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Wireless Communication Elec 534 Set I September 9, 2007

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### Wireless CommunicationElec 534Set ISeptember 9, 2007

Behnaam Aazhang

The Course

- Light homework
- Team project
- Individual paper presentations
- Mid October
- Team project presentations
- Early December

Multiuser Network

- Multiple nodes with information

Outline

- Transmission over simple channels
- Information theoretic approach
- Fundamental limits
- Approaching capacity
- Fading channel models
- Multipath
- Rayleigh
- Rician

Outline

- Transmission over fading channels
- Information theoretic approach
- Fundamental limits
- Approaching achievable rates
- Communication with “additional” dimensions
- Multiple input multiple (MIMO)
- Achievable rates
- Transmission techniques
- User cooperation
- Achievable rates
- Transmission techniques

Outline

- Wireless network
- Cellular radios
- Multiple access
- Achievable rate region
- Multiuser detection
- Random access

Why Information Theory?

- Information is modeled as random
- Information is quantified
- Transmission of information
- Model driven
- Reliability measured
- Rate is established

Information

- Entropy
- Higher entropy (more random) higher information content
- Random variable
- Discrete
- Continuous

Communication

- Information transmission
- Mutual information

Channel

Useful Information

Maximum

useful information

Noise; useless information

Wireless

Interference

- Information transmission

Channel

Useful Information

Maximum

useful information

Randomness

due to channel

Noise; useless information

Multiuser Network

- Multiple nodes with information

References

- C.E. Shannon, W. Weaver, A Mathematical Theory Communication, 1949.
- T.M. Cover and J. Thomas, Elements of Information Theory, 1991.
- R. Gallager, Information Theory and Reliable Communication, 1968.
- J. Proakis, Digital Communication, 4th edition
- D. Tse and P. Viswanath, Fundamentals of Wireless Communication, 2005.
- A. Goldsmith “Wireless Communication” Cambridge University Press 2005

References

- E. Biglieri, J. Proakis, S. Shamai, Fading Channels: Information Theoretic and Communications, IEEE IT Trans.,1999.
- A. Goldsmith, P. Varaiya, Capacity of Fading Channels with Channel Side Information, IEEE IT Trans. 1997.
- I. Telatar, Capacity of Multi-antenna Gaussian Channels, European Trans. Telecomm, 1999.
- A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity, Part I. Systemdescription,” IEEE Trans. Commun.,Nov. 2003.
- ——, “User cooperation diversity. Part II. Implementation aspects andperformance analysis,” IEEE Trans. Commun.,Nov. 2003.
- J. N. Laneman, D. N. C. Tse, and G. W. Wornell,“Cooperative diversityin wireless networks: Efficientprotocols and outage behavior,” IEEETrans. Inform. Theory, Dec. 2004.
- M.A. Khojastepour, A. Sabharwal, and B. Aazhang, “On capacity of Gaussian ‘cheap’ relay channel,” GLOBECOM, Dec.2003.

Reading for Set 1

- Tse and Viswanath
- Chapters 5.1-5.3, 3.1
- Appendices A, B.1-B.5
- Goldsmith
- Chapters 1, 4.1,5
- Appendices A, B, C

Single Link AWGN Channel

- Model

where r(t) is the baseband received signal, b(t) is the information bearing signal, and n(t) is noise.

- The signal b(t) is assumed to be band-limited to W.
- The time period is assumed to be T.
- The dimension of signal is N=2WT

Signal Dimensions

- A signal with bandwidth W sampled at the Nyquist rate.
- W complex (independent) samples per second.
- Each complex sample is one dimension or degree of freedom.
- Signal of duration T and bandwidth W has 2WT real degrees of freedom and can be represented 2WT real dimensions

Signals in Time Domain

- Sampled at Nyquist rate
- Example: three independent samples per second means three degrees of freedom

Voltage

1 second

time

1/W

Model

- There are N orthonormal basis functions to represent the information signal space.
- For example,
- The discrete time version

Noise

- Assumed to be a Gaussian process
- Zero mean
- Wide sense stationary
- Flat power spectral density with height
- Passed through a filter with BW of W
- Samples at the rate W are Gaussian
- Samples are independent

Noise

- Projection of noise
- Projections, nionto orthonormal bases fi(t) are
- zero mean
- Gaussian
- Variance

Noise

- The samples of noise are Gaussian and independent
- The received signal given the information samples are also Gaussian

Model

- The discrete time formulation can come from sampling the received signal at the Nyquist rate of W
- The final model
- The discrete time model could have come from projection or simple sampling

Statistical Model

- Key part of the model
- The discrete time received signals are independent since noise is assumed white

Example

- A Gaussian random variable with mean and variance
- The differential entropy is
- If complex then it is
- Among all random variables with fixed variance Gaussian has the largest differential entropy

Proof

- Consider two zero mean random variables X and Y with the same variance
- Assume X is Gaussian

Variance of X

Gibbs’ Inequality

- The KL distance is nonnegative

Capacity

- Maximum reliable rate of information through the channel with this model.
- In our model

Mutual Information

- Information flow

Channel

Useful Information

Maximum

useful information

Noise; useless information

Capacity

- In this model

the maximum is achieved when information vector has mutually independent and Gaussian distributed elements.

AWGN Channel Capacity

- The average power of information signal
- The noise variance

AWGN Capacity

- The original Shannon formula per unit time
- An alternate with energy per bit

Achievable Rate and Converse

- Construct codebook with
- N-dimensional space
- Law of large numbers
- Sphere packing

Sphere Packing

- Number of spheres (ratio of volumes)
- Non overlapping
- As N grows the probability

of codeword error vanishes

- Higher rates not

possible without overlap

Achieving Capacity

- The information vector should be mutually independent with Gaussian distribution
- The dimension N should be large
- Complexity
- Source has information to transmit
- Full buffer
- Channel is available
- No contention for access
- Point to point

Achieving Capacity

- Accurate model
- Statistical
- Noise
- Deterministic
- Linear channel
- Signal model at the receiver
- Timing
- Synchronization

Approaching Capacity

- High SNR:
- Coded modulation with large constellation size
- Large constellation with binary codes
- Low SNR:
- Binary modulation
- Turbo coding
- LDPC coding

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