Part 3: Channel Capacity

1. Part 3: Channel Capacity ECEN478 Shuguang Cui

2. Shannon Capacity • Defined as the maximum mutual information across channel (need some background reading) • Maximum error-free data rate a channel can support. • Theoretical limit (usually don’t know how to achieve) • Inherent channel characteristics • Under system resource constraints • We focus on AWGN channel with fading

3. AWGN channel capacity, bandwidth W (or B), deterministic gain: Per dimension: Bits/s/Hz 0.5 Total: Bits/s If average received power is watts and single-sided noise PSD is watts/Hz, g[i]=1 is known and fixed Goldsmith, Figure 4.1 AWGN Channel Capacity

4. Power and Bandwidth Limited Regimes Bandwidth limited regime capacity logarithmic in power, approximately linear in bandwidth. Power limited regime capacity linear in power, insensitive to bandwidth. If B goes to infinity?

5. Capacity Curve

6. What is the minimum SNR per bit (Eb/N0) for reliable communications? for small Shannon Limit in AWGN channel Where:

7. Capacity of Flat-Fading Channels • Capacity defines theoretical rate limit • Maximum error free rate a channel can support • Depends on what is known about channel • CSI: channel state information • Unknown fading: • Worst-case channel capacity • Only fading statistics known • Hard to find capacity

8. For l-th coherence time period, we have roughly the same gain: Average capacity over L period: Capacity of fast fading channel : Flat Rayleigh, receiver knows. Unit BW, B=1. Fast fading, with a certain decoding delay requirement, we can transmit time duration LTc (L>>1), i.e., L coherence time periods. The received SNR: The capacity (Rx knows CSI):

9. As L goes large: Less than AWGN Fast fading, only Rx knows CSI This is so called Ergodic Capacity. Achievable even only receiver knows the channel state.

10. Example • Fading with two states • Ergodic capacity • AWGN counterpart • Capacity

11. Fading Known atboth Transmitter and Receiver • For fixed transmit power, same as only receiver knowledge of fading, but easy to implement • Transmit power can also be adapted • Leads to optimization problem:

12. Subject to: An equivalent approach: power allocation over time Channel model: Notation:

13. To define the water level, solve: Optimal solution Use Lagrangian multiplier method, we have the water-filling solution:

14. Asymptotic results As L goes to infinity, we have: The solution converges to be the same as the textbook approach!

15. Example • Fading with two states • Water-filling Where is the water level? Three possible cases for

16. Water-filling over time

17. Implementation with discrete states Goldsmith, Fig 4.4 We only need N sets of optimal AWGN codebooks. (We need feedback channel to know the channel state.)

18. Performance Comparison At high SNR, waterfilling does not provide any gain. Transmitter knowledge allows rate adaptation and simplifies coding.

19. Time Invariant Frequency Selective Channel We have multiple parallel AWGN channels with a sum power constraint! Yes, water-filling!

20. Multicarrier system in ISI channel

21. OFDM-discrete implementation of multi-carrier system Transmitter

22. OFDM receiver FFT matrix:

23. Two-dimension Water-filling! Time Varying Frequency Selective Channel Maximize: s. t.:

24. Summary of Single User Capacity • Fast fading channel: • Ergodic capacity: achievable with one fading code or multiple sets of AWGN codes • Power allocation is WF over distribution • Frequency selective fast fading channel: • Ergodic capacity is achieved with 2-D WF