First we define the bivariate spectrum of the continuous 
surface as follows : 
Fey v, = | | z.(x,y) e (ux «ie y) dx dy (8) 
Where : ad ; 
zy) "n J J Ft, 95) eC x39, y) do dg (9) 
Using equation(7) it is possible to write : 
Za (xy) "37 | EXC e 49, Az jo, 423) dg, dg, (10) 
Equation (10) can be manipulated into the form of an in- 
verse Fourier transform for discrete signals.It is pos- 
sible to substitute w; =; 8; and wa= uy, 52 to get the 
exponential terms into the correct form (Duágeon, 1979) . 
oo eo 
1 1 ; (jwx | 
2453) "z73| [pF eun eim Y quan, (11) 
8» 
-00 = 
The double integral over the entire range of w,;,w, can 
be considered as an infinite number of integrals , each 
of which is over a given square area of (2Tx2T). 
If S(1,,1;) represents the square {-T+2 T1, <w<T+2T 1; 
and -TT42T 124, 44,4 T427T 1;], then eqn.(11) can be written 
in ihe form : 
1 1 w (je, xj; y) 
Z , = — > -—— T (181,44 JP XI Y 
aum ml 1; s, Lan MCN d 
(12) 
Replacing Wi by wı-2T 1, and wz by W2-2 11, it becomes pos= 
sible to change the limits of the integrals as follows : 
TT 
Ya M un-2T 1, 2-27 Lay) 
2, (x,7) m] Zn A, Ar) 
  
aliwrx+jwzy) -(32T1 x +32T 1,y) (13) 
The second exponential element of eqn.(13) is equal to 
unity for all values of the integer veriables x,1,,y,1; . 
Equation(13) now is in the form of an inverse Fourier 
transform , therefore , it is possible to conclude that : 
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