sampling theorem. This theorem states that a continuous band-limited function 



of time may be reconstructed from equally spaced digitized samples provided 



that you have at least two samples per shortest period. The reconstruction is 



accomplished by convolving the digital values with a Sine function 



/ Sin x\ . 

 (— ) '• e " 



00 Sin 2itA (t - £■) 



«» - * f (£) 7777 nf (23 > 



n = -oo 2irA ^ t - 2^J 



where the band limits are ±2irA and the sampling rate is 2A. See Papoulis (1962) 

 for a proof of this theorem. Although it is theoretically impossible for a 

 function of finite length to be band limited, a reasonably accurate estimate 

 of A may be obtained for a small area by application of a two-dimensional 

 Fourier transform. 



In one dimension, the direct Fourier transform of an arbitrary function, 

 g(x), is defined as 



CO 

 G(f x ) = / g(x)e" i2irfxX dx, (24) 



-CO 



with f x equal to the frequency. By means of this formula, it is possible to 

 transform information from the space domain into the frequency domain. G(f x ) 

 is, in general, a complex number of the form a(f ) + ib(f x ) or equivalently 



(a 2 + b 2 ) 1/2 e' arCtan a where (a 2 + b 2 ) 1/2 = | G(f ) | is called the 



b X 



amplitude spectrum and arctan — = 9 (f ) is termed the phase spectrum. For 



our application of the Fourier transform, we will be primarily concerned with 

 the amplitude spectrum. In the context in which it is used here, the term 

 frequency refers to spatial frequency with the units being either cycles per unit 

 distance or normalized to cycles per data interval . 



15 



