Lecture Notes 11: Direct-Sequence Spread-Spectrum

filter isDirect-Sequence Spread-Spectrum with Multipath InterferenceiTZ iT 2a t cos ωct dtTrt i 1T LZ iTConsider a direct-sequence system over a channel with multipath fading the received signal ismodeled asτjj 2wherentiTτj a tst2 T cos 2π f ct dt αjIj α js trt Ij1 LEbii 1TiT j 1 αj where τ j is the delay of the j-th path, α j is the amplitude and n t is white Gaussian noise.Below we analyze the performance of two different receivers. The first receiver ignores themultipath interference and uses a filter matched to a single path to make a decision. Thesecond receiver uses a bank of filters matched to the various paths and combines the filteroutputs to make a decision. The can be implemented by a single filter and a tapped delay line.Because the structure of the receiver looks like a garden rake it is called the rake receiver. Therake receiver usually requires amplitude and phase estimation of the various paths and is thusmore complex than a single branch receiver.2P cos 2π f c t i 1Tτj a tτj a t2 T cos 2π f ct dt iTτj a tτ j a t dt i 1T bt P T cos 2π f c τ j αj τiT . Theni 1 T τjτ j a t dt P T cos 2π f c τ jbi 2 a t αj Ij Now assume that 0 i 1T iTAssuming that the receiver is matched to the first path and τ1τi b t 0 the output of the matched i 1 T τjbi 1 a tτ j a t dt XI-29 where The output due to the desired signal then is given byE Z T b0τ a t dt1 of the the interference is determined as follows. L bi 2 R τ j cos 2π f c τ j E α j R̂ τ j Ij bi 2 R τ j cos 2π f c τ j bi 1 R̂ τ j αjE bi 1 α 1Eα1 The conditional variance Var Z T b0LetThusZ iT1 τat T 1T 0andR̂ τ τ a t dt at τ1TRτcausing 2π f c τ j to vary over many multiples of 2π. Thus for each very small range of τ j ,2π f c τ j will vary over many multiples of 2π. Thus when computing the expectation we cancos 2π f c τ j randomness from the R τ j , R̂ τ j randomness. Inseparate out the cos φ jfact, we will treat φ j and τ j as independent random variables. bi 1 R̂ τ j α j PT cos 2π f c τ j bi 2 R τ jXI-30j 2 ThenThus the channel experiences some intersymbol interference. If we model the intersymbolinterference as Gaussian noise, the variance of the interference (with random delays andphases) can be determined.LVar Z T b0 E I 2jj 2Now consider the case where the delays are uniformly distributed over the intervalTc T Tc . In most cases the phase variable cos 2π f c τ j will be independent of τ j . This is1. Thus when τ j varies only slightly, f c τ j will vary considerably. Whentrue because f c τ jcomputing averages then we can think of only slightly varying τ j without changing R τ j butEα2jT Tc E I 2jN02 1 2 N 2T 1τ j TcR̂2 τ jR2 τ j d τ j XI-31The averaging above is with respect to the delays of the multipath signal. Substituting in forXI-32

If we define the parameter r asN 1T Tc2 Nα2jEτ j TcN 1r k 2 τjN 1R̂ τ jk 1 TcTcC kNC kLN021N1Nthen the mean square interference issC k12Var Z T b0NsC k1s2N2ds2C kErTc36T 3 1 2 NC k Tc216N 3N α2jj 2Er1 2 NL α2jj 2N02 k 2 s 01 sC kTcs sC k C k1C kNC k1The signal-to-noise ratio is defined as the squared mean output divided by the variance and isgiven asNC kNdsC kC kSNR1 k 1NC k1NC2 k1C2 k1NEα21Er Li 2 α2i6N 3 1 2 N C kE 2 Z T b0Var Z T b011 C k2N 21 C2 k1N 22 NC2 k N 1s2 C 2 k6T 3 11 sC k2 TcEα2j Tc3C kC k Tck 2 s 0TcEα j2T 3 1 2 NN 2 NC2 k2 2T 3 1kC2 kR τ j dτ j2 C2k 2 Eα2jR2 τ j d τ jkTc 2T 1R̂2 τ j 2 N 2T 1α2j E E I 2j the definition of the partial correlation functions we obtainN02 For example the spreading sequence of length 7 has parameter r82. The length 31 XI-33XI-34 m-sequence has r 618 and the spreading sequence of length 127 has r 11106. Considerthe case of negligible background noise. The signal-to-noise ratios for these differentspreading sequences are NNow consider the case of a receiver that uses a filter matched to each delay. The usual methodto combine the different filter outputs is by weighting each component by the strength of thepath it is matched to. Below we show a block diagram of such a receiver. The received signalis first mixed to baseband by a pair of mixers with 90 degrees phase offset for the locallygenerated reference. We represent this by a complex mixing. The double lines correspond tocomplex signals. The baseband signal is then filtered with a filter matched to the basebandtransmitted signal. The output of the baseband filter enters a tapped delay line. Differentdelays are weighted by different amounts. The magnitude of the weighting corresponds to themagnitude of a particular path while the phase compensates for any phase change so that thedesired multipath component after the gain has zero phase or in other words a purely real part.Of course there will be some interference from other paths that contribute to the imaginarypart but this will be ignored by the receiver.SNR d B 7 SNRPerformance with a Rake Receiverα21 6N 3 1 2 Nr Li 2 α2i12 5α21 Li 2 α2iα210 log10 L 1 α2 i 2 iα210 log10 L 1 α2 i 2 i10 log10 3123 5 127 30 3 In the homework it is shown that the signal-to-noise ratio averaged over all possible spreadingsequences increases linearly in N. Notice that the signal-to-noise ratio decreases as thenumber of paths increase. This is because the receiver is treating all the paths except one asinterference. The direct-sequence receiver reduces the effect of these interfering paths by afactor of N because of the processing gain. A receiver which makes uses of these extra pathsis discussed next. XI-35XI-36

1.5rt ht DELAY LINE1αL α1j2π f ct0.5 Real[ ]y1(t) x1(t)*h(t) expDEC0 0.5 1Figure 140: Rake Receiver 1.501234time5678 Below we show the output of a matched filter for a baseband signal with three paths withdelays 0, 0 3T and 0 8T with relative amplitudes 1, 0.7 and 0.3. The signal is spread by afactor of 31. The output of the rake which delays the signal by 0 8T and weights by 1, delaysby 0 5T weights by 0.7 and adds these to an undelayed version weighed by 0.3. Figure 141: Matched Filter Output XI-37XI-38 where3τj T In the absence of background thermal noise1Lτ j dt cos ωc τlτj τl a t τl a tbt τj l 1lτj T αl ETj α j Eb0 0 Zj z1(t) y1(t)*hr(t)τ j dt τ j cos ωc t τj 22atTrt Zj 1 α j Eb0L Ij ll 1l j 2The decision statistic Z due to the desired users is 3L235678 α2jEj 1To compute the variance of the interference we postulate the following model of the delays.The delays are random variables distributed over disjoint intervals of length 2Tc . Furthermore,the minimum separation between delays is also 2Tc . That is min τ j τl2Tc . This is doneso that the paths that the receiver is able to lock onto are in fact distinguishable. For exampleconsider the case where τl is uniform over the interval 4 l 1 Tc 4 l 1 Tc 2Tc andassume that 4 L 1 Tc 2c T Tc . Furthermore assume that the delays are independent.Figure 142: Rake Receiver Output The receiver computes the following decision statistic for bit b 0L α jZ j Z1 4time 1 E Z b0 0 j 1 XI-39XI-40

Lj 1 α2j 2 EL l 1 l j α2l E 3N 1 Lj 1 α2j SNRτj τ j cos ωc τl b0 R̂ τl τj2 j lN τjIf we define τj 21αj Lj Ll α jV j 21 αjEα 2 E 3NN0 2 whereNotice that if α j is a constant then the signal-to-noise ratio does not decrease as of the numberof paths in the channel increases. This is in contrast to the single filter receiver in which theperformance degrades the more paths there are in the channel.L 2 j lN LjSNRj 1VjN0 2thenLVar1 The variance isVar Z21 l j α l E 2NL j 1 α2j α 2 cos ωc τlT τj b1 R̂ τl T τj α l E b 0 R τl Ij l Thus the signal-to-noise ratio isα l E b 1 R τl For τlN0 2 l 1l j j 12 j lNα2l E 3N 1 τlτlIj lL τ j dt cos ωc τ j τl a t τl a t For τ jbtτjL α2j Var Zτj T αlIj l1ETIt is straightforward (but very lengthy) to show that Then the variance of the interference can be calculated as follows. First let I j l be the effect ofthe l-th multipath on Z j . ThenIj l l 1l j XI-41XI-42 b1 t2P cos ωctb2 tDelay τ2 a2 tIn this section we consider the performance a direct-sequence system with multiple-accessinterference (also know as code division multiple access (CDMA). Each user is given a codesequence. The receiver for a particular user demodulates the signal by match filtering thereceived signal with a filter that is matched to the transmitted signal of the desired user. Weshould point out that this is not the optimal receiver but is one that is currently being used inpractical systems.2P cos ωct aK tIn our analysis we would like to determine the average probability of error. The averaging isrespect to the data bits that the other users are transmitting, the relative delays of the otherusers and the relative phase of the other users. Delay τK bK t a1 t Direct-Sequence Spread-Spectrum Multiple-Access (DS-SSMA) Delay τ12P cos ωctFigure 143: Block Diagram of a Direct-Sequence System kbl pT tlT l bk t There are numerous different modulation formats that can be used in a direct-sequence systemincluding BPSK, QPSK, MSK. For our purposes we will just consider BPSK. XI-43XI-44

kal pTc t lTc l bibi 001111 tZ iT 2 T cos ωctnt τka1 T K sk t htThe received signal consist of the delayed versions of all of the users and additive whiteGaussian noise.rtiT trt2Pak t bk t cos 2π f ctsk t ak t k 1Figure 144: Direct-Sequence Spread-Spectrum ReceiverAt receiver 1 the received signal is first mixed down to baseband by multiplying the receivedsignal by cos 2π f ct and then filtered with a filter matched to the spreading sequence of user1. Equivalently (except with respect to generating timing information) the received signalafter the mixer can be correlated with a local replica of the spreading sequence to produce adecision statistic. We will assume that the receiver is perfectly synchronized to the transmittedsignal (both timing and phase) so that without loss of generality we can assume that τ 1 0. The filter for user 1 is matched to the spreading code of user 1. The output of the filter for user1 contains the desired signal, interference from other users and noise. Below we show thematched filter output for a single user, two users and three users with spreading sequences oflength 31. XI-45XI-461.5110.50.5y1(t) (x1(t) x2(t))*h1(t)y1(t) (x1(t) x2(t) x3(t))*h1(t) 1.50 0.5 1 1.500 0.5 11234time567 1.508Figure 145: Matched filter output for a single user with N311234time56Figure 146: Matched filter output for two users with N831 with τ2τ10. XI-477XI-48

1.5401y1(t) (x1(t) x2(t) x3(t))*h1(t)3020Z(t)100 10 200.50 0.5 1 30 1.50 400123time/T45Figure 147: Matched filter output for two users with N1234time631 with τ2τ1567831 with τ3Figure 148: Matched filter output for three users with Nτ1 0.0.τ2 XI-49XI-502π f c τk and where φk The matched filter output at the sampling time is given byiTτk b k tThe term due to other users can be written in terms of the crosscorrelation of the differentusers sequences. n t a1 t cos 2π f ct dt τkk10 kb0 iTkτk a 1 t d tk b0 T R̂k 1 τk b 1 T R k 1 τk ηiτk Ikwhere the functions Rk 1 and R̂k 1 are given byηi 1 Ebii 1Tτk a 1 t d tak tτk a 1 t d t K k 2τk b k t ak t 1Tak tT E cos φk τk a 1 t d t ηiτk b K1τk b k t a1 t d t τk cos φkk 2 τk b k tak t T ak t a1 t b 1 ti 1Tτk a 1 t d t 0k 2Kτk b k tak t 0τkτk a 1 t d t ηiτk b k t τk a1 t cos 2π f ct dtak t sk t s1 t i 1TT K τki 1T k 1iTEbi K sk tiT2T Z 1 iT i 1TiT2Twhere ηi is a Gaussian random variable with mean zero and variance N0 2.PTτk a 1 t d t ak t i 1T 1T E cos φk r t a1 t cos 2π f ct dt Z 1 iTIk iT2T τak tτ a1 t d t 0 XI-511T Rk 1 τk 2XI-52

T The cross correlation functions Rk 1 and R̂k 1 can be written in terms of the aperiodic crosscorrelation of the spreading sequences given by τ a1 t d tak tτ 1T R̂k 1 τ l0 otherwise a0 1τ Tc 1NNCk 1 l 1 lTc Ck 1 lCk 1 l lTc Ck 1 l τ R̂k 1 τ1Ck 1 l N TcT1Ck 1 l Tc τT Rk 1 τtτka1 t0N For l1 aN 21 k aN1 kl kaNkaN l 1 kaN l 2 ka1N ka01 kaN2 aN kl kaN kaN l 1 Ck 1 lkb011 l k0 am l aml k1τk bτk b k tak tN m0l b0 R̂k 1 τk1am kb 1 R k 1 τk1 l k0 am kE cos φk IkN m Thus 1 11aN 1 1aN 2 al1 al11 al 1a1 1a0 E1E R2k 1 τk2E R̂2k 1 τk E Ik2Figure 149: Received Signal t T1 Tc l lTc 2Tc Tc 0 The variance of the interference (which has zero mean) can be determined for random phaseskkkasφk and random data b0 and b 1 and delays τk and spreading sequences ai XI-53N 1N 1l 1 Tc R̂2k 1 τ dτN 1τ21 τCk2 1 llTc l2NN 1τ1 Tc l1τ21 TcN τN Ck 1 l1N τlTclTc l1 Tcτ dτCk2 1 l2N 11NCk2 1 l1N Ck 1 l1E Ik2 1N11NN Ck 1 l1Nl 01N0 2If the output of the matched filter due to other users signals is modeled as a Gaussian random Ck 1 l 1 Ck 1 l Ck 1 lE Kk 2 rk 1E6N 3 Ck2 1 lVar Z T1b0 N1 Ck2 1 l1E 2 Z T b0 SNR1 Ck2 1 llErk 16N 3The signal-to-noise ratio isNThe parameter rk 1 defined below captures the effect of different spreading sequences on thesignal-to-noise ratio.N 1mThe variance of the multiple-access term becomes Ck 1 l 1 Ck 1 l Ck 1 lCk k l Ci i l Ck2 1 l1 l 1 N Ck2 1 llml 0 Ck2 1Ck k l C1 1 lN 1Ck i l Ck i ll 1 N 2Ck 1 lrk 12Ck k l C1 1 lThe last line follows from the identity Ck2 1 l Ck2 1 1 Ck 1 l τlTc 1 Tc l Ck2 1 l2Ck 1 lE6N 31l 1 N lTcCk 1 l Ck 1 l l 1 Tc l 0R2k 1 τ2Ck2 1 ll 1 N N 1 E2T 3lTc l 0 E2T0 R̂2k 1 τ dτ R2k 1 τ T E2TXI-54 XI-55XI-56

K 13N N02EThe error probability is then approximated by1 SNR variable and we consider only random spreading sequences then QSNR 1 0 03â41 02â4 13â3 011 â4 2âi 1 2 âi.with initial values 513âi 3 0 ai 30 â4 14 3â331 ai 3 1âi3 3â22 with initial values51 â4 13 âi âi3â132â30 1 132â213â22â3 âi. 32 âi2 1 âi 2â1User 3: âi3â0âi21âi1 1 1 2â02 User 2: âi212â211â3then the signal-to noise ratio becomes SNR 15 47dB. (These are the initial states thatmaximize and minimize the signal-to-noise ratio). On the other hand for random sequencesthe signal-to-noise ratio is 16.67dB. Thus by choosing the appropriate starting points of thespreading sequences (even with a given feedback connection) we can affect thesignal-to-noise ratio. However for large N the difficulty in finding the optimal starting statesbecome computationally intractable.1 The actual spreading sequence is obtainedafter the usual conversion of 0 to 1 and 1 to -1. ai2 11 â413â1 11 â312â111â2 5 with initial values10 â211â1 11 â1 31 âi 1 âi3â0 1â012â0 1â0As an example consider the case of three users with sequence of length 31. The sequences forthe different users are m-sequences derived from the following feedback shift registerconnections.User 1: âi17 73dB. If we change the initial values to noise) is calculated to be SNRPe.With these sequences and initial states the signal-to-noise ratio (in the absence of background XI-57Now consider reindexing the data bits according to the above ordering. We will let the index ldesignate which data bit from l 0 to l JK 1. The l 0 data bit corresponds to data bit 0of user 1. The l 1 data bit corresponds to data bit 0 of user 2. In general if l mK k 1where 0 m J 1 then data bit l corresponds to the m-th data bit of the k-th user. Thereceived signal can be written as Optimal Multiuser DetectionXI-58 rt where τk τk cos 2π f c t K12K12xl nK1bJ1xn l bJ 1 bJ1. b2 b2 b2 b1 Note the following about the correlations. 2b1cl t c n t d t mT pT t 1b1mT k Kb0 xl n bmAnother critical assumption is that the receiver knows exactly the delays of all the users aswell as the spreading signals and received powers. Consider the data sequence2τk p T tWe now define the following correlation valuesm 0b02Pk ak t cl t J 1bk t1ntl 0where each user could possible have different received power Pk and delay τk . Assume thatthe data sequence of user K is finite and of length J. Assume also that the users are labeledτK T . A critical assumption that we will make is that eachsuch that 0 τ1 τ2user employs rectangular pulse shapes. This effectively limits the effect of a single data bit toa time interval of duration T . Thus we will assume thatb02Pl cl t bl ntk 1b JK 1τk τk cos 2π f c t τk b k t 2Pk ak t K rt In this section we consider the problem of optimally detecting the data sequences transmittedby K users to minimize the probability of chosing the wrong set of sequences. The setup is thesame as the previous section. XI-59XI-60

yy0yJK 1 and uses that as a sufficient statistic in order to compute the optimaldecision rule. The vector y can be obtained from a bank of K matched filters, matched to theindividual spreading signals of different users. 2. xl l0j j KThis is due to the fact that the data pulses are rectangular pulses of duration T .Now we can simplify the metric computation as follows.JK 1 ΛJK 1 cl t bl dtΛ l 0Λ Λ bl xl n bn2 JK 1 JK 1 bl xl n bnl 0 n 0n l where b2l xl lJK 1 l 12JK 1bl yl b2l xl lJK 1 l2 bl xl lj bl jl 0 j 1l 0JK 1 bl xl n bnl 0 n 0l 0JK 12l 0l 0 n 0l 0b2l xl lJK 1 K 1 bl xl l JK 1 JK 1bl ylJK 1bl yll 0 cl t c n t d tl 0 n 0JK 12 JK 1 l 0r t cl t d t2l 0JK 1 JK 1bl l 0JK 1 JK 12JK 1bl yl 2Equivalently the receiver should choose the data sequence b to maximizeΛ2l 0 rt Now the goal is to minimize the probability of choosing the wrong sequence b. To do this weneed to find the sequence b to minimizebl yll 0b2l xl l2j bl jl 0 j 1where we have assumed that xl m 0 if m 0 or l m K. The form for Λ is essentially thesame as the form for the metric for MLSE with intersymbol interference. Now it is clear wecan apply (as in the ISI case) dynamic programming (Viterbi Algorithm) to determine the yl r t cl t d t The conclusion from the above equation is that the optimal receiver computes the vector2yl 11 bl 1blK 1b2l xl lblK 12bl1 xl1 l 1 j bl jj 1JK 1 Λλl σ l σ l12

Lecture Notes 11: Direct-Sequence Spread-Spectrum Modulation In this lecture we consider direct-sequence spread-spectrum systems. Unlike frequency-hopping, a direct-sequence signal occupies the entire bandwidth continuously. The signal is obtained by starting with a narrowban