Random Signals for Engineers using MATLAB and Mathcad Copyright 1999 Springer Verlag NY Example 7.12 Spectral Shaping In the example we will determine the transfer function of a filter with the property that when we apply white noise the shape the output power density spectrum will be as desired. This procedure can be used to generate a family of power density spectrums that can be synthesized by applying white noise to linear systems. We begin by defining the desired spectrum syms om s S=(om^2+1)/(om^4+16); pretty(S) 2 om + 1 -------4 om + 16 The output power density spectrum, S(), for a system transfer function G() using Equation 7.6-5 and assuming a unity input white noise spectrum S ( ) G(i ) G( i ) in order to solve for G(j ) we must factor the numerator and denominator polynomial. For the numerator omn=solve('om^2+1') omn = [ i] [ -i] and the denominator roots using symbolic solutions omd=solve('om^4+16') omd = [ 2^(1/2)+i*2^(1/2)] [ -2^(1/2)-i*2^(1/2)] [ 2^(1/2)-i*2^(1/2)] [ -2^(1/2)+i*2^(1/2)] Writing the polynomials that have roots in the left half plane for G(i ) GNom=om+omn(1) GDom=expand((om+omd(1))*(om+omd(3))) GNom = om+i GDom = om^2+2*om*2^(1/2)+4 We have identity the numerator and denominator polynomial that belong to G(i ) and G(-i ). We take the roots or poles and zeros of G(i ) in the left half plane corresponding to stable poles and non minimum phase zeros with G(s). We obtain Gom=GNom/GDom; pretty(Gom) om + i ------------------2 1/2 om + 2 om 2 + 4 and similarly with the minus sign need because the substitution must be made for i Gomr=subs(Gom,om,-om); pretty(-Gomr) -om + i - ------------------2 1/2 om - 2 om 2 + 4 We may verify the result by computing G (i ) 2 G (i ) G ( i ) simplify(expand(-Gom*Gomr)) ans = (om^2+1)/(om^2+2*om*2^(1/2)+4)/(om^2-2*om*2^(1/2)+4) num=expand((om^2+2*om*2^(1/2)+4)*(om^2-2*om*2^(1/2)+4)) num = om^4+16 The last result was obtain by isolation the denominator and expanding the expression rather that the complete term. This verifies the synthesis.