The upper limit of the summation in equation(5) is made Tl 
rather than co because; for a discrete process measured at 
unit intervals of distances there is no loss of generality 
in restricting w to the range (0, ) since : 
COS w X for m even 
cCos(xXwW+MTxXx) = 
cos( TT -+w)x form odd 
Where x and m are integers . Therefore, variations at 
frequencies higher than Tr cannot be distinguished from 
variations at the corresponding frequency in the range (O;T) 
The upper limit of the integral in eqn.(5) is called the 
Nyquist frequency for a discrete process with unit inter- 
vals . However, if the sampling interval is Ax , the 
Nyquist frequency will be dT/Ax. 
If ^x is chosen to be too large , then a phenomenon 
called Aliasing may occur. Aliasing means that there is 
& multitude of different surfaces that have the same 
sample values , i.e. surfaces can not be correctly re- 
constructed from the sampled discrete elevations . 
Now , we consider & continuous terrain profile Z(x) 
which has a spectrum PF (02) where O < w «oo , This 
continuous profile may © be sampled at equal intervals 
of length A x . The resulting discrete elevations will 
have a spectrum PF.(w) defined over 0 £L w << T/ax. 
The method contrivéd in this paper depends on finding 
a relationship between F (o and F aC) from which the 
necessary conditions of Cequating the two spectra can 
be extracted . Fortionately , this relationship already 
exists and can be developed as follows : 
It is clear from equations (3) and (5) that when k takes 
an integer vslue than : 
a 
co 
^e | F(w) coswk dw = 1 Fal(w) coswk dw 
o 0 
The left hand side of this equation can be manipulated 
as follows : 
  
eo 2T(n+1)/8% 
SO 
1 F (w) cosuk dw => J F (w) coswk dw 
9 C n29 o«n/ e 
S 
ow 2 | 
= > 1 PF (wU +2Tn/Ax) coswk dw 
ns0 c 
0 
= 2T 2T (n«1) 
a to n n+l) _ 
zi lw AF} + DS w)}coswk dw 
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