Direct-Sequence Spread-Spectrum

Transcription

1 Chapter 3 Direct-Sequence Spread-Spectrum In this chapter we consider direct-sequence spread-spectrum systems. Unlike frequency-hopping, a direct-sequence signal occupies the entire bandwidth continuously. he signal is obtained by starting with a narrowband signal and directly modulating a high bandwidth signal. As with frequency hopping direct-sequence has advantages when the channel contains a jamming like signal. he jamming could be intentional and hostile, self jamming (multipath), and multiuser jamming. In the remained of this chapter we examine the capabilities of direct-sequence is these three environments.. Introduction Below we show the transmitter and receiver for a direct-sequence system. he data sequence b t consists of a sequence of data bits of duration. he data sequence is multiplied with a binary spreading sequence a t which has N components called chips per data bit. b t s t a t Pcos ω c t Figure 3.: Block Diagram of Direct-Sequence Spread-Spectrum ransmitter b t b l p t l l b l a t a l p c t l c a l l c N Usually a i is a periodic sequence with period. In some cases for each period of the sequence a i one data bit is transmitted, i.e. c. When the sequence a i is periodic the spreading signal is a periodic waveform so 3-

2 3- CHAPR 3. DIRC-SQUNC SPRAD-SPCRUM that a t a t l for any integer l. In other cases c, that is, many data bits are transmitted before the sequence repeats. In this case it is useful to model a i as a sequence of independent, identically distributed binary random variables equally likely to be. In any case it is nearly always true that c N is an integer. his is usually called the processing gain. It is the factor by which the signal is spread. s t Pa t b t cos π f c t he transmitted signal has power P. Below we show a data signal and the result of multiplying by a spreading signal with 3 chips per bit. he sequence was generated by a linear feedback shift register such that â i mod â i 3 â i 5 where â denotes a, sequence. he actual sequence a is found via the usual transformation of,. he initial loading of the shift register is [ ]. he receiver consist of a mixer followed by a filter Data waveform b(t) time/ User waveform s(t) time/ Figure 3.: Waveforms b t and a t b t matched to the spreading code of the transmitter. Consider the case where N, that is there is exactly one period r t Filter t i Z i b i b i cos ω c t Figure 3.3: Direct-Sequence Spread-Spectrum Receiver of the spreading sequence per data bit. (It is easy to see how to modify the results for N. For N for each data bit we either transmit a t or a t depending on the sign of the data bit with appropriate delays. he matched filter has impulse response given by h t a t t he filter output is given by Z t cos π f cτ r τ h t τ dτ

3 . 3-3 t t cos π f cτ r τ a t τ dτ hus the filter does a running correlation of the received signal (mixed down to baseband) with the spreading sequence. hat is, at each time instance the filter output is the correlation of the received signal over the past seconds with the spreading signal a t. From the below figure we can visualize the output of the filter at time t as the integral of the product of the received signal with a shifted version of the spreading code. In this sense the matched filter provides a running correlation of the received signal over the past seconds with the spreading signal. Consider the filter output during the time interval t due to the transmitted signal alone, r t Pa t b t cos π f c t hen Z t P t a τ b τ a τ t P b a τ a τ t P b ˆR t P b R t t dτ t dτ t Pb a τ a τ t dτ where ˆR t t a τ a τ t t dτ a u a u t du a u a u t du t and R t t t a τ a τ a τ a τ t dτ t dτ he last step in each of the above two equations follows because of the periodicity of the spreading signal (one period of the spreading signal per data bit). We can write these correlation functions in terms of the spreading sequences as follows. For t k c s with s c. R t c s a N ka a N k a a N a k s a N k a a N ka a N a k k k c s a l k N a l s a l k N a l l l c s C k N sc k N

4 3-4 CHAPR 3. DIRC-SQUNC SPRAD-SPCRUM where C k N k l a l a l k k N N k l a l ka l N k Similarly, for k c t k c otherwise ˆR t c s a a k a a k a N ka N s a a k a a k a N ka N c s C k sc k Notice that both R t and ˆR t vary linearly with t for t k c k c for every k. r τ b b a k a k a N a N a a a k a τ t kc a k a k a N a N k c a τ t a a a N k a N k a N k a N a N t τ Figure 3.4: Received Signal Consider the spreading sequence a he aperiodic correlation function is k C k It is useful to state a few properties of the aperiodic autocorrelation function.. he partial or aperiodic autocorrelation functions are symmetric. C k C k. he full autocorrelation is the sum of two aperiodic or partial autocorrelation functions. θ k N C a l a l k k C k N k N l C k C k N N k he function θ k is called the autocorrelation function of the sequence a.

5 If we consider the spreading sequences to be a sequence of independent, identically distributed random variables then the following expectation with respect to the spreading sequences can be computed. Assume k and n. hen C k C n N k l N n a l a l k m k n N k k n N k n a m a m n For the sequence above the autocorrelation function has the following values. k θ k 7 he output of the filter with impulse response matched to the spreading sequence is shown below for a sequence of length 3. If two consecutive bits have the same sign (b b ) then the output of the filter during the interval [,] is given by Z t P b c s θ k sθ k On the other hand if b b then the output during the interval is given by Z t P b c s ˆθ k sˆθ k where the function ˆθ k is called the odd autocorrelation and is defined as ˆθ k N k a l a l k l N a l a l k C k C k N k N l N k he odd autocorrelation function differs from the standard autocorrelation function in that part of the terms in the summation have a negative sign. At the sampling time (t i ) the filter output due to the desired signal alone is given by Z i i i a τ a τ i dτ his result leads to the correlator implementation of the optimal (for AWGN) receiver. It is useful (for synchronization) to know the outputs of the matched filter at other times besides the time that we make a decision. he filter output is sampled at multiples of. he output at the sampling times is (s k ) Z i P c N b i Pb i b i. Direct-Sequence Spread-Spectrum with one Interference Consider a direct-sequence system with a jammer whose signal is an unmodulated tone at the same phase and frequency of the direct-sequence user. j t J cos ω c t he jamming signal has power J. he ratio J P is called the jammer-to-signal power ratio. he received signal is r t s t j t he receiver is similar to that for BPSK.

6 3-6 CHAPR 3. DIRC-SQUNC SPRAD-SPCRUM 4 Data waveform 3 Z(t) time/ Figure 3.5: Output of Matched Filter for a Sequence of ength 3 Demodulator r t PF t i b i b i Z i cos ω c t a t b t s t r t a t Pcos ω c t j t Figure 3.6: Block Diagram of Direct-Sequence Spread-Spectrum ransmitter

7 . 3-7 Figure 3.7: Distribution of Interference for n 7 Figure 3.8: Distribution of Interference for n 3 he decision statistic for bit b i is Z it i Z i i r t b i η i a t cos ω ct dt where η i is the output due to the jamming signal. he output due to the jamming signal can be written as (ignoring double frequency terms) η i i J a t dt i Since a t in a i p c t j c t i i j i N η i J c J N in j i N in a i j i N a i Random Sequence Model If we model the sequence a i as i.i.d. binary random variables then in j i N a i is a binomial distributed random variable with mean and variance N. hus η i is zero mean with variance J N. he signal-to-noise ratio, SNRout, at the output of the demodulator is then SNRout P J N Since the signal-to-noise ratio, SNR in, at the input to the receiver is the system is said to have a processing gain of N. SNR in P J rror Probability with one Interference he probability of error for the tone jammer (with perfect phase information) is P e P e P e P P J N N a i a i J c Figure 3.9: Distribution of Interference for n 7

8 3-8 CHAPR 3. DIRC-SQUNC SPRAD-SPCRUM Figure 3.: Distribution of Gaussian Interference Figure 3.: Comparison of Distributions where the sum extends over N values of the index i. For large N, a i N is approximately Gaussian with mean zero and variance (central limit theorem). he error probability can then be approximated by P e Q J c Q P J N Note that the jamming power is effectively reduced by a factor of N. Another way of expressing the error probability is in terms of an effective jamming noise power density. Since BPSK with spreading by a factor N has noise bandwidth of c the effective jamming noise spectral density N J is defined as N J Using this in the expression for error probability yields P e Q J c J c Notice that this is a factor of (3dB) worse than Gaussian noise of the same spectral density. he reason is that we have assumed the jammer has perfect phase information so that no power is wasted in the quadrature component of the signal. If the jammer had a random phase the performance would be better by 3dB and equivalent to the performance in Gaussian noise of the same power. N J 3. Direct-Sequence Spread-Spectrum with Multipath Interference Consider a direct-sequence system over a channel with multipath fading the received signal is modeled as r t α j s t τ j n t j 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. he first receiver ignores the multipath interference and uses a filter matched to a single path to make a decision. he second receiver uses a bank of filters matched to the various paths and combines the filter outputs to make a decision. he 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. he rake receiver usually requires amplitude and phase estimation of the various paths and is thus more complex than a signly branch receiver. Assuming that the receiver is matched to the first path and τ the output of the matched filter is Z i Z i i i b i r t j a t cos ω ct dt I j η i

9 where I j α j i α j i i i s t τ j a t cos π f c t dt α j P cos π f c τ j i Pcos π f c t τ i b t τ j a t τ j a t cos π f c t dt i b t τ j a t τ j a t dt and η i is a Gaussian random variable with mean zero and variance N. Now assume that τ i. hen where and hus Z i b i i a t τ j a t dt i b i τ i a t τ j a t dt j α j P cos π f c τ j b i R τ j b i ˆR τ j I j α j P cos π f c τ j i τ j b i α R τ ˆR τ τ a t τ a t dt a t τ a t dt τ α j b i j ˆR τ j b i R τ j cos π f c τ j hus the channel experiences some intersymbol interference. If we model the intersymbol interference as Gaussian noise, the variance of the interference (with random delays and phases) can be determined. Now consider the case where the delays are uniformly distributed over the interval c c. In most cases the phase variable cos π f c τ j will be independent of τ j. his is true because f c τ j. hus when τ j varies only slightly, f c τ j will vary considerably. When computing averages then we can think of only slightly varying τ j without changing R τ j but causing π f c τ j to vary over many multiples of π. hus for each very small range of τ j, π f c τ j will vary over many multiples of π. hus when computing the expectation we can separate out the cos φ j cos π f c τ j randomness from the R τ j, ˆR τ j randomness. In fact, we will treat φ j and τ j as independent random variables. he output due to the desired signal then is given by Z b he conditional variance Var Z b of the the interference is determined as follows. et hen I j α α j ˆR τ j b i R τ j cos π f c τ j η i Var Z b j I j N I j α j N c τ j c ˆR τ j R τ j dτ j

10 3- CHAPR 3. DIRC-SQUNC SPRAD-SPCRUM he averaging above is with respect to the delays of the multipath signal. Substituting in for the definition of the partial correlation functions we obtain I j α N j c ˆR τ j τ j R τ j dτ j c α N j α j 3 N N k N k k c τ j k c ˆR τ j R τ j dτ j c s c s C k sc k c s C k N sc k N ds α j N 3 N c k s c s C k C k N c s s C k C k C k N C k N s C k C k N ds α j c 3 N 6 3 N C k C k N C k C k k C k N C k N C k C k N If we define the parameter r as r N k C C k C k N C k C k C k N C k N k C k N then the mean square interference is Var Z b r 3 c 6 3 N r 6N 3 N j j α j α j N N he signal-to-noise ratio is defined as the squared mean output divided by the variance and is given as SNR Z b Var Z b α r i α i 6N 3 N N For example the spreading sequence of length 7 has parameter r 8. he length 3 m-sequence has r 68 and the spreading sequence of length 7 has r 6. Consider the case of negligble background noise. he signal-to-noise ratios for these different spreading sequences are SNR α 6N3 N r i α i

11 3. 3- N SNR db α 7 5 log i α i α log i α i α log i α i In the homework it is shown that the signal-to-noise ratio averaged over all possible spreading sequences increases linearly in N. Notice that the signal-to-noise ratio decreases as the number of paths increase. his is because the receiver is treating all the paths except one as interference. he direct-sequence receiver reduces the effect of these interferring paths by a factor of N because of the processing gain. A receiver which makes uses of these extra paths is discussed next.. Performance with a Rake Receiver Now consider the case of a receiver that uses a filter matched to each delay. he usual method to combine the different filter outputs is by weighting each component by the strength of the path it is matched to. Below we show a block diagram of such a recevier. he received signal is first mixed to baseband by a pair of mixers with 9 degrees phase offset for the locally generated reference. We represent this by a complex mixing. he double lines correspond to complex signals. he baseband signal is then filtered with a filter matched to the baseband transmitted signal. he output of the baseband filter enters a tapped delay line. Different delays are weighted by different amounts. he magnitude of the weighting corresponds to the magnitude of a particular path while the phase compensates for any phase change so that the desired 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 imaginary part but this will be ignored by the receiver. r t h t DAY IN exp jπ f c t β β Real[ ] DC Figure 3.: Rake Receiver

12 3- CHAPR 3. DIRC-SQUNC SPRAD-SPCRUM Figure 3.3: Matched Filter Output Figure 3.4: Rake Receiver Output Below we show the output of a matched filter for a baseband signal with three paths with delays, 3 and 8 with relative amplitudes,.7 and.3. he signal is spread by a factor of 3. he output of the rake which delays the signal by 8 and weights by, delays by 5 weights by.7 and adds these to an undelayed version weighted by.3. he receiver computes the following decision statistic for bit b where τ Z j j r t τ j In the absence of background thermal noise α Z j α j b l l l j he decision statistic Z due to the desired users is Z j α j Z j a t τ j cos ω c t τ j dt τ j τ j b t τ l a t τ l a t τ j dt cos ω c τ l τ j Z b o compute the variance of the interference we postulate the following model of the delays. he delays are random variables distributed over disjoint intervals of length c. Furthermore, the minimum separation between delays is also c. hat is min τ j τ l c. his is done so that the paths that the receiver is able to lock onto are in fact distinguishable. For example consider the case where τ j is uniform over the interval 4 l c 4 l c c and assume that 4 c c c. Furthermore assume that the delays are independent. hen the variance of the interference can be calculated as follows. First let I j l be the effect of the l-th multipath on Z j. hen I j l α l τ j τ j j α j b t τ l a t τ l a t τ j dt cos ω c τ j τ l For τ j τ l For τ l τ j I j l α l b R τ l τ j b ˆR τ l τ j cos ω c τ l τ j I j l α l b R τ l τ j b ˆR τ l τ j cos ω c τ l τ j he variance is where Var Z Var V j j l l j I j l α j V j

13 It is straightforward (but very lengthy) to show for random spreading codes that Var Z hus the signal-to-noise ratio is j α j α l l 3N l j l N N j If we define SNR α j α j j α j l l j α 3N l j l N N j α j l l j α l j l N j α j then SNR j α j α 3N N Notice that if α j is a constant then the signal-to-noise ratio does not decrease as the number of paths in the channel increases. his is in contrast to the single filter receiver in which the performance degrades the more paths there are in the channel. 4. Direct-Sequence Spread-Spectrum Multiple-Access (DS-SSMA) In this section we consider the performance a direct-sequence system with multiple-access interference (also know as code division multiple access (CDMA). ach user is given a code sequence. he receiver for a particular user demodulates the signal by match filtering the received signal with a filter that is matched to the transmitted signal of the desired user. We should point out that this is not the optimal receiver but is one that is currently being used in practical systems. In our analysis we would like to determine the average probability of error. he averaging is respect to the data bits that the other users are transmitting, the relative delays of the other users and the relative phase of the other users. here are numerous different modulation formats that can be used in a direct-sequence system including BPSK, QPSK, MSK. For our purposes we will just consider BPSK. b k t a k t s k t b k l p t l l a k l p c t l c l Pa k t b k t cos π f c t he received signal consist of the delayed versions of all of the users and additive white Gaussian noise. r t K s k t τ k n t k At receiver the received signal is first mixed down to baseband by multiplying the received signal by cos π f c t and then filtered with a filter matched to the spreading sequence of user. quivalently (except with

14 3-4 CHAPR 3. DIRC-SQUNC SPRAD-SPCRUM b t Delay τ b t b K t a t a t Pcos ω c t Delay τ Pcos ω c t Delay τ K + a K t Pcos ω c t Figure 3.5: Block Diagram of a Direct-Sequence System respect to generating timing information) the received signal after the mixer can be correlated with a local replica of the spreading sequence to produce a decision statistic. We will assume that the receiver is perfectly synchronized to the transmitted signal (both timing and phase) so that without loss of generality we can assume that τ. he filter for user is matched to the spreading code of user. he output of the filter for user contains the desired signal, interference from other users and noise. Below we show the matched filter output for a single user, two users and three users with spreading sequences of length 3. he matched filter output at the sampling time is given by Z i i r t a i t cos π f c t dt i i K k i s i t s k t τ k n t a t cos π f c t dt K s k t τ k a t cos π f c t dt η i k where η i is a Gaussian random variable with mean zero and variance N. Z i P i b i i a t b t K k cos φ k K a k t τ k b k t τ k cos φ k a t dt η i k i a i k t τ k b k t τ k a t dt η i

15 r t h t a t Z i t i b i b i cos ω c t Figure 3.6: Direct-Sequence Spread-Spectrum Receiver Figure 3.7: Matched filter output for a single user with N 3 where φ k π f c τ k and I k b i K k cos φ k I k η i i a i k t τ k b k t τ k a t dt he term due to other users can be written in terms of the crosscorrelation of the different users sequences. where the functions R k and ˆR k are given by τ a k t τ k b k t τ k a t dt k a k t τ k b k t τ k a t dt a k t τ k b k t τ k a t dt b k τ k τ k a k t τ k a t dt b k a k t τ k a t dt τ k b k R k τ k b k ˆR k τ k R k τ ˆR k τ τ a k t τ a t dt a k t τ a t dt τ hus I k cos φ k b k R k τ k b k ˆR k τ k he cross correlation functions R k and ˆR k can be written in terms of the aperiodic cross correlation of the Figure 3.8: Matched filter output for two users with N 3 with τ τ.

16 3-6 CHAPR 3. DIRC-SQUNC SPRAD-SPCRUM N k l a Figure 3.9: Matched filter output for three users with N 3 with τ 3 τ τ. a k t τ k b k t τ k b k a N k l a N k a N k a k a k a N k l N k l a b k a N k l a N k a N k a a t τ k t a a a l a l a l a N a N c c l c l c t Figure 3.: Received Signal spreading sequences given by C k l N l m a k m a m l l N N l m a m k la m N l otherwise For l τ c R k τ ˆR k τ C k l N c τ l c C k l N C k l N C k l c τ l c C k l C k l he variance of the interference (which has zero mean) can be determined for random phases φ k and random data b k and b k i as and delays τ k and spreading sequences a k Ik 3 6N 3 R k τ k ˆR k τ k N l N l R τ k ˆR τ k dτ l c R τ k ˆR τ k dτ l c l c C l k l c τ Ck l τ l c l c C k l C k l τ l c l c τ Ck l N l c τ Ck l N τ l c C k l N C k l N τ l c l c τ dτ N C l l k Ck l Ck l N Ck l N

17 C k l C k l C k l N C k l N he parameter r k defined below captures the effect of different spreading sequences on the signal-to-noise ratio. r k N C l l k Ck l Ck l N Ck l N C k l C k l C k l N C k l N N C l l k C k l C k l N N C k k l C l C k k l C l l N he last line follows from the identity N C k i l C k i l m l N N C k k l C i i l m l N he variance of the multiple-access term becomes I k r k 6N 3 he signal-to-noise ratio is SNR Z b Var Z b K k r k 6N 3 N If the output of the matched filter due to other users signals is modeled as a Gaussian random variable and we consider only random spreading sequences then SNR he error probability is then approximated by N K 3N P e Q SNR As an example consider the case of three users with sequence of length 3. he sequences for the different users are m-sequences derived from the following feedback shift register connections. i 3 User : â i â i 5 â with initial values â â â â 3 â spreading sequence is obtained after the usual conversion of to + and to -. a User : â i â a i â i. User 3: â 3 i â 3 a 3 3 i â i. i â i â i 3 i â i 3 3 â i 3 4 i â 4 he actual i. â i 5 with initial values â â â â 3 â 4 â i with initial values â â 3 â 3 â 3 3 â 3 4

18 3-8 CHAPR 3. DIRC-SQUNC SPRAD-SPCRUM With these sequences and initial states the signal-to-noise ratio (in the absence of background noise) is calculated to be SNR 7 73dB. If we change the initial values to â â â â 3 â 4 â â â â 3 â 4 â 3 â 3 â 3 â 3 3 â 3 4 then the signal-to noise ratio becomes SNR 5 47dB. (hese are the initial states that maximize and minimize the signal-to-noise ratio). On the other hand for random sequences the signal-to-noise ratio is 6.67dB. hus by choosing the appropriate starting points of the spreading sequences (even with a given feedback connection) we can affect the signal-to-noise ratio. However for large N the difficulty in finding the optimal starting states become computationally intractable. 5. Optimal Multiuser Detection In this section we consider the problem of optimally detecting the data sequences transmitted by K users to minimize the probability of chosing the wrong set of sequences. he setup is the same as the previous section. r t K k P k a k t τ k b k t τ k cos π f c t τ k n t where each user could possible have different received power P k and delay τ k. Assume that the data sequence of user K is finite and of length J. Assume also that the users are labeled such that τ τ τ K. A critical assumption that we will make is that each user employs rectangular pulse shapes. his effectively limits the effect of a single data bit to a time interval of duration. hus we will assume that b k t J b k m p t m m Another critical assumption is that the receiver knows exactly the delays of all the users as well as the spreading signals and received powers. Consider the data sequence b b b b K b b b K b b b K b J b J b K J Now consider reindexing the data bits according to the above ordering. We will let the index l designate which data bit from l to l JK. he l data bit corresponds to data bit of user. he l data bit corresponds to data bit of user. In general if l mk k where m J then data bit l corresponds to the m-th data bit of the k-th user. he received signal can be written as where r t JK l P l c l t b l n t c l t P k a k t τ k p t m τ k cos π f c t τ k We now define the following correlation values x l n c l t c n t dt Note the following about the correlations.. x l n x n l

19 x l l j j his is due to the fact that the data pulses are rectangular pulses of duration. Now the goal is to minimize the probability of choosing the wrong sequence b. o do this we need to find the sequence b to minimize r t K JK c l t b l dt l quivalently the receiver should choose the data sequence b to maximize where Λ JK b l r t c l t dt l JK JK b l y l l l JK l JK b l x l nb n n y l r t c l t dt JK c l t c n t dt n he conclusion from the above equation is that the optimal receiver computes the vector y y y JK and uses that as a sufficient statistic in order to compute the optimal decision rule. he vector y can be obtained from a bank of K matched filters, matched to the individual spreading signals of different users. Now we can simplify the metric computation as follows. Λ Λ Λ Λ JK l JK l JK b l y l b l y l b l y l l JK b l y l l JK l JK l JK b l x l l b l x l l b l x l l l JK b l l x l l JK JK l JK l l n JK l JK l n n l l b l x l nb n b l x l nb n b l x l l jb l j j K b l x l l jb l j j where we have assumed that x l m if m or l m K. he form for Λ is essentially the same as the form for the metric for MS with intersymbol interference. Now it is clear we can apply (as in the ISI case) dynamic programming (Viterbi Algorithm) to determine the optimal sequence b. We define the state to be the last K data bits σ l b l K b l K λ l σ l σ l y l b l b l x l l b l x l l jb l j j Λ JK λ l σ l σ l b y x b l he complexity of the optimal detector is proportional to the number of states in the Viterbi algorithm. For binary signaling this is K. If the pulses were not time limited to duration but of longer duration the memory of the channel would grow and the number of states would also grow.

20 3- CHAPR 3. DIRC-SQUNC SPRAD-SPCRUM 6. Problems. Show that for random spreading sequences the average aperiodic correlation functions has the following averages. C k C n N k l N n a l a l k m k n N k k n N k n a m a m n. Derive the mean square value of the interference at the output of a matched filter due to a multipath signal with delay distributed uniformly on c c in a direct-sequence spread-spectrum system with N chips per bit and random spreading sequences. Compare to the mean square interference if the delay is uniformly distributed on. 3. Derive the mean square value of the multiple-access interference with random spreading sequences. Use this to determine the signal-to-noise ratio for random sequences. 4. Derive the mean square value of the multiple-access interference for k 3 and N 3 with the sequence shown below. Sequence one derived via the shift register with x i x i 3 x i 5. Sequence two derived via the shift register with x i x i x i x i 3 x i 5. Sequence three derived via the shift register with x i x i x i 3 x i 4 x i 5. Verify that these sequences (of length 3) are not cyclic shifts of each other. Assume they all start in state (). 5. A direct-sequence spread-spectrum system transmits a binary modulated signal s t using a spreading sequence a t. he data sequence is a sequence of M bits b M b b b b M where b t s t M l M b l p t l Pa t b t a t NM a m p c t m c m NM he spreading sequence is periodic with period N and c N, thus there is exactly one period of the spreading sequence per data bit. he received signal is r t s t αs t τ n t where τ is a delay known to the receiver and α is a known constant. Determine an optimal receiver that has small complexity (not exponential in M). he optimal receiver if only one data bit (the one-shot problem) is transmitted (M ) is a filter matched to one period of a t αa t τ. However, this receiver can also be used when multiple bits are transmitted. he receiver samples the matched filter appropriately and makes a decision about the individual data bits at each sampling time. Find the average value of the desired signal output and the variance of the interference from multipath (if any) for this receiver. Assume random spreading sequences. Find an expression for the bit error probability with this receiver.

21 Derive the signal-to-noise ratio for each user in a direct-sequence spread-spectrum system that has K 3 users and N 3 with the sequences shown below. here is no fading but only white Gaussian background noise. Sequence one derived via the shift register with â i â i 3 â i 5. Starting state=(). a a Sequence two derived via the shift register with â i â i â i â i 3 â i 5. Starting state=(). a a Sequence three derived via the shift register with â i â i â i 3 â i 4 â i 5. Starting state=(). Compare the results to the signal-to-noise ratio for random spreading codes. 7. Derive the signal-to-noise ratio for the rake receiver for random spreading sequences in multipath fading. Assume that τ l is uniformly distributed on the interval 4 l c 4 l c c. First show the following equality for the model discussed in the notes. Second show that I j l I j l V j V j Use the above to derive the signal-to-noise ratio α l 3N j j l l α j α j 4 j l N 3N l j l j otherwise α l l l j 3N j j α j α j 4 j j N 3N j j SNR j α j j α j α l l l j 3N j l N N 8. Consider a direct-sequence spread spectrum (DSSS) system with N chips per bit and random spreading codes. Assume a pulsed jammer with power J transmitting Gaussian noise part of the time and not transmitting the other part of the time. et ρ be the fraction of time the jammer is on and ρ be the fraction of time the jammer is off. When the jammer is on the power is J ρ so that the average power is J. Assume the jammer is on (or off) for a whole bit interval. Assume the receiver can coherently demodulate the received signal. (a) Determine the average error probability as a function of ρ. (b) Determine the worst case error probability (maximum over ρ. (c) Compare to a BPSK system with Rayleigh fading. 9. Consider a direct-sequence spread-spectrum system with N 3 chips per bit. Assume the spreading code is from a maximal-length shift register sequence with feedback connection as follows. he sequence is derived via the shift register with feedback x i x i 3 x i 5 and starting state=(). (a) Plot the output of the matched filter as a function of time for a sequence of 5 bits (= ) Now consider a second transmitter with a sequence derived as follows. Sequence two derived via the shift register with x i x i x i x i 3 x i 5. Starting state=(). (b) Determine the output of the matched filter (matched to sequence one) but due to a signal using sequence two with the same sequence of data as in part (a).