Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

27-6 Passive, Active, and Digital FiltersThe ridge function gives a complex trigonometric oscillatory constant along the direction ~u. Takingthe inner product of it with a function f ( ~ j) gives^f (v, ~u) ¼hf ( ~ j), c v ( ~ j ~u)i ¼ðf ( ~ j)e iv( ~j~u) d ~ j: (27:6)This is exactly the d-dimensional Fourier transform expressed in polar coordinate form, or just simplypolar Fourier transform. The conversion to Cartesian coordinates can be done by separating the radialfrequency into a frequency vector:~v ¼ v~u: (27:7)With a slight rearrangement of Equation 27.6, the equivalence becomes obvious:^f (v, ~u) ¼ððf ( ~ j)e i~ j(v~u) d ~ j ¼ f ( ~ j)e i~ j~v d ~ j ¼ ^f (~v): (27:8)Meanwhile, the definition for the continuous Radon transform in d-dimensions isRf (t,~u) ¼ðf ( ~ j)d( ~ j ~u t)d ~ j: (27:9)The relationship between the Radon transform and the polar Fourier transform is stated as the FourierSlice Theorem [22]:THEOREM 27.1(Fourier Slice Theorem)The 1-D Fourier transform with respect to t of the projection Rf (t,~u) is equal to a central slice, at a givenorientation ~u, of the higher dimensional Fourier transform of the function f ( ~ j), that is,dRf(t, ~u) ¼ ^f (v, ~u): (27:10)If some boundary condition is to be posed at each polar orientation, instead of being defined on R, theRadon slice R ~ u f (t) is assumed to be an even function R ~u f (t) ¼R ~u f ( t), then its Fourier transform canbe written as follows:rffiffiffiðd 2Rf(t,~u) ¼pþ1 0Rf (t,~u) cos (vt)dt: (27:11)It is noted that only the real cosine part remains in the above equation due to the even boundaryassumption on the slices. This is equivalent to a cosine tranform with ridge-type basis functionsdefined asrffiffiffic C v (~ 2j ~u) ¼ cos (vp~ j ~u): (27:12)