EE359 – Lecture 8 Outline Capacity of Flat-Fading Channels Fading Known at TX and RX Optimal Rate and Power Adaptation Channel Inversion with Fixed Rate Capacity of Freq.-Selective Fading Channels Digital Modulation Review Geometric Signal Representation Passband Modulation Tradeoffs Linear Modulation Analysis

Review of Last Lecture Multipath Intensity Profile Doppler Power Spectrum Capacity of Flat-Fading Channels Theoretical Upper Bound on Data Rate Unknown Fading: Worst Case Capacity Fading Statistics Known: Capacity Hard to Find Fading Known at Receiver Only

Fading Known at Transmitter and Receiver For fixed transmit power, same as with only receiver knowledge of fading Transmit power S(g) can also be adapted Leads to optimization problem

Optimal Adaptive Scheme Power Adaptation Capacity Waterfilling 1 g g0

Channel Inversion Fading inverted to maintain constant SNR Simplifies design (fixed rate) Greatly reduces capacity Capacity is zero in Rayleigh fading Truncated inversion Invert channel above cutoff fade depth Constant SNR (fixed rate) above cutoff Cutoff greatly increases capacity Close to optimal

Frequency Selective Fading Channels For TI channels, capacity achieved by water-filling in frequency Capacity of time-varying channel unknown Approximate by dividing into subbands Each subband has width Bc (like MCM). Independent fading in each subband Capacity is the sum of subband capacities 1/|H(f)|2 Bc P f

Review of Digital Modulation: Geometric Signal Representation Transmit symbol mi{m1,…mM} Want to minimize Pe=p(decode mj|mi sent) mi corresponds to signal si(t), 0tT Represent via orthonormal basis functions: si(t) characterized by vector si=(si1, si2,…, siN) Vector space analysis s3 s2 d s1 s4 s8 s5 s6 s7

Decision Regions and Error Probability ML receiver decodes si closest to x Assign decision regions: Zi=(x:|x-si|<|x-sj| all j) xZim=mi Pe based on noise distribution Signal Constellation s3 s2 s1 Z3 Z2 Z4 Z1 x s4 ^ Z5 Z8 s8 s5 Z6 Z7 s6 s7 dmin

Passband Modulation Tradeoffs Want high rates, high spectral efficiency, high power efficiency, robust to channel, cheap. Linear Modulation (MPAM,MPSK,MQAM) Information encoded in amplitude/phase More spectrally efficient than nonlinear Issues: differential encoding, pulse shaping, bit mapping. Nonlinear modulation (FSK) Information encoded in frequency Continuous phase (CPFSK) special case of FM Bandwidth determined by Carson’s rule (pulse shaping) More robust to channel and amplifier nonlinearities Our focus

Linear Modulation Bits encoded in carrier amplitude or phase Pulse shape g(t) typically Nyquist Signal constellation defined by (an,bn) pairs Can be differentially encoded M values for (an,bn)log2 M bits per symbol Ps depends on Minimum distance dmin (depends on gs) # of nearest neighbors aM Approximate expression:

Main Points Capacity with TX/RX knowledge: variable-rate variable-power transmission (water filling) optimal Almost same capacity as with RX knowledge only This result may not carry over to practical schemes Channel inversion practical, but should truncate Capacity of ISI channel obtained by breaking channel into subbands (similar to OFDM) Linear modulation more spectrally efficient but less robust than nonlinear modulation Pe depends on constellation minimum distance Pe in AWGN approximated by