SYSC 4607 – Slide Set 8 Outline

1. SYSC 4607 – Slide Set 8 Outline • Review of Last Lecture • Capacity of Wireless Channels • Capacity of Flat-Fading Channels • Fading Known at RX • Fading Known at TX and RX (Optimal Rate and Power Adaptation)

2. Capacity of Wireless Channels • Pioneered by Claude Shannon in 1948 • Shannon capacity is defined as the maximum Mutual Information of the channel (IT definition) • Capacity defines theoretical rate limit: Maximum error free rate a channel can support (Operational definition) • Theoretical limit (not achievable) • A property of channel (bandwidth, noise). Does not depend on design techniques

3. Capacity of Flat-Fading Channels Depends on what is known about the channel • Unknown Fading: • Worst-case channel capacity • Fading Statistics Known • Hard to find capacity • Fading Known at Receiver Only • Fading known at Transmitter and Receiver

4. Capacity of Flat-Fading ChannelsReceiver CSI Only • Channel Side Information (CSI) is channel power gain • Fading value known at receiver only; fading statistics known at both transmitter and receiver • Shannon capacity (ergodic capacity) This is average capacity which is lower than capacity with average γ.

5. Capacity of Flat-Fading ChannelsReceiver CSI Only • Jensen’s inequality: • Shannon capacity with receiver CSI only is smaller than channel capacity of AWGN channel with the same average γ. • Fading reduces capacity with the receiver CSI only

6. Capacity of Flat-Fading ChannelsTransmitter and Receiver CSI • S: a collection of Discrete Memoryless Channel states, each state denoted by s (s in S) • p(s) : probability of being in State s • Cs = capacity of channel in state s • Cγ= Blog2(1 + γ) is capacity of channel with average SNR γ • Without power adaptation, same capacity as “receiver CSI only”

7. Capacity of Flat-Fading ChannelsTransmitter and Receiver CSI • Power adaptation policy: Transmit at higher powers (and hence data rates) when channel is good (has high SNR γ) • Transmit power P(g) subject to average power constraint • Leads to optimization problem

8. 1 g g0 g Capacity of Flat-Fading ChannelsTransmitter and Receiver CSI Waterfilling • Optimal Power Allocation (Power Adaptation) Follows a “water-filling” scheme • Capacity • Cutoff threshold γ0 depends on fading distribution p(γ) and is obtained using power constraint formula

9. Capacity Comparisons At high SNR water-filling does not provide much gain

10. Capacity Comparisons Water-filling provides significant improvement at low SNR

11. Main Points • Capacity depends on degree of channel knowledge: CDI only, CSI at receiver only, CSI at both receiver and transmitter • Capacity with TX/RX knowledge: variable-rate variable-power transmission (water filling) optimal