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Communications II

Communications II. Prof. Dr.-Ing. Karl-Dirk Kammeyer Department of Communications Engineering. 4. Optimal Receiver for AGN Optimal AWGN Receiver Generalization for coloured noise Symbol / Bit Error Probability 5. Equalization Linear Equalization Decision Feedback Adaptive Equalization

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Communications II

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  1. Communications II Prof. Dr.-Ing. Karl-Dirk Kammeyer Department of Communications Engineering

  2. 4.Optimal Receiver for AGN Optimal AWGN Receiver Generalization for coloured noise Symbol / Bit Error Probability 5.Equalization Linear Equalization Decision Feedback Adaptive Equalization Optimal Receiver for ISI Conditions Forney-Receiver (MLSE) Viterbi-Algorithm 3. Error Probability at Viterbi Detection 4. Influence of Channel Impulse Response 5. Channel Estimation (Least Squares) Mobile Radio Channels Multipath Propagation Doppler Spreading Multiple Access Mobile Radio Transmission Concepts The OFDM System Principles of Code Multiplex (CDMA) Communications Part II Contents

  3. 4.1 Optimal Receiver for AWGN Channels received signal: for time discrete representation the following vectors can be defined: this results in Optimal Receiver

  4. A-priori-probability Joint probability density of received signal Joint probability density of received signal under the condition that was transmitted This expression has to be calculated for all M possible hypotheses for . MAP (Maximum a-posteriori) Criterion task: getting an optimal decision criterion for transmitted elementary signal probability for transmission of impulse on condition of reception of signal vector has to be maximized. with the Bayes rule we get with Optimal Receiver

  5. The maximum leads to the signal that was transmitted with greatest probability. Maximum-Likelihood-Criterion Thus the MAP-Criterion is The joint probability density of a Gaussian process can be described as where is the autocorrelation matrix of the noise process with conjugate elements Optimal Receiver

  6. Maximum-Likelihood-Criterion when transmitting we get which is the joint probability density of a Gaussian process with modified mean now the MAP-criterion can be expressed as Optimal Receiver

  7. Maximum-Likelihood-Criterion maximizing this expression yields with the first term being a constant the MAP-criterion is Optimal Receiver

  8. Maximum-Likelihood-Criterion The last term contains a-priori-prohability of the transmitted data. If this is unknown at the receiver, we get Maximum-Likelihood-Decision if  equal results for ML- and MAP-Criterion Optimal Receiver

  9. is energy of - th transmit impulse a-priori-probability requires knowledge of noise power Correlation Receiver for AWGN additive white Gaussian noise: MAP-criterion: !  generally unknown Optimal Receiver

  10. Correlation Receiver for AWGN  evaluate ML-Criterion ! this presumes equal a-priori-prohability  in many cases realistic assumption further simplification: if energies of all transmit impulses are the same not needed  Optimal Receiver

  11. with Correlation Receiver for AWGN MC-Criterion includes correlation of complex envelope of received signal with all possible transmit impulses • in time discrete representation: scalar product of two vectors • in time continous representation: intergration over interval Optimal Receiver

  12. in Matched-Filter-Receiver for AWGN we can define with and • general form of Matched-Filter-relation for complex signals • in case of Gaussian-distributed noise maximizing the SNR and maximizing the ML-criterion yields the same solution Optimal Receiver

  13. Matched-Filter-Receiver for AWGN if  sampling at of all preceding and following data symbols enables decision of i-th data symbol independent Optimal Receiver

  14. Matched-Filter-Receiver for AWGN block diagram: Optimal Receiver

  15. real and QPSK-Signal, Energies of all elementary pulses are equal with Example: QPSK Transmit impulses and Matched-Filter impulse responses Bit Mapping is Gray coded Optimal Receiver

  16. Optimal QPSK receiver; requirements: • Nyquist I • memoryless channel; AWGN MC-Criterion with and Example: QPSK for the four hypotheses this results in comparing all hypothesis we find Optimal Receiver

  17. auto-correlation-matrix of noise no longer identity matrix Colored noise modified impulses Correction term can be omitted if for Correlation-Receiver for colored noise Optimal Receiver

  18. has to have minimum phase Solution for Matched Filter: use decorrelation filter Matched-Filter: combined receiver filter: Matched Filter for colored noise Define colored noise as spectraly shaped white noise. Spectral shaping by filter Optimal Receiver

  19. Where is the noncausal representation of the matchen filter and is the autocorrelation matrix Matched-Filter-condition: Matched Filter for colored noise In time domain: For use the expression Optimal Receiver

  20. Error Probability for M-PSK: Examples (AWGN) bit error probability symbol error probability Optimal Receiver

  21. Error Probability for M-QAM: Examples (AWGN) bit error probability symbol error probability Optimal Receiver

  22. Error Probability for Differential PSK Optimal Receiver

  23. Bit Error Probability for M-DPSK: Examples (AWGN) noncoherent coherent Optimal Receiver

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