SAMPLED-DATA SYSTEMS 101 
contour encloses the origin on the function plane is equal to the difference 
between the number of zeros and poles so enclosed. 
For the problem at hand, it is necessary to choose a contour on the 
z plane which encloses the entire region outside the unit circle in order to 
study the function 1 + GH(z). Such a contour is sketched in Fig. 5.9, 
where the outer radius RF is made to approach infinity. The map of this 
contour on the [1 + GH(z)] plane will indicate by its enclosures of the 
origin the difference between the zeros and poles of this function. It is 

Fic. 5.9. Contour used to enclose poles Fic. 5.10. Typical map of T on [1 + 
outside the unit circle. GH (z)] plane. 
seen that the poles of 1 + GH(z) are the same as those of the function 
GH(z). Thus, if GH(z) is a stable function, that is, if it does not contain 
poles outside the unit circle, then neither does 1 + GH(z). In such an 
event, the enclosure of the origin by the map of I indicates the number of 
zeros or roots of the characteristic equation. Such enclosures or lack of 
enclosures indicates a condition of instability or stability respectively. 
A typical map resulting from application of this procedure to a practical 
system is shown in Fig. 5.10, where it is seen that the origin is enclosed 
once. If the open-loop pulse transfer function is stable, this enclosure 
indicates that one zero lies outside the unit circle and that the closed-loop 
system is unstable. As in the case of continuous systems, it 1s con- 
venient to shift the imaginary axis to the point (1,0), as shown in Fig. 
5.11. With this shifted axis it is necessary to plot only GH(z) and to 
observe the enclosure of the critical point (—1,0) instead of the origin. 
When plotted on this modified plane, the map is referred to as the pulse 
transfer locus. Essentially, then, the pulse transfer locus is a map of the 
unit circle only, since the function GH (z) vanishes as z approaches infinity 
in practical, physically realizable systems. This is shown more clearly in 
the illustrative example which follows. 
