Real time DSP

1. Real time DSP Professors: Eng. Diego Barral Eng. Mariano Llamedo Soria Julian Bruno

2. Filters • conventional filters • time-invariant • fixed coefficients • adaptive filters • time varying • variable coefficients • adaptive algorithm • function of incoming signal • exact filtering operation is unknown or is non-stationary!

3. Random Processes • random != deterministic • concepts • realization • ensemble • ergodic • tools • mean • variance • correlation/autocorrelation • stationary processes & WSS

4. Adaptive Filters • parts • digital filter • adaptive algorithm • filter • FIR • IIR (stability problems are difficult to handle)

5. Adaptive Filters • d(n) desired signal • y(n) output of the filter • x(n) input signal • e(n) error signal

6. FIR Filter • wl(n) adaptive filter coefficients

7. Performance Function • coefficients are updated to optimize some predetermined performance criterion • mean-square error (MSE) • for FIR • R: input autocorrelation matrix • p: crosscorrelation between d(n) and x(n)

8. Performance Function • MSE surface • One global minimum point!

9. Gradient Based Algorithms • properties • convergence speed • steady-state performance • computation complexity • method of steepest descent • greatest rate of decrease (negative gradient) • iterative (recursive)

10. LMS Algorithm • statistics of d(n) and x(n) are unknown • estimation of MSE • avoids explicit computation of matrix inversion, squaring, averaging or differentiating

11. Performance Analysis • stability constraint • μ controls the size of the incremental correction • λmax is the largest eigenvalue of the autocorrelation matrix R • Px input signal power • large filters => small μ • strong signals => small μ

12. Performance Analysis • convergence speed • large μ => fast convergence • λ => relation between stability and speed of convergence • estimation

13. Performance Analysis • excess mean-square error • the gradient estimation prevents w from staying at wo in steady state • w varies randomly about wo • trade-off between the excess MSE and the speed of convergence • trade-off between real-time tracking and steady-state performance

14. Modified LMS Algorithms • normalized LMS algorithm • μ varies with input signal power • optimize the speed of convergence and maintain steady-state performance • independent of reference signal power • c is a small constant • μ(n) is bounded • 0 < α < 2

15. Modified LMS Algorithms • leaky LMS algorithm • insufficient spectral excitation may result in divergence of the weights and long term instability • where v is the leakage factor • 0 < v ≤ 1 • equivalent of adding low-level white noise • degradetion in performance • (1 - v) < μ

16. Applications • operate in an unknown enviroment • track time variations • identification • inverse modeling • prediction • interference canceling

17. Applications • adaptive system identification • experimental modeling of a process or a plant

18. Applications • adaptive linear prediction • provides an estimate of the value of an input process at a future time • in y(n) appear the highly correlated components of x(n) • i. e. speech coding and separating signals from noise • output is e(n) for spread spectrum corrupted by an additive narrowband interference

19. Applications • adaptive linear prediction

20. Applications • adaptive noise cancellation (ANC) • most signal processing techniques are developed under noise-free assumptions • the reference sensor is placed close to the noise source to sense only the noise, because noise from primary sensor and reference sensor must be correlated • the reference sensor can be placed far from the primary sensor to reduce crosstalk, but it requires a large-order filter • P(z) represents the transfer function between the noise source and the primary sensor • uses x(n) to estimate x’(n)

21. Applications • adaptive noise cancellation (ANC)

22. Applications • adaptive channel equalization • transmission of data is limited by distortion in the transmission channel • channel transfer function C(z) • design of an equalizer in the receiver that counteracts the channel distortion • training of an equalizer • agreed sequence by the transmitter and the receiver • Decision device

23. Applications • adaptive channel equalization

24. Implementation considerations • finite-precision effects • prevent overflow • scaling of coefficients (or signal) • quantization & roundoff • => excess MSE • => stalling of convergence • depends on μ • threshold of e(n) -> LSB