Wireless CommunicationElec 534Set ISeptember 9, 2007 : Wireless CommunicationElec 534Set ISeptember 9, 2007 Behnaam Aazhang The Course : The Course Light homework
Individual paper presentations
Team project presentations
Early December Multiuser Network : Multiuser Network Multiple nodes with information Outline : Outline Transmission over simple channels
Information theoretic approach
Fading channel models
Rician Outline : Outline Transmission over fading channels
Information theoretic approach
Approaching achievable rates
Communication with “additional” dimensions
Multiple input multiple (MIMO)
Transmission techniques Outline : Outline Wireless network
Achievable rate region
Random access Why Information Theory? : Why Information Theory? Information is modeled as random
Information is quantified
Transmission of information
Rate is established Information : Information Entropy
Higher entropy (more random) higher information content
Continuous Communication : Communication Information transmission
Mutual information Channel Useful Information Noise; useless information Maximum
useful information Wireless : Wireless Information transmission Channel Useful Information Noise; useless information Maximum
useful information Interference Randomness
due to channel Multiuser Network : Multiuser Network Multiple nodes with information References : 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 : 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 and performance analysis,” IEEE Trans. Commun., Nov. 2003.
J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. 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 : Reading for Set 1 Tse and Viswanath
Chapters 5.1-5.3, 3.1
Appendices A, B.1-B.5
Chapters 1, 4.1,5
Appendices A, B, C Single Link AWGN Channel : 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 : 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 : Signals in Time Domain Sampled at Nyquist rate
Example: three independent samples per second means three degrees of freedom time Voltage 1/W 1 second Signal in Frequency Domain : Signal in Frequency Domain Bandwidth W at carrier frequency fc frequency Power W Carrier frequency fc Baseband Signal in Frequency Domain : Baseband Signal in Frequency Domain Passband signal down converted
Bandwidth W frequency Power W Sampling : Sampling The baseband signal sampled at rate W
Sinc function is an example of expansion basis Model : Model There are N orthonormal basis functions to represent the information signal space.
The discrete time version Noise : Noise Assumed to be a Gaussian process
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 : Noise Projection of noise
Projections, ni onto orthonormal bases fi(t) are
Variance Noise : Noise The samples of noise are Gaussian and independent
The received signal given the information samples are also Gaussian Model : 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 : Statistical Model Key part of the model
The discrete time received signals are independent since noise is assumed white Entropy : Entropy Differential entropy
Differential conditional entropy
with Example : 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 : Proof Consider two zero mean random variables X and Y with the same variance
Assume X is Gaussian Variance of X Proof : Proof Kullback-Leibler distance
Due to Gibbs inequality! Gibbs’ Inequality : Gibbs’ Inequality The KL distance is nonnegative Capacity : Capacity Formally defined by Shannon as
where the mutual information
with Capacity : Capacity Maximum reliable rate of information through the channel with this model.
In our model Mutual Information : Mutual Information Information flow Channel Useful Information Noise; useless information Maximum
useful information Capacity : Capacity In this model
the maximum is achieved when information vector has mutually independent and Gaussian distributed elements. AWGN Channel Capacity : AWGN Channel Capacity The average power of information signal
The noise variance AWGN Capacity : AWGN Capacity The original Shannon formula per unit time
An alternate with energy per bit Achievable Rate and Converse : Achievable Rate and Converse Construct codebook with
Law of large numbers
Sphere packing Sphere Packing : Sphere Packing Number of spheres (ratio of volumes)
As N grows the probability
of codeword error vanishes
Higher rates not
possible without overlap Achievable Rate and Converse : Achievable Rate and Converse Construct codebook with
bits in N channel use Achieving Capacity : Achieving Capacity The information vector should be mutually independent with Gaussian distribution
The dimension N should be large
Source has information to transmit
Channel is available
No contention for access
Point to point Achieving Capacity : Achieving Capacity Accurate model
Signal model at the receiver
Synchronization Approaching Capacity : Approaching Capacity High SNR:
Coded modulation with large constellation size
Large constellation with binary codes
LDPC coding Constellations and Coding : Constellations and Coding