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Wireless Communication Elec 534 Set III October 11, 2007. Behnaam Aazhang. Reading for Set 3. Tse and Viswanath Chapters 5.4,6 Appendices B.8,B.9 Goldsmith Chapters 4. Model. A simple discrete time model where h is a complex Gaussian distributed fading coefficient

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## Wireless Communication Elec 534 Set III October 11, 2007

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**Wireless CommunicationElec 534Set IIIOctober 11, 2007**Behnaam Aazhang**Reading for Set 3**• Tse and Viswanath • Chapters 5.4,6 • Appendices B.8,B.9 • Goldsmith • Chapters 4**Model**• A simple discrete time model where h is a complex Gaussian distributed fading coefficient • Channel distribution information (CDI) at transmitter and receiver • Channel state information at receiver (and CDI) • Channel state information at transmitter and receiver (and CDI)**Channel Distribution Information (CDI)**• Achievable rate • Finding the maximizer is non trivial • For Rayleigh independent channel coefficients • Maximizing input is discrete with finite number of mass points • Mass at zero**Channel Distribution Information (CDI)**• Achievable rate computed numerically • Maximizing input distribution computed numerically • Not much to discuss—little analytical results**CSI Model**• State of the channel S (a function of h ) • Known to the receiver as V • Known to the transmitter as U**CSIR**• Channel state as a part of channel output since fading (or more precisely CSIR) is independent of the channel input r b v n**CSIR**• Proof**Ergodic**• The achievable rate when CSIR but no CSI at transmitter**Ergodic Capacity**• The model • Perfect channel state information at receiver**Ergodic**• The achievable rate is not a variable in time • If channel gain changes instantaneously the rate does not change • The rate is achieved over a long long codebook across different realizations of the channel • Long long decoding delay**Fading**• Fading does not improve Ergodic capacity • The key to the proof is Jensen’s inequality**Example**• A flat fading (frequency nonselective) with independent identically distributed channel gain as**Example**• CSIR no CSIT • Other system assumptions**Example**• The three possible signal to noise ratios**Example**• Ergodic capacity**Example**• Average SNR • The capacity of AWGN channel with the average SNR**CSITR**r b v u n**CSITR**• The mutual information • Capacity when there is CSI at transmitter and receiver • The original definition is not applicable • Define fading channel capacity**CSITR**• A result for multi-state channel due to Wolfowitz capacity for each state • Applied to CSITR**Ergodic Capacity**• Channel state information at transmitter and receiver • Power adjusted with constraint**Achievable Rate with CSITR**• Constraint optimization • Solving via differentiation**Power Control**• The solution is • Temporal water filling • Variable rate and variable power • Different size code books • Multiplexing encoders and decoders**Power Control**Minimum Channel Quality Threshold Allocated Power Better Channels**Capacity with CSITR**• The maximized rate • The threshold not a function of average power limit**Example**• A flat fading (frequency nonselective) with independent identically distributed channel gain as**Example**• CSITR • Other system assumptions**Example**• The three possible signal to noise ratios**Example**• Calculate the threshold • If the weakest channel is not used a consistent threshold emerges**Example**• Ergodic capacity**Example**• Average SNR • The capacity of AWGN channel with the average SNR**Probability of Outage**• Achieving ergodic channel capacity • Codewords much be longer than coherence time • Slow fading channels have long coherence times • Ergodic capacity more relevant in fast fading cases**Outage**• A burst with signal to noise ratio • Probability of outage • Capacity with outage • Information sent over a burst • Limited decoding delay • Nonzero probability of decoding error**Outage**• The minimum required channel gain depends on the target rate. • When instantaneous mutual information is less than target rate depends on the channel realization**Outage**• Probability of outage (CSIR) • Fading channel (CSIR)**CSI at Transmitter and Receiver**• Use CSITR to meet a target rate • Channel inversion • Minimize outage • Truncated channel inversion**Outage**• Probability of outage with CSITR • Fading channel with CSITR**Power Control**• Outage minimization • The solution for CSITR • Truncation with channel inversion**Power Control**Minimum Channel Quality Threshold Rayleigh PDF Allocated Power Better Channels**Minimum Channel Quality Threshold**Power Control Realization**Outage Capacity**• Target probability of outage • Fixed power • The outage capacity**Frame Error Rate**• An appropriate performance metric • In many examples, probability of outage is a lower bound to Frame Error Rate**Frequency Selective**• Recall the input output relationship**Capacity for Frequency Selective Channels**• Consider a time invariant channel • CSI is available at transmitter and receiver • Block frequency selective fading • An equivalent parallel channel model**CSITR: Frequency Selective**• The sum of rates • The power distribution**Power Control**• The power distribution threshold • Spectral water filling • Variable rate and variable power across channels • Different size code books • Multiplexing encoders and decoders**Achievable Rate**• The rate**Frequency Selective Fading**• Continuous transfer function • Power distribution across spectrum

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