at a constant sampling rate f 3 requires the assumption con- 

 tained in the sampling theorem (for the time domain), if 

 unique representation of the equivalent continuous function 

 is to be realized through equally spaced sampling. If a 

 function g(t) contains no frequencies higher than f m cycles 

 per unit time, it is completely determined by giving its 

 ordinates at a series of points l/2f m units of time apart, 

 the series extending throughout the time domain: 3 



A«= ±/2f m = 1/f 



(8) 



For equally spaced data points, condition (8) limits the 

 frequencies to be considered to the range 



[-0-5/, = / 



0.5/. 



(9) 



Frequencies greater than | 0.5/ s | cannot be distinguished 

 from those within the range given by (9)- This phenomenon 

 is known as aliasing-. 



It is important to note that once the frequency/,,, has been 

 specified and the sampling rate f g determined by equation 

 (6), frequency components above 0.5/s in absolute value 

 must be filtered out prior to the sampling operation, and 

 if left lit the data contribute errors due to spectrum 

 folding (aliasing). 



The sufficient condition that filtering functions do not 

 shift the phase of any frequencies is obtained by requiring 

 that the weighting function shall be even, namely |v(t)| = |y(-£ ] 

 When this condition is imposed on equation (6) the imaginary 

 part vanishes, and (6) reduces to 



R(f) = 2 j J/(t)cos(2jt/*) dt 



(10) 



and the frequency response becomes a pure, real quantity 

 equal to the gain of the filter. For equally spaced data 

 relative to time extending over a finite range, equation 

 (lO)is approximated as follows: 



N 



J?(/)=yo+2\ J/fccos(2jt/$ 



(11) 



k=x 



