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Use of A( , ). bandpass filtering

Filtering/smoothing. Use of A( , ). bandpass filtering. Suppose X(x,y)   j,k  jk exp{i(  j x +  k y)} Y(x,y) = A[X](x,y)   j,k A(  j ,  k )  jk exp{i(  j x +  k y)} e.g. If A(  ,  ) = 1, |  ±  0 |, |  ±  0 |  

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Use of A( , ). bandpass filtering

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  1. Filtering/smoothing. Use of A(,). bandpass filtering Suppose X(x,y)  j,kjk exp{i(j x + k y)} Y(x,y) = A[X](x,y) j,k A(j,k) jk exp{i(j x + k y)} e.g. If A(,) = 1, | ± 0|, |±0|  = 0 otherwise Y(x,y) contains only these terms Repeated xeroxing

  2. Approximating an ideal low-pass filter. Transfer function A() = 1 ||  Ideal Y(t) =  a(u) X(t-u) t,u in Z A() =  a(u) exp{-i  u) - <  a(u) =  exp{iu}A()d / 2 = |lamda|<Omega exp{i u}d/2 = / u=0 = sin u/u u  0

  3. Bank of bandpass filters

  4. Fourier series. How close is A(n)() to A() ?

  5. By substitution

  6. Error

  7. Convergence factors. Fejer (1900) Replace (*) by Fejer kernel integrates to 1 non-negative approximate Dirac delta

  8. General class. h(u) = 0, |u|>1  h(u/n) exp{-iu} a(u) =  H(n)() A(-) d (**) with H(n)() = (2)-1 h(u/n) exp{-iu} h(.): convergence factor, taper, data window, fader (**) = A() + n-1H()d A'() + ½n-22H()d A"() + ...

  9. Lowpass filter.

  10. Smoothing/smoothers. goal: retain smooth/low frequency components of signal while reducing the more irregular/high frequency ones difficulty: no universal definition of smooth curve Example. running mean avet-kst+k Y(s)

  11. Kernel smoother. S(t) =  wb(t-s)Y(s) /  wb(t-s) wb(t) = w(t/b) b: bandwidth ksmooth()

  12. Local polynomial. Linear case Obtain at , bt OLS intercept and slope of points {(s,Y(s)): t-k  s  t+k} S(t) = at + btt span: (2k+1)/n lowess(), loess(): WLS can be made resistant

  13. Running median medt-kst+k Y(s) Repeat til no change Other things: parametric model, splines, ... choice of bandwidth/binwidth

  14. Finite Fourier transforms. Considered

  15. Empirical Fourier analysis. Uses. Estimation - parameters and periods Unification of data types Approximation of distributions System identification Speeding up computations Model assessment ...

  16. Examples. 1. Constant. X(t)=1

  17. Inversion. fft()

  18. Convolution. Lemma 3.4.1. If |X(t)M, a(0) and |ua(u)| A, Y(t) =  a(t-u)X(u) then, |dYT() – A() dYT() |  4MA Application. Filtering Add S-T zeroes

  19. Periodogram. |dT ()|2

  20. Chandler wobble.

  21. Interpretation of frequency.

  22. Some other empirical FTs. 1. Point process on the line. {0j <T}, j=1,...,N N(t), 0t<T dN(t)/dt = j (t-j) Might approximate by a 0-1 time series Yt = 1 point in [0,t) = 0 otherwise j Yt exp{-it}

  23. 2. M.p.p. (sampled time series). {j , Mj } {Y(j )} j Mj exp{-ij} j Y(j ) exp{-ij}

  24. 3. Measure, processes of increments 4. Discrete state-valued process Y(t) values in N, g:NR t g(Y(t)) exp{-it} 5. Process on circle Y(), 0   <  Y() = k k exp{ik}

  25. Other processes. process on sphere, line process, generalized process, vector-valued time, LCA group

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