 1 11 1 0

1 0 0 1

Powers of D Poly over GF Sequ over GF

DD DD DD DD D DD DD D DD

++ +

++



Example : An example of GF(24) , generated from p(D)=1+D+D4 with D4=1 +D.


§4.5 Binary pseudo-random sequences

4.5.1 Generation of binary pseudo-random sequencesWe use shift register (Linear Feedback Shift Register, LFSR) to perform multiplication and division of polynomials over GF(2).Consider the simple feedback shift registers shown in Figure 4.1where the initial states of the r-stage shift registers are (ar−1, ar−2, . . ., a1, a0) and the feedback function f(x0, x1, . . . , xr−1) is a binary function.


4.5.1 Generation of binary pseudo-random sequence

Circuit for polynomial multiplication input code sequence : generator polynomial.

with hj=1 implies connection and hj=0 implies no connection

0( ) nn

nA x a x

∞

=

=∑

0( )

rn

nn

h x h x=

=∑

D-FF register

Br(x)

B1(x)



Once all registers are loaded with zeros (i.e. A(x)=0), the generator could not change it’s state. Therefore an all-zero state is not allowed.The output of the jth adder is Bj(x) :

1 1 2 2 1

1

0 1

( ) ( ) ( ) , ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

r r r

j r j j

r r

B x A x h A x xh B x A x h B x x

B x A x h B x x

B x A x h B x x

− −

− −

−

= + = +

= +

= +

0 1 0 1 2

20 1 2 0 1

0

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

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

r r rr

r rr

jj

j

B x B x A x h B x x A x h A x h B x x xA x h A x xh B x x A x h A x xh A x x h

A x h x A x h x

− −

−

=

= = + = + +

= + + = + + +

= =∑



Example :

4 50 0 1 0 0 1 3 1 2 2 3 2( ) ( ) ( )

( ) ( )b D a h a h a h D a h a h D a h D

a D h D= + + + + + +

=

22 1 0

3 23 2 1 0

( )

( )

h D h D h D ha D a D a D a D a

= + +

= + + +

0 1

0 0 2 0 1 0 0

1 1 2 1 1 0 2 1 0 0 1

2 2 2 2 1 1 2 2 0 1 1 0 2

3 3 2 3 1 2 2 3 0 2 1 1 2

3 2 3 1 2 2

3 2

( ) ( ) ( )0 1 0 0 0

0123

0 4 00 5 0 00 6 0 0 0

input time b D b D b D

a a h a h a ha a h a h a h a h a ha a h a h a h a h a h a ha a h a h a h a h a h a h

a h a h a ha h

−

+ ++ + ++ + +

+



The maximum period of the binary sequence generated by the r-stage shift register is limited to 2r −1. A binary sequence which achieves this maximum period is called maximal-length sequence or simply m-sequence. M-sequence is obtained by using primitive polynomials in Table 4.1 as generator polynomials. It must be emphasized that the period of the generated sequence depends on the choice of h(x) and only connections based on these primitive polynomials are capable of generating sequences of length 2r −1.



Circuit for polynomial division

1 2( ) ( ) ( ) ( ) ( )B x A x h x A x k x= +



Suppose that k0 =0 so that the connection between multiplier k0 and the corresponding adder is disconnected and that A2(x) is taken from the output, i.e. A2(x)=B(x). Let us define the polynomial g(x) such that k(x)=g(x)+1.

Therefore from (4.12)

Generally A1(x) sets the initial state of the registers contentsand can be represented by a finite polynomial given by: A1(x)= a0 +a1x+a2x2+…+ar-1xr-1

0( ) ( ) 1, ( . ., 1, , 2 )i ig x k x i e g g k i r= + = = =

1 1

11

( ) ( ) ( ) ( )[ ( ) 1] ( ) ( ) ( ) ( ) 0( ) ( )( ) ( ) ( ) ( ) ( )

( )

B x A x h x B x g x A x h x B x g xA x h xA x h x B x g x B x

g x

= + + ⇒ + =

⇒ = ⇒ =



Figure 4.3b Multiplication by h(x) and division by g(x)

Figure 4.4 Galois linear feedback shift register sequence generator



Let . Find B(D).Solution:

Therefore, using long division, it is found that 6 7 10 11 12( )B D D D D D D= + + + + +

6 2 3 61( ) , ( ) 1 , ( ) 1h D D g D D D D D A D= = + + + + =

61

2 3 6( ) ( )( )

( ) 1A D h D DB D

g D D D D D= =

+ + + +



Output sequence with initial states: When the registers are loaded with sequence A1(x) and h(x)=xr

The circuit in Figure 4.3b can be simplified to that shown in Figure 4.4. The loading process takes r time units and while theregisters are loading, the output B(x) is zero for these r time units. Therefore, B(x) starting at time r is:

2 11 0 1 2 1

1 1

( )

( )( ) ( ) ( )( )

( ) ( )

rr

r

r

A x a a x a x a xh x x

A x h x A x xB xg x g x

−−= + + + +

=

= =

1( )( )( )

A xB xg x

=



There are two methods of constructing LFSR sequence generators for a given generator polynomial:

Galois feedback implementation where the output bits are feedback through the connection polynomialFibonacci feedback generator where the output bits are feedback into the shift register directly.

Consider Fig. 4.4 that B(x)g(x)=A1(x)=a0+a1x+a2x2+…+ar-1xr-1. So the coefficients of xi, i>r, in B(x)g(x) should be zero.

OrSince g0=1, we get

1 1 0 0, r i r r i r ig b g b g b i r− − − ++ + + = >

1 11 1

11 1 0 0

( ) ( ) ( )

( )

i i i r i ri i i r i r

rr r ir r n i nn

B x g x b x b x b x b x

g x g x g x g g b x

− − + −− − + −

−− −=

= + + + + + +

+ + + + = + +∑

1 1 1 1 1, ri r i r r i r i m i mmb g b g b g b g b i r− − − + − −== + + + = >∑



Fibonacci feedback generator

Therefore, the coefficient for xi, i>r, will be satisfied by

Figure 4.5 Fibonacci linear feedback shift register sequence generator

21 2

10 1

1 2 21 1 2 2

( ) ( ) ( ) ( )

[.. ..] [.. ..] .. [.. ..]

rr

r ir i

i i i r ri i i r r

B x B x xg B x x g B x x gb b x b x b x

b x xg b x x g b x x g− − −− − −

= + + +

+ + + + + + =

+ + + + + + + + +

1 1 1 1 1, ri r i r r i r i m i mmb g b g b g b g b i r− − − + − −== + + + = >∑



Maximum period of shift register generation Given nonzero initial state, the shift registers (Galois, Fibonacci generator) will never reach an all-zeros state.r-state shift register has at most 2r-1 nonzero states. That is maximum period= 2r-1.The same circuit may generate many different output sequences depending on which initial state is used.



Procedure of finding maximum period : for an LFSR with the generator connection defined by the polynomial g(x) is computed as follows:

Find the reciprocal polynomial of g(x), , which is also a polynomial of degree r. E.g., g(x)= g0 +g1x+ g2x2+…+grxr, gr(x)=gr+gr-1x+…+g1xr-1+g0xr

Find the smallest integer N, such that xN+1 is divisible by gr(x)N is the maximum period of the sequence.

1( ) ( )rrg x x g x−=



Example 4.5: Consider the sequence generator shown in Figure 4.4 with the generator polynomial g(x) is given by: g(x)=1+x2+x3+x4. Assume the initial load of the register be 0001.

i. Find the output periodic sequence.ii. What would the maximum possible period be?



Example 4.5: i. Find the output periodic sequence.

ii. What would the maximum possible period be?

Compute (xN+1)/gr(x)

maximum period=7Note that the maximum possible period for the output sequence given by this sequence generator is 24 −1=15.

2 3 7 9 10 1412 3 4

( ) 1( ) 1( ) 1

A xB x x x x x x xg x x x x

= = = + + + + + + ++ + +

1 4 2 3 4 4 2( ) ( ) (1 ) 1rrg x x g x x x x x x x x− − − −= = + + + = + + +

7 73

4 2

1 1 ( 1)( ) 1r

x x x xg x x x x+ +

= = + ++ + +

1011000 1011000 101 . . . . .



Output sequence with initial states of Fibonacci generatorGiven initial state of Fibonacci generators A1(x), what does the output B(x) become? Since Galois and Fibonacci feedback generators are equivalent, we want to find an equivalent initial state of Galois circuit, A1’(x) to produce the output sequence of Galois circuit B’(x)B’(x)~=B(x).

Procedure : Let equivalent initial state of Galois circuit be A1’(x). Let the output of Galois circuit be equal to the output of the rightmost shift register in Fig. 4.5. including the initial state, it becomes A1(x)+xr B(x). Compute A1’(x) by :



Procedure : Compute A1’(x) by :

Then compare the coefficients of both sides to find A1’(x) .B(x)= x-r (-A1(x)+A1’(x)/g(x))~= x-r (A1’(x)/g(x))

1

1

1 1 1 12 1

0 1 2 1

( )'( ) is the Galois circuit output assume both are the same( )

( ) ( ) is the Fibonacci circuit output

( )( ( ) ( )) ( ) ( ) ( ) ( )

( )

r

r

rr

A xB xg x

A x x B x

g x A x x B x A x A x g x A xa a x a x a x −−

⎫′= ⎪

⎬⎪+ ⎭

′ ′+ = ⇔ =

′ ′ ′ ′+ + + + =2 2 1

0 1 2 0 1 2 1 ( )( )r r

r rg g x g x g x a a x a x a x−

−+ + + + + + + +



Example 4.6: Find the sequence at the output of the generator shown in Figure 4.5 with polynomial given by: g(x)=1+x2+x3+x4. Assume the initial load of the register be 0001.Solution

2 3 2 3 41 1 0 1 2 3

2 30 1 1 3 1

( ) ( ) ( ) ( ) (1 )(1)

1, 0, 1, 1 ( ) 1

A x g x A x a a x a x a x x x x

a a a a A x x x

′ ′ ′ ′ ′= ⇔ + + + = + + +

′′ ′ ′ ′= = = = ⇒ = + +

The output polynomial B(x) is the same as that generated from generator shown in Figure 4.4 in the previous example.


§ 4.5.2 Maximal-length sequences

The m-sequence is a sequence for an r-stage LFSR generator connected according to a primitive polynomial of degree r selected from Table 4.1.The m-sequences have a maximum period N=2r−1 due to

If g(x) is primitive polynomial of order r, the reciprocal polynomial gr(x) is also a primitive polynomial of order r. (1972, Peterson, “error correcting code”) The maximum period of the sequence is the smallest integer N, such that xN+1 is divisible by gr(x)Irreducible polynomial of degree r is primitive if it divides [1+xn] for which the smallest positive integer n=2r −1..

The m-sequences are their two-valued autocorrelation functions

D. V. Sarwate and M. B. Pursley, "Cross correlation properties of pseudorandom and related sequences", Proc. IEEE, vol. 68, pp. 593-619, May 1980.



The periodic cross-correlation function between any pair of m-sequences of the same period can be relatively large. A list of the peak magnitude for the periodic cross-correlation between pairs of m-sequences for 3≤r≤12 is shown in Table 4.2.



The m-sequences have the following well-known properties:There are exactly N non-zero sequences representing the N different phases of the m-sequence. If the m-sequence is x=(x0, x1, x2, . . . , xN−1), then the non-zero sequences are (x1, x2, x3, . . . , xN−1, x0), (x2, x3, x4, . . . , xN−1, x0, x1), (x3, x4, x5, . . . ,xN−1, x0, x1, x2), etc.Let T be a phase shift operator such that Tx= (x1, x2, x3, . . . , xN−1, x0), T2x= (x2, x3, x4, . . . , xN−1, x0, x1), T3x=(x3, x4, x5, . . . ,xN−1, x0, x1, x2), etc.Shift-and-add property of the m-sequences suggests that the modulo-2 sum of an m-sequence and any phase shifted version of itself is another phase of the same m-sequence.The Hamming weight of an m-sequence is (N+1)/2. This is because the number of ones in an m-sequence is (N+1)/2 . The number of zeros is of course (N-1)/2 .



The m-sequences have the following well-known properties:The periodic autocorrelation function of an m-sequence is a two-valued function given by R(τ) = N for τ = jN, otherwise R(τ) =0 where j is any integer. A plot of the autocorrelation for an m-sequence with chip duration Tc and time period NTc is shown in Figure 4.6.

Figure 4.6 Auto-correlation function for an m-sequence.



The m-sequences have the following well-known properties:A run is defined as a set of identical symbols within the m-sequence. The length of the run is equal to the number of these symbols in the run. For any m-sequence generated by r-stage shift registers, it has the following statistics:

1 run of ones of length r1 run of zeros of length r−11 run of ones and one run of zeros of length r−22 runs of ones and 2 runs of zeros of length r−34 runs of ones and 4 runs of zeros of length r−48 runs of ones and 8 runs of zeros of length r−52r−3 runs of ones and 2r−3 of zeros of length 1.

For example the m-sequence 000100110101111 contains a total of eight runs as follows: one run of four 1s, one run of three 0s, one run of two 0s, two runs of s single 1, two runs of a single 0.


§ 4.5.3 Decimation of m-sequences

Consider sequence u=u0, u1, u2, u3, . . . . Then sequence v is denoted by u[q] if performing the decimation by q of u where q is a positive integer, i.e., taking every qth bit of the sequence u.Let u be an m-sequence and v = u[q] =u0, uq, u2q, u3q, . . . ..Property of decimation of m-sequence1) If u is an m-sequence with period N and generator (primitive) polynomial

h(x), then u[q] has period Nv where Nv=N/gcd(N,q). The new sequence u[q] can be generated using LFSR with generator polynomial ˆh(x).

2) When the decimation yields an m-sequence, it is called proper decimation(但長度不知道) and if gcd(N, q)=1, sequence v=u(q) is also an m-sequence of period N. Proper decimation guarantees that sequence v=u(q) is an m-sequence and the polynomial ˆh(x) is primitive.

D. V. Sarwate and M. B. Pursley, "Cross correlation properties of pseudorandom and related sequences", Proc. IEEE, vol. 68, pp. 593-619, May 1980.



Property of decimation of m-sequence3) The decimation of any phase of sequence u will give a certain phase of v,

i.e., Tiu[q]= Tjv.4) Among the N phase sequences generated by h(x), there is exactly one ũ

satisfies ũi= ũ2i. This unique sequence ũi is called a characteristic phase of m-sequence u. It was shown that ũ=ũ[2].

E.g., u=100,010,011,010,111,100,010,011,010,111 is an m-sequence of length 15.ũ=00,010,011,010,111,1

5) When proper decimation is achieved by odd integer q, then u[2jq]= u[2jq mod N] represents different phases of the same m-sequence u[q].

6) Decimating u by q=(N−1)/2 and by N-1 produces the same m-sequence with different phases and u[N-1] will be a certain m-sequence generated by the reciprocal polynomial of h(x), hr(x)=?



Example 4.7: A primitive polynomial h(x) of degree 5, given by the octal number 45, is used to generate an m-sequence u. Decimation of u by 3 generates the m-sequence 75 and decimation by 5 produces the m-sequence 67. Consider every possible decimation in the range 1≤q≤N−1, find the m-sequences that can be formulated by each decimation.

Note : It is convenient and conventional to represent a polynomial h(x)=h0+h1x+…+ hm-1xm-1+hmxm by a binary vector h=(hm,hm-1,…,h1,h0 ) , and to express this vector in octal notation. For example, the polynomials x4+x+1 and x5+x+1 are represented by the binary vectors 10011 and 100101, respectively, and the octal notation for these polynomials is 23 and 45, respectively. Its reciprocal polynomial is hr(x)= h0xm+h1xm-1+…+hm-1x+hm



Sequence u is generated by primitive polynomial h0(x) given by the octal number 45 where h0(x) = 1+x2+x5. Decimation of u by 2jq where j≥0 with q=1, that is 1, 2, 4, 8, 16 produces different phases of u.

u[3] is generated by primitive polynomial h3(x) given by the octal number 75 which is equivalent to [111101] in binary. h3(x) = 1+x2+x3+x4+x5. Decimating the sequence u by 2jq where j≥0 and q=3, that is 3, 6, 12, 24, 17,results in phases of m-sequence given by h3(x).

Decimating the sequence u by 5 generates an m-sequence with primitive polynomial h5(x) given by the octal number 67 is equivalent to [110111] in binary where: h5(x)=1+x+x2+x4+x5. Similarly, decimating 5, 10, 20, 9, 18produces the m-sequence given by polynomial 67.



Consider decimating u by 7. This decimation will generate the same primitive polynomial as decimating by 14, 28, 25, 19. Now decimation by 14 is equivalent to decimation by 14+N=45 which is the same as decimating u(3) by 15. Decimation by 15= (N−1 )/2 results in an m-sequence generated by the reciprocal polynomial of h3(x) (75). The octal number 75 is [111101] in binary and the reciprocal polynomial h7(x) is given by [101111] that is the octal number 57. h7(x) = 1 + x + x2 + x3 + x5

Consider decimating u by q=11. The same primitive polynomial is used when decimation by 22, 13, 26, 21. Now the decimation by 13 is equivalent to decimating by 13+2N=75 which is the same as decimating u(5) by 15. Thus the m-sequence is produced by the reciprocal polynomial 67=[110111] and the reciprocal polynomial is given by 73=[111011]. Thus, the primitive polynomial h11(x) is: h11(x) =1+x+x3+x4+x5



Lastly, consider decimating u by 15. The same polynomial corresponds to decimation by 30, 29, 27, 23. The primitive polynomial is the reciprocal polynomial of h0(x) which octal number is 45 or [100101] in binary. The reciprocal polynomial h15(x) is [101001], which is 51 in octal format. h15(x)=1+x3+x5 . Summary of the sequences: The decimation of u generates a total of six m-sequences for primitive polynomials of degree 5. These m-sequences have the following primitive polynomials:

h0(x) = 1 + x2 + x5 generates m-sequence u.h3(x) = 1 + x2 + x3 + x4 + x5 generates u(3)h5(x) = 1 + x + x2 + x4 + x5 generates u(5)h7(x) = 1 + x + x2 + x3 + x5 generates u(7)h11(x) = 1 + x + x3 + x4 + x5 generates u(11)h15(x) = 1 + x3 + x5 generates u(15)

Decimation relations for m-sequences of period 31. When traversed clockwise, solid lines and dotted lines correspond to decimations by 3 and 5, respectively.


§ 4.5.4 Preferred pairs of m-sequences

The periodic autocorrelation of m-sequence is a two-valued function. However, the cross-correlation between two m-sequences generated by two different primitive polynomials can be three-valued, four-valued, or possibly many valued.preferred pair is a pair of m-sequences which has a three-valued cross-correlation function. The designated pair could be selected as the m-sequence u and its decimated version v=u[q] for some q.Let u be m-sequence with period N=2r–1. Conditions for preferred pairs, u and v=u[q], are in the following:1. r =0 mod 4, that is r is odd or r=2 mod 4.2. q is odd and satisfies either one condition :

q =2k + 1 orq=22k − 2k + 1

3. k in condition #2 is given bygcd (r, k) = 1 for r odd gcd (r, k) = 2 for r = 2 mod 4



Three-valued cross-correlation: The preferred pairs of m-sequences have three-valued cross-correlation function defined as [−1, −t(r), t(r)−2] where 1

2

22

1 2 ,( )

1 2 , 2mod 4

r

r

r oddt r

r

+

+

⎧+ ∈⎪= ⎨

⎪ + =⎩

Table 4.3 Maximum cross-correlation associated with preferred pair of m-sequences



A connected set of m-sequence is a collection of m-sequences that has the property that each pair in the set is a preferred pair. The largest possible connected set is called the maximal connected set and the size is denoted by Mr.



Example 4.8: Consider the m-sequence u generated by a primitive polynomial of degree r=5. Construct the maximal connected set of preferred pairs of m-sequencesproduced by the decimation of u. What is the size of this set?Solution: A preferred pair of m-sequences must satisfy conditions i, ii, iii.It is easy to see that Mr = 3, and that each triangle on Fig. 4 corresponds to a maximal connected set. Notice that there are eight maximal connected sets, and that each m-sequence belongs to four of them.

Figure 4.7 Set of preferred pairs of m-sequence.


§ 4.5.5 Gold sequences

If [u, v] is any preferred pair of m-sequences generated by primitive polynomials h(x) and ˆh(x) and each of degree n and period N=2n −1, then a set of Gold sequences G[u, v] is defined by {u, v, u♁v, u♁Tv, u♁T2v, u♁T3v, . . ., u♁TN-1v}

where Tiv represents m-sequence v phase shifted by i symbols with i=0, 1, 2, . . . , N−1.

The Gold set of sequences contains N+2 sequences and is generated by polynomial given by h(x) and ˆh(x).A typical Gold generator can be constructed using the preferred pair of m-sequences {u, u[3]} where:

h0(x) = 1 + x2 + x5 gives m-sequence u.h3(x) = 1 + x2 + x3 + x4 + x5 gives u[3].



A typical Gold generator using m-sequences u, u[3]

Figure 4.8 Block diagram of Gold generator.



Property of Gold code.Autocorrelation : autocorrelation functions are not two-valued except u and v. In fact, it takes three values as cross-correlation.Cross-correlation :

The lower bound on the peak cross-correlation (max) between any pair of sequences of period N in a set of M sequences is given by Welch bounds (Welsh, 1974) as:

The maximum cross-correlation between the preferred sequences of a Gold sequence is:

max1 For large values of N and M1

MN NNM

−Φ ≥ ≈

−

12 2

max 22 2

1 2 22 2 for ( )

1 2 22 2 for

m m

m m

N m oddt m

N m even

+

+

⎧+ ≈ ≈ ∈⎪Φ = = ⎨

⎪ + ≈ ≈ ∈⎩


§ 4.5.6 Kasami sequences

A set of Kasami sequences can be generated using two different types

i. a small set of Kasami sequences.ii. a large set of Kasami sequences.

Generating a small set of Kasami sequences.Let u be an m-sequence generated by a primitive polynomial hu(x) with period N=2n −1 where n is an even number.Generate a sequence v using primitive polynomial hv(x) by decimating u by 2n/2 +1; that is v=u[2n/2 +1].It has been proven that v is an m-sequence with period Nv=2n/2 -1.The small set of Kasami sequences is generated by the primitive polynomial h(x)=hu(x)hv(x) using a module addition of u with all possible phases of v; that is:



The small set of Kasami sequences is generated by the primitive polynomial h(x)=hu(x)hv(x) using a module addition of u with all possible phases of v; that is: {u, u♁v, u♁Tv, u♁T2v, u♁T3v, . . ., u♁TNv-1vThe small set of Kasami sequences contains 2n/2 sequences.Cross-correlation of small set of Kasami sequences: with three-valued correlation function [−1, −s(n), s(n)−2] where s(n)= 2n/2+1. the peak value is about

The maximum magnitude of correlation acquired is s(n) and it is approximately one half of the maximum magnitude value achieved by Gold set.

2 2max ( ) 1 2 2

n n

s n NΦ = = + ≈ ≈



Example: The small set of Kasami sequence with N=63, n=6 and hu(x)=1+x+x6, hv(x)=1+x2+x3

Need some modification



Generating a large set of Kasami sequences.: This set contains an m-sequence of period N, N=2n-1, n∈even, and the related Gold sequence as well as the related small set of Kasamisequence.

Assume that m-sequence u is generated by primitive polynomial hu(x) of degree n and has a period NSequence v is the decimation of u by s(n), i.e. v=u[s(n)=2n/2+1] generated by the primitive polynomial hv(x) of degree n/2 and has period 2n/2 −1. (u,v) produces a small set Kasami sequence.Sequence w=u[t(r)] is generated by a polynomial hw(x) of degree n with period N where t(r)=1+2(n+2)/2 in (4.25) where n≣2mod4Then the large set of Kasami sequences KL(u) is generated by primitive polynomial h(x)=hu(x) hv(x) hw(x).



KL(u) = u ♁ v ♁ w, and has a period N=2n −1. The size of KL(u) is

2n/2(2n +1) for n≣2 mod 4, and 2n/2 (2n +1)−1 for n≣0 mod 4 (using Gold-like sequence).

The correlation function for KL(u) is many-valued with values chosen from the set {−1, −t(r), t(r)−2, −s(n), s(n)−2}. The maximum magnitude of correlation is t(r)=1+2(n+2)/2.

Table 4.5 Comparison between Kasami and Gold sequences



Example: The large set of Kasami sequence with N=63, n=6 and hu(x)=1+x+x6 , hv(x)=1+x2+x3 , hw(x)=1+x+x2+x5+x6

Need some modification


§ 4.5.7 Walsh sequences

Walsh code sequences are obtained from the Hadamard matrix where each row in the matrix is orthogonal to all other rows, and each column in the matrix is orthogonal to all other columns. Walsh code is an orthogonal code.Generation of Hadamard matrix

Each column or row in the Hadamard matrix corresponds to a Walsh code sequence of length n

[ ]1 2 4

2

0 0 0 00 0 0 1 0 1

00 1 0 0 1 1

0 1 1 0

: , 2N N nNN N

Hadamard Matrix where N

⎡ ⎤⎢ ⎥⎡ ⎤ ⎢ ⎥= ⇒ = ⇒ =⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦

⎡ ⎤= =⎢ ⎥⎣ ⎦

H H H

H HH

H H



Hardware implementationThe input to the generator is eight bits from the clock 01010101, so the output from the first T-FF is 00110011 and from the second T-FF is 00001111. The binary variables u2 u1 u0 represent a Walsh code index



Walsh sequence: used to separate users in the same channel of synchronization systems. (the zero correlation properties of Walsh sequences will be destroyed in asynchronization system)row3 and row4 will be indistinctive in asynchronous system H8.

8

0 0 0 0 0 0 0 00 1 0 1 0 1 0 10 0 1 1 0 0 1 10 1 1 0 0 1 1 00 0 0 0 1 1 1 10 1 0 1 1 0 1 00 0 1 1 1 1 0 00 1 1 0 1 0 0 1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

H



Alternative representation : amplitude level, 0 1, 1 -1


§ 4.5.8 Multi-rate orthogonal codes

Wideband CDMA is designed to support a variety of data service from low to high bit rates. Since the spreaded signal bandwidth is the same for all users, multiple-rate transmission needs multiple spreading factors (SF) in the physical channels.

Spreaded signal bandwidth is the same for all users, i.e., the chip period is constant (1.28M or 3.84M cps)Multiple-rate transmissions are used (bit-period is varied) and need multiple spreading factor in the physical channels.The spreading sequences for multiple spreading factors still need to satisfy the orthogonal requirement. Orthogonal variable spreading factor (OVSF) code is designed to achieve these goals.

E. H. Dinan and B. Jabbari, "Spreading Codes for Direct Sequence CDMA and Wideband CDMA Cellular Networks," IEEE Communications Magazine, Vol.36 Issue 9, Sep 1998, pp. 48 –54.



Let Rmin is the minimum bit rate and SF=N=2n

Modified Hadamard transformation : Let CN be a matrix of size N x N and {CN(n)}, n=1,..,N denote the set of N binary spreading codes of N-chip length, where CN(n) is the row vector of N elements and N = 2n . {CN(n)} are the N Walsh orthogonal sequences (but they are different to Walsh sequences in order).

min1

min2

min

min

2

2 2 Therefore we need orthogonal codes with different length.4 2

2 2

n

n

n

k n k

R SFR SFR SFR SF

−

−

−

⎫⇒ =⎪

⇒ = ⎪⇒⎬⇒ = ⎪

⎪⇒ = ⎭



CN is generated from CN/2 as

Code tree for OVSF codeVariable length orthogonal codes that preserve orthogonality between different rates can be generated recursively with root value 1 by using a tree structureNotice that the codes generated in this way are exactly the same as the rows of Hadamard matrices, only the order is differentSo user data rate determines the spreading factor which determines the level of the tree

/ 2 / 2

/ 2 / 2

/ 2 / 2

/ 2 / 2

(1) (1)(1)(2) (1) (1)

( 1) ( / 2) ( / 2)( ) ( / 2) ( / 2)

N NN

N N N

N

N N N

N N N

C CCC C C

CC N C N C N

C N C N C N

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= =⎢ ⎥⎢ ⎥

− ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦



Code tree for OVSF code



Orthogonal propertyCodes in the same layer are orthogonalCodes of different layers may be not orthogonal if one code is amother code of the other. Otherwise they are orthogonal.Therefore a code can be used in a channel if only if no other code is used in the same channel on the path from the specific code to the root of the tree or subtree.

Example: If C8(1) is used. Then C16(1), C16(2), C32(1), C32(2), C32(3), C32(4) can not be assigned to users requesting lower rates, and C4(1), C2(1) can not be assigned to users requesting high rates



Example:


§4.6 HomeWorks

第四章重點整理、讀書心得報告 (手寫)Problems: 4.4, 4.5, 4.7, 4.8


Chapter 5: Time Synchronization of Spread Spectrum Systems

1. Introduction2. Code acquisition3. Analysis of serial acquisition system in AWGN channels4. Sequential detection acquisition system5. Matched filter acquisition system6. Effects of frequency errors on the acquisition detector

performance7. Code tracking in AWGN channels8. Dither early-late noncoherent tracking loop9. Time synchronization of spread spectrum systems in

mobile fading channels


§5.1 Introduction

Synchronization Code acquisition : coarse synchronizationCode tracking : fine synchronization.

Figure 5.1 Direct sequence spread-spectrum code time synchronization.


§5.2 Code acquisition

Code acquisition (coarse synchronization) : search a region of time and frequency uncertainty to get the right time-delay (phase) and frequency offset.

Time-delay (phase) :Propagation delay due to propagation distanceRelative clock instability between Tx/Rx

Frequency offsetDoppler frequency due to relative movement between Tx/RxRelative oscillator instability between Tx/Rx

Optimum acquisition: Construct a 2-D phase/frequency uncertainty regionSearch phase/frequency uncertainty region and compare their output energy.According to the ML estimator, the point with the largest outputenergy is the answer, ^Td, ^wo.


§5.2.1 Optimum acquisition

Let r(t) be the received signal, a(t) be the reference spreading waveform. Then

( ) ( )

( ) ( )cos( ) ( )

( ) 2 ( )cos( )

( ) ( ) ( ) | ( ) ( )cos( ( ) ) '( )

If and ,

x(t) will have the maximum energy at the energy detector output

d o

d o IF

BPF d d IF o o

o o d d

r t c t T w t n t

a t c t T w t w t

x t r t a t c t T c t T w t w w t n t

w w T T

= − +

= − +

= = − − + − +

= =

r(t)

a(t) x(t)


§5.2.2 Sub-optimum acquisition system

Sub-optimal performance is achieved by assuming a negligible frequency uncertainty. This assumption will develop the 2D search into a simple 1D search system along the code phase.The synchronization detectors can be grouped into either coherent or non-coherent according to the information available concerning the carrier phase offset.

Figure 5.3 Sub-optimum acquisition system.


§5.2.2 Sub-optimum acquisition system

Figure 5.4 A block diagram for the code synchronization of a coherent detector used for QPSK system

BPF


§5.2.3 Search strategies

Conventionally, either serial or parallel search algorithms are employed to search the uncertainty region and to acquire the code phase.Parallel search

Figure 5.6 Parallel search circuit.



Serial search: Reduce the complexity, size and cost over parallel acquisitionCcompare the radiometer output with a preset threshold,

If radiometer output < threshold, then the delay of PN local code generator is incremented by △. The comparison is reexamined.If radiometer output > threshold, then the PN code is assumed tohave been acquired. Code tracking is initiated thereafter.

Worst case acquisition time: let Ti =NTc, △=Tc/2 thenTacq= (NTc /△)Tc=2NTc

Two-stage detection system:The correlation functions are calculated in 3 successive points When no comparator output exceeds the threshold, the sequences are advanced over 3Tc.When the threshold is exceeded, the largest correlation output is chosen.Worst case acquisition time: Tacq= (NTc /3△)Tc=2NTc /3



Serial search:

Two-stage detection system:


§ 5.7 Code tracking in AWGN channels

Having acquired the received code phase within less than one chip, the receiver has to track any changes in the code phase using code phase trackingloops. Code phase tracking loops, which is similar to PLL, are used to track the timing delay error between the received code and the locally generated code.Tracking the delay errors is based on the correlation between the received code and two different replicas of the received code: one is an early versionand the other is late version of the locally generated code.Code tracking loops can be grouped in several ways:

Coherent loops that make use of an available carrier phaseNon-coherent loops without carrier phase information.DLL (delay-locked loop): full time early-late tracking loop (two independent correlators)TDL (tau-dither loop): time-shared early-late tracking loop (single correlator). Loops that use two independent correlators are known as full-time early–late loops and loops that share a single correlator


§ 5.7.1 Optimum code tracking

Ignoring the AWGN effect.With m-sequence of period N=7, we have the output of the correlator for ^Td>Td and |^Td-Td|


§ 5.7.1 Optimum code tracking

The output of the lowpass loop filter (averaged DC component) can be computed by

The step in time delay for each iteration is ∆Td=-αD, i.e., TdTd +∆Td= Td –αD, where α is a condtant

[ ]

[ ]

( ) ( )

1 and

1 and

0

o d d

d d d d cc

o d d d d cc

d d c

dD E y E s t T s t Tdt

N if T T T T TNTNE y if T T T T TNT

if T T T

⎡ ⎤= = − −⎢ ⎥⎣ ⎦+⎧ > −


§ 5.7.2 Baseband early–late tracking loop

Early–late tracking loop: K1=1


§ 5.7.3 DLL in noiseless channels

Define the delay error (td −ˆtd) normalized with respect to chip duration Tc as δ:

The early correlator output, y1(t) is:

the late correlator output, y2(t) is:

Delay-lock discriminator output ε(t) is :

where ε(t) contains DC-component used for code tracking.Time-varying component behaved as code self-noise which is ignored since it usually falls outside the bandwidth of loop filter.

( ) /d d ct t Tδ = −

2 1( ) ( ) ( ) / 2 ( ) ( ) ( )2 2d d c d ct y t y t P c t T c t t T c t t Tε Δ Δ⎡ ⎤= − = − − − − − +⎢ ⎥⎣ ⎦

1( ) / 2 ( ) ( )2d d cy t P c t t c t t TΔ= − − +

2 ( ) / 2 ( ) ( )2d d cy t P c t t c t t TΔ= − �

Documents

Spread Spectrum Communicationsscholar.fju.edu.tw/課程大綱/upload/049469/scheme/971... · 2008-09-09 · FJU-EE – YUJL - Spread Spectrum Communications– Outline - Page 3 TEXT