use a correlation receiver to determine whether a +1 or a,1 We transmitted at a time instant m. We assume that the was is connected to an information

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1 a binary discrete-time communication system with the Consider signal a time sequence fr m g 1 m=0 dened by received r m = Eb m + w m b m is the sequence of information symbols (antipodal binary, where m 2 f1g) and w m is additive zero-mean white Gaussian noise b Direct-Sequence Spread Spectrum (AWGN), i.e., E[w m ] = 0; E[w m w m+l ] = 2 (l): Paul Flikkema/USF 1

2 use a correlation receiver to determine whether a +1 or a,1 We transmitted at a time instant m. We assume that the was is connected to an information source which outputs transmitter and,1's with equal probability. In this case, such a receiver +1's r m = y m is known as the decision variable. Its statistics Here the performance of the receiver. It is easy to see that it determine a normal (Gaussian) random variable with mean Eb m and is 2. variance is a simple level detector: r m 0 : decide that + 1 was sent r m < 0 : decide that, 1 was sent Paul Flikkema/USF 2

3 either c 0 ;c 1 ; :::; c or, c 0 ;,c 1 ; :::;,c ; each bit of duration T is coded into a sequence of N chips of Thus T c = T=N: The increase in signaling rate spreads the duration received sequence can be described as (dropping the subscript The for clarity) m E c = E N and E[w2 n] = 2 N : where consider modulating each symbol with another 1-valued Now (a spreading sequence) fc n g n= such that each symbol sequence b m results in the transmission of depending on the value of b m. spectrum of the transmitted signal by a factor of N. r n = E c bc n + w n ; n = 0; 1; :::; N, 1: Paul Flikkema/USF 3

4 let us specify two properties of the spreading sequence fc n g: it Now a mean value of approximately zero, i.e., has 8< N; : conditions are ideal, but can be closely approached in These These properties give sequences of this type a noise-like practice. c n 0; and an autocorrelation given by i = 0 c n c n+i 0; otherwise. appearance; thus the name pseudo-noise (PN) sequences. Paul Flikkema/USF 4

5 the correlation receiver performs the following operation to Then, the decision variable y: obtain r n c n y = or y = (E c bc n + w n )c n ; which yields, based on the properties of the spreading sequence, y = N E c b + w n c n : Paul Flikkema/USF 5

6 our decision variable is normal with mean N E c b = Eb and Hence 2. Comparing this result with the non-spread system variance shows that spreading yields no improvement in the ideal above channel. This can be seen intuitively by noting that the AWGN rate is increased by a factor of N, but this also increases signaling signal bandwidth, and therefore the noise power, by a factor of the we will see, the power of spreading comes from its eect on As or correlated signals. These include interference, narrowband N. multipath, or signals from other transmitters the network. Paul Flikkema/USF 6

7 suppose the channel contains an interferer: an unknown Now is added to the received signal. Then we have constant i n = I, a real constant. Then our correlation receiver where the decision variable produces w n c n r n = E c bc n + i n + w n ; n = 0; 1; :::; N, 1: (E c bc n + i n + w n )c n y = which becomes y = N E c b + I c n + or y N E c b w n c n ; Paul Flikkema/USF 7

8 again a decision variable with mean N E c b = Eb and yielding 2, so the interference is suppressed by the despreading variance contrast, the decision variable in non-spread system would have In mean of Eb + I, which will render the system unusable for jij a Notice that the recovery of the signal requires that the Remark: own copy of the spreading sequence be synchronized with receiver's received version. This is a key requirement in spread spectrum the design. system From now on, we normalize to E c = 1 (unit chip energy) for Note: so that the bit energy is N. simplicity, (correlation) operation. suciently large. Paul Flikkema/USF 8

9 consider a multipath channel, with a direct path and a Now (reected) path which causes another copy of the signal to specular 8< b mc n + b m,1 c N,l+n ; n = 0; :::l, 1 : we assume that l < N, i.e., the delay is less than one symbol where a duration. the specular path causes interference from both a delayed Notice of the desired symbol and the previously transmitted version arrive at a delay of l with unknown attenuation : r n = b m c n + b m c n,l ; n = l; :::N, 1; b m,1. symbol This assumption can be relaxed using straightforward methods. a Paul Flikkema/USF 9

10 multipath signal is suppressed by the despreading. In the case The the unspread system, this channel could cause severe ISI, of Here, l,1 = Nb m + b m,1 m y c n + b m N,l+n c c n,l c n + w n c n ; n=l which becomes y m Nb m w n c n : resulting in a performance loss. Paul Flikkema/USF 10

11 assume there are K users (transmitters), where the kth Now modulates its data with the spreading sequence fc (k) transmitter set of signature sequences or spreading codes has the This property crosscorrelation (k) c(j) 8 N; k <: = j; i = 0 n+i n c we have a set of K sequences with zero crosscorrelations and Thus autocorrelations. (Note that this includes the impulse-valued Direct-Sequence Code Division Multiple Access n g. 0; otherwise earlier autocorrelation as a special case.) Paul Flikkema/USF 11

12 that all K users simultaneously transmit, and we are Assume only in the signal from user k = 1. Assuming time interested K k=1 (k) m c(k) n b + w n follows that the correlation receiver for user 1 generates the It variable decision (1) m y K k=2 Nb(1) m n c (1) n : w synchronization between users, the received signal is r n = (1) m = b (1) m y (c (1) n ) 2 + (k) m b (k) n c(1) n + c n c (1) n w Using the crosscorrelation property, Paul Flikkema/USF 12

13 the crosscorrelation property of the sequences allows Therefore, transmissions in the same channel to be successfully simultaneous This property allows spread spectrum to be used as a detected. method (like TDMA or FDMA); it is usually called multiple-access code-division multiple access, or CDMA. Note: Normally, the dierent transmitters would not be time-synchronized, so all signals would be received with dierent relative delays. However, due to the crosscorrelation property, it is easy to show that the same result is true when the other K, 1 signals have arbitrary delays relative to the desired signal. Paul Flikkema/USF 13

14 that the multipath and multi-user interferences are additive. Notice the previous results can be combined to show the spread Therefore, DS Spread Spectrum { Summary spectrum provides resistance combinations of multipath multi-user interference Considerations: The sequence properties used here are idealized; degradation occur in practice. Solutions include coding, power can control, and multi-user receivers. The sequences are periodic; the period may be greater than a symbol duration to prevent performance loss. Paul Flikkema/USF 14

Revision of Lecture Eighteen

Revision of Lecture Eighteen Previous lecture has discussed equalisation using Viterbi algorithm: Note similarity with channel decoding using maximum likelihood sequence estimation principle It also discusses