EE359 – Lecture 8 Outline

1. 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

2. 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

3. 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

4. 1 g g0 g Optimal Adaptive Scheme Waterfilling • Power Adaptation • Capacity

5. 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

6. P Bc 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 f

7. d 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 s1 s4 s8 s5 s6 s7

8. 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 Z2 Z3 x Z1 Z4 s4 ^ Z8 Z5 s8 Z7 s5 Z6 s6 s7 dmin

9. Our focus 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

10. 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:

11. 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 • Pedepends on constellation minimum distance • Pein AWGN approximated by