Communications Part II

1. 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

2. : complex element of vector 5. Equalization: Definitions vector algebra: Equalization

3. gRx(t) gTx(t) c(t) e(k) 5.1 Linear Equalization: Conditions digital transmission system (symbolrate 1/T) equalizer downsampling channel  w w:oversampling factor impulse response: Linear Equalization

4.   equalizer order: Linear Equalization: Conditions zero forcing solution: !  1st Nyquist condition  NB conditions for calculating n+1 equalizer coefficients: ! Linear Equalization

5. with Linear Equalization: T-Equalizer w = 1: no exact solution (with n < ) for meeting 1st Nyquist Criterion possible • approximation: minimizing the energy of an error i (least squares) F• e=i + i in vector algebra: n+m+1 n+1 Linear Equalization

6. Linear Equalization: T-Equalizer zero forcing solution (least squares) for equalizer coefficient vector e : error vector energy : problem with zero forcing: equalizer coefficients may become very great  equalizer amplifies noise Linear Equalization

7. equalizer state vector: : Data : Noise equalizer coefficient vector: Linear Equalization: T-Equalizer  MMSE (Minimum Mean Square Error) - solution considers noise equalizer input: minimizing the power of (equalizer output – original data): equalizer output: Linear Equalization

8. Linear Equalization: T-Equalizer ACF-matrix (with conjugate complex elements): CCF-vector of data and equalizer input:  in vector algebra: Linear Equalization

9. MMSE - solution for equalizer coefficient vector e:  and needed for calculation of e : estimated from received data zero forcing = MMSE-solution no noise: contains original data  trainings sequence necessary Linear Equalization: T-Equalizer error energy: Linear Equalization

10. Linear Equalization: Examples (w = 1) Es/N0=15dB Zero Forcing power of MMSE power of Linear Equalization

11. n2+1 Linear Equalization: T/2-Equalizer (w = 2) zero forcing: F1• e=i 1st Nyquist cond. considers symbolrate 1/T, not (w1/T) n2+m2+1 Linear Equalization

12. n2+1 n2+1 Linear Equalization: T/2-Equalizer  w = 2: leave out every 2nd row: F2• e =i2 zero forcing solution (exact) for equalizer coefficient vector e : Linear Equalization

13. Linear Equalization: T/2-Equalizer drawback: equalizer coefficients may become very large • choosing equalizer order greater as necessary ( > n2 ) and minimizing the coefficient energy cost function : zero forcing solution (least squares) for equalizer coefficient vector e : Linear Equalization

14. Linear Equalization: Examples (w = 2)  ISI  no ISI • less noise amplification • great noise amplification • much more coefficients Linear Equalization

15. Signal Space for QPSK (no noise) Linear Equalization

16. Signal Space for QPSK (ES/N0=15 dB) Linear Equalization

17. Influence of Sampling Time Offset Linear Equalization

18. : data : impulse response of channel and filter ; 5.2 Nonlinear Equalization: Decision Feedback received signal, sampled with symbolrate1/T:  decided data Nonlinear Equalization

19. Nonlinear Equalization: Decision Feedback decision + e - FIR pre-equalizer without pre-equalizer:  decision: Nonlinear Equalization

20. Nonlinear Equalization: Decision Feedback problem: if f0 << f1, ... , fm : error propagation possible solution: FIR pre-equalizer kills samples of f(k) from 0 to time (i0-1)T following DF equalizer kills samples from (i0+1)T to end choosing equalizer coefficients e and b: MMSE-solution Nonlinear Equalization

21. MMSE-solution for uncorrelated data, : Nonlinear Equalization: Decision Feedback Nonlinear Equalization

22. 4 crucial zeros  nb = 4 Nonlinear Equalization: Example (FIR-DF) Nonlinear Equalization

23. Nonlinear Equalization: Example (FIR-DF) Nonlinear Equalization

24. 5.3 Adaptive Equalization equalizer coefficients are based on channel impulse response problem:channel impulse response is unknown at receiver  equalizer must be able to find coefficients automatically (adaptive)  transmission of a pilot sequence (known at the receiver) Adaptive Equalization

25. : step size : cost function determines convergence of algorithm : slow convergence, small error small : fast convergence, greater error (gradient noise) great Adaptive Equalization: LMS algorithm iterative calculation of equalizer coeffcient vector e use of gradient algorithm: Adaptive Equalization

26. Adaptive Equalization: LMS algorithm demonstration: gradient algorithm Adaptive Equalization

27. Adaptive Equalization: LMS algorithm is unknownat time iT problem:  using instead of  stochastic gradient algorithm:  Least Mean Square (LMS) – algorithm: Adaptive Equalization

28. Adaptive Equalization: LMS algorithm adaptive equalizer: block diagram Adaptive Equalization

29. Adaptive Equalization: Example (T-Equalizer, n=24, LMS) Convergence of largest equalizer coefficient Adaptive Equalization

30. Adaptive Equalization: Decision Feedback Adaptive Equalization

31. Adaptive Equalization: Decision Feedback adaptive FIR-DF equalizer: block diagram FIR, w=1 DF Adaptive Equalization