16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde 2jT F(SF(s)Si(s)ds We know that S(o) is even. If it is a rational function of @, and we will work exclusively with rational spectra, it is then a rational function of a. So only even powers of o appear in S(o) and thus Sa= which we may call S(s)is derived from S(o) by replacing o by -s F'(s)is the ordinary transfer function of the system -the Laplace transform of its weighting function. Because w(0)=0, I<0 We shall drop the primes from now on F(S)F(sS.(sds in S(s) Integrating the output spectrum auchy Residue Theorem s-plan fF(s)ds=2r Z(residues of F(s) at the poles enclosed in the contour C) If F(s)has a pole of order m at ==a Res(a) (m-1) (s-a)"F(s) F(s) has a pole of order m at s=a if m is the smallest integer for which Page 5 of 616.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 5 of 6 2 1 2 1 () ( ) () 2 j xx j xx s s s ds y FF S j j jj F s F s S s ds j π π ∞ − ∞ ∞ −∞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ = − ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ = − ′′ ′ ∫ ∫ We know that ( ) xx S ω is even. If it is a rational function of ω , and we will work exclusively with rational spectra, it is then a rational function of 2 ω . So only even powers of ω appear in ( ) xx S ω and thus xx s S j ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ which we may call ( ) xx S s is derived from ( ) xx S ω by replacing 2 ω by 2 −s . F s ′( )is the ordinary transfer function of the system – the Laplace transform of its weighting function. Because wt t ( ) 0, 0 = < . We shall drop the primes from now on. 2 2 2 4 4 1 () ( ) () 2 in ( ) j xx j xx y F s F s S s ds j s S s s π ω ω ∞ − ∞ = − = − ⎫⎪ ⎬ = ⎪⎭ ∫ Integrating the output spectrum General method Cauchy Residue Theorem ( ) 2 residues of ( ) at the poles enclosed in the contour C ( ) C >∫ F s ds j F s = π ∑ If F s( ) has a pole of order m at z a = , ( ) 1 1 1 Res( ) ( ) ( ) 1 ! m m m s a d a s a Fs m ds − − = ⎧ ⎫ = − ⎨ ⎬ ⎡ ⎤ ⎣ ⎦ − ⎩ ⎭ F s( ) has a pole of order m at s a = if m is the smallest integer for which