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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