It should be noticed that in the practical calculations
of the discrete spectrum , it is not necessary to determine
the spectrum values for the whole range of frequencies.
As it is required to check the values of Fa(w ) at the
Nyquist frequency , only those values in the reagon of this
frequency should be calculated and tested .
Throughout the present investigation fictitious data are
used to illustrate the contrived method and test the de-
veloped algorithm . As an example of the obtained results
Fig. (1) shows a continuous terrain profile that is repre-
sented by a sine wave of the form :
2(m) = sin ( 4m / 30 ) + 4
This function is then sampled at different sampling inter-
vals as indicated in the corresponding plots. A high samp-
ling density as that shown in Fig.(l)a , results in a set
of discrete elevations that represents the terrain profile
sufficiently acéurate . The spectrum of such discrete data
approaches zero far beyond the Nyquist frequency which is
equal to TT in this case . Fig.(1)b shows how the original
profile is distorted by increasing the sampling interval
(taking every fourth elevation) and hence the spectrum is
not any more equals to zero at w > 1/3 , which is the
Nyquist frequency for this case. As the sampling interval
increases in Figures(l)c & 4 , the data are becoming more
and more unrealistic and the spectra shows large aliasing
values at frequencies higher than the Nyquist frequency.
4. APPLICATION IN THE BIVARIATE MODE
The same principles illustrated in the univariate mode
can be applied to DEM data in two dimensional grids.
The most popular mode of sampling in DEM is the rectan-
gular sampling in which observations are taken in two
dimensional grids. Usually these grids are square homo-
genious ones.
If Zc(x,y) is & continuous surface , the discrete
elevations Zg(x,y) obtained from it by rectangular
sampling are given by :
24(x,) = Z,(O1x , 627) (7)
Where A; and Â2 are positive real constants known as the
two dimentional sampling intervals.
It is now required to determine the relationship between
the spectrum of Ze (x3) and the discrete spectrum of
Zà(x,y) as we did in the profile case.
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