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Ali_Abbas_Ali_Thesis_199000018

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					                        q
           P~(f)    -   ~      Rxx(k) exp(-j2rrfkdt)                  (2.3.36)
                        k=-q


   Hence, if only an                  estimate to the MA spectra is required
and if (q+l) lags                     of the autocorrelation function are
available, then the                   use of equation (2.3. 36) can achieve
that. But if the MA                  model parameters are required, then we
need    to   solve                   the    nonlinear   set   of   equations
-Equ. (2.3.35)-, and                  so, we need to determine the model
order.

   Chow [101 suggested (when only the data samples are
available) the use of the unbiased estimate - Equ. (2.1.5) -
for the autocorrelation lags. He stated that the MA model
order is that for which the autocorrelation lags approaches
zero rapidly. So having obtained the model order, we can use
equation (2.3.11) -repeated here- to compute the moving
average model spectra.



                                      q                     2
  P
      KA
           (f) -   (j21lt
                    w          1 +   ~ b k exp(-j2rrfkdt)            (2.3.37)
                                     k=l



2.3.2.4.           PRONY'S method:
                   Though Prony's method is not a spectral estimation
technique in the usual sense, a spectral interpretation can
be provided for it. Originally it is a technique for
modelling data of equally spaced samples by a linear
combination of exponentials ( P exponentials, each has
arbitrary amplitude A ,                    phase 8 k , frequency fk and damping
                     k
factor cx ).
               k


                                               2-34
              T
   Let X = xO,xl' ..... ,xN _ l be, as before, the observation
-or data samples- vector to which Prony's method tries to
fit the model is given by :


                      p
        ""
        xn -      L
                  k=l
                                    n
                               b k zk              for     0::5 n       ::5   N-l          (2.3.38)


where
                      b
                           k       -   AkeXp(jBk )


and                   zk - exp [ (a k + j2rrfk ) At ]                                      (2.3.39)


   Equation (2.3.38) represents a set of nonlinear equations
in the unknown b    parameters. In matrix form, it can be
                  k
written as •           .
        ""                                                                                 (2.3.40)
        X -       ~B



where             ""
                  X -              [ Xo xl x 2     .....    x N- 1            JT

                                       1      1      · .........                   1
                                       zl     z2     ·. .... ....                  Z
                                                                                       P
                  ~    -

                                               N-l                              N-l
                                        N-l
                                       zl     z2     ·. .. . .. ...            zp


and               B -              [ bl b2 b3      .....    b       ]    T
                                                                P


      In order to find the exponential model parameters ( Ak ,
B      f  and a   ), we need to minimize the squared error c,
 k'     k                      k
defined as :




                                                          2-35
                  N-1
       C     =    LI
                  n=O
                               Xn
                                                  2
                                                                                          (2.3.41)




which is a difficult nonlinear least squares minimization.
There are a                lot of methods to do this                          job,    such as that
suggested by McDonough and Huggins [39].                                      But    Prony~s   method
is    simpler             and           provides      satisfactory           solution    though    it
doesn't minimize equation (2.3.41).                                    It can be developed as
shown below;

     Let    ~(z)         be a polynomial defined as :


                                                            p
                               p
                l/J(z)    - n(
                           k=l
                                         z - zk )
                                                       -L   i=O
                                                                  a. z p-i
                                                                   ~
                                                                                           (2.3.42)




      Using equation (2.3.38),                            the (n-m) sample estimates can
be written as :



                           p
           x n-m    -L    1=1
                                   bi                                                      (2.3.43)




      Multiplying both sides by am and summing over the past
(p+1) products gives:


            p                               p         p

           L
           m=O
                                        -L L
                                           1=1
                                                 bi
                                                      m=O
                                                                n-m
                                                            am z1                          (2.3.44)



                         for ps n sN-1



                                                            2-36

				
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posted:3/13/2013
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