PERFORMANCE OF COHERENT DIRECT SEQUENCE SPREAD SPECTRUM FREQUENCY SHIFT KEYING

Transcription

1 PERFORMANCE OF COHEREN DIREC SEQUENCE SPREAD SPECRUM FREQUENCY SHIF KEYING A thesis presented to the faulty of the Russ College of Engineering and ehnology of Ohio University In partial fulfillment of the requirements for the degree Master of Siene Sudhir Kumar Sunara Novemer 5

2 his thesis entitled PERFORMANCE OF COHEREN DIREC SEQUENCE SPREAD SPECRUM FREQUENCY SHIF KEYING y SUDHIR KUMAR SUNKARA has een approved for the Shool of Eletrial Engineering and Computer Siene and the Russ College of Engineering and ehnology y David W. Matola Assoiate Professor of Eletrial Engineering and Computer Siene Dennis Irwin Dean, Russ College of Engineering and ehnology

3 SUNKARA, SUDHIR KUMAR. M.S. Novemer 5. Eletrial Engineering and Computer Siene Performane of Coherent Diret Sequene Spread Spetrum Frequeny Shift Keying 83pp. Diretor of hesis: David W. Matola We analyze the performane of a multi-user oherently-deteted diret sequene spread spetrum frequeny shift eying modulated system. Detetion is done with respet to an aritrary user. We derive it error proaility expressions for oth synhronous and asynhronous systems. For the synhronous system, we analyze different variations of the system in order to ensure orthogonality. Orthogonality is ahieved via oth frequeny separation and orthogonal odes. In the asynhronous system, orthogonality is ahieved only through frequeny separation. he hannel impairments onsidered were additive white Gaussian noise and multi-user interferene. We then extend this to a Rayleigh fading hannel, and provide analytial and simulation results for single user and multiuser ases. We also analyzed the spetral harateristis of these systems. Approved: David W. Matola Assoiate Professor of Eletrial Engineering and Computer Siene

4 Anowledgements I would lie to than my thesis advisor Dr. David W. Matola, for his exellent guidane and support during the ourse of this thesis. his wor would not have een possile without his insightful suggestions, and I feel privileged to have wored with him. I would also lie to than Dr. Jeffrey Dill, Dr Mehmet Celen and Dr. Sergio Lopez for agreeing to e on my ommittee. None of this wor would have een possile without the support of my GA supervisor, Issam Khoury. I am very muh grateful to him for all the help he has extended to me during my stay at OU. I than my parents and rother for their love and support. Finally, I lie to than all my friends who inspired me during my studies at Ohio University.

5 5 ale of Contents Astrat...3 Anowledgements...4 List of Figures...7 CHAPER : INRODUCION...9. Digital Communiations...9. Spread Spetrum Communiations....3 Digital Modulation ehniques....4 hesis Sope...3 CHAPER : SYSEM MODEL...5. Frequeny Shift Keying...5. Literature Review ransmitter Struture Reeiver Struture System Variations....4 Power Spetrum... CHAPER 3: ANALYSIS Single User Bit Error Proaility Deision Statistis System Variations Asynhronous System...4

6 6 3.5 Rayleigh Fading Channel...48 CHAPER 4: SIMULAIONS Simulation Model Simulation Results...54 CHAPER 5: SUMMARY, CONCLUSIONS AND FUURE WORK Summary and Conlusions Suggestions for Future Wor...63 REFERENCES...64 APPENDIX...65

7 7 List of Figures Figure.. Blo diagram of a typial digital ommuniation system... Figure.. BFSK modulated waveform for transmitted sequene CPFSK...6 Figure.. BFSK modulated waveform for transmitted sequene non-cpfsk.6 Figure.3. ransmitter lo diagram...8 Figure.4. Reeiver lo diagram for first user... Figure.5. Power spetrum for a 3 user BFSK system...3 Figure.6. Power spetrum of single user DS-SS-BFSK...4 Figure.7. Power spetrum for two user ase....6 Figure.8. Power spetrum of two user ase: different PG suh that BW is equal...8 Figure.9. P vs. E /N, two users, different PG, same main-loe andwidth...9 Figure 3.. BER vs. E /N performane omparison analytial of oherent and nonoherent BFSK...3 Figure 3.. Simulated and analytial performane BER vs. E /N of a single user oherent BFSK system on the AWGN hannel...3 Figure 3.3. Single user BER performane for DS-SS-BFSK modulated system...33 Figure 3.4. Chip timing relationship etween users p and, L 4, ε Figure 3.5. Single user x and hannel struture for a DS-SS-BFSK system over Rayleigh faded hannel...49 Figure 4.. Shemati desription of MALAB simulation model...53

8 8 Figure 4.. P vs. E /N, synhronous, different spaing, different odes, PG 5, 3 users ase...55 Figure 4.3. P vs. E /N, synhronous, different spaing, same odes PG5, 3 users ase...56 Figure 4.4. P vs. E /N, synhronous same spaing, different odes, PG3, users ase Figure 4.5. P vs. E /N, synhronous, same spaing, same odes PG3, users ase Figure 4.6. P vs. E /N, asynhronous, PG 3, users...59 Figure 4.7. P vs. E /N, Rayleigh fading hannel, PG3...6 Figure 4.8. P vs. E /N, multi-user Rayleigh fading hannel, PG 5, users...6

9 9 Chapter INRODUCION In this hapter, we review asi onepts of digital ommuniation systems used in this researh. We present a rief overview of the spread spetrum varieties of digital ommuniation systems. hen, we present a rief desription of various digital modulation shemes. Finally, we present the sope of this thesis.. Digital Communiations Digital ommuniations is a ranh of ommuniations that utilizes disontinuous signals, i.e., signals whih appear in disrete steps, for example and for inary. It is different from analog ommuniations, whih uses ontinuous waveforms for transmitting data. he advantages of digital ommuniations tehniques inlude the following: greater data proessing options and flexiilities; roustness to transmission impairments suh as noise; and the aility to use error-detetion and error-orretion odes whih further improve the performane. he disadvantages are that it requires somewhat omplex equipment, may in some instanes have limited transmission speed, and for some ases may require more andwidth. he prinipal feature of a digital ommuniation system is that during a finite interval of time, the transmitter sends a waveform from a finite set of possile waveforms, in ontrast to an analog ommuniation system, whih sends a waveform from an infinite variety of waveform shapes with theoretially infinite resolution []. In a digital ommuniation system, the ojetive at the reeiver is not to reprodue a

10 transmitted waveform with preision; instead, the ojetive is to determine from a noisepertured signal whih waveform from the finite set of waveforms was sent y the transmitter. A lo diagram of a digital ommuniation system is shown in Figure. []. Information Soure Soure Enoder Channel Enoder Digital Modulator Channel Information Sin Soure Deoder Channel Deoder Digital Demodulator Figure.. Blo diagram of a typial digital ommuniation system. he upper row in Figure. depits the various signal transformations from the information soure to the transmitter output. Soure enoding is performed primarily to remove redundant information, whereas the hannel oding is done in order to derease the proaility of it error in the presene of random noise. Modulation is neessary to onvert the symols to waveforms that are ompatile with the transmission hannel. We provide more details aout modulation shemes in Setion.3. he lower row of los denotes the signal transformations from the reeiver input to the information sin. It an

11 e oserved that the proesses undergone from the reeiver input to information sin are asially opposite to the transformations undergone y the signal from the soure to the transmitter output. A detailed desription and funtionality of eah lo in Figure. an e found in [].. Spread Spetrum Communiations Over the last 5 years, a lass of digital modulation tehniques alled "Spread Spetrum" SS has een developed. he SS investigation was motivated primarily y the desire to ahieve highly seure digital ommuniation. hese tehniques are alled spread spetrum eause the transmission andwidth employed is muh greater than the minimum andwidth required to transmit the information. Spreading is aomplished y means of a spreading signal, often alled a ode signal or pseudo-random PR or pseudo-noise PN signal. One important parameter of a SS system is the proessing gain, defined as the ratio of transmission andwidth to the information andwidth. he advantages of SS systems inlude the following: suppression of interferene suh as that whih omes from multipath propagation; resistane to jamming; redution of energy density; and use in multiple aess tehniques suh as ode-division multiple aess CDMA. An SS system an primarily e implemented in one of two varieties: diret sequene DS and frequeny hopping FH. In a DS-SS system, the information sequene is multiplied y a high-speed PN signal for spreading. he PN signal is independent of the data and from a spetral perspetive, it has noise-lie properties. At the reeiver, a replia of the PN signal from the transmitter is used for de-spreading and

12 susequent data reovery. he wideand signal required for a SS system is generated in a different manner in a FH system. he FH system taes the data signal and modulates it with a arrier signal whose enter frequeny hops from frequeny to frequeny over a wide and. he speifi order in whih frequenies are oupied is a funtion of the ode signal, and the rate of hopping from one frequeny to another is a funtion of the information rate. In this wor, we fous on DS-SS ommuniation systems. We do this for two reasons: DS-SS tehnology is in popular use today in many appliations, and the use of FH-SS for FSK modulation has een thoroughly studied..3 Digital Modulation ehniques Digital modulation is the proess y whih digital symols are transformed into waveforms that are ompatile with the harateristis of the hannel. In the ase of aseand modulation, these waveforms tae the form of shaped pulses. Bandpass modulation is the proess y whih an information signal is impressed upon a sinusoidal waveform; for digital modulation, suh a sinusoid of duration is referred to as a digital symol. he sinusoid has just three features that an e used to distinguish it from other sinusoids: amplitude, frequeny, and phase. hus, andpass modulation an e defined as the proess wherey the amplitude, frequeny, or phase of a radio frequeny arrier, or a omination of them, is varied in aordane with the information to e transmitted. he general form of the arrier wave is given as follows: t A t os π f t t s θ. where, At is the time varying amplitude, f is the arrier frequeny, and θ t is the time varying angle.

13 3 Some of the types of andpass modulation tehniques ommonly employed are Phase Shift Keying PSK, Frequeny Shift Keying FSK, Amplitude Shift Keying ASK, Continuous Phase Modulation CPM, et. When the reeiver exploits nowledge of the arrier s phase to detet the signals, the proess is alled oherent detetion; when it does not utilize a phase referene, suh a proess is alled nonoherent detetion. Nonoherent systems are typially less omplex than oherent ones, ut oherent systems have etter it error rate performane..4 hesis Sope Non-oherent detetion of FSK, and of SS-FSK modulated systems has een investigated [3], [4]. Coherent FSK is has some appliations in optial ommuniation systems [6], [7], [9]. he main ojetive of this thesis is to develop analytial results for oherent detetion of SS-FSK modulated systems, and orroorate these results with omputer simulations. he spetral harateristis of FSK waveforms are first desried. We investigated the power spetra of oth unspread and DS-spread FSK waveforms. We have also derived a set of relationships that allow different system users to have the same signal andwidth, ut different proessing gains, a variation heretofore not explored. o derive the proaility of it error expressions, we egin y onsidering a synhronous system. Synhronism refers to synhronism at the hip level. An example is a ase to ellular transmitter. We onsidered different variations of this system, in terms of frequeny separation and odes, to he for orthogonality. Proaility of it error expressions were developed for all orthogonal ases. We then onsidered an

14 4 asynhronous system, with all user signals eing asynhronous with the user signal with respet to whih we are deteting. he hannel impairments we onsidered were additive white Gaussian noise AWGN, and multi-user interferene. We also extended the analysis to a flat Rayleigh faded hannel, and derived proaility of it error expression for a multi-user system. We provide simulation results to orroorate our analysis.

15 5 Chapter SYSEM MODEL In this hapter, we disuss the asis of frequeny shift eying modulation and then riefly disuss previous and related wor. We then desrie the system model employed in this thesis. After this, we present a desription of the different ases we have onsidered to ensure orthogonality of the signals. Finally, we desrie the power spetra of spread and unspread FSK waveforms.. Frequeny Shift Keying Frequeny Shift Keying FSK is a digital modulation tehnique in whih the symols are represented y unique frequeny signals that are transmitted through the hannel. he general analytial expression for an M-FSK modulated signal is given y []. E s i t os ω t φ i t, i,..,m. where ω i πf i, the term f i is the tone frequeny, in Hz, orresponding to the transmitted symol, and is a funtion of arrier frequeny f and the transmitted its. he term φ is the phase assoiated with the frequeny f i ; often this is modeled as random. For M-FSK, f i taes M disrete values. he parameter E is the symol energy of the signal and is the symol time. he symol time is the time required to transmit one symol. he arrier frequeny is related to the it/symol time, and is typially muh greater than the reiproal of the it/symol time. Figure. shows a ontinuous phase inary FSK CPFSK modulated waveform for an example transmitted sequene.

16 6 Amplitude time Figure.: BFSK modulated waveform for transmitted sequene CPFSK. he it time is.s, and the frequeny separation is /. he hoie of frequenies in Figure. yields a ontinuous phase waveform, whih is not typially the ase for FSK. For the same transmitted sequene, and different spaing etween the frequeny tones we otain a non ontinuous phase waveform as shown in Figure.. he it time is again.s, and the frequeny separation is.5/. Amplitude time Figure. : BFSK modulated waveform for transmitted sequene non-cpfsk.

17 7. Literature Review As mentioned in Chapter, non-oherent detetion of SS-FSK modulated signals has een studied reently, y Yang and Hanzo [3]. In this paper, they onsider a spread spetrum multiple aess system where the DS spread signals are only orthogonal over the symol duration and not over the hip duration. herefore, the frequeny and of a spread FSK tone may e fully or partially overlapping with other spread signals. hese authors develop an estimate of the variane of the multiple-aess interferene, with the aim of analyzing performane in this multi-user environment. hey onsider nonoherent demodulation at the reeiver. heir paper onludes that for a given system andwidth, the system s BER it error rate an e optimized y ontrolling the amount of spetral overlapping. In our researh, we are also assuming that the DS spread signals are orthogonal over the symol duration only. Ryu et.al., in [4], extended this analysis y onsidering multi-tone jamming as an additional impairment. Geraniotis [5] analyzed the performane of non-oherent DS spread multiple aess systems. Coherent demodulation was not onsidered eause of its more omplex implementation. Yet as noted in the first hapter, oherent FSK is now finding some appliation, in partiular in optial ommuniation systems [6], [7], thus we have analyzed the performane of a oherently deteted SS-BFSK modulated system..3 ransmitter Struture We are onsidering a multi-user system with additive white Gaussian noise AWGN, and multi-user interferene MUI as the impairments. Later, we also address

18 8 hannel fading. Eah user generates its it sequene, and the it sequene is then BFSK modulated, i.e., at symol it transitions, there is a shift in the frequeny of the modulated signal. he modulated waveforms are then spread using diret sequene spreading odes. We onsider different ases when all the users have the same spreading odes and all of them have different spreading odes. he spreading odes employed are pseudo-random odes, for the most part this is disussed susequently. All the transmitted signals are then transmitted over the AWGN hannel. Figure.3 shows the transmitter diagram. A possile appliation for this ind of transmitter struture is a ellular ase station. t FSK Modulation t s t t t FSK Modulation.. FSK Modulation t t... s t t s n t r t Figure.3: ransmitter lo diagram. he transmitted signal for the th user is given as follows: s t { π [ f t ] t θ } P tos.

19 9 where P is the power of the th user transmitted signal, and t is the signature spreading waveform onsisting of a sequene of retangular pulses g, where eah hip pulse is of duration. he spreading waveform is defined as t P t g,.3 g g where {±}, and the waveform P t-nz is a retangular pulse equal to unity from g tnz to tnz, and zero elsewhere. he th user data waveform t onsists of a sequene of mutually independent inary random variales with unit amplitude, or -, orresponding to retangular pulses of duration : t P t n,.4 n n with n {±}. he frequenies of the tones are at f n, with i / 4, where is the symol it time, and i is a fixed integer, whih speifies the spaing etween the frequeny tones for the th user. he variale θ is the aritrary phase introdued y the th FSK modulator. nt is the AWGN..4 Reeiver Struture he reeiver we employ is a onventional orrelator reeiver, and we detet with respet to an aritrary user denoted the first user. he remaining users are onsidered interfering users. he lo diagram for the reeiver is shown in Figure.4.

20 tos { π f t } θ rt Z Z- Selet largest value of Z & Z - ˆ t { π f t θ } tos Fig..4: Reeiver lo diagram for first user. he reeived signal, whih is otained after adding all the user transmitted signals and the AWGN, is multiplied with the spreading ode of the first user, and then demodulated with respet to the frequeny tones of the first user. Detetion is oherent. We assume perfet nowledge of the phase throughout this wor. he outputs of the multipliers are integrated over one it time and the outputs of the orrelators are then ompared. he larger of the two is assigned to a inary. he output of this omparator is then ompared to the nown transmitted its of the first user, to determine the it error ratio BER for this system. We have onsidered different variations at the transmitter end to ensure orthogonality among signals, and we initially onsider a synhronous system. his synhronism refers to synhronism at the hip level. Suh a setting is a model for a point-to-multipoint lin, where all user signals are generated at a ommon ase station, e.g., a terrestrial ellular forward lin. hese ases are now desried.

21 .5 System Variations Case : Users are separated in frequeny and have different spreading odes he user signals are separated in frequeny y virtue of the spaing etween the unspread signals, and signals have different random spreading odes. Orthogonality is ahieved via the frequeny separation and not y the odes. he odes are used here for spetral spreading and signal onfidentiality only. o ensure orthogonality for a oherent FSK system, the minimum separation etween the frequeny tones of a given user should e equal to /, where is the it time. In general, orthogonality an e maintained y employing any integer multiple of this minimal separation. he spreading odes, as mentioned, are randomly generated. he frequeny spaing etween the users signals is etter illustrated in Fig..5, whih shows the unspread power spetra for a 3- user ase. he innermost user s frequeny tones are separated y user s frequeny tones are separated y are spaed y /. he seond /, and the outermost user s frequeny tones 3 /. With this hoie of frequeny separation, all the user signals are orthogonal with respet to eah other. Case : Users are separated in frequeny with the same spreading ode he user signals are separated in frequeny in the same manner as in ase. he only differene here is that all user waveforms are spread using the exat same spreading ode. Case 3: Users have the same frequeny separation and different spreading odes. In this ase, orthogonality of the system is ensured y hoosing orthogonal spreading odes for the users. wo spreading sequenes are said to e orthogonal if their

22 orrelation is zero. We employ the popular Walsh-Hadamard odes for this orthogonal ode set, whih requires that all signals e synhronized, at the hip level. All the users have the exat same frequeny separation etween their frequeny tones, and the signals are orthogonal y virtue of the orthogonal spreading odes employed to differentiate etween the users. Case 4: Users have the same frequeny separation and the same spreading ode. Even though the frequeny tones of individual user signal are orthogonal, the user signals are not mutually orthogonal in this ase. Sine all users have the same spreading ode, the signals are not orthogonal, and in fat, the signals are fully orrelated. his ase is mentioned only for ompleteness, as fully orrelated signals an not e employed in a multi-user appliation. We then extend the analysis to an asynhronous system. In an asynhronous system, ode orthogonality is not employed, as the signals are asynhronous at the hip level, and no odes are nown that maintain orthogonality over an aritrary range of relative delays. Orthogonality of suh a system is ahieved via frequeny separation of the tones..6 Power Spetrum he Power Spetral Density PSD of a signal represents the distriution of power in the frequeny domain. FSK is a form of frequeny modulation and the spetra of FM signals are very omplex due to the fat that the modulating wave doesn t just ause a hange in the frequeny of the arrier. he FM proess reates many sideands. FSK

23 3 spetra are thus muh more ompliated to predit than those of linear modulations suh as PSK. We have plotted power spetra for oth un-spread and spread BFSK ases, otained via omputer simulation using the periodogram method [8]. he power spetra for multi-user BFSK signals are as expeted. Figure.5 shows a set of example power spetra for a 3 user BFSK modulated system unspread. - - Power,dB frequeny Figure.5: Power spetrum for a 3 user BFSK system. In this figure, the frequeny tones of the innermost user are separated y /, the seond user tones are separated y 4/, and the 3 rd user tones are separated y 6/. For plotting, the it time is normalized to unity. he arrier frequeny is set to 9R /. his value was hosen so that the sideloes are lear and evident in the plot. he sampling frequeny f 4 f. he power spetra were generated using s

24 4 realizations, eah of 8 its. he signal parameter values were maintained for all users, and the frequeny tones of the users have different separations, as noted. he power spetrum of a spread BFSK signal has also een plotted. he input inary its are first BFSK modulated, and this waveform is then multiplied with the spreading waveform with hip rate R. he arrier frequeny is a funtion of R. he spaing etween the frequeny tones is still a funtion of R. Figure.6 shows the power spetrum of a single user s spread BFSK waveform. For this ase, the proessing gain is large enough so that the spetrum appears to e essentially that of the random inary spreading waveform. Power Spetrum, DS-SS-BFSK, PG3,R,f s 4*R Frequeny/f s Figure.6: Power spetrum of single user DS-SS-BFSK, R, PG3, f s 4R. his figure was plotted for a proessing gain of 3. he arrier frequeny was f 4R, where R PG R. he sampling frequeny s f 6 f 4R. he spaing etween the two frequeny tones is /.

25 5 Figure.7 shows the DS-SS-FSK power spetra for two users. We have used different spreading odes on the two users. he dotted waveform represents the power spetra of user and the solid line waveform represents user. he speifiations for user s signal are as follows: it rate R, proessing gain PG 3, hip rate R PG R, arrier frequeny f 4R. he frequeny tones are at f ± R. he first nulls on either side of f our at f R R and f R R. he speifiations for user s signal are as follows: it rate R, proessing gain PG 3, hip rate f 8R R PG R, arrier frequeny f 4R. he frequeny tones are at and ±. he first nulls on either side of f our at f R 4R R R f 4. Not apparent in this plot is the slightly different andwidths of these two signals. his is due to the different values of frequeny separations for the two tones of eah user signal. For proessing gains very large with respet to these frequeny separations, this andwidth differene will e minor, ut for smaller proessing gains, the signal andwidths may differ sustantially. In order to have all signals oupy the same andwidth ~ main loe, we also investigated a setting wherein y hanging the proessing gain of the various user signals, we ould have all the signal spetra oupy the same spetral and. Speifially, we analyzed the ase where the spetral nulls of the main loes of all the user spetra ourred at the same frequenies.

26 6 Power Spetrum, DS-SS-BFSK -5 - User User Frequeny/f s Figure.7: Power spetrum for two user ase. For a generi ase with two users with the same it rate positive side of arrier frequeny our at the following frequenies: R, the nulls on the User : User : f f R R i R i R.5 where here, i,i are integers that speify the spaing etween the frequeny tones of user and user, respetively. Similarly, the nulls on the negative side of the arrier frequeny our at the following frequenies: User : User : f f R R i R i R.6 For the first nulls to e equal, we have from Eq..5 and Eq..6

27 7 R i R f R i R f Using the nowledge that R PGR, we get i i R PG R R PG Caneling out R, we get i i PG PG.7 Similarly, for the first nulls on the negative side to oinide we have i i PG PG.8 herefore the ondition, from either.7 or.8, that will ensure that two users will have the same main-loe andwidth is i i PG PG.9 We have to ensure that i i is an integer multiple of for the proessing gains to e integers. Using this ondition, we show in Figure.8 that the main-loe andwidths of oth the users are indeed the same when this ondition is met. In this figure, we used i, and i 8. he proessing gain of user is, and therefore from Eq..9, we have the proessing gain of the seond user is 8-/ 4.

28 8 Power Spetrum, DS-SS-BFSK -5 User User Frequeny/f s Figure.8: Power spetrum of two user ase: different PG suh that BW is equal. Figure.9 plots the simulated proaility of it error performane for this ase of unequal proessing gains, on the AWGN hannel. From the figure, we see that orthogonal signaling performane is otained for this ase. Sine the proessing gains are different, we employ different spreading odes. he odes are synhronous at the hip level. Figure shows the performane of user in the presene of AWGN and the seond user. he BER performane of the seond user is similar to that of the first user in idential onditions.

29 9 Analytial P Simulated P P E /N, db Figure.9: P vs. E /N, two users, different PG, same main-loe andwidth.

30 3 Chapter 3 ANALYSIS In this hapter, we analyze the performane of the DS-SS-BFSK system for oth the synhronous and asynhronous ases. We analyze the different variations of the system, disussed in Chapter. 3. Single User Bit Error Proaility When the reeiver exploits nowledge of the arrier s phase to detet the signals, the proess is alled oherent detetion; when it does not utilize a phase referene, suh a proess is alled non-oherent detetion. For two FSK tones to e orthogonal, the minimum separation etween them has to e / s if the detetion is oherent, and the minimum separation has to e / s if the detetion is non-oherent, where s is the symol/it time. he general expression for the proaility of it error for inary oherent signals is given as []: P ρ E / N π u exp du 3. where ρ osθ is the ross orrelation oeffiient etween signals s t and s t that are defined y. and θ is the angle etween signal vetors s and s. For inary antipodal signals suh as BPSK, θ π, so ρ. For orthogonal signals suh as BFSK, θ π /, sine the vetors s and s are perpendiular to eah other and so ρ. hus, 3. an e written as

31 3 P E / N π u exp du Q E N 3. where Qx, the omplementary error funtion, is defined as Q x, e dt x, π x gives the area under the tail of the Gaussian proaility density funtion PDF. he proaility of it error for inary non-oherent signals is given as follows []: E P exp 3.3 N o illustrate these systems performane, Figure 3. shows the BER versus energy per it to noise density ratio E /N of oth oherent and non-oherent BFSK in AWGN. t Coherent Non-Coherent - -4 P E /N Figure 3.: BER vs. E /N performane omparison analytial of oherent and nonoherent BFSK

32 3 he simulated performane of a single user oherently deteted BFSK modulated system is ompared with the analytial result in Figure 3.. he simulation is arried out with onventional BFSK for a waveform that is not spread, transmitted over an AWGN hannel. he it frequeny, R, is normalized to one. he arrier frequeny f 5R. he separation etween the frequeny tones is.5r thus ensuring orthogonality. Analytial and simulation results are in very good agreement, validating the simulation auray. Analytial P Simulated P P E /N Figure 3.: Simulated and analytial performane BER vs. E /N of a single user oherent BFSK system on the AWGN hannel. Now, if we spread the BFSK modulated waveform using a spreading ode with hip rate R, we still get orthogonality if the FSK tones are separated y / s. he

33 33 spread signals do not need to e separated y /, where is the hip time, for the system to e orthogonal. If they are separated y /, the FSK tone spetra do not overlap. Figure 3.3 shows the analytial and simulated BER results for a oherently deteted DS-SS-BFSK system with a single user. he proessing gain is PG 3, and the it rate R is normalized to one. he hip rate, R PGR, and the arrier frequeny f 4R. he frequeny tones are spaed y.5 f. One again, for this spread ase, simulation and analytial results agree very well. Figure 3.3: Single user BER performane for DS-SS-BFSK modulated system.

34 34 3. Deision Statistis he transmitted signal for the th user in a synhronous system as desried in Figure.3 is given as s t { π [ f t ] t θ } P tos 3.4 he parameters in 3.4 are desried in.3 and.4. From Figure.4, we have the reeived signal as K r t s t n t 3.5 he deision statisti for the zero th it without loss of generality is then otained as Z m r t tos[π f m t] dt 3.6 where m or -. Using Eq. 3.5 then yields Z m K s t n t tos{ π f m t}, whih an e expanded as Z m K P tos{ π f t θ } n t tos{ π f m t} dt 3.7 After applying trigonometri identities to 3.7 and performing some algera, the doule frequeny term an e ignored eause f >>. hen 3.7 an e roen up into the desired signal part, an interfering signal part, and the noise omponent. Speifially we have Z m K P Dm I m N m 3.8

35 35 where, m osθ m D m δ, 3.9 is the desired signal omponent, δ, m for m, otherwise, m the phase angle of the referene reeiver loal osillator signal when m. δ, and θ, m is he noise omponent of the deision statisti is N m P n t tos[π f m t] dt 3. he Gaussian noise has a mean of, so E [ ]. We next determine the noise variane. his is omputed as follows: Var[ N m ] E ni tos I π P Rearranging the terms on the right side, we have N m Var [ Nm] E[ ni t ni y ] os π f m t π f m t dt n yos f m y os π f m y P dy dtdy R t yos π f m t n P Sine noise is white, its autoorrelation funtion R t y σ δ t y, then Var[ N m ] σ n P os π P σ n f m t dt dt n n P N

36 36 hus we have Var [ N m ] E 3. N sine the it energy E P. For our synhronous ase, the interferene term is given y I m P P I { m t θ} t tos π dt, 3. where P I is the power of the interfering th user, and P is the power of user the user whose its we are deteting. Let x m, where is the it transmitted y the th user, and m or. We next analyze the interferene term for eah of our ases of interest. his term will e zero mean due to the zero mean value of the random spreading odes. 3.3 System Variations Case : When f is different for all users and different odes are used for eah user, user separation is oth y ode and f. he integral over in 3. an e replaed y a sum of N integrals over orresponding to the DS hip duration. I m N l PI l l P l l osπ x t θ dt 3.3 P P I N l πx os[ πx l θ ] sin l l πx

37 37 where N is the proessing gain of the system defined as N R /R. Sine we will e modeling the multiuser interferene as a Gaussian random variale, we need to alulate the variane of the interferene term, whih is E PI [ I m ] sin πx P E N l N p l l d p N N PI d sin π x E[ l p ] P p os[ os[ os E os { πx l } θ ] { } πx p θ ] { πx l p } { } θ πx l l p p where sinxsinx/x. When taing the ensemle average, the expetation of the osine term with θ is zero, eause this phase term is a random variale uniformly distriuted in [, π. For random odes, we have equal proailities that ±, and ±, hene E l ] equals one when lp and zero otherwise. hus we otain [ p l d p P I Var [ I m ] sin πx N P P I sin x P N π, 3.4 where N is the proessing gain, and x m. he transmitted it sequene onsists of a series of ± terms eah ourring with equal proaility. herefore {, } x taes values either in the set {, }, when the term m-, and, respetively, with equal proaility. So, the variane of the interfering term an e alulated y averaging 3.4 over all the possile four values of x. We therefore have or

38 38 Var [ I ] m PI PI sin π x 4 x P N x P m m πx sin N P I { sin π sin π } 4 P N P { i i N i i N } I sin / sin π / 4 P N π 3.5 where i and i are integers that speify frequeny spaing for the th and the st users. he mean value of the square of the desired signal term is ½, and thus using 3.5 and 3., the SNIR is given as SNIR E N 4 PI P N / { sin π i i / N sin π i i / N } 3.6 Approximating the MUI as Gaussian, the proaility of it error is now given as: SNIR P Q 3.7 o orroorate our analysis we provide simulation results in Chapter 4 for eah of the ases. Case : Here, f is different for all users, with the same spreading ode for all the users. When we are onsidering the same spreading ode for all the users, the MUI term in 3. an e written as I m P P I t tosπ x t θ dt N P I l os π x t θ l P l dt

39 39 he produt is equal to one sine the spreading ode is idential for all users. l l Using this, and evaluating the integral, we have I m P P I sin πx N πx l os [ πx l θ ] 3.8 he losed form solution for the summation term in general is given y the following simpler form [] { β [ G / ] α} sin{ G α / } sin{ α / } os α 3.9 G os l β l From 3.8, we set α 4πx, and β πx θ, therefore the summation term in 3.8 an now e written as N l os { 4πx } sin{ N πx } sin{ πx } os πx θ.5 [ ] N π l θ x hen if we let πx θ.5 N 4πx, we otain P A sin πx os A sin{ N πx } πx sin{ πx } I I m P PI P N sin N πx os A N πx os A PI sin 3. P Next, we need to alulate the variane of this interferene term. E P I [ I m ] E sin N π x os A P P P [ ] I sin N π x E os A

40 4 he term A is a funtion of the uniformly distriuted variale θ, and therefore the mean of osa is zero. hus, Var PI [ I ] sin N πx m 3. P As mentioned, the variane of the interferene is found y averaging the right side of 3. over all four possile values of x. his yields Var [ I ] m PI PI sin Nπ x 4 x P x P m m sin Nπx PI { sin Nπ sin Nπ } 4P PI { sin i i / sin π i i / } 4P π 3. hen for this ase, our SNIR is given as SNIR E N PI K { sin π i i / sin π i i / } 4P / 3.3 and the it error proaility expression is P Q E N / PI K 4P i sin sin π i i / / π i 3.4 Case 3: In this ase we have all user signals. he multiuser interferene term is f the same for all users, with different spreading ode on

41 4 I m P P I t tosπ x t θ dt N P I l os x t θ l P l π dt 3.5 We employ orthogonal spreading odes to ensure orthogonality amongst user signals. wo spreading sequenes are said to e orthogonal if their orrelation is zero. We employ the popular Walsh-Hadamard odes for this orthogonal ode set. Sine the spreading odes are orthogonal, their produt equals zero. Mathematially, we have N l 3.6 l l herefore the interferene term in 3.5 would also equal zero, leaving us with only noise. he proaility of it error is given y the expression for unspread oherent FSK, i.e., y / E P Q Q 3.7 E N N Case 4: For this ase, we have f the same for all users and the same spreading ode on all users. As mentioned previously, this is the worst ase senario, and is generally not appliale for a multiuser system. he signals are not orthogonal, and therefore the BER performane will e very poor ompared to the other systems.

42 4 3.4 Asynhronous System All the analysis so far has een done assuming the system to e synhronous at the hip level. We now extend this analysis to the asynhronous ase. In an asynhronous system, with aritrary delays among user signals, we an t ahieve ode orthogonality, sine no odes are nown that maintain orthogonality over an aritrary range of relative delays. Orthogonality of suh a system is ahieved via frequeny separation of the tones. he reeived signal rt is given y K r t s t τ n t 3.8 where nt is the hannel noise and is again assumed to e a zero-mean stationary Gaussian proess with doule-sided spetral density of N /. he term τ is the time delay relative to the referene signal. he th transmitted signal is expressed as s t τ P t τ os{π [ f t τ ] t ψ }, K 3.9 where ψ θ π [ f t τ ] τ. he deision variale is given y Z m r t tos[π f m t] dt ; m or -; From 3.8, we have Z K s τ { t}dt 3.3 t n t t os π f m m Upon simplifiation, we get Z m K P Dm I m N m 3.3

43 43 δ θ where Dm, m os m is the desired signal omponent,, m δ otherwise, m Also, the noise term is δ for m,, and θ m is the phase angle of the referene signal when m. N m P n t tos[π f m t] dt. he interferene term in 3.3 is given y the following expression with { R [ τ,, ψ ] Rˆ [ τ,, ψ ]} I m d d R d 3.3 τ [ τ, ψ ] t t τ os{ π [ m ] t ψ }, dt 3.33 [,, ψ ] t t τ os{ π [ m ] t ψ } R ˆ d τ dt 3.34 where [ τ ψ ] and ˆ [, ψ ] Rd,, τ R, d τ are the partial ross orrelation terms orresponding to the transmitted it sequenes. We now proeed to find the variane of the interferene term to alulate the it error proaility. We first alulate the variane of the ross-orrelation term R d given y We quantize the delay τ into an integer numer of hips plus a frational-hip part as follows: τ L ε 3.35 where {,,... N } L and ε. Figure 3.4 illustrates the hip timing relationship for two users.

44 user p p p p p 3 p 4 p 5 user ε ε τ ime ε 3 ε Figure 3.4: Chip timing relationship etween users p and, L 4, ε. 5 By reaing down 3.33 into a sum of integrals over eah frational hip period, we have R d L [ τ, x] l L l osπx t ψ dt Lp m l l ε l osπx t ψ l l ε dt where x m. Upon integration of the osine terms, we have L sinπx ε Rd [ τ, x] l L l os[πx ε ψ ] l πx l L p l sin[πx ε ] πx l L l os[πx l ε ] ψ 3.36 We now tae the expetation of the square of he terms otained y multiplying two different-hip-indexed terms will e equal to zero sine the hips are independent of eah other. So we have to tae the expetation of the sum of the squares of the two terms on the right side of 3.36.

45 45 [ ] [ ] [ ] [ ] [ ] os os os os, L p L l L p L l d p x p L p l x l L l B p x p L p l x l L l A E x R E ψ ε π ψ ε π ψ ε π ψ ε π τ 3.37 where A x ε π ε sin, and B x ε π ε sin. Rearranging the terms in 3.37, we have [ ] { } { } os os os os, L l L p L l L p d p l x D p L p l L l E B p l x C p L p l L l E A x R E π ψ π ψ τ 3.38 When taing the ensemle average in 3.38, the expetation of the osine term ontaining ψ is zero, eause it is a random variale uniformly distriuted from [, π. For equal proaility that ± l, and ± p, ] [ p l E equals one when lp and zero otherwise. hus we otain the muh simpler expression [ ] d x x L x R E sin sin, ε π ε ε π ε τ 3.39

46 46 In 3.39, x taes the values in { }, when m-, and respetively, with equal proaility. So, the variane of the interfering term an e alulated y averaging 3.39 over the two possile values of x. We therefore have E [ R ] d π ε π ε π ε sin π ε sin ε L sin τ 3.4 sin ε We next evaluate the expetation of the square of the seond partial ross orrelation term given y By reaing down 3.34 into a sum of integrals over eah frational hip period, we have Rˆ d l ε L l osπx t ψ dt N l τ l 3.4 l l L L p m osπx t ψ dt l ε B [, x] Simplifying the aove equation similar to the proedure desried for the first partial orrelation funtion, we otain E [ ] x Rˆ ε sin π ε d, x N B L In 3.4, τ 3.4 ε sin πx ε x taes the values in {, }, when m-, and respetively, with equal proaility. So, the variane of the interfering term an e alulated y averaging 3.4 over the two possile values of x. We therefore have

47 47 [ ] d L N R E sin sin sin sin ˆ ε π ε π ε ε π ε π ε τ 3.43 From 3.3, we now have, [ ] I m N P P I E sin sin sin sin ε π ε π ε ε π ε π ε 3.44 Simplifying 3.44 using the nowledge that N / and i /, we get [ ] N i i N i i N i i N i i N P P I E I m / sin / sin / sin / sin 4 ε π ε π ε ε π ε π ε 3.45 When the delay τ is zero, we see that the aove result is idential to that of the synhronous ase. We now average the interferene term over the random variales ε and L. We model L as a uniformly distriuted integer over [,N-], and ε as a real uniform variate over [,. Let N i i y N i i y,. Averaging 3.45 over ε, we now have [ ] sin sin sin sin 4 y y y y y y y y N P P I E I m π ε π π ε π π ε π π ε π dx 3.46 Solving the integral in 3.46 we get

48 48 E [ I m ] P I sin 4πy sin 4P N πy πy 4πy 3.47 Simplifying 3.47 y sustituting for y and y, we finally otain E { sin 4π i i / N } P [ ] { π i i } I I m N 4 P { sin 4π i i / N } { π i i } 3.48 his expression is used in the SNIR expression analogous to 3.6, and then into the BER expression of 3.7 to estimate performane for this asynhronous ase. 3.5 Rayleigh Fading Channel Moile ommuniation hannels an e haraterized y two types of fading effets: large-sale fading and small-sale fading. Large-sale fading represents the average signal power attenuation or path loss due to motion over large areas []. he statistis of large-sale fading provide a way of omputing an estimate of path loss as a funtion of distane. Small sale fading is a loal phenomenon, in that it is nearly independent of distane etween the transmitter and reeiver. Small-sale fading refers to the dramati hanges in signal amplitude and phase that an e experiened as a result of small hanges in the spatial positioning etween a reeiver and transmitter. Smallsale fading is often modeled as Rayleigh fading if there are multiple refletive paths that are large in numer and of approximately equal amplitude in the asene of a line-ofsight signal omponent. he fading we onsider is flat, or frequeny non-seletive.

49 49 his type of fading arises when all the multipath delays are muh smaller than the shortest aseand signal duration here, the hip time. In the system we are onsidering, fading is enountered after all the transmitted signals are summed, when this aggregate signal is sent through the hannel. he AWGN is then added to this signal at the reeiver. We now proeed to desrie the it error rate omputations for single user and multi-user Rayleigh faded hannels. Figure 3.5 shows a lo diagram of the transmitter and hannel struture for this Rayleigh fading ase illustrating a single user DS-SS-BFSK transmission. t BFSK s t rt t α t nt Figure 3.5: Single-user x and hannel struture for a DS-SS-BFSK system over Rayleigh faded hannel In a synhronous multi-user environment, all the spread waveforms are spread and then transmitted over the hannel. he resultant signal is Rayleigh faded and AWGN is then added. he reeived signal is now given as K r t α s t n t 3.49 where α is the fading amplitude of the hannel and nt is the AWGN with mean zero and variane N /. We have without loss of generality set the hannel propagation delay to zero.

50 5 Using the inary minimum error proaility detetor formulation developed in [], the proaility of it error for a BFSK modulated system an e written as P γ γ Q 3.5 where the effetive signal to noise ratio at the reeiver is γ α E / N. In order to otain an expression for the it error ratio BER on a Rayleigh fading hannel, P γ must e averaged over the PDF of γ. When α is Rayleigh distriuted, α has a hi-square proaility distriution with two degrees of freedom. Consequently, γ is also hi-square distriuted with two degrees of freedom, and its PDF is given y [] p γ γ e γ / γ, γ 3.5 where γ is the average signal-to-noise ratio, defined as γ E α E / N. he expression for BER is then otained y averaging P γ over pγ P P p γ dγ γ 3.5 Q γ γ e γ / γ dγ Using integration y parts, and solving 3.5, we get P γ γ 3.53 We assume that the mean square value of the Rayleigh variale is one for simulations. In the multi-user setting, we model the multi-user interferene as Gaussian. he phase of eah user at the transmitter is different and random, α is Rayleigh distriuted,

51 5 so we an model the multi-user interferene onditioned on the fading variale as Gaussian. he proaility of it error is then given as P γ u γ u 3.54 where γ u is the average signal-to-noise plus interferene ratio, defined as α E γ u E N I where I is the multi-user interferene term derived for different ases in the previous setions.

52 5 Chapter 4 SIMULAIONS In this hapter, we present the simulation model used in the thesis and also the results otained for different variations of the system. 4. Simulation Model A DS-SS-BFSK simulation model was developed for this researh. One of the methods used for the performane evaluation of digital systems is estimation of it error proaility. We employ the Monte Carlo method for this estimation. he Monte Carlo method is a numerial method for statistial simulation whih utilizes sequenes of random numers to perform the simulation. he simulation omputes an estimate for the it error proaility. he simulation model was developed in MALAB. At the transmitter end, random inary data is generated using the rand funtion in MALAB. he rand funtion generates random numers hosen from a uniform distriution on the interval,. he generated its are then BFSK modulated. he frequeny tones are at f ± R i, where f is the arrier frequeny. Spreading odes are then generated similar to the way data its were generated. he length of the spreading ode is PG, the proessing gain. he hip rate is R PG R. he BFSK modulated waveform is then multiplied with the spreading ode to otain the spread BFSK waveform. he it rate R is normalized to one for onveniene.

53 53 Random Binary Soure Compare P Estimate FSK Mod Selet largest as inary Spreading ode Repeated for eah user Correlate and integrate Correlate and integrate Sum all users transmitted signals Rayleigh fading generator Despreading AWGN Generator Figure 4.: Shemati depition of MALAB simulation model he impairments present in the hannel are AWGN, MUI, and Rayleigh fading. he AWGN is generated y using the randn funtion in MALAB. he funtion randn generates random numers hosen from a normal Gaussian distriution with mean zero and variane one. By eeping energy per it E onstant and hanging the variane of AWGN, desired values of E /N are otained.

54 54 he Rayleigh fading is generated y using the fat that the square root of the summation of two squared Gaussian random variales is a Rayleigh random variale, i.e., if X and Y are two Gaussian random variales, then z X Y, is a Rayleigh random variale, and it is haraterized y the single parameter Ez, the mean square value of z where E. is the expetation operator. As noted, we have used a mean-square value of for the Rayleigh distriution in the simulations. he unit mean-square value of Rayleigh fading energy eeps the average E /N ratio onstant for a onstant value of transmitted it energy. he signal that enters the reeiver is despread and downonverted y the omposite ode signal. he signal is then integrated over the symol time, and hard it deisions are made on the deision variales. he it error rate estimation is performed y omparing these reeived it estimates with transmitted its. he input parameters to these simulations are as follows R : it rate, f : Carrier frequeny f s : Sampling frequeny PG: Proessing gain, E /N range Spaing etween the frequeny tones f 4. Simulation Results In this setion, we present the results of the simulations we otained for different variations of the system we studied. Figure 4. plots the simulated it error rate vs.

55 55 analytial it error rate for a 3-user synhronous DS-SS-BFSK system in an AWGN hannel. his plot is for the ase where we have different spaings etween the frequeny tones, and also different spreading odes for eah user. Orthogonality amongst the users is ahieved via frequeny separation. he first user s frequeny tones are at f ±. 5 f, the seond user s tones are at f ± f, and the third user s are at f ± 3 f. he ommon input parameters onsidered were: R, proessing gain, PG 5, arrier frequeny f 4R, and sampling frequeny f s 4f. Analytial P Simulated P - P E /N Figure 4.: P vs. E /N, synhronous, different spaing, different odes, PG 5, 3 users ase. It is evident from the figure that simulations yield orthogonal performane. We get similar BER performane for the other two users also. For the same speifiations used in the plots of Figure 4., we plot the simulated performane of the synhronous system when all the user signals are spread using the

56 56 same spreading ode Case. Here again, orthogonality is ahieved via frequeny separation. Figure 4.3 shows the simulated performane is in very lose agreement with the analysis and orthogonal performane is ahieved. Again, we get similar performane for the other two users. Analytial P Simulated P P E /N, db Figure 4.3: P vs. E /N, synhronous, different spaing, same odes PG5, 3 users ase. Figure 4.4 plots the simulated performane of the synhronous system when orthogonality is ahieved through spreading odes. For this simulation, we employ Walsh-Hadamard odes of length 3, i.e., the proessing gain used is 3. We ran this simulation for a two user system. Both the users have their frequeny tones at f ±. 5. he other parameters remain the same as in the previous two figures. As f disussed in Chapter 3, we again have orthogonal signaling performane, and simulation