MISCELLANEOUS APPLICATIONS OF SAMPLED-DATA THEORY 297 
The contour I is the map of the truncated imaginary axis shown in Fig. 
11.11la on thez plane. The map is the unit circle on the z plane, as shown 
in Fig. 11.116. The poles of F(s) are shown to lie on the left half of the s 
plane, and these map inside of the unit circle of the z plane. The integral 
of (11.24) is recognized to be the standard inversion integral for a pulse 
transfer function M(z) given by 
WAG) = ae ( in 2) (11.25) 
It is relatively difficult to evaluate (11.24) because M(z) is a transcen- 
dental function rather than the ratio of polynomials in z which permits 
numerical inversion by the method of long division. 
To facilitate the inversion of M(z), In z will be approximated by a con- 
venient series as follows: 
Inz = 2(u + du? + du®+ -- -) (11.26) 
h See 
where WS a 
This particular series converges rapidly for regions of the z plane where z 
is large. To apply this approximation, the Laplace transform will be 
expressed as a polynomial in s“!, the latter variable peing more desirable, 
as will be seen. From the denntten B= Bh 
si silj/se = Time (C277) 
Substituting the series expression for In z from (11.26) into (11.27), 
g T/2 
ut u3/8 + u2/5 + + -: 
Expanding (11.28) into powers of u by the simple process of long division, 
st = T/2(1/u — u/3 — 4u3/45 — 44u5/945 — -- -) (11.29) 
To obtain the general term s~*, (11.29) is raised to the power k and is 
expanded by means of the binomial theorem, where 1/u is considered the 
first term and the remainder of the expression the second term. For 
instance, taking k = 2, 
s 


(11.28) 
s-? = T?/4[(1/u) — (u/3 + 4u?/45 + - - -))? (11.30) 
which expands to 
s-? = T?/4[(1/u)? — (2/u)(w/3 + 4u3/45 — ---) +--+] (11.31) 
Taking only the first term of the series contained in the third parenthesis 
of (11.31) and ignoring all other terms, 
s-? = T?/4[(1/u?) — (2/3)] (11.32) 
