Adaptive Filters Application of Linear Prediction Gerhard Schmidt - PowerPoint PPT Presentation

Adaptive Filters – Application of Linear Prediction Gerhard Schmidt Christian-Albrechts-Universität zu Kiel Faculty of Engineering Electrical Engineering and Information Technology Digital Signal Processing and System Theory Slide 1

Contents of the Lecture Today:  Repetition of linear prediction  Properties of prediction filters  Application examples  Improving the convergence speed of adaptive filters  Speech and speaker recognition  Filter design Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 2 Slide 2

Repetition Structure Consisting of an Prediction Filter and of an Inverse Prediction Filter Prediction filter Prediction error filter Prediction filter Inverse prediction error filter Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 3 Slide 3

Repetition Design of a Prediction Filter Cost function: Minimizing the mean squared error Solution: Yule-Walker equation system Robust and efficient implementation: Levinson-Durbin recursion Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 4 Slide 4

Repetition Levinson-Durbin Recursion Initialization:  Predictor:  Error power (optional): Recursion:  PARCOR coefficient:  Forward predictor:  Backward predictor:  Error power (optional): Termination: If , use the coefficients of the previous  Numerical problems: step and stop the recursion.  Final order reached: If has reached the desired order stop the recursion. Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 5 Slide 5

Repetition Impact of a Prediction Error Filter in the Frequency Domain – Part 1 Estimated power spectral densities Input signal (speech) Decorrelated signal (filter order = 16) Frequency in Hz Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 6 Slide 6

Repetition Impact of a Prediction Error Filter in the Frequency Domain – Part 2 Prediction filter Power adjustment Prediction error filter Prediction filter Inverse prediction error filter Inverse power adjustment Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 7 Slide 7

Repetition Impact of a Prediction Error Filter in the Frequency Domain – Part 3 Inverse prediction error filter (order = 1) Inverse prediction error filter (order = 2) Power adjusted filter Power adjusted filter Power spectral density of the input signal Power spectral density of the input signal Inverse prediction error filter (order = 4) Inverse prediction error filter (order = 8) Power adjusted filter Power adjusted filter Power spectral density of the input signal Power spectral density of the input signal Inverse prediction error filter (order = 16) Inverse prediction error filter (order = 32) Power adjusted filter Power adjusted filter Power spectral density of the input signal Power spectral density of the input signal Frequency in Hz Frequency in Hz Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 8 Slide 8

Prediction Error Filter Properties – Part 1 Minimization without restrictions (included in the filter structure)  Cost function: The resulting filter has minimum phase:  An FIR filter is computed with all its zeros within the unit circle.  Signals can pass the filter with minimum delay.  The inverse prediction filter is stable, since all zeros become poles and the zeros are located in the unit circle. Normalized filters are generated – Part 1:  Frequency response of the filter:  Frequency response of the inverse: Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 9 Slide 9

Prediction Error Filter Properties – Part 2 Normalized filters are generated (true for the prediction filter as well as for the inverse filter) – Part 2:  Frequency response of the prediction filter:  Frequency response of the inverse filter:  Type of normalization: Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 10 Slide 10

Prediction Error Filter Properties – Part 3 Normalized filters are generated (true for the prediction filter as well as for the inverse filter) – Part 3: Prediction error filter (FIR, filter order = 16) Inverse prediction error filter (IIR, filter order = 16) Frequency in Hz Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 11 Slide 11

Inverse Prediction Error Filter Estimation of the Spectral Envelope Parametric estimation of the spectral envelope:  Reducing the amount of parameters required to describe the specral envelope (compared to short-term spectrum)  Independence of other signal properties (such as the pitch frequency) Short-term spectrum of a vowel Spectrum of the corresponding inverse prediction error filter Frequency in Hz Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 12 Slide 12

Applications of Linear Prediction Applications of Linear Prediction – Part 1 Improving the Speed of Convergence of Adaptive Filters Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 13 Slide 13

Applications of Linear Prediction Improving the Speed of Convergence of Adaptive Filters – Part 1 Excitation Simulation example:  Excitation: colored noise Samples PSD (first 1000 samples) PSD (second 1000 samples) (power spectral density [PSD] of the excitation is changed dB after 1000 samples)  Distortion: white noise Normalized frequency Normalized frequency  Monitoring the error power Distortion and the system distance Samples System distance dB Error power Samples Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 14 Slide 14

Applications of Linear Prediction Improving the Speed of Convergence of Adaptive Filters – Part 2 Time-invariant decorrelation: Inverse prediction Prediction error filter error filter Decorrelated signal domain Prediction error filter Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 15 Slide 15

Applications of Linear Prediction Improving the Speed of Convergence of Adaptive Filters – Part 3 Simplified time-invariant decorrelation:  The adaptive filter has to model the (unknown) system in series with the inverse prediction error filter (the convolution of both impulse responses) Prediction error filter  Wiener solution: Decorrelated signal domain Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 16 Slide 16

Applications of Linear Prediction Improving the Speed of Convergence of Adaptive Filters – Part 4 Time-variant decorrelation  Every 10 to 50 ms the prediction filters are updated.  With the update also the signal memory of the adaptive filters needs to be corrected.  This can be realized in an efficient manner Prediction Prediction by using a so-called error filter error filter double-filter structure. Decorrelated signal domain Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 17 Slide 17

Applications of Linear Prediction Improving the Speed of Convergence of Adaptive Filters – Part 5 Convergence runs (averaged over Without decorrelation several simulations, Time-invariant dec. (1. order) speech was used as Time invariant dec.(2. order) Time-variant dec. (10. order) excitation): Time-variant dec. (18. order) System distance in dB Time in seconds Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 18 Slide 18

Applications of Linear Prediction Application of Linear Prediction – Part 2 Speech and Speaker Recognition Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 19 Slide 19

Applications of Linear Prediction Basics of Speaker Recognition – Part 1 Basic Principle:  To recognize a speaker, first features are extracted out of the signal, e.g. the spectral envelope. This is performed every 5 to 30 ms.  After extracting the feature vector it is compared with all entries of a codebook and the entry with minimum distance is detected.  This has to be done for several codebooks, each belonging to an individual speaker.  For each codebook the minimum distances are accumulated.  The accumulated minimum distances determine which speaker is the one with the largest likelihood.  Models for known speakers are competing with “universal” models.  Often the winning codebook is adapted according to the new features. Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 20 Slide 20

Applications of Linear Prediction Basics of Speaker Recognition – Part 2 Codebook of the first speaker „Best“ entry of the first codebook Current spectral envelope dB Codebook of the second speaker Frequency „Best“ entry of the second codebook Digital Signal Processing and System Theory| Adaptive Filters | Applications of Linear Prediction Slide 21 Slide 21