Parametric Spectral Estimation

Transkript

1 Parametric Spectral Estimation Exercises for Digital Signal Processing II Exercise Stefan Goetze / Volker Mildner

2 Infos about the examination Diploma students: Oral examinations on March, 29 th.- 3 th. 26 Duration: 2-3 Minutes Master students: Written examination on Wednesday, March 29 th 26 You can choose to write about DSPI or DSPII or DSPI&II (choose at your registration at the master office) hour for each part Consultation hours will be announced on our webpage. Stefan Goetze / Volker Mildner 2

3 Parametric Spectral Estimation Methods MA-model qk () z z b b m All-zero-model (no recursive paths) Analysis filter (transmitting site) + MA X () k AR-model qk () + X AR () k All-pole-model (recursive) + Synthesis filter (receiving site) a n a z z ARMA-model z z b b m (Auto Regressive Moving Average) X ARMA () k a m z z a Stefan Goetze / Volker Mildner 3

4 Parametric Spectral Estimation Methods Yule-Walker-Equation / Lattice structure (estimate of the model coefficients from the acf of the process) Estimation on basis of a finite set of samples of the process Yule-Walker- / Autocorrelation-Approach: biased estimate / leads to stable synthesis filter ( ) Covariance-Method unbiased estimate / may lead to unstable synthesis filter Burg Algorithm (joint minimization of forward and backward prediction error) unbiased estimate / leads to stable synthesis filter Stefan Goetze / Volker Mildner

5 Yule Walker & Burg (Parametric Spectral Estimation Methods Exercise ) Creating a process with a desired power spectral density The acf of the resulting process (after filtering white noise with is limited to. Now we try to estimate this power spectral density by an AR-model. r xx (k) k Stefan Goetze / Volker Mildner

6 Yule Walker & Burg (Parametric Spectral Estimation Methods Exercise ) LDS von YW-AR() Pole von YW-AR() LDS von YW-AR(8) Pole von YW-AR(8) LDS von YW-AR(32) Pole von YW-AR(32) Estimation gives bad results, because we chose the wrong model Stefan Goetze / Volker Mildner 6

7 Yule Walker & Burg (Parametric Spectral Estimation Methods Exercise ) LDS von Burg-AR() Pole von Burg-AR() LDS von Burg-AR(8) Pole von Burg-AR(8) LDS von Burg-AR(32) Pole von Burg-AR(32) 28 Real estimation from a data series leads to the same results Stefan Goetze / Volker Mildner 7

8 Yule Walker & Burg (Parametric Spectral Estimation Methods Exercise ) LDS von YW-AR() Pole von YW-AR() LDS von YW-AR(8) Pole von YW-AR(8) LDS von YW-AR(32) Pole von YW-AR(32) 32 Here the underlying process can be modeled correctly by the AR estimation Stefan Goetze / Volker Mildner 8

9 Yule Walker & Burg (Parametric Spectral Estimation Methods Exercise ) LDS von Burg-AR() Pole von Burg-AR() LDS von Burg-AR(8) Pole von Burg-AR(8) LDS von Burg-AR(32) Pole von Burg-AR(32) Same results for real estimation by the BURG algorithm Stefan Goetze / Volker Mildner 9

10 Yule Walker (Parametric Spectral Estimation Methods Exercise 2) Creating a process with a desired power spectral density Impulsantwort des AR(6)-Modells Pole des AR(6)-Modells h(k) k The output process can assumed to be stationary after 6 samples. Stefan Goetze / Volker Mildner

11 Yule Walker (Parametric Spectral Estimation Methods Exercise 2) (e jω ) in db LDS von YW-AR(6); N= Exakt Mittel. 2σ-Gr Koeffizienten AR(6)-Koeffizienten; N= Exakt Mittel./E{e 2 }=2.89e-2 2σ-Gr./ σ= i For the estimation based on samples we get a strong bias and a high variance of the filter coefficients. (e jω ) in db - - LDS von YW-AR(6); N=26 Exakt Mittel. 2σ-Gr. Koeffizienten.. -. AR(6)-Koeffizienten; N=26 Exakt Mittel./E{e 2 }=.3e- 2σ-Gr./ σ=. The bias and the variance decreases for samples i Stefan Goetze / Volker Mildner

12 Burg Algorithm (Parametric Spectral Estimation Methods Exercise 3) (e j*ω ) in db LDS von Burg-AR(2), N=892 exaktes LDS geschaetztes LDS Pole von Burg-AR(2) (e j*ω ) in db LDS von Burg-AR(6), N=892 exaktes LDS geschaetztes LDS Pole von Burg-AR(6) (e j*ω ) in db exaktes LDS geschaetztes LDS LDS von Burg-AR(32), N=892 Pole von Burg-AR(32) 26 If the assumed order of the model is too low, the algorithm can not estimate the model Stefan Goetze / Volker Mildner 2

13 Burg Algorithm (Parametric Spectral Estimation Methods Exercise 3) (e j*ω ) in db LDS von Burg-AR(2), N=892-6 exaktes LDS geschaetztes LDS Pole von Burg-AR(2) (e j*ω ) in db LDS von Burg-AR(6), N=892-6 exaktes LDS geschaetztes LDS Pole von Burg-AR(6) (e j*ω ) in db exaktes LDS geschaetztes LDS LDS von Burg-AR(32), N=892 Pole von Burg-AR(32) AR-model estimation does not work well if the model actually is a MA-model Stefan Goetze / Volker Mildner 3