Part 3: Channel Capacity

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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

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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

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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?

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Capacity Curve

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Shannon Limit in AWGN channel

What is the minimum SNR per bit (Eb/N0) for reliable communications?

for small

Where:

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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

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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:

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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

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Example

Fading with two states

Ergodic capacity

AWGN counterpart

Capacity

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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:

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An equivalent approach: power allocation over time

Channel model:

Subject to:

Notation:

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Optimal solution

Use Lagrangian multiplier method, we have the water-filling solution:

To define the water level, solve:

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Asymptotic results

As L goes to infinity, we have:

The solution converges to be the same as the textbook approach!

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Example

Fading with two states

Water-filling

Where is the water level? Three possible cases for

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Water-filling over time

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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.)

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Performance Comparison

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

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Time Invariant Frequency Selective Channel

We have multiple parallel AWGN channels with a sum power constraint!

Yes, water-filling!

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Multicarrier system in ISI channel

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OFDM-discrete implementation of multi-carrier system

Transmitter

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OFDM receiver

FFT matrix:

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Time Varying Frequency Selective Channel

Maximize:

s. t.:

Two-dimension Water-filling!

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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