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5. Capacity of Wireless Channels. Information Theory. So far we have only looked at specific communication schemes. Information theory provides a fundamental limit to (coded) performance.

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Information theory
Information Theory

  • So far we have only looked at specific communication schemes.

  • Information theory provides a fundamental limit to (coded) performance.

  • It succinctly identifies the impact of channel resources on performance as well as suggests new and cool ways to communicate over the wireless channel.

  • It provides the basis for the modern development of wireless communication.

Historical perspective
Historical Perspective

Gugliemo Marconi

Claude Shannon

  • Wireless communication has been around since 1900’s.

  • Ingenious but somewhat adhoc design techniques



  • Information theory says every channel has a capacity.

  • Many recent advances based on understanding wireless channel capacity.

New points of views arise.

Multipath fading a modern view
Multipath Fading: A Modern View


Classical view: fading channels are unreliable

Modern view: multipath fading can be exploited to increase spectral efficiency.

Capacity of awgn channel
Capacity of AWGN Channel

Capacity of AWGN channel

If average transmit power constraint is watts and noise psd is watts/Hz,

Power and bandwidth limited regimes
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.

Frequency selective channel
Frequency-selective Channel

's are time-invariant.

OFDM converts it into a parallel channel:

where is the waterfilling allocation:

with  chosen to meet the power constraint.

Can be achieved with separate coding for each sub-carrier.

Fading channels
Fading Channels

Different notions of capacity, depending on scenarios:

  • Slow fading channel: outage capacity

  • Fast fading channel: “ergodic” capacity under infinitely deep time interleaving

  • Capacity also depends on whether channel knowledge is known at the transmitter and/or receiver.

Slow fading channel
Slow Fading Channel

h random.

There is no definite capacity.

Outage probability:

-outage capacity:

Outage for rayleigh channel
Outage for Rayleigh Channel

Pdf of log(1+|h|2SNR)

Outage cap. as fraction of AWGN cap.

Receive diversity
Receive Diversity

Diversity plus power gain.

Time diversity i
Time Diversity (I)

Coding done over L coherence blocks, each of many symbols.

This is a parallel channel. If transmitter knows the channel, can do waterfilling.

Can achieve:

Time diversity ii
Time Diversity (II)

Without channel knowledge,

Rate allocation cannot be done.

Coding across sub-channels becomes now necessary.

Fast fading channel
Fast Fading Channel

Channel with L-fold time diversity:


Fast fading channel has a definite capacity:

Tolerable delay >> coherence time.

Capacity with full csi
Capacity with Full CSI

Suppose now transmitter has full channel knowledge.

What is the capacity of the channel?

Fading channel with full csi
Fading Channel with Full CSI

This is a parallel channel, with a sub-channel for each

fading state.


is the waterfilling power allocation as a function of the fading state, and  is chosen to satisfy the average power constraint.

Can be achieved with separate coding for each fading state.


At high SNR, waterfilling does not provide any gain.

But transmitter knowledge allows rate adaptation and

simplifies coding.

Performance low snr
Performance: Low SNR

Waterfilling povides a significant power gain at low SNR.

Waterfilling vs channel inversion
Waterfilling vs Channel Inversion

  • Waterfilling and rate adaptation maximize long-term throughput but incur significant delay.

  • Channel inversion (“perfect” power control in CDMA jargon) is power-inefficient but maintains the same data rate at all channel states.

  • Channel inversion achieves a delay-limited capacity.


  • A slow fading channel is a source of unreliability: very poor outage capacity. Diversity is needed.

  • A fast fading channel with only receiver CSI has a capacity close to that of the AWGN channel. Delay is long compared to channel coherence time.

  • A fast fading channel with full CSI can have a capacity greaterthan that of the AWGN channel: fading now provides more opportunities for performance boost.

  • The idea of opportunistic communication is even more powerful in multiuser situations, as we will see.