1. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 1 DSSS Multiple Access Channel Objectives Definition of DSSS Definition of spectral bit rate Computing approximated SIR with DSSS in cellular network Spectral efficiencies with random-spreading Outline Understand DSSS MAC mechanism Understand spectral bit rate efficiency Realize the conditions that effect DSSS spectral efficiency

2. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 2  With orthogonal code origin data can fully be recovered (up to noise):  Requires full chip and phase synchronization; and  ideal channel (no signal fading nor distortion)  Semi-orthogonal codes  Code orthogonality at the RCV is hard to preserve – on one hand  Semi-orthogonal codes are faster to generate – on the other hand  A common semi-orthogonal code is the Pseudorandom Noise (PN) Semi-Orthogonal Sequences

3. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 3 Pseudorandom Noise (PN)  A PN sequence of length N >> G is a sequence of where n = 1,2,…,N.  Every user k has its own unique sequence (signature)  Nearly equal number of 0’s and 1’s within any long subsequence  A run of length r chips with the same sign occurs with prob. 2 -r  Low correlation between any two shifted versions  Emulates white Gaussian noise  Can be generated with an m-Stage Feedback Shift Register (Rappaport Ch. 6.11.1)  The sequence is periodic with period (in IS-95 it is equivalent to several tens of years)

4. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 4 PN Correlation Functions  For very large PN periods N, every subsequence of length G is statistically random taking each value with equal probabilities, and independent of any shift of it. Thus, for large G we get:  Since each user gets a unique sequence, we also get (9.1) (9.2) (9.3)  Recall that good sequences have and for all  So, let’s compute the correlation function of PN sequence.

5. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 5 PN Correlation Functions (cont.)  For the corresponding signal waveforms where,  Thus, for very large N and G, the PN sequences are nearly orthogonal within any symbol period T (uncorrelated along any symbol period)

6. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 6 Exercise 9.1: Compute the auto and cross correlation functions over the sequence period, N. Exercise 9.2: The aim of this exercise to evaluate the correlation properties of sequences generated by a LFSR with a finite period N. Consider the following Linear Feedback Shift Register (LFSR) with 5 stages initialized with a seed value 16 (i.e., (1,0,0,0,0) ). a) Compute the generated sequence after replacing the 0’s with –1’s. b) Is the sequence periodic? What is the period? c) How many 0’s and 1’s are there? d) Compute the number of runs (within one period) of length 1,2,3 and 4. Compare their frequencies with the probabilities expected from a truly random sequence. e) Compute the auto-correlation of this sequence f) Compute the sequence resulting from a seed value 16 (i.e., (0,1,0,0,0) ). g) How does the sequences from a) and f) relate to each other? What conclusions can you draw about the distribution of the sequence from f)? h) Compute the cross correlation between the sequences from a) and f) at shift 0. i) Can you find the relation between the sequence from a) and any other sequence generated from a non-zero seed. What can you draw about the period a sequence with a random seed? tap

7. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 7 Exercise 9.3: Consider the Linear Feedback Shift Register (LFSR) from exercise 9.2 but with 3 stages and a single tap you are free to choose. a) What is the seed that generates the sequence (1,1,1,0,0,1,0)? b) Compute the auto-correlation of the sequence for every time shift (replace 0 by –1). c) What is the cross-correlation between any two sequences that can be generated? d) Compute the auto-correlation of the sequence (1,1,1,1,0,0,0) (replace 0 by –1). e) Is the sequence from d) feasible with your LFSR? Exercise 9.4: The aim of this exercise to evaluate the correlation properties of sequences generated by more general Feedback Shift Registers. Consider the two following families of Feedback Shift Register (FSR) with n stages. a) Experiment with the cases of 3 stages and several combinations of 2 taps. b) Do you get any insight? What is it? FibonacciGalois

8. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 8 Exercise 9.5: The aim of this exercise is to learn what tap arrangements make good PN sequences. For a shift register of fixed length M, the number, and duration of the sequences that it can generate, are determined by the number, and position of taps used to generate the “parity” feedback bit. The combination of taps and their location is often referred to as a polynomial, and expressed e.g., a) A polynomial is primitive if it cannot be factored (i.e. it is prime), and if it is a factor of (i.e. can evenly divide) X N +1, where N = 2 n -1 (the length of the n-sequence) and n is the number of stages b) If the LFSR polynomial is primitive then the generated sequence is of a maximum length. c) Consider the Galois LFSR with n=3 and a single tap after R1. What is the corresponding polynomial? Is it primitive? Check out the generated sequence. d) Where are the taps that correspond to the polynomial g(x)=x+1 ? e) Generate the sequence that corresponds to g(x)=x+1. Assume that the “parity” register is filled with 0 (alternatively 1).What is the sequence length is this case? Galois

9. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 9 Intra-Cell SIR value with PN  Assume  Intra-cell interference only  Equal received power at BTS  User’s TRX-RCV are in sync  No multi-path fading  Ideal rectangular pulse shape  AWGN noise with PSD /symbol  Received signal from K intra-cell users by RCV i (during symbol period)  Received signal of all users after de-spreading with the PN of user i where Data symbol User i Data symbol User j (9.4) element wise

10. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 10 Signal Power - Refresh  The instantaneous power of a signal s(t) at time t is given by  The average power of a signal s(t) during an interval [t,t+T] is given by watts  The average power of signal s(t) that can be represented by a stationary r.v. S with a zero mean is given by watts  The avg power of uncorrelated signals is the sum of their avg powers

11. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 11  Due to “random” PNs and equal received power, I is an average of i.i.d r.v.’s, and therefore can be approximated by a Gaussian r.v.  Users PNs are selected randomly - thus, can be regarded as r.v.’s. Given the user data symbols,, we get: 1/G Uncorrelated signals element wise The Power of the MAC Interference I (9.5)

12. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 12 The Power of the MAC Interference I (cont.)  We use the Gaussian r.v. representation of I to computes its power using its variance as given in (9.5).  Call this method “The Variance Method”.

13. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 13  Thus, the MAC interference with “random” PNs, I, is roughly white noise with power (per chip period) equals to total received power divided by the processing gain G. That is, the chip power vector is  The MAC interference power during the symbol period,, is the sum of powers in all symbol chips, namely, where is the received power of data symbol j  If the received power of all users are the same and equal, then The Power of the MAC Interference I (cont.) where (9.6)

14. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 14  To compute the power of the subject signal we “The Direct Method”. That is, we follow the definition. The Power of the Subject User signal

15. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 15  The chip power vector of during the symbol period is given  The power of during the symbol period,, is given by The Power of the Subject User signal (cont.) Exercise 9.2 : Compute the power of I using the “direct method”, and the power of using the “variance method”. (9.7)

16. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 16  Assuming perfect power control (PC) that equalizes the received power, let be the symbol received power. From (9.4)  NOTE: Recall that we ignore  Inter-cell interference, actual symbol shape, non-perfect PC, multi-path fading, imperfect PN randomness, imperfect sync (9.8) Bandwidth  For (9.8) we invoked Eqs. (9.6) and (9.7) and used the fact that white noise PSD (density function over the frequency band) per symbol is

17. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 17  Further assuming  Stationary and ergodic channels  Cell independency  Equal average received power from every uplink (intra & inter) - Adjacent Interfering Cell Factor - Voice Activity Factor - Coefficient rising from non-rectangular pulse shape (9.10) Eq. (9.10) expresses the SIR of channel i as function of:  the number of chips per symbol, G  the number of users per cell, K  the topology of adjacent cells  recovered signal power at the receiver  added white noise spectral density

18. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 18 DSSS MAC Spectral Properties - Summary  Channel introduces noise, ISI, Narrowband and MAC interference  Spreading has no effect on AWGN noise  ISI delayed by more than T c is reduced by code auto-correlation  Narrowband interference is reduced by de-spreading at RCV  MAC interference is reduced by code cross-correlation  Frequency diversity – overcomes frequency selective impairments  Welsh-Hadamard code has low cross and auto correlation  reduces multi-path signals delay spread of more than one chip  Sync between TRX and RCV is essential  Orthogonality between spreading codes enhances MAC capacity  TRX power control to resolve ”near-far problem”  Macro-Diversity, Voice-Activity-Detection, Soft-handoff are integral parts

19. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 19 DSSS MAC Spectral Properties – Summary (cont.)  Diversity (see Rappaport Sec. 7.10)  A technique to compensates fading channels impairment  Implemented by using multiple diverse receiving antennas  The receivers select the strongest signal(s) among the multiple propagation paths  Micro-Diversity is aimed to handle small-scale fading (deep and rapid signal fluctuations, e.g., Rayleigh or Rician fading distributions). Uses antennas separated by a fraction of a meter  Macro-Diversity is aimed to handle large-scale fading (caused by, shading, e.g., Log-normal fading distribution). Uses antennas that are sufficiently to overcome shadowing. E.g., multiple BTS per user for the uplink channel

20. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 20 DSSS MAC Spectral Properties – Summary (cont.)  Soft-handoff (see Rappaport Sec. 3.4.2)  A technique to hand off a mobile from one BTS to another w/o changing the physical radio channel. Feasible with CDMA by only sync the parties on the same spreading codes.  The MSC simultaneously evaluates the mobile received signals from its controlled BTS and selects the best instantaneous received signal to pass on. Done w/o mobile assistance nor awareness.  Facilitated by reporting to the MSC (several hundreds time per second) each channel quality from every serving BTS.  Exploits the Macro-Diversity  Used in IS-95A  In contrast to hard-handoff where the physical radio channel is changed and the mobile assists with the hand-off decision

21. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 21  Eq. (9.10) above doesn’t provide the channel capacity yet  The channel capacity is expressed by Shannon’s coding Theorem and is expressed as a function of the received SNR - Received signal bandwidth (Hz) - Average received signal power (watts/transmission) (9.11) Bits/transmission - One-sided Gaussian noise spectral level ( )  C is the total number of bits/sec that can be transmitted arbitrarily reliably with optimal coding  Eq. (9.11) holds for channels with AWGN (Gaussian channels)  Here, SNR is given by The Capacity–SNR Connection

22. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 22

23. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 23

24. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 24

25. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 25 Hence,

26. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 26 Fixed

27. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 27 Bits / transmission / Hz Bits / transmission (also called spectral efficiency) watts / transmission bits / transmission

28. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 28 Maximum bit rate/Hz - AWGN channel - Unbounded bandwidth - Single user (no spreading) - Optimal coding Resolved from the equation: or (9.12) Shannon Fixed-Point Equation

29. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 29 Shannon Fixed Point for capacity  Eq. (9.12) provides Shannon’s fixed point for the spectral bit rate R  In a similar way we can Shannon’s fixed point for channel the capacity C (as expressed in Eq. (9.11)) Exercise 9.6 : Derive Shannon’s fixed point for channel the capacity C as a function of the bit energy and noise PSD

30. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 30

31. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 31 Power efficient modulation schemes are most often used in this case

32. Lecture 9: DSSSAdv. Wireless Comm. Systems - MAC - 32 Modulation Bandwidth efficient modulation schemes are most often used in this case