Spread Spectrum Communications

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(1)Spread Spectrum Communications Jung–Lang Yu Department of Electrical Engineering Fu Jen Catholic University Taipei, Taiwan Tel: +866-2-29052102 Fax:+866-2-29042638 e-mail : [email protected]. FJU-EE – YUJL - Spread Spectrum Communications– Outline - Page 1. OUTLINE 1. 2. 3. 4. 5. 6. 7. 8.. Introduction (p1~p50) Introduction to Digital Communications (p51~p152) Fundamentals of Spread-Spectrum Techniques (p153~p194) Pseudo-Random Code Sequences for Spread-Spectrum Systems (p195~p252) Time Synchronization of Spread-Spectrum Systems (p253~p336) Cellular Code Division Multiple Access (CDMA) Principles (p337~p400) Multi-User Detection in CDMA Cellular Radio (p401~p472) CDMA Wireless Communication Standards (p473~p504). FJU-EE – YUJL - Spread Spectrum Communications– Outline - Page 2.

(2) 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~Chap7 3. S.G. Glisic, “Adaptive WCDMA,” 2003. 全華書局 4. S. Verdu, “Multi-user Detection,” 1998 5. V.P. Ipatov, “Spread spectrum and CDMA, Principles and Applications,”, 2005 6. A.J. Viterbi,”Principles of spread spectrum communication,” 1995. FJU-EE – YUJL - Spread Spectrum Communications– Outline - Page 3. GRADE Exercise 40% [重點整理、讀書心得報告 (手寫)] Midterm Exam. : 11 Nov. 30% Term project and presentation 30% (duplicate its simulation results) : 06~13 Jan. 2010 Due date z Topic of term project : on 23 Nov.. FJU-EE – YUJL - Spread Spectrum Communications– Outline - Page 4.

(3) Project topics: . Topics : z Multi-user detection z RAKE receivers z Channel Estimation z Synchronization in spread spectrum systems z Power control in CDMA systems z Handover techniques z Wireless LANs based on spread spectrum technology z Advanced wireless techniques: OFDM, UWB, etc. z Any topics related CDMA are OK after discussing with Dr. Yu.. . References : z IEEE Transaction on Communication z IEEE Transaction on Vehicle Technology z IEEE Transaction on Signal Processing z IEEE Transaction on Wireless Communication z Signal Processing. FJU-EE – YUJL - Spread Spectrum Communications– Outline - Page 5.

(4) Chapter 1 : Introduction. 1. 2. 3. 4. 5. 6. 7.. Mobile communications Development of CDMA 3G development history International Telecommunication Union 4G possible techniques Important Research Topics Reference. FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 1. §1.1 Mobile communications 1st Generation: analog voice service (9.6Kbps) z z z z. AMPS(USA), Advances Mobile Phone Service, IS-54 NMTS(Europe), Nordic Mobile Telephone System TACS(England), Total Access Communication System NAMTS(Japan), NEC Advances Mobile Telephone System. 2nd Generation: voice and lower-rate data service (9.6Kbps) z z z z z. D-AMPS(USA), Digital-AMPS, IS-136 GSM(Europe), Global System for Mobile Communication DCS(England), Digital Cellular System PDC(Japan), Personal Digital Cellular CDMA(North American), IS-95. FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 2.

(5) §1.1 Mobile communications 2.5 Generation: enhanced data service for GSM z GPRS for packet switching system (9k, 13.4k, 15.6k, 21.4k/slot, 8 slots/channel) z HSCSD for high-speed circuit switching data (14.4k/slot, 8 slots/channel) z 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) FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 3. §1.1 Mobile communications Quality of Service in old generations: z Voice Quality (improved) , Coverage (world-wide seamless access) & Costs (low) z Quality of Service aspects : low BER and low delay time. New Services and Capabilities in new generations z Enabling new voice and data service that are not currently available with 1G and 2G technology z High bandwidth services (data, image, multimedia). FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 4.

(6) §1.1 Mobile communications. FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 5. §1.2 Development of CDMA 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).. FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 6.

(7) §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 z 144K bps in fast moving speed z 384K bps in walking, slow moving speed z 2M bps in standstill environment Proposals for 3G Standards z Wideband-CDMA (Europe) z CDMA-2000 (North American) z TD-SCDMA (China). FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 7. §1.3 3G development history Seamless World-wide Access. FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 8.

(8) §1.3 3G development history 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-2000 It is an extension of narrowband CDMA (IS-95) z CDMA one, integration of IS-95 in 1997, 8 voices, 64K bps/channel z CDMA-2000 1X, wideband service, 307K bps in 1.25M Hz BW z CDMA-2000 3X, wideband service 2M bps in 5M Hz BW z CDMA 2000 1X EV-DO, 2.5M downlink /307K uplink bps in 1.25M Hz BW FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 9. §1.3 3G development history TD-SCDMA It is the combination of TDMA system and synchronization CDMA, which is proposed by the Simens (Germany) and Datang (China) in 1999.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 10.

(9) §1.3 3G development history. 3G Telecommunication licenses. z z z. 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 CDMA2000 Korean: 3 for W-CDMA and 1 for CDMA2000. FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 11. §1.3 3G development history Network Operators. http://www.cdg.org & http://www.umtsworld.com/umts/links.htm FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 12.

(10) §1.4 International Telecommunication Union IMT-2000 specification : ←→ ITU z International Mobile Telecommunications 2000 z the time schedule for the first trial system : year 2000 z the frequency range to be used : around 2000 MHz. The International Telecommunication Union (ITU) is responsible for the IMT-2000 specification. z The requirements for the 3G standardisation have been discussed under the term FPLMTS (Future Public Land Mobile Telecommunications System) since the early 1990s. z In the mid 1990s the term FPLMTS was changed to the term IMT-2000.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 13. §1.4 International Telecommunication Union UMTS (WCDMA) ←→ ETSI z UMTS stands for Universal Mobile Telecommunications System z UMTS is a member of the ITU‘s IMT-2000 global family of 3G mobile communication systems z The European Telecommunication Standards Institute (ETSI) is responsible for the UMTS standardization z UMTS is the successor standard to the second generation GSM. z 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. FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 14.

(11) §1.4 International Telecommunication Union Air Interfaces for 3G : WCDMA WCDMA GSM. FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 15. §1.4 International Telecommunication Union Air Interfaces for 3G : WCDMA. FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 16.

(12) §1.4 International Telecommunication Union Air Interfaces for 3G : CDMA2000. FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 17. §1.4 International Telecommunication Union Spectrum Allocation for 3G. FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 18.

(13) §1.5 4G possible techniques . W-CDMA with OFDM technique → Multi-Carrier CDMA LAS-CDMA (large area synchronization CDMA by China) Position CDMA UWB (ultra wideband) technique 4G standards will be proposed in 2010.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 19. §1.6 Important Research Topics 1. PN sequences 2. Code acquisition / Code tracking 3. Modulation/demodulation 4. Power control 5. Handover techniques 6. RAKE receivers 7. Channel Estimation 8. Adaptive CDMA networks 9. Radio fading channel 10. Multiuser detection 11. Advanced CDMA systems, MC-CDMA, OFDM FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 20.

(14) §1.7 Reference 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.. CDMA development group, http://www.cdg.org 3G Partnership Project 2，http://www.3gpp2.org/ 3 G Partnership Project，http://www.3gpp.org/ UMTS World，http://www.umtsworld.com/umts/links.htm 3G Today - IMT-2000 Standard，http://www.3gtoday.com/index.html CDMA2000，http://www.ericsson.com/ Cellular Online，http://www.cellular.co.za/main.htm Cellular Technologies Of The World，http://www.cellular.co.za/main.htm Philips Consumer Communications， http://www.wca.org/dgibson/index.htm TDD White Paper，http://www.tddcoalition.org/ TD-SCDMA White Paper，http://www.siemens-mobile.com/mobile Wireless Web Features - TD-SCDMA and W-CDMA make ideal partners for 3G，http://wireless.iop.org/ UMTS World，http://www.umtsworld.com/umts/links.htm. FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 21. §1.8 HomeWorks 第一章重點整理、讀書心得報告 (一~二頁). FJU-EE – YUJL - Spread Spectrum Communications– Chapter1 - Page 22.

(15) Chapter 3: Fundamentals of SpreadSpectrum Techniques 1. Historical 2. Benefits of spread-spectrum 3. Principles of spread-spectrum communications (Scholtz, 1977) 4. Most common types of spread-spectrum systems 5. Processing gain 6. Correlation functions (Sarwate and Pursley, 1980) 7. Performance of spread-spectrum systems (Pursley, 1977) .. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 1. §3.1 Historical. 3G Telecommunication licenses. z z z. 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 CDMA2000 Korean: 3 for W-CDMA and 1 for CDMA2000. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 2.

(16) §3.1 Historical Network Operators. http://www.cdg.org & http://www.umtsworld.com/umts/links.htm FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 3. §3.2 Benefits of spread-spectrum Avoiding interception z The successful interceptor usually measures the transmitted power in the allocated frequency band. z Spreading the transmitted power over a wider band undoubtedly lowers the power spectral density, and thus hides the transmitted information within the background noise. z Because of its low power level, the spread spectrum transmitted signal is said to be a Low Probability of Interception (LPI) signal.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 4.

(17) §3.2 Benefits of spread-spectrum Privacy of transmission z The transmitted information over the spread-spectrum system cannot be recovered without knowledge of the spreading code sequence. z Thus, the privacy of individual user communications is protected in the presence of other users.. Resistance to fading z The resistance of the spread-spectrum signals to multipath fading is brought about by the fact that multipath components are assumed to be independent. z 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.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 5. §3.2 Benefits of spread-spectrum . . Accurate low power position finding z The 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). z 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 scheme z Multiple access schemes are designed to facilitate the efficient use of a given network resource by a group of users. z Frequency Division Multiple Access (FDMA), Time Division Multiple Access (TDMA) and Code Division Multiple Access (CDMA) are commonly used schemes for multiple access systems. z 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.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 6.

(18) §3.2 Benefits of spread-spectrum Anti-jamming ability. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 7. §3.2 Benefits of spread-spectrum Anti-jamming ability. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 8.

(19) §3.3 Principles of spread-spectrum communication. 補充 Power Spectral Density z Formula for PSD-- Wiener Khinchin theorem, (ref. Ziemer Chapter 5, textbook p44, eq(1.115) ) ∞ z If v(t) is a stationary R.P. and v(t ) = ∑ i =−∞ ai g (t − iT ), ai ∈ R.V . where g(t) is the specified waveform, then the PSD of v(t) is given by S ( f ) = 1 S ( f ) G ( f ) 2 , v. a. T. Sa ( f ) =. ∞. ∑ R (m)exp(−2π fmT ). m=−∞. a. Ra (m) = E [ ai am+i ] , G ( f ) = FT {g (t )}. z If u (t ) = v (t ) cos(2π f 0t ) , then the PSD of u(t) is given by Su ( f ) =. 1 [ Sv ( f − f o ) + Sv ( f + f o ) ] 4. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 9. §3.3 Principles of spread-spectrum communication Example : Baseband d (t ) =. ∞. t. ∑ b p(t − iT ), b ∈{±1}, p(t ) = ∏ (T ). i =−∞. i. i. Sb ( f ) = 1; P( f ) = Tsinc( fT ) PSD{d (t )} =. 1 2 Sb ( f ) P( f ) = Tsinc2 ( fT ) D( f ) T. Example : Passband sd (t ) = 2 Pd (t ) cos( wot ) 2P [ D( f − fo ) + D( f + fo )] 4 1 = PT ⎡⎣sinc 2 ( f − f o )T + sinc 2 ( f + f o )T ⎤⎦ 2. Sd ( f ) =. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 10.

(20) §3.4 Types of spread-spectrum systems Direct-sequence (DS) spread spectrum Frequency-hoping (FH) spread spectrum Hybrid DS/FH spread spectrum. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 11. §3.4.1 Direct-sequence spread spectrum BPSK : Transmitter (Model I) Oversampling should be executed. m(t ) =. ∞. ∑ m δ (t − iT );. i =−∞. i. sn (t ) = m(t )C (t ) =. b. C (t ) =. ∞. ∑ c δ (t − jT );. j =−∞. ∞. j. c. N −1. ∑ m ∑ c δ (t − iT. i =−∞. y (t ) = sn (t ) ∗ h(t ) =. i. j =0. ∞. j. − jTc ). b. N −1. ∑ m ∑ c h(t − iT. i =−∞. i. j =0. ss (t ) = y (t ) cos( wt ) FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 12. j. b. − jTc ). ci , mi ∈ {±1} If periodic. spreading code is used, then Tb=NTc and C(t) is periodic with period N Data stream Spreading code Pulse shaping filter.

(21) §3.4.1 Direct-sequence spread spectrum BPSK : Transmitter (Model II) Shaping filter is included in c(t)and m(t). m(t ) =. ∞. ∑ m p (t − iT ); i. i =−∞. b. C (t ) =. b. ∞. ∑c. j =−∞. j. pc (t − jTc ); ci , mi ∈ {±1}. sn (t ) = m(t )C (t ); pb (t ) = Π (t / Tb ), pc (t ) = Π (t / Tc ) y (t ) = sn (t ) =. ∞. N −1. ∑ m ∑c. i =−∞. i. j =0. j. pc (t − iTb − jTc ). If periodic spreading code is used, then Tb=NTc and C(t) is a periodic waveform with period NTc Data waveform Spreading waveform. ss (t ) = y (t ) cos( wt ) FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 13. §3.4.1 Direct-sequence spread spectrum BPSK : Receiver A(t). D(T). B(t). C (t ) = ∑ c j pc (t − jTc ) j. A(t ) = 2 ss (t ) cos( wt ) = 2 y (t ) cos( wt ) cos( wt ) = 2m(t )C (t ) cos 2 ( wt ) B (t ) = A(t )C (t ) = 2m(t )C (t )C (t ) cos 2 ( wt ) = 2m(t ) cos 2 ( wt ) D (T ) = ∫. KTb. ( K −1)Tb. B (t ) dt = ∫. KTb. ( K −1)Tb. m(t )(1 + cos(2 wt ))dt = ∫. if mK = 1 ⎧T, mK dt = ⎨ b ( K −1)Tb ⎩−Tb , if mK = −1. D (T ) = ∫. KTb. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 14. KTb. ( K −1)Tb. m(t ) dt.

(22) §3.4.1 Direct-sequence spread spectrum Waveforms. Periodic spreading code. BW=?. aperiodic spreading code. m(t). C(t) y(t). BW=? FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 15. §3.4.1 Direct-sequence spread spectrum Waveforms. Figure (a) Product signal y(t) = C(t)m(t). (b) Sinusoidal carrier. (c) DS/BPSK signal. Ss(t). FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 16.

(23) §3.4.1 Direct-sequence spread spectrum. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 17. §3.4.1 Direct-sequence spread spectrum QPSK :. Figure 3.5 (a) Quadrature spreadspectrum modulator; (b) Quadrature spreadspectrum receiver.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 18.

(24) §3.4.1 Direct-sequence spread spectrum 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. z i. Determine the baseband transmitted signal. z ii. Express the signal received. Ignore the background noise. z iii. Assuming negligible noise, determine the detected signal at the output of the receiver.. Solution. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 19. §3.4.1 Direct-sequence spread spectrum Ans i z Let the data stream be denoted as m(t). The baseband spreadspectrum data mS(t) can be represented as: z mS(t)=m(t)C(t)=[01101001, 10010110, 01101001, 01101001]. Ans ii z the received signal mr(t)= mt(t)+ I(t)=mt(t)+cos(ωCt+30)= mS(t)cos(ωCt)+ cos(ωCt+30). Ans iii z The demodulated signal is mb(t)= mr(t)2(cosωCt). Therefore: mb(t)=(mS(t)cos(ωCt)+ cos(ωCt+30))2(cosωCt) ~= mS(t)+cos30 z The de-spread signal md(t) is md(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) FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 20.

(25) §3.4.2 Frequency-hoping (FH) spread spectrum Concept:. for slow FHSS , for fast FHSS ,. Th > Tb Th < Tb. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 21. §3.4.2 Frequency-hoping (FH) spread spectrum Coherent FHSS : Transmitter. N=2k subbands in FH systems. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 22.

(26) §3.4.2 Frequency-hoping (FH) spread spectrum Coherent FHSS : Receiver. N=2k subbands in FH systems. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 23. §3.4.2 Frequency-hoping (FH) spread spectrum Noncoherent FHSS: FH/MFSK combines the FH technique with Noncoherent M-ary FSK demodulation This method is applied in BlueTooth®. M=2L subcarriers, L bits/subcarrier. N=2k subbands in FH systems FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 24.

(27) §3.4.2 Frequency-hoping (FH) spread spectrum k=2, L=2 and Th = 4Ts 使用MFSK的慢速FHSS 00 00. 10. 01. 11 00. 11. 10. 01. 01 01. 10. 01. 10. 10 00. 01. 10. 11. PN序列 0 0二進位資料. 頻率. Wd. There are M=2L subcarriers. The bandwidth is equal to. Wd WS Wd. BW = Wd = M Δf = 2 L Δf Wd. 時間. T. BWtotal = Ws = 2k Wd = NWd Wd : BW before spreading. TS TC FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 25. BWtotal : BW after spreading .. §3.4.2 Frequency-hoping (FH) spread spectrum k=2, L=2 and Ts = 4Th 使用MFSK的快速FHSS. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 26. ..

(28) §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 :. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 27. .. §3.4.3 Hybrid DS/FH spread spectrum Receiver :. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 28. ..

(29) §3.5 Processing gain The effectiveness of the processor is measured with a factor called the processing gain Gp defined as: Modified signal parameter at processor output Signal parameter at input. In spread-spectrum systems, the processing gain (Gp) expresses the bandwidth expansion factor. Gp =. Bs (signal spectrum at the output ) Bb (signal spectrum at the input ). For a DS-SS system: Gp =. Rc (code sequence rate) 1/ Tc Tb = = =N Rb (data bit rate ) 1/ Tb Tc. For a FH-SS system: Gp=N=2k FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 29. .. §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. z i. Find the required chip rate. z ii. If the speech was transmitted by an FH-SS system, what would be the number of hopping channels?. Solution z i. the PCM bit rate=Rb =n×6.8=54.4 k bits/sec c Processing gain=23dB=199.53=Gp =Rc/Rb d Substituting for Rb gives Rc =10854.2 k chip/sec. z ii. the number of FH channels=N≈200. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 30. ..

(30) §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 an and bn that can be real or complex such that: {a} = {a0 , a1 , a2 ," , aN −1}. {b} = {b0 , b1 , b2 ," , bN −1} We assume the two sequences to be periodic in Sec. 3.6.1 with long period N and aperiodic in Sec.3.6.2. .. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 31. §3.6.1 Periodic correlation function The periodic correlation function Ra,b(τ) of N-element sequences {a} and {b} is defined by: N −1 Ra ,b (τ ) = ∑ n=0 anbn*+τ Periodic Auto-Correlation Function [PACF]: Ra,a(τ) Periodic Cross-Correlation Function [PCCF]: Ra,b(τ) The normalized correlation function: Ra,b(τ) /N The periodic correlation Ra,b(τ) can be expressed by modulo operation in eq(3.18), ((.))N Ra ,b (τ ) = ∑ n=0 anbn*+τ = ∑ n=0 anb((* n+τ )) N N −1. N −1. Ra,b(τ) can be separated into two parts:. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 32. ..

(31) §3.6.1 Periodic correlation function Ra,b(τ) can be separated into two parts:. Ra ,b (τ ) = ∑ n=0 anbn*+τ = ∑ n=0 anb((* n+τ )) N N −1. N −1. = a0bτ* + " + a N −1−τ bN* −1 + a N −τ b0* + " + aN −1bτ*−1 = Ra′ ,b (τ ) + Ra′′,b (τ ) Ra′ ,b (τ ) = a0bτ* + " + a N −1−τ bN* −1 = ∑ n=0 anbn*+τ N −1−τ. Ra′′,b (τ ) = aN −τ b0* + " + a N −1bτ*−1 = ∑ n= N −τ anb((* n+τ )) N N −1. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 33. .. §3.6.1 Periodic correlation function 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.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 34. ..

(32) §3.6.1 Periodic correlation function Periodic autocorrelation functions of sequence {a}. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 35. .. §3.6.1 Periodic correlation function Periodic autocorrelation functions of sequence {b}. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 36. ..

(33) §3.6.1 Periodic correlation function Periodic cross-correlation function of sequences {a} and {b}. .. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 37. §3.6.2 Aperiodic correlation function The aperiodic correlation function between sequence {a} and {b} is defined by Ca,b(τ) Ca ,b (τ ) = ∑ n=0 anbn*+τ = ∑ n=0 anbn*+τ , 0 ≤ τ ≤ N − 1 N −1−τ. N −1. = ∑ n=0 an−τ bn* , 1 − N ≤ τ ≤ 0 N −1+τ. = 0, τ ≥ N. If {a}={b} , the expression Ca,b (τ) represents the Aperiodic Auto-Correlation Function [AACF]. When {a}~= {b} , the expression defines the Aperiodic Cross-Correlation Function [ACCF]. Geometric N −1−τ * * * Ra′ ,b (τ ) = a0bτ + " + aN −1−τ bN −1 = ∑ n=0 anbn+τ ≡ Ca ,b (τ ) interpretation Ra′′,b (τ ) = aN −τ b0* + " + aN −1bτ*−1 = ∑ n=0 aN −τ + nbn* ≡ Ca ,b (τ − N ) τ −1. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 38. ..

(34) §3.6.5 Interference rejection capability Interference can be caused by an external transmitter tuned to a frequency 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: r (t ) = S s (t ) + j (t ) + n(t ) = m(t )C (t ) cos( wt ) + j (t ) + n(t ). The reference signal used by the matching filter receiver is given by: rref (t ) = 2C (t − τ ) cos( wt + θ ). FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 39. .. §3.6.5 Interference rejection capability The signal component at the matched filter output is:. 請多2倍. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 40. ..

(35) §3.6.5 Interference rejection capability 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 r (t ) = 2 PC (t )m(t ) cos (ω0t ) + 2 J cos (ω0t ) At the receiver r(t) is multiplied with a reference signal 2C(t)cos(wt) (=despreading+demodulation) y (t ) = 2 Pm(t )C (t )C (t )2 cos 2 (ω0t ) + 2 J C (t )2 cos 2 (ω0t ) = 2 Pm(t )[1 + cos ( 2ω0t )] + 2 J C (t )[1 + cos ( 2ω0t )] FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 41. .. DS spectra in tone jamming (cont.) W.W. Ali-Ahmad, “The CDMA Receiver System in an IS-98-A Standard”, Electronics Engineer, July 2000. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 42.

(36) DS spectra in tone jamming (cont.) Despreading distributed the jammer power in frequency:. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 43. DS spectra in tone jamming (cont.) Receiver filtering suppresses the jammer power:. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 44.

(37) Characterizing SS systems Code gain, for BPSK G p = Bs / Bb = Tb / TC. z 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 spreadspectrum 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.. .. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 45. Characterizing SS systems PSD=J/Bs for interference PSD=No/2 for AWGN. Bs. (SNR)i. Bb. (SNR)o. The signal power to noise power ratio at the input of the receiver is: Pr ( SNR )i =. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 46. No Bs + J 2 ..

(38) Characterizing SS systems The MF acts like a low-pass filter with BW=Bb. The power of noise and interference at the MF are given by. Pn =. No J Bb , Pj = Bb 2 Bs. The ratio of output signal power to noise power, (SNR)0 is expressed as: ( SNR )o =. Pr No J Bb + Bb Bs 2. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 47. =. Bs Pr = G p ( SNR )i Bb N o B + J s 2. .. Characterizing SS systems 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 <1. Includes noise Simplified SS block diagram and jammer. A. J. Viterbi, “Spread spectrum communications - myths and realities,” IEEE Communication Magazine, May 1979. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 48. ..

(39) Characterizing SS systems Jamming margin: Mj =. Pj + n Pr. =. Gp 1 1 ⇒Mj = = ( SNR )i ( SNR )i ( SNR ) o. Related to specified BER =Q(sqrt(SNRo)). M j (dB ) = −( SNR )i ( dB ) = G p ( dB ) − ( SNR )o (dB ) 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.. Let L be system losses between transmitter and receiver. We include the system loss into the jamming margin 1 ⇒ M j (dB ) = −( SNR )i ( dB ) − L( dB ) Pr L ( SNR )i L ⇒ M j (dB ) = G p (dB ) − ( SNR )o ( dB ) − L( dB ). Mj =. Pj + n. =. A. J. Viterbi, “Spread spectrum communications - myths and realities,” IEEE Communication Magazine, May 1979. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 49. .. Characterizing SS systems Jamming margin: for example G p = 30dB,available code gain L = 2dB, margin for system losses. SNRo = 10dB, required SNR after despreading (at the RX) ⇒ M j = 18dB,limit for additional interference and noise. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 50. ..

(40) §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. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 51. §3.7 Performance of spread-spectrum systems 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. Pursley has shown that for a large community of users (N>>1), the worst-case probability of error Pmax is given by: ⎛ 2C 2 Eb ⎞ Pmax ≤ 1 − Φ ⎜ [1 − ( K − 1)( c )] ⎟ N No ⎠ ⎝. where Φ(.) is cdf for N(0,1) and Cc is the maximum magnitude of the aperiodic cross-correlation given by: Cc = max{ Ck ,i (τ )}, 1 − N ≤ τ ≤ N − 1 Ck ,i (τ ) is the magnitude of the cross-correlation between code. sequences that belong to users k and i.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 52.

(41) §3.7 Performance of spread-spectrum systems 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: Qa =. PI ( K − 1) = Ps 3N. The signal-to-noise ratio for the ith channel,−1SNRi, is given by: −1 SNRi =. ⎛P P ⎞ ⎛ N Ps ( K − 1) ⎞ =⎜ n + I ⎟ =⎜ o + Pn + PI ⎝ Ps Ps ⎠ 3 N ⎟⎠ ⎝ 2 Eb. The probability of error Pe is given by: Pe = Q ( SNRi ) for K users FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 53. §3.8 HomeWorks 第三章重點整理、讀書心得報告 (一~二頁) Problems: 3.3, 3.4, 3.8. FJU-EE – YUJL - Spread Spectrum Communications– Chapter3 - Page 54. SNR=Ps/Pn=2Eb/No in AGWN.

(42) Chapter 4: Pseudo-Random Code Sequences for Spread-Spectrum Systems 1. 2. 3. 4. 5. 6.. Introduction Basic Algebra concepts Arithmetic of binary polynomial Computing elements of GF(2m) Binary pseudo-random sequences Complex sequences. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 1. §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: z z z z. Basic binary algebra maximal-length sequences or simply m-sequences. decimation and the preferred pairs of the m-sequence Gold, Kassami and Walsh sequences. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 2.

(43) §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.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 3. §4.2 Basic Algebra concepts . . . 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: z Commutative property z Associative property z Distributed property z Inverse property: additive inverse and multiplication inverse z Closure property. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 4.

(44) §4.2 Basic Algebra concepts 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. z Mod-2 addition z Mod-2 multiplication z Mod-2 subtraction. z Mod-2 division. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 5. §4.2 Basic Algebra concepts The arithmetic rules for Galois field GF(3) are: z Subtraction: Since 2+1=0, then −0=−0, −1=2 and −2=1 z Division: This is accomplished by multiplying by the multiplicative inverse. 1−1 =1 and 2−1 =2. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 6.

(45) §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…bm1 ,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) : z Element b0b1…bm-1∈GF(2m) can be represented by a polynomial of degree m-1, b( x) = b0 + b1 x1 + b2 x 2 + " + bm−1 x m−1 z There are 2m possible polynomials corresponds to the elements of the extension field GF(2m) 2 z E.g. m = 2 ⇒ GF (2 ) = {bi + b j x | bi , b j ∈ 0,1} = {0,1, x,1 + x} m = 3 ⇒ GF (23 ) = {bi + b j x + bk x 2 | bi , b j , bk ∈ 0,1} FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 7. §4.3 Arithmetic of binary polynomial 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). z z z z. P1(x)+P2(x)=1+x+x3 +x+x2+x3=1+x2 since x+x=0 and x3 +x3 =0 P1(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).. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 8.

(46) §4.3 Arithmetic of binary polynomial z z z z. 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. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 9. §4.3 Arithmetic of binary polynomial 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.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 10.

(47) §4.3 Arithmetic of binary polynomial Example 4.4: Find the elements that arise from the addition and multiplication of each pair of elements of the polynomials in GF(22). z 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. z Addition:. z Multiplication: we choose the following primitive polynomial, h(x) of degree 2 : h(x) = 1+x+x2. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 11. §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 z b(x) = b0 +b1x+b2x2+…+bm-1xm-1 z Alternative : c Select a primitive polynomial p(x) of degree m which is primitive over GF(2m). p(x)=p0 +p1x+p2x2+…+xm d GF(2m)={0, xk , k=0,1,…, 2m -2, with modulo-p(x) } element = Mod ( x k , p ( x)). For example: the element xm corresponds to the polynomial mod(xm , p(x))= p0 +p1x+p2x2+…+pm-1xm-1. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 12.

(48) §4.4 Computing elements of GF (2m) 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 modulop(x) }. Powers of x Poly. over GF (8) Sequence. over GF (8) 0 0 0 0 0 x0 1 0 0 1 x1 x2 x3. x x2 x +1 x2 + x x 2 + x +1 x2 +1. x4 x5 x6 x7. 1. 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 1. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 13. §4.4 Computing elements of GF (2m) Example : An example of GF(23) , generated from p(D)=1 +D2+D3 with D3=1 +D2. Powers of D Poly. over GF (8) Sequ. over GF (8) 0 D0. 0. D1. D. D2 D3 D4. 1 D2 D2 D2. D5 D6. 0 0 0. D2. D7 FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 14. 0 0 1 0 1 0. +1 + D +1 D +1 +D. 1 0 0 1 0 1 1 1 1 0 1 1 1 1 0. 1. 0 0 1.

(49) §4.4 Computing elements of GF (2m) Example : An example of GF(24) , generated from p(D)=1 +D+D4 with D4=1 +D.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 15. §4.5 Binary pseudo-random sequences 4.5.1 Generation of binary pseudo-random sequences We 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 4.1 where 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.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 16.

(50) 4.5.1 Generation of binary pseudo-random sequence . Circuit for polynomial multiplication z z z. input code sequence : generator polynomial.. r. ∞. A( x) = ∑ an x n. h( x) = ∑ hn x n. n=0. n=0. D-FF register. z. with hj=1 implies connection and hj=0 implies no connection B1(x) Br(x). FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 17. 4.5.1 Generation of binary pseudo-random sequence . . 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) : B1 ( x) = A( x) hr −1 + A( x) xhr ,. B2 ( x) = A( x)hr − 2 + B1 ( x) x. % B j ( x) = A( x)hr − j + B j −1 ( x) x % Br ( x) = A( x)h0 + Br −1 ( x) x B ( x) = Br ( x) = A( x)h0 + Br −1 ( x) x = A( x)h0 + [ A( x)h1 + Br − 2 ( x) x] x = A( x) h0 + A( x) xh1 + Br − 2 ( x) x 2 = A( x) h0 + A( x) xh1 + " + A( x) x r hr r. = A( x)∑ h j x j = A( x) h( x) j =0. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 18.

(51) 4.5.1 Generation of binary pseudo-random sequence . Example : h( D) = h2 D 2 + h1 D + h0 a ( D) = a3 D 3 + a2 D 2 + a1 D + a0 input time b0 ( D ) b1 ( D ) 0 a0 a1 a2 a3 0 0 0. −1 0 1 2 3 4 5 6. 0 a0 h2 a1h2 a2 h2 a3 h2 0 0 0. 0 a0 h1 a1h1 + a0 h2 a2 h1 + a1h2 a3 h1 + a2 h2 a3 h2 0 0. b( D ) 0 a0 h0 a1h0 + a0 h1 a2 h0 + a1h1 + a0 h2 a3 h0 + a2 h1 + a1h2 a3 h1 + a2 h2 a3 h2 0. b( D ) = a0 h0 + ( a1h0 + a0 h1 ) D + " + ( a3 h1 + a2 h2 ) D 4 + a3 h2 D 5 = a ( D )h( D ) FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 19. 4.5.1 Generation of binary pseudo-random sequence . . 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 z z. 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.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 20.

(52) 4.5.1 Generation of binary pseudo-random sequence. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 21. 4.5.1 Generation of binary pseudo-random sequence. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 22.

(53) 4.5.1 Generation of binary pseudo-random sequence . Circuit for polynomial division. B ( x) = A1 ( x)h( x) + A2 ( x) k ( x). FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 23. 4.5.1 Generation of binary pseudo-random sequence . 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. g ( x) = k ( x) + 1, (i.e., g 0 = 1, gi = ki , i = 2" r ). . Therefore from (4.12) B ( x) = A1 ( x)h( x) + B ( x)[ g ( x) + 1] ⇒ A1 ( x)h( x) + B( x) g ( x) = 0 ⇒ A1 ( x)h( x) = B ( x) g ( x) ⇒ B( x) =. . A1 ( x)h( x) g ( x). Generally A1(x) sets the initial state of the registers’ contents and can be represented by a finite polynomial given by: A1(x)= a0 +a1x+a2x2+…+ar-1xr-1. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 24.

(54) 4.5.1 Generation of binary pseudo-random sequence. Figure 4.3b Multiplication by h(x) and division by g(x). Figure 4.4 Galois linear feedback shift register sequence generator FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 25. 4.5.1 Generation of binary pseudo-random sequence Let h( D ) = D 6 , g ( D ) = 1 + D + D 2 + D 3 + D 6 , A1 ( D ) = 1. Find B(D). D6 Solution: B( D) = A1 ( D)h( D) = g ( D). 1 + D + D 2 + D3 + D 6. z Therefore, using long division, it is found that B( D) = D 6 + D 7 + D10 + D11 + D12 + ". FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 26.

(55) 4.5.1 Generation of binary pseudo-random sequence Output sequence with initial states: When the registers are loaded with sequence A1(x) and h(x)=xr A1 ( x ) = a0 + a1 x + a2 x 2 + " + ar −1 x r −1 h( x) = x r A1 ( x ) h( x ) A1 ( x ) x r B( x) = = g ( x) g ( x). 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 the registers are loading, the output B(x) is zero for these r time units. Therefore, B(x) starting at time r is: B( x) =. A1 ( x) g ( x). FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 27. 4.5.1 Generation of binary pseudo-random sequence There are two methods of constructing LFSR sequence generators for a given generator polynomial: z Galois feedback implementation where the output bits are feedback through the connection polynomial into shift registers z Fibonacci feedback generator where the output bits are feedback from the shift registers directly into the first one.. 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. B ( x ) g ( x ) = (" + bi x i + bi −1 x i −1 + " + bi − r +1 x i − r +1 + bi − r x i − r + ") ( g r x r + g r −1 x r −1 + " + g1 x + g 0 ) = " + ∑ n = 0 g n bi − n x i + " r. z Or g r bi − r + g r −1bi − r +1 + " + g 0bi = 0, i > r z Since g0=1, we get r bi = g r bi − r + g r −1bi − r +1 + " + g1bi −1 = ∑ m =1 g m bi − m , i > r FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 28.

(56) 4.5.1 Generation of binary pseudo-random sequence Fibonacci feedback generator. C(x)+xrB(x). Figure 4.5 Fibonacci linear feedback shift register sequence generator. B ( x ) = B ( x ) xg1 + B ( x ) x 2 g 2 + " + B ( x ) x r g r b0 + b1 x1 + " + br x r + " + bi x i + " = [.. + bi −1 x i −1 + ..] xg1 + [.. + bi − 2 x i − 2 + ..] x 2 g 2 + .. + [.. + bi − r x i − r + ..] x r g r. Therefore, the coefficient for xi, i>r, will be satisfied by bi = g r bi − r + g r −1bi − r +1 + " + g1bi −1 = ∑ m =1 g m bi − m , i > r r. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 29. 4.5.1 Generation of binary pseudo-random sequence Maximum period of shift register generation z Given nonzero initial state, the shift registers (Galois, Fibonacci generator) will never reach an all-zeros state. z r-state shift register has at most 2r-1 nonzero states. That is maximum period= 2r-1. z The same circuit may generate many different output sequences depending on which initial state is used.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 30.

(57) 4.5.1 Generation of binary pseudo-random sequence Procedure of finding maximum period : for an LFSR with the generator connection defined by the polynomial g(x) is computed as follows: z Find the reciprocal polynomial of g(x), g r ( x) = x r g ( x −1 ) , which is also a polynomial of degree r. E.g., g(x)= g0 +g1x+ g2x2+…+grxr, gr(x)=gr+gr-1x+…+g1xr-1+g0xr z Find the smallest integer N, such that xN+1 is divisible by gr(x) z N is the maximum period of the sequence.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 31. 4.5.1 Generation of binary pseudo-random sequence 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. z i. Find the output periodic sequence. z ii. What would the maximum possible period be?. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 32.

(58) 4.5.1 Generation of binary pseudo-random sequence Example 4.5: i. Find the output periodic sequence. B( x) =. A1 ( x) 1 = = 1 + x 2 + x 3 + x 7 + x 9 + x10 + x14 + " 2 3 4 g ( x) 1 + x + x + x 1011000 1011000 101 . . . . .. ii. What would the maximum possible period be? r −1 4 −2 −3 −4 4 2 z g r ( x ) = x g ( x ) = x (1 + x + x + x ) = x + x + x + 1 z Compute (xN+1)/gr(x). x7 + 1 x7 + 1 = 4 = ( x 3 + x + 1) 2 g r ( x) x + x + x + 1. z maximum period=7 z Note that the maximum possible period for the output sequence given by this sequence generator is 24 −1=15. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 33. 4.5.1 Generation of binary pseudo-random sequence Output sequence with initial states of Fibonacci generator z Given initial state of Fibonacci generators A(x), what does the output B(x) become? z Since Galois and Fibonacci feedback generators are equivalent, we want to find an equivalent initial state of Galois circuit, C(x) to produce the output sequence of Galois circuit D(x) z D(x)~=B(x).. Procedure : z Let equivalent initial state of Galois circuit be C(x) . ( cf. D(x)= C(x)/g(x) ) z Let the output of Galois circuit be equal to the output of the rightmost shift register in Fig. 4.5 of Fibonacci circuit including the initial state. The output is written as A(x)+xr B(x). z Compute C(x) by the following procedure: FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 34.

(59) 4.5.1 Generation of binary pseudo-random sequence Procedure : z Compute C(x) by : C ( x) D( x ) = is the Galois circuit output g ( x). ⎫ ⎪ ⎬ assume both are the same A( x ) + x r B ( x ) is the Fibonacci circuit output ⎪⎭ C ( x ) = g ( x ) D ( x ) = g ( x )( A( x ) + x r B ( x )) ⇔ C ( x ) = g ( x ) A( x ) ( c0 + c1 x + c2 x 2 + " + cr −1 x r −1 ) =. Ignore the terms with power >=r. ( g0 + g1 x + g 2 x 2 + " + g r x r )( a0 + a1 x + a2 x 2 + " + ar −1 x r −1 ) z Then compare the coefficients of both sides to find C(x) . z B(x)= x-r (D(x)-A(x))= x-r (C(x)/g(x)-A(x))~= x-r (C(x)/g(x)) = x-r D(x) FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 35. 4.5.1 Generation of binary pseudo-random sequence 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 C ( x ) = g ( x ) A( x ) ⇔ ( c0 + c1 x + c2 x 2 + c3 x 3 ) = (1 + x 2 + x 3 + x 4 )(1) c0 = 1, c1 = 0, c3 = 1, c3 = 1 ⇒ C ( x ) = 1 + x 2 + x 3. 0001. The output polynomial B(x) is the same as that generated from generator shown in Figure 4.4 in the previous example. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 36.

(60) § 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 z 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”) z The maximum period of the sequence is the smallest integer N, such that xN+1 is divisible by gr(x) z 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. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 37. § 4.5.2 Maximal-length sequences The periodic cross-correlation function between any pair of msequences 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.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 38.

(61) § 4.5.2 Maximal-length sequences The m-sequences have the following well-known properties: z 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. z 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. z 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. z 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 . FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 39. § 4.5.2 Maximal-length sequences . The m-sequences have the following well-known properties: z The periodic autocorrelation function of an m-sequence is a two-valued function given by R(τ) = N for τ = jN, otherwise R(τ) =-1 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. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 40.

(62) § 4.5.2 Maximal-length sequences The m-sequences have the following well-known properties: z 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: c 1 run of ones of length r d 1 run of zeros of length r−1 e 1 run of ones and one run of zeros of length r−2 f 2 runs of ones and 2 runs of zeros of length r−3 g 4 runs of ones and 4 runs of zeros of length r−4 h 8 runs of ones and 8 runs of zeros of length r−5 i 2r−3 runs of ones and 2r−3 of zeros of length 1.. z 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. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 41. § 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-sequence 1) 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). If gcd(N, q)=1 further, 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. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 42.

(63) § 4.5.3 Decimation of m-sequences Property of decimation of m-sequence 3) 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 ũ is called a characteristic phase of m-sequence u. It was shown that ũ=ũ[2]. ũi is i-th z E.g., u=100,010,011,010,111,100,010,011,010,111 is an m-sequence element of length 15. of ũ 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) Let the m-sequence u be generated by polynomial h(x)=h0+h1x +…+hrxr. Decimating u by q=(N−1)/2 will be an m-sequence generated by the reciprocal polynomial of h(x), hr(x)=hr+hr-1x+…+ h1xr-1+h0xr FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 43. § 4.5.3 Decimation of m-sequences 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+h1xm1+…+h m-1x+hm. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 44.

(64) § 4.5.3 Decimation of m-sequences . 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, 18 produces the m-sequence given by polynomial 67.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 45. § 4.5.3 Decimation of m-sequences . 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. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 46.

(65) § 4.5.3 Decimation of m-sequences 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 msequences have the following primitive polynomials: z z z z z z. 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.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 47. § 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 crosscorrelation 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 : c q =2k + 1 or d q=22k − 2k + 1 3. k in condition #2 is given by c gcd (r, k) = 1 for r odd d gcd (r, k) = 2 for r = 2 mod 4 FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 48.

(66) § 4.5.4 Preferred pairs of m-sequences Three-valued cross-correlation: The preferred pairs of msequences have three-valued cross-correlation function defined as [−1, −t(r), t(r)−2] where ⎧ r +1 ⎪ 1 + 2 2 , r ∈ odd t(r) = ⎨ r +2 ⎪1 + 2 2 , r = 2 mod 4 ⎩. Table 4.3 Maximum cross-correlation associated with preferred pair of m-sequences. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 49. § 4.5.4 Preferred pairs 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.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 50.

(67) § 4.5.4 Preferred pairs of m-sequences . . 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-sequences produced 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 msequence belongs to four of them.. Figure 4.7 Set of preferred pairs of m-sequence.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 51. § 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} z 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= 2n+1 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: z h0(x) = 1 + x2 + x5 gives m-sequence u. z h3(x) = 1 + x2 + x3 + x4 + x5 gives u[3].. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 52.

(68) § 4.5.5 Gold sequences A typical Gold generator using m-sequences u, u[3]. Figure 4.8 Block diagram of Gold generator.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 53. § 4.5.5 Gold sequences Property of Gold code. z Autocorrelation : autocorrelation functions are not two-valued except u and v. In fact, it takes three values as cross-correlation. z Cross-correlation : c 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: M −1 ≈ N For large values of N and M NM − 1 d The maximum cross-correlation between the preferred sequences of a Gold sequence is: m +1 m ⎧ 2 ⎪1 + 2 ≈ 22 2 ≈ 2 N for m ∈ odd Φ max = t ( m ) = ⎨ m+2 m ⎪ 1 + 2 2 ≈ 22 2 ≈ 2 N for m ∈ even ⎩ Φ max ≥ N. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 54.

(69) § 4.5.6 Kasami sequences A set of Kasami sequences can be generated using two different types z i. a small set of Kasami sequences. z ii. a large set of Kasami sequences.. Generating a small set of Kasami sequences. z Let u be an m-sequence generated by a primitive polynomial hu(x) with period N=2n −1 where n is an even number. z Generate a sequence v using primitive polynomial hv(x) by decimating u by 2n/2 +1; that is v=u[2n/2 +1]. z It has been proven that v is an m-sequence with period Nv=2n/2 -1. z 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: FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 55. § 4.5.6 Kasami sequences z 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-1v} z The small set of Kasami sequences contains 2n/2 sequences. z Cross-correlation of small set of Kasami sequences: with threevalued correlation function [−1, −s(n), s(n)−2] where s(n)= 2n/2+1. the peak value is about n n Φ max = s(n ) = 1 + 2 2 ≈ 2 2 ≈ N z The maximum magnitude of correlation acquired is s(n) and it is approximately one half of the maximum magnitude value achieved by Gold set.. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 56.

(70) § 4.5.6 Kasami sequences 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. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 57. § 4.5.6 Kasami sequences 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 Kasami sequence. z Assume that m-sequence u is generated by primitive polynomial hu(x) of degree n and has a period N z Sequence 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. z Sequence w=u[t(n)] is generated by a polynomial hw(x) of degree n with period N where t(n)=1+2(n+2)/2 in (4.25) where n=2mod4 z Then the large set of Kasami sequences KL(u) is generated by primitive polynomial h(x)=hu(x) hv(x) hw(x). FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 58.

(71) § 4.5.6 Kasami sequences z KL(u) = u ♁ v ♁ w, and has a period N=2n −1. z The size of KL(u) is c 2n/2(2n +1) for n≣2 mod 4, d and 2n/2 (2n +1)−1 for n≣0 mod 4 (using Gold-like sequence). z The correlation function for KL(u) is many-valued with values chosen from the set {−1, −t(n), t(n)−2, −s(n), s(n)−2}. The maximum magnitude of correlation is t(n)=1+2(n+2)/2.. s(n). t(n). Table 4.5 Comparison between Kasami and Gold sequences FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 59. § 4.5.6 Kasami 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. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 60.

(72) § 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 ⎡0 ⎢0 ⎡0 0 ⎤ H1 = [ 0] ⇒ H 2 = ⎢ ⇒ H4 = ⎢ ⎥ ⎢0 ⎣0 1 ⎦ ⎢0 ⎣ ⎡H H2N = ⎢ N ⎣H N. 0 1 0 1. 0 0 1 1. 0⎤ 1⎥ ⎥ 1⎥ 0 ⎥⎦. HN ⎤ : Hadamard Matrix, where N = 2n ⎥ HN ⎦. Each column or row in the Hadamard matrix corresponds to a Walsh code sequence of length n FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 61. § 4.5.7 Walsh sequences Hardware implementation z The 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 TFF is 00001111. z The binary variables u2 u1 u0 represent a Walsh code index. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 62.

(73) § 4.5.7 Walsh sequences 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. ⎡0 ⎢0 ⎢ ⎢0 ⎢ 0 H8 = ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎣0. 0 0 0 0 0 0 0⎤ 1 0 1 0 1 0 1⎥ ⎥ 0 1 1 0 0 1 1⎥ ⎥ 1 1 0 0 1 1 0⎥ 0 0 0 1 1 1 1⎥ ⎥ 1 0 1 1 0 1 0⎥ 0 1 1 1 1 0 0⎥ ⎥ 1 1 0 1 0 0 1⎦. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 63. § 4.5.7 Walsh sequences Alternative representation : amplitude level, 0Æ1, 1Æ-1. FJU-EE – YUJL - Spread Spectrum Communications– Chapter4 - Page 64.