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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