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Extra info for Advances in Neural Information Processing Systems 2
C NH=N/2 MH=M/2 DO I=1,MH-NH YR(I)=0. YI(I)=0. ENDDO J=l DO I=MH-NH+1,MH+NH YR(I)=XR(J) YI(I)=XI(J) J=J+1 ENDDO DO I=MH+NH+1,M YR(I)=0. YI(I)=0. ENDDO C C INVERSE FOURIER TRANSFORM. C CALL F F T 1 D ( Y R , Y I , M , 1 ) C C WRITE OUTPUT.
T h es a m p l i n g t h e o r e m o f t h es a m p l e d o f t h e c o m b f u n c t i o n f u n c t i o n is o b t a i n g(x). m u s t T o d o first b e 43 Quantitative coherent imaging obtained. We must therefore evaluate the integral oo comb(x) exp(—ikx)dx J The key to evaluating this integral lies in expressing the comb function in terms of a Fourier series (not a transform). This is given by oo comb(x) = an exp(i2wnx/X) N= — OO where the coefficients a are obtained by computing the integral n X/2 an = J comb(a:) exp(—i2nnx/X)dx -X/2 Substituting the definition for the comb function into the equation above and noting that comb(x) = δ(χ) in the interval [ - X / 2 , X / 2 ] , we get X/2 an I f = — / δ(χ) exp(—i2nnx/X)dx 1 = — -X/2 Hence, we can represent the comb function by the Fourier series 1 00 comb(x) = — X exp(i2nnx/X) N— — OO The Fourier transform of the comb function can therefore be written as 7 00 1 / — ^2 1 00 — 7 / exp[—ix(k — N= — 00 44 β χ ρ ( ζ 2 π η χ / Χ ) exp(—ikx)dx 2πη/Χ)]άχ Fourier transforms = Υ Σ Η= S(k-2im/X) — ΟΟ Hence, we obtain the important result (crucial to the proof of the sampling theorem) oo ~ oo ΤΙ— — oo Η— — oo Proof of the sampling theorem Suppose we sample a function at regular intervals of δχ.
Thus, multiplying F(k) by exp(ik · r') and integrating over k, we have oo J oo 2 F ( k ) exp(ik · r > k = J —oo oo 2 2 d r/(r) j —oo exp[ik · (r' - r)]d k —oo oo 2 = j 2 2 d rf(r)(2n) 6 (r'-r) — OO = 2 / (2π) /(Γ ) Hence, oo l / ( r ) = F2~ F(k) = (2^)2 / In Cartesian coordinates, oo f(x,y) = j FM exp(*k · 2 r)d k — OO oo j F(kx,ky)exp(ikxx)exp(ikyy)dkxdky — oo —oo 2D Convolution The convolution theorem is the same in two dimensions. 9 T H E SAMPLING T H E O R E M A N D SINC INTERPOLATION Digital signals are usually obtained from analogue signals by converting them into a sequence of numbers (a digital signal) so that digital computers can be used to process them.