part 3: channel capacity ecen478 shuguang cui. ecen478, cui shannon capacity defined as the maximum...
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ECEN478, Cui
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
ECEN478, Cui
AWGN Channel Capacity
Goldsmith,Figure 4.1
AWGN channel capacity, bandwidth W (or B), deterministic gain:
g[i]=1 is knownand fixed
Total: Bits/s
If average received power is watts and single-sided noise PSD is watts/Hz,
Per dimension: Bits/s/Hz0.5
ECEN478, Cui
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?
ECEN478, Cui
Shannon Limit in AWGN channel
What is the minimum SNR per bit (Eb/N0) for reliable communications?
for small
Where:
ECEN478, Cui
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
ECEN478, Cui
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:
ECEN478, Cui
Fast fading, only Rx knows CSI
This is so called Ergodic Capacity.Achievable even only receiver knows the channel state.
As L goes large:
Less thanAWGN
ECEN478, Cui
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:
ECEN478, Cui
An equivalent approach: power allocation over time
Channel model:
Subject to:
Notation:
ECEN478, Cui
Optimal solution
Use Lagrangian multiplier method, we have the water-filling solution:
To define the water level, solve:
ECEN478, Cui
Asymptotic results
As L goes to infinity, we have:
The solution converges to be the same as the textbook approach!
ECEN478, Cui
Example
Fading with two states
Water-filling
Where is the water level? Three possible cases for
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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.)
ECEN478, Cui
Performance Comparison
At high SNR, waterfilling does not provide any gain. Transmitter knowledge allows rate adaptation and simplifies coding.
ECEN478, Cui
Time Invariant Frequency Selective Channel
We have multiple parallel AWGN channels with a sum power constraint!
Yes, water-filling!