Three Lessons Learned

1. Three Lessons Learned • Never discard information prematurely • Compression can be separated from channel transmission with no loss of optimality • Gaussian noise is worst case. Optimal signal in presence of Gaussian noise has Gaussian distribution. So self-interference should be designed as Gaussian.

2. Realities • Never discard information prematurely • Use soft-decisions and sequence detectors, if complexity okay. • Compression can be separated from channel transmission • For time-invariant single-user channels only. • Self-interference should be designed as Gaussian • Based on Viterbi’s argument, this represents a saddle (not optimal) point. • If the self-interference is not treated as interference, then Gaussian signaling is suboptimal (by Shannon theory).

3. MAC and Broadcast Channel Capacity • User Capacity • How many users can be accommodated in the channel given performance specs. • Assumes identical users and white noise model for interference • Shannon Capacity Region • Upper bound on rate vector that all users can achieve simultaneously • No complexity or delay constraints. • Optimal signaling and reception (unless constraints are added) • Asymptotically small error probabilty. • Signals from other users not treated as interference

4. User Capacity • Applicable to CDMA, since TDMA and FDMA have fixed capacity (# of channels). • S/(N+I(M)) determined based on the total number of users M and the system model. • Can be deterministic or random (fading). • Interference I(M) modeled as AWGN • Based on the modulation, coding, channel model, etc., we find the probability of bit error Pe=f[S/(N+I(M))] • For a given performance Pewe invert the above expression to get the maximum possible M. • Often set N=0 to simplify inversion, implies an interference-limited system.

5. Probability of Error • Coherent BPSK Modulation: for m users, and a spreading gain G: • m is typically random. For L total users each with probability p of active transmission and voice activity factor a: Note that Pe is concave in m

6. Pe Approximation • By concavity of Peand Jensen’s inequality: Use RHS as approximation for Pe ``Spread spectrum for mobile communications”, Pickholtz, Milstein, Schilling

7. Effective Energy/Symbol • M is average number of active users. • r is the code rate • K is the out-of-cell interference ratio (equals zero for a purely MAC channel) • a is the voice activity factor • N=G is the number of chips per symbol • Factor of 2/3 assumes rectangular pulses, will decrease for other shapes. • Assumes no ISI, flat-fading, or diversity gain.

8. Required Es/N0 • Target Pe • Invert target Pe to get required Es/N0 • Example: DPSK Often cannot get greqd in closed form: Must use numerical techniques or obtain from BER curve.

9. User Capacity • Total number of users the MAC channel can support: • A rougher approximation Note: Channel coding and interference mitigation techniques increase user capacity

10. Multiuser Channel Capacity in Fading R3 R2 R1 Goal: Maximize the rate region {R1,…,Rn} through dynamic allocation of channels, power, and rate as the user channels and requirements change.

11. Spectral Sharing • Time-Division (TD) and Frequency-Division(FD) • Channels are divided orthogonally. • Reduces the multiuser channel to single-user channels. • Dynamic allocation of time, bandwidth, rate, and power.* • Code-Division (CD) • Orthogonal codes: compromised by fading. • Semi-orthogonal codes: introduce co-channel interference. - Reduced by multiuser detection. • Dynamic allocation of codes, rate, and power.* *Requires Channel Side Infomation and Adaptation

12. AWGN Broadcast Channel Capacity • Model • One transmitter, two receivers with spectral noise density n1, n2: n1<n2. • Transmitter has average power S and total bandwidth B. • Single User Capacity • Set of achievable rates includes (C1,0) and (0,C2), obtained by allocating all resources to one user.

13. Rate Regions • Time Division (Constant Power) • Fraction of time t allocated to each user is varied • Time Division (Variable Power) • Fraction of time t and power siallocated to each user is varied • Frequency Division • Bandwidth Biand power Siallocated to each user is varied. Note: Equivalent to TD for Bi=tiB and Si=tisi.

14. Code Division • Superposition Coding • Coding strategy allows better user to cancel out interference from worse user. • DS spread spectrum with spreading gain G and cross correlationr12= r21 =G: • By concavity of the log function, G=1 maximizes the rate region. • DS without interference cancellation