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. Optimal Adaptive Scheme
Power Adaptation
Capacity
Waterfilling 1 g g0

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. 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

7. 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

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

9. 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

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
Pe depends on constellation minimum distance
Pe in AWGN approximated by