2 
leads to (s°) > (Sy), and to a set of parameters 
N°2 
{a° A } - Continuing the process yields the mono- 
tone nondecreasing sequence 
< (S,) Save (27) 
which converges to Max se. The values of the 
api 
parameters that yield this maximum form an 
optimum set of parameter values, 
IMPLEMENTATION OF THE ADAPTIVE 
CORRELATOR 
The block diagram implementation of the cor- 
relation principle discussed above is shown in 
Fig. 1, Random process g(t) is applied to the 
input of the first set of orthogonal filters. 
The resulting outputs -- vz(t), vo(t),..., vy(+) 
--are multiplied by the second random process, 
go(t), by means of analogue multipliers; and the 
respective products are passed through low-pass 
filters, to give the averages Ad, Moy eee, Ans 
At the same time a second set of identical 
orthogonal filters receives a repetitive square- 
wave input, producing outputs aj(t),..., ay(t). 
Each of these responses is multiplied by the cor- 
responding average, and the products are summed, 
Finally, the sum is differentiated to obtain 
gy(t), the approximate expansion of the correla- 
tion function, 
Fig. 2 shows the block diagram of the 
mechanization of a correlator capable of self- 
adjustment to achieve an optimum orthogonal ex- 
pansion in the minimum-integral-square-error 
sense, The quantity Sf, obtained by squaring 
each of the averages Aj,..., Ay and summing the 
squares, is applied as an input to an optimiza- 
tion computer and parameter controller, which 
iteratively determines the optimum settings of 
the parameters of the orthogonal filters by the 
routine of Fig, 3. After storing the set of 
initial parameter values, f« 4 } , and the initial 
value of the sum of the squares of the averages, 
(Sy)g> the computer determines the set of ap- 
proximate initial parameter sensitivities, 
{234} . 
by aenie each by the absolute value of the 
largest, Po. Each parameter is then varied 
over its range in steps proportional to its 
normalized sensitivity. The set of parameter 
values fot 5 } » yielding the largest value of Sy 
is compared to the initial set fo}. rf all 
corresponding members of the two sets differ 
by less than preassigned amounts, the process 
is terminated; if not, the initial parameter 
values are replaced by the values giving the 
greatest ch and the process is repeated, Each 
It then normalizes these sensitivities 
351 
parameter is thereby iteratively adjusted to its 
optimum value, The optimization computer and 
parameter controller is amenable to mechaniza- 
tion by hybrid-computer and step-servo 
techniques, 
CONC LUSTONS 
In this paper we have considered certain theo- 
retical and practical aspects of a correlator 
that approximates correlation functions by series 
of orthogonal time functions and iteratively 
adjusts itself to minimize the approximation 
error, Since operational-amplifier methods are 
employed to synthesize the orthogonal filters 
and analogue multipliers are used to obtain the 
required signal products, the device is re- 
stricted to the correlation of signals contain- 
ing no significant energy at frequencies greater 
than 100 ke, With amplifiers and multipliers 
capable of operating at higher frequencies, how- 
ever, the range could be increased, 
A method for synthesizing a class of linear 
orthogonal filters has been presented, Typical 
members of this class are the well-known 
Laguerre and Kautz filters, Other members are 
filters with transfer functions 
21S wo, S 
H,(S) = (28) 
1 
S*42_ S+ * 
iL ai al 
and 
bers 
H_(S) 2 ee an 
z 2 Tn-Mn-1 i 
S-2 w Sw 
n=) n=. n-1 H,-1(S) 
Tee nEsnoe. == 
S“+ 25, w, 5+, 
(29) 
The impulse responses of such filters contain 
exponentially damped cosinusoids and are there- 
fore applicable to the expansion of correlation 
functions with periodic components, 
REFERENCES 
1. Y. W. Lee, Statistic Communica- 
tion, John Wiley & Sons, Inc,, New York, 
1960, 
2. A. A. Wolf and J, H, Dietz, "A Device for 
Measuring Correlation Functions and 
Spectral Density Functions," Technical 
Report No, 1, and "A Study of White-Noise 
Fault Diagnosis in Linear Passive Systems," 
Technical Report No, 2, Stromberg-Carlson 
Company (now General Dynamics/Electronics) 
Applied Mathematical Studies Department, 
Rochester, N.Y., Nov., 1960, 
