MISCELLANEOUS APPLICATIONS OF SAMPLED-DATA THEORY 303 
(11.46) is interpreted as a z transform relation which is directly converted 
to a recursion formula between the variables. This is best shown by 
means of an example. 
EXAMPLE 
The time-varying differential equation of first order is expressed as 
follows: 
d?y d 
de ap di (ty) = u(t) 
_ where u(t) is the unit step function. The equation is reduced to the 
form of (11.40): 
¥(s) +2 Gils) = 5 UUs) 
where G(s) is £[ty(t)] and U(s) is the Laplace transform of the unit 
step function. Since the Laplace transform of the independent vari- 
able is, in this case, in the form of 1/s*, it is possible to apply the 
z forms directly to the right side of the equation. Reducing the equa- 
tion to the form of (11.45) with the slight change that the z form 1/s* is 
used on the right side, the relation is obtained that 
Ta + 27) 
2(1 — 27) 
ee (Cau + ome) 
Cues ae 
TGi(z) = 
This relation represents a difference equation or recursion formula 
between samples. To obtain this relation, both sides of the equation 
are multiplied by (1 — 27)’, and, canceling out 7’, there results 
@ — 2-})*Y(z) + T/20 +29) — 2 )Gi(z) = T?/2(21 + 2”) 
The recursion relationship represented by this equation is, after com- 
bination of terms, 
Baia Lonnl)) — (oy yi rain — 1) 2] 
s> [(GN@ = AYP = We = 2) 20) = [Pale = 8) = TMG == 3) 0) 
= T’u(n — 1)T + T?u(n — 2)T 
where gi(nT’) is from the definition of Gi(s) given by 
gi(nT) = nTly(nT)] 
Thus, the recursion relation can be written 
ase LCI) GOLD) — (0 > IG ae ey ele 
+ [6 — T(n — 2)Tly(n — 2)T — [2 — Tir — 3)Tly(n — 3)T 
= T’u(n — 1)T + T?u(n — 2)T 
For instance, if 7 is unity, the first ordinate y(0) is zero and the other 
