Adaptive Filters Wiener Filter

Adaptive Filters – Wiener Filter

Gerhard Schmidt

Christian-Albrechts-Universität zu Kiel Faculty of Engineering Institute of Electrical and Information Engineering Digital Signal Processing and System Theory

Contents of the Lecture

•Today Contents of the Lecture:  Introduction and motivation  Principle

of orthogonality  Time-domain solution  Frequency-domain solution  Application example: noise suppression

Digital Signal Processing and System Theory | Adaptive Filters | Wiener Filter

Slide 2

Basics

•History and Assumptions Filter design by means of minimizing the squared error (according to Gauß) Independent development

1941: A. Kolmogoroff: Interpolation und Extrapolation von stationären zufälligen Folgen, Izv. Akad. Nauk SSSR Ser. Mat. 5, pp. 3 – 14, 1941 (in Russian)

1942: N. Wiener: The Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications, J. Wiley, New York, USA, 1949 (originally published in 1942 as MIT Radiation Laboratory Report)

Assumptions / design criteria:  Design of a filter that separates a desired signal optimally from additive noise

 Both signals are described as stationary random processes  Knowledge

about the statistical properties up to second order is necessary


Slide 3

Application Examples – Part 1

•Noise Suppression Application example:

Wiener filter

Speech Noise

(No echo components)

Model: Speech (desired signal)

+ Noise (undesired signal) The Wiener solution if often applied in a “block-based fashion”. Digital Signal Processing and System Theory | Adaptive Filters | Wiener Filter

Slide 4

Application Examples – Part 2

•Echo Cancellation Application example:

Model: Echo cancellation filter

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The echo cancellation filter has to converge in an iterative manner (new = old + correction) towards the Wiener solution. Slide 5

Generic Structure

•Noise Reduction and System Identification Wiener filter

Error signal

Wiener filter

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Linear system Generation of a desired signal Echo cancellation

Noise suppression

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Slide 6

Literature Hints

•Books Main text:  E. Hänsler / G. Schmidt: Acoustic Echo

and Noise Control – Chapter 5 (Wiener Filter), Wiley, 2004

Additional texts:  E. Hänsler: Statistische Signale: Grundlagen

und Anwendungen – Chapter 8 (Optimalfilter nach Wiener und Kolmogoroff), Springer, 2001 (in German)  M. S.Hayes: Statistical Digital Signal Processing and Modeling – Chapter 7 (Wiener Filtering), Wiley, 1996  S. Haykin: Adaptive Filter Theory – Chapter 2 (Wiener Filters), Prentice Hall, 2002

Noise suppression:  U. Heute: Noise Suppression,

in E. Hänsler, G. Schmidt (eds.), Topics in Acoustic Echo and Noise Control, Springer, 2006


Slide 7

Principle of Orthogonality

•Derivation Derivation during the lecture …


Slide 8

Principle of Orthogonality

•A Deterministic Example Derivation during the lecture …


Slide 9

Wiener Solution

•Time-Domain Solution Derivation during the lecture …


Slide 10

Time-Domain Solution

•Example – Part 1 Desired signal: Sine wave with known frequency but with unknown phase, not correlated with noise

FIR filter of order 31, delayless estimation at filter output

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Noise:

White noise with zero mean, not correlated with desired signal


Slide 11


•Example – Part 2 Wiener solution:

Desired signal and noise are not correlated and have zero mean:

Simplification according to the assumptions above:

Wiener solution (modified):


Slide 12


•Example – Part 3 Input signals: Excitation: sine wave Noise: white noise

Assumptions:  Knowledge

of the mean values and of the autocorrelation functions of the desired and of the undesired signal  Desired signal and noise are not correlated  Desired signal and noise have zero mean  32 FIR coefficients should be used by the filter


Slide 13


•Example – Part 4  After a short initialization time the noise suppression

performs well (and does not introduce a delay!)


Slide 14

Error Surface

•Derivation – Part 1 Derivation during the lecture …


Slide 15

Error Surface

•Derivation – Part 2 Error surface for: 



Properties:  Unique minimum (no local minima)  Error

surface depends on the correlation properties of the input signal


Slide 16

Frequency-Domain Solution

•Derivation Derivation during the lecture …


Slide 17

Applications

•Noise Suppression – Part 1 Frequency-domain Wiener solution (non-causal):

Desired signal = speech signal:

Desired signal and noise are orthogonal:


Slide 18

Applications

•Noise Suppression – Part 2 Frequency-domain solution:

Approximation using short-term estimations:

Practical approaches:  Realization using a filterbank system (time-variant attenuation of subband signals)

 Analysis filters with length of about 15 to 100 ms  Frame-based processing with frame shifts between

1 and 20 ms  The basic Wiener characteristic is usually „enriched“ with several extensions (overestimation, limitation of the attenuation, etc.)


Slide 19

Applications

•Noise Suppression – Part 3 Processing structure: Analysis filterbank

Synthesis filterbank

Input PSD estimation

Filter characteristic Noise PSD estimation

PSD = power spectral density Digital Signal Processing and System Theory | Adaptive Filters | Wiener Filter

Slide 20

Applications

•Noise Suppression – Part 4 Power spectral density estimation for the input signal:

Power spectral density estimation for the noise:

Estimation schemes using voice activity detection(VAD)

Tracking of minima of short-term power estimations


Slide 21

Applications

•Noise Suppression – Part 5 Schemes with voice activity detection:

Tracking of minima of the short-term power:

Constant slightly larger than 1 Bias correction

Constant slightly smaller than1


Slide 22

Applications

•Noise Suppression – Part 6 Problem:  The short-term

power of the input signal usually fluctuates faster than the noise estimate – also during speech pauses. As a result the filter characteristic opens and closes in a randomized manner, with results in tonal residual noise (so-called musical noise).

Simple solution:  By inserting a fixed overestimation

the randomized opening of the filter can be avoided. This comes, however, with a more aggressive attenuation characteristic that attenuates also parts of the speech signal.

Enhanced solutions:  More

enhanced solutions will be presented in the lecture “Speech and Audio Processing – Audio Effects and Recognition” (offered next term by the “Digital Signal Processing and System Theory” team).


Slide 23

Applications

•Noise Suppression – Part 7 : Microphone signal : Output without overestimation : Output with 12 dB overestimation


Slide 24

Applications

•Noise Suppression – Part 8 Limiting the maximum attenuation:  For

several application the original shape of the noise should be preserved (the noise should only be attenuated but not completely removed). This can be achieved by inserting a maximum attenuation:

 In addition, this attenuation limits can be varied slowly over

time (slightly more attenuation during speech pauses, less

attenuation during speech activity).


Slide 25

Applications

•Noise Suppression – Part 9 : Microphone signal : Output without attenuation limit : Output with attenuation limit


Slide 26

Applications

•Noise Suppression – Part 10


Slide 27

Adaptive Filters – Wiener Filter

•Summary and Outlook This week:  Introduction and motivation  Principle of orthogonality  Time-domain solution  Frequency-domain solution  Application example: noise suppression

Next week:  Linear Prediction


Slide 28

Adaptive Filters Wiener Filter

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