THE Z-TRANSFORM ANALYSIS OF LINEAR SAMPLED-DATA SYSTEMS 59 
where the only contribution to the value of the integral is made by f(nT), 
the coefficient of the z~! term of the series in (4.17). A word is to be 
included concerning the contour I which must be employed in the 
integration of (4.20). As in all contour integrals, the value of the integral 
is determined by the singularities in 
the form of poles of the integrand. 
Thus, if (4.20) is to have the correct 
value, it is necessary that T include 
all the poles of z”—!F (z) for all inte- 
gralsn. The integral (4.20) is thus 
the inversion theorem when the 
contour I satisfies the stated condi- 
tion. It will be further discussed 
in detail, but it should be noted now 
that the poles of F(z) are all con- 
tained inside or on the unit circle 
for stable systems, so that T is usu- 
ally taken as the unit circle. The Fie. 4.3. Map showing equivalence be- 
integral can be evaluated by the tween integration along unit circle and 
: general contour. 
usual residue methods. 
It is illuminating sometimes to evaluate the inversion integral by first 
expanding the z transform F(z) into partial fractions and then evaluating 
term by term using residue methods. In this manner, the contributions 
from each of the poles of the function F(z) are stated more explicitly. 
For example, a typical term found on expansion into partial fractions is 
F 1(2), 

z-plane 
(4.21) 
This function contains a pole in the z plane at e?7. Rewriting (4.21) 
in another form, and substituting into (4.20), 
i soi bee Az” 
= le F1(z) dz = = | an (4.22) 
this integral has a residue at e~*7, which, upon multiplication by 27j 
gives the result 
nol) = 2g)" (4.23) 
It is seen that if e-°? has a magnitude less than unity, fi(n7') will 
tend toward zero as n approaches infinity. While not complete, this is a 
condition for stability. Thus, stable functions F(z) will have all their 
poles inside or, in the limiting case, on the unit circle in the z plane. It is 
for this reason that the contour I which is used in the inversion theorem 
is generally the unit circle. 
