Lecture 4-1 Parametric Model Method Prof.N Rao
Lecture 4-1 Parametric Model Method Prof. N Rao
Basic Ideas Known the observed signal x(n) w(n) x(n) H(Z) h(n)
Basic Ideas Known the observed signal x(n)
Model transfering function:H(Z) 9 H2) X() w(n) x(n) W(z) H(Z) ) k=0 x(e吃a,:=w(e24: Assume a=1 dox(n)=-ax(n-k)+>b.w(n-k).do=I x(n)=-ax(n-k)+>bw(n-k).do=1
Model transfering function: H(Z) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0 0 1 0 ( ) ( ) ( ) 1 , 1 q k k k p k k k p q k k k k k k p q k k k k b z X z H z W z a z X z a z W z b z Assume a a x n a x n k b w n k a − = − = − − = = = = = = = = = − − + − = ∑ ∑ ∑ ∑ ∑ ∑
Three kinds of models x(n)=->dzx(n-k)+>bw(n-k).do=1 -0 1.Moving average (MA)model,denoted by MA(g). Except ao-1,all other ax=0,there are only zero points in H(. a=1 x(n)=∑b:w(n-k) () 得立
Three kinds of models 1. Moving average (MA) model, denoted by MA( q). Except a 0 =1, all other a k =0, there are only zero points in H( z). ( ) ( ) ( ) ( ) ( ) 0 0 0 1 q k k q k k k a x n b w n k X z H z b z W z = − = = = − = = ∑ ∑
Three kinds of models x(n)-->ax(n-k)+>bw(n-k).ao-1 k=0 (2)Autoregressive (AR)model,denoted by AR(p). Except bo=1,all other b-0,there are only pole points in H(. a0=1 x(n)=->ax(n-k)+w(n),b=1 H2) X()1 W(=)1+2a=
Three kinds of models (2) Autoregressive (AR) model, denoted by AR(p). Except b 0 =1, all other b k =0, there are only pole points in H( z). ( ) ( ) 0 0 1 1 1 ( ), 1 ( ) 1 ( ) ( ) 1 p k k p k k k a x n a x n k w n b X z H z W z a z = − = = = − − + = = = + ∑ ∑
Three kinds of models x(m)）=-2ax(n-k)+之bw(n-ka=l k=0 (3)Autoregressive moving average (ARMA) model,denoted by ARMA(p,q). Both ak and bk are not equal to zeros, h H2)= X( 、 k=0 W(z) k=0
Three kinds of models (3) Autoregressive moving average (ARMA) model, denoted by ARMA(p , q). Both a k and b k are not equal to zeros, 0 0 ( ) ( ) ( ) q k k k p k k k b z X z H z W z a z − = − = = = ∑ ∑
Power spectral estimation using parameter model method The parameters in the model are key to estimate the power spectral of a signal.Assume that the parameters of model are known,we have: w(n) x() H(Z) h(n) 5.()-5.(=)H()-H(=)s.(=) Let z=eio
Power spectral estimation using parameter model method The parameters in the model are key to estimate the power spectral of a signal. Assume that the parameters of model are known, we have: ( ) ( ) ( ) ( ) ( ) 2 * * 1 xx ww ww j S z S z H z H H z S z z Let z e ω = = =
Power spectral estimation based on MA model s.()=s()u()h=u(es(e) Let z=elo ao 1 x(n)=∑b4w(n-k) u(e)0昌三A:
Power spectral estimation based on MA model ( ) ( ) ( ) ( ) ( ) 2 * * 1 xx ww ww j S z S z H z H H z S z z Let z e ω = = = ( ) ( ) ( ) ( ) ( ) 0 0 0 1 q k k q k k k a x n b w n k X z H z b z W z = − = = = − = = ∑ ∑
Power spectral estimation based on AR model 5.()=s(e)Ha)H（日=(es(a Let z=eio ao =l x(n)=-∑ax(n-k)+w(n,b。=l H(a) X(2 2-1 w(日)1+2a
Power spectral estimation based on AR model ( ) ( ) ( ) ( ) ( ) 2 * * 1 xx ww ww j S z S z H z H H z S z z Let z e ω = = = ( ) ( ) 0 0 1 1 1 ( ), 1 ( ) 1 ( ) ( ) 1 p k k p k k k a x n a x n k w n b X z H z W z a z = − = = = − − + = = = + ∑ ∑
Power spectral estimation based on ARMA model 5.(=)=s..()H()H()s.(=) Let z=eio X(2) he* H(a)= k=0 W(z) k=0
Power spectral estimation based on ARMA model ( ) ( ) ( ) ( ) ( ) 2 * * 1 xx ww ww j S z S z H z H H z S z z Let z e ω = = = 0 0 ( ) ( ) ( ) q k k k p k k k b z X z H z W z a z − = − = = = ∑ ∑