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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.

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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).

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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

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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 P e =f[S/(N+I(M))] For a given performance P e we invert the above expression to get the maximum possible M. –Often set N=0 to simplify inversion, implies an interference-limited system.

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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 : Note that P e is concave in m

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P e Approximation By concavity of P e and Jensen’s inequality: Use RHS as approximation for P e ``Spread spectrum for mobile communications”, Pickholtz, Milstein, Schilling

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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) – 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.

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Required E s /N 0 Target P e Invert target P e to get required E s /N 0 Example: DPSK Often cannot get reqd in closed form: Must use numerical techniques or obtain from BER curve.

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

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Multiuser Channel Capacity in Fading Goal: Maximize the rate region {R 1,…,R n } through dynamic allocation of channels, power, and rate as the user channels and requirements change. R1R1 R2R2 R3R3

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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

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AWGN Broadcast Channel Capacity Model –One transmitter, two receivers with spectral noise density n 1, n 2 : n 1 <n 2. –Transmitter has average power S and total bandwidth B. Single User Capacity Set of achievable rates includes (C 1,0) and (0,C 2 ), obtained by allocating all resources to one user.

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Rate Regions Time Division (Constant Power) –Fraction of time allocated to each user is varied Time Division (Variable Power) –Fraction of time and power i allocated to each user is varied Frequency Division –Bandwidth B i and power S i allocated to each user is varied. Note: Equivalent to TD for B i = i B and S = i i.

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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 correlation 12 = =G: –By concavity of the log function, G=1 maximizes the rate region. DS without interference cancellation

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