THE Z-TRANSFORM ANALYSIS OF LINEAR SAMPLED-DATA SYSTEMS 83 
The pulse sequence for positive time, n = 0, is obtained by taking 
the residues for all poles inside the unit circle, in this case only one pole, 
PY? ae Bo OLDS 
ails @) 
MGI) = a 
Wide aad = @) 
=iq" n= 0 

Now, for negative time, n S 0, the pulse sequence is obtained by 
taking the residues at the poles outside the unit circle. In this case, 
there is only one, at q-"!. Thus, 

= ) 
Nay HO = @) 
= GF ° ns0 
The negative sign resulting from a direct application of the residue 
method is caused by the fact that the enclosure outside the unit circle 
is counter to that inside the unit circle, and an additional negative sign 
must be applied to take this into account. Thus, the correct value of 
r(nT) is 
AGL) = or? n=<0 
In some cases, an equivalent of the method of partial fractions can be 
applied to the inversion of two-sided z transforms. This can be done 
directly on the form used in introducing the problem. The first frac- 
tion, having a pole outside the unit circle, represents the sequence for 
negative time only. Hence, it should be expanded into a power series 
in z. The second term represents the pulse sequence for positive time 
only and hence should be expanded into a power series in z—!. Doing 
so, there results 
+. +e 
> & non ng—n __ | (20 
RZ) Die + Y, gre 1.02 
Inverting this summation term by term, the resulting pulse sequence 
becomes 
+2 fe 
i) gro(t + nT) + q’6(t — nT) — 1.06(t) 
The pulse sequence is seen to cover the entire range of time. This 
approach has the same advantages as the partial-fraction method and 
the long-division method of inversion and is often applicable to prob- 
lems where the z transforms are ratios of polynomials in z. 
4.10 Summary 
As in the case of linear continuous systems, transform methods greatly 
simplify the solution of problems in the analysis and synthesis of linear 
