D. G. Lampard, "A New Method of Determining 
Correlation Functions of Stationary Time 
Series," J, Inst, Elec, Eng., part IIT, 
Now 72), pp. 343°f., sept., 1954. 
W. H. Kautz, "Transient Synthesis in the 
Time Domain," Trans, IRE PGCT, vol, CT-1, 
No, 3, pp. 29-39, 1954. 
T. L. Saaty, Mathematical Methods of Opera- 
tions Research, McGraw-Hill Book Co,, Inc., 
New York, 1959 
APPENDIX 
Proof of Theorem 
Since each h,(t) is square-integrable, it 
follows that 
[ee) 
f h(t) h(t) at < o (30) 
6) 
for alln,m 1, Since in the Laplace trans- 
forms H,(s) the terms C,(s) are polynomials in 
s, the transforms contain only poles in the left 
half-plane, One may therefore write: 
ee) 
| h(t) by(t) at = Al Hm(-8) Hy(s) ds 
re) He 
e Ora J sales) H(s) ds (31) 
in which Br and Br! denote Bromwich contours in 
the s-plane enclosing the poles of H(s) and 
H,(s), respectively. 
Noting that the repeated application of 
(10) leads to 
n-1 
TT ¢,,(-8) 
Hy(s) = K, By(s) +, nz 2 (32) 
I} c,(s) 
k=1 
we consider the following three cases in which 
n 7m: 
(1)n=1,m> 2 
Substituting (9) and (32) in (31) gives 
m-1 
af 
air Br 
0, (6) 
KK, B,(s) By(-s) Ke x 
teh ee) 
k=1 
(33) 
354 
because the denominator term Cy(s) is cancelled 
by Cy(s) in the numerator, leaving no terms in 
the denominator with zeros in the left half- 
plane, 
(2)n>2,m<n 
iol 
ig C (s) 
k 
oa | K, Km Byl-s) By(s) a x 
. Br! Tal cs) 
k=1 
= 
7 C,(-s) 
kel ds = 0 (34) 
m 
TT of) 
k=1 
because the denominator terms Cy(s) Co(s) 
Cn 8) are cancelled. again leaving no terms in 
the denominator with zeros in the LHP. 
(3) 
m2 2, n€m 
ais B (-s) =e 
Br 
n-1 
Th °x¢-3) 
k=1 
T1°x6s) 
k=1 
ds = 6 (35) 
because the denominator terms C,(s) C,(s) 
C,(s), having zeros in the LHP, are cancelled, 
For n =m, (11) applies. Hence 
feo) 
Lanes im 
i) hy(t) Ay(t) dt { (36) 
0 Of nm 
and the theorem is proved, 
