电子科技大学：《现代数字信号处理理论与算法 Modern theory and algorithm of digital signal processing》课程教学资源（课件讲稿）01 Introduction to ADSP - Theory, Algorithm and Application

CH1 Introduction to ADSP:Theory, Algorithm and Application “I hear,I forget; I see,I remember; Ido,I understand.” A Chinese Philosopher

CH1 Introduction to ADSP: Theory, Algorithm and Application “ I hear，I forget； I see，I remember； I do，I understand.” A Chinese Philosopher

Contents o S1.Filters:devices of signal processing .Classic Filter Optimal Filter Adaptive Filter o S2.Adaptive filtering applications 0 S3.Stochastic processes Partial characteristic:Autocorrelation Matrix (ACM) ●PSD Linear parametric model o S4.Mean Square Information Space 2020-01-18 2

2020-01-18 2 Contents  S1. Filters: devices of signal processing  Classic Filter  Optimal Filter  Adaptive Filter  S2. Adaptive filtering applications  S3. Stochastic processes  Partial characteristic: Autocorrelation Matrix (ACM)  PSD  Linear parametric model  S4. Mean Square Information Space

S1.Filters:devices of signal processing o Application ● signal of interest can be discerned more effectively in the filter output o Reduce additive noise or interference o Reveal the useful information o Theory ● Deals with linear filters,where the filter output is a (possibly time- varying)linear function of the filter input. 2020-01-18 3

2020-01-18 3 S1. Filters: devices of signal processing  Application  signal of interest can be discerned more effectively in the filter output  Reduce additive noise or interference  Reveal the useful information  Theory  Deals with linear filters, where the filter output is a (possibly time￾varying) linear function of the filter input

(1)Two distinct theoretical approaches o Classical approach Aimed at designing frequency-selective filters such as lowpass/bandpass/notch filters etc. Based on knowledge of the gross spectral contents of both the useful signal and the noise components. It is applicable mainly when the signal and noise occupy clearly different frequency bands. o Optimal filter design Based on optimization theory,where the filter is designed to be "best"(in some sense). 2020-01-18 4

2020-01-18 4 (1) Two distinct theoretical approaches  Classical approach  Aimed at designing frequency-selective filters such as lowpass/bandpass/notch filters etc.  Based on knowledge of the gross spectral contents of both the useful signal and the noise components.  It is applicable mainly when the signal and noise occupy clearly different frequency bands.  Optimal filter design  Based on optimization theory, where the filter is designed to be “best” (in some sense)

(2)Optimal filter o The signal and noise are viewed as stochastic processes CDF,PDF,Partial character ●Parametric Model o Statistical criterion Minimizes the effects of the noise at the filter output according to some statistical criterion. o Based on minimizing the mean-square value of the difference between the actual filter output and some desired output 2020-01-18 5

2020-01-18 5 (2) Optimal filter  The signal and noise are viewed as stochastic processes  CDF, PDF, Partial character  Parametric Model  Statistical criterion  Minimizes the effects of the noise at the filter output according to some statistical criterion.  Based on minimizing the mean-square value of the difference between the actual filter output and some desired output

Desired signal o If such a thing is available why bother with the filter? o It is usually possible to obtain a signal that is sufficient for the purpose of controlling the adaptation process. o Examples will be given in the context of ‘adaptive filtering Data based:do not assume knowledge of the stochastic parameters But is based on a very similar idea. 2020-01-18 6

2020-01-18 6 Desired signal ?  If such a thing is available why bother with the filter?  It is usually possible to obtain a signal that is sufficient for the purpose of controlling the adaptation process.  Examples will be given in the context of ‘adaptive filtering’  Data based: do not assume knowledge of the stochastic parameters  But is based on a very similar idea

A priori design o Wiener Filter Dates back to the work of Wiener in 1942 and Kolmogorov in 1939. o‘A priori'design ● Based on a priori statistical information. Be optimal when the statistics of the signals at hand truly match the a priori information on which the filter design was based. When such a priori information is not available, it is not possible to design a wiener filter. 2020-01-18 7

2020-01-18 7 A priori design  Wiener Filter  Dates back to the work of Wiener in 1942 and Kolmogorov in 1939.  ‘A priori’ design  Based on a priori statistical information.  Be optimal when the statistics of the signals at hand truly match the a priori information on which the filter design was based.  When such a priori information is not available, it is not possible to design a Wiener filter

(3)Adaptive filter:self-designing o A priori information is not available o The signal and/or noise characteristics are nonstationary. Statistical parameters vary with time 0 Although the Wiener theory still applies,it is difficult to apply it in practice o An alternative method is to use an adaptive filter .Data based,Self-designing 2020-01-18 8

2020-01-18 8 (3) Adaptive filter: self-designing  A priori information is not available  The signal and/or noise characteristics are nonstationary.  Statistical parameters vary with time  Although the Wiener theory still applies, it is difficult to apply it in practice  An alternative method is to use an adaptive filter  Data based, Self-designing

Prototype adaptive filtering scheme filter input adaptive filter parameters filter filter output error desired signal 2020-01-18 9

2020-01-18 9 Prototype adaptive filtering scheme

Adaptation algorithm o Monitor the environment and vary the filter transfer function accordingly. o Starts from a set of initial conditions,that may correspond to complete ignorance about the environment o Find the optimum filter design based on the actual signals received ● Stationary:the filter is expected to converge to the Wiener filter. Nonstationary:the filter is expected to track time variations and vary its filter coefficients accordingly. 2020-01-18 10

2020-01-18 10 Adaptation algorithm  Monitor the environment and vary the filter transfer function accordingly.  Starts from a set of initial conditions, that may correspond to complete ignorance about the environment  Find the optimum filter design based on the actual signals received  Stationary: the filter is expected to converge to the Wiener filter.  Nonstationary: the filter is expected to track time variations and vary its filter coefficients accordingly