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

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

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. 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.
For l-th coherence time period, we have roughly the same gain:
The received SNR:
The capacity (Rx knows CSI):
Average capacity over L period:

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

10. Example Fading with two states Ergodic capacity AWGN counterpart

11. Fading Known at both 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. An equivalent approach: power allocation over time
Channel model:
Notation:
Subject to:

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

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

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