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Digital frequency measurement for sine waves is based on counting the number of pulses during a period T, with subsequent use of an appropriate algorithm and computer device [1, 2]. The measurement t i m e in this case does not exceed a single period. However, a single measurement gives low accuracy, and one needs to apply stat i s t i c a l methods to process the results. A t i m e NT is required to process statistically N independent equally accurate measurements, which makes the t o t a l t i m e requirement excessive. Only low accuracy is obtained from the analysis of the structural properties of signals and associated d i g i t a l methods and devices for determining frequency or periods of sine-wave signals [3, 4]. Here we consider frequency determination for sine waves in which there is no relationship between the time of measurement and the period of the signals, while the accuracy is substantially increased. Consider a harmonic curve represented in the form

g (t) = a sin (cot + r

(1)

where a is amplitude, w is circular frequency, and ~ is i n i t i a l phase. The following r e l a t i o m define the equidistant ordinates of the signal of (1) with separations q: y~ = a sin (c0~ + Tx) n = 0 , 1, 2 . . . . ,

(2)

where ~?i is the phase at the start of the t i m e reading q . If a digitizing device is employed, the determination of the equidistant ordinate can be referred to the case of a single measurement [5], in which the measured quantity y(t~) is assigned the digital equivalent YK =EL, where A is the level quantization step and n is the number of pulses in a measurement. The error of a single measurement 5 n is considered as the difference between the a c t u a l measurement result Yn = n A and the true value y(t n) of the measured quantity, so

8~ =

y~--

y (tD.

(3)

If we bear in mind that 5~ had a normal distribution in direct measurements, we have if there is no bias that

M 8~ = 0.

(4)

Consider the following sum: m--I

m--I

m--1

~r 0

~;~0

~0

(5) Translated from I z m e r i t e r n a y a Tekhnika, No. 11, pp. 71-73, November, 1973.

9 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00.

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From (2), (4), and (5) we get m--1

ZI=

m--1

(6)

~ YK = ~ asin(c~ g=0 g=0

~*7~hose~ right part in (6) is the sum of trigonometric gl~c~~ arguments constitute an a r i t h m e t i c progress i o n we have Fig. 1

m--1

z,=Ey~c=asinE~_t_(m_l)_~__lsinm(o~q/2)sin(coql2 )

(7)

If the sum of (5) is produced at arbitrary t i m e intervals t~ and t3, then for Z z and Z 3 we get analogous expressions with (ol replaced by ~2 and r respectively. We assume that the start of the readings in generating the sums for z2 and z s is taken from the condition t,--t2

= t~--tl

= to.

(8)

Then for the phase relationships we have % = qh + coto,

(9) % = ~1 + 2o)to.

We solve (7) in the formulas fo~ z2 and z3, which are analogous to (7), for the frequency of the harmonic signal to get [6] that

COS (010 = (Z1 @

Z3)I2z2.

(10)

If cot0 << 1, which can always be obtained by appropriate choice of to we expand (10), neglect s m a l l terms above the second, and get

1 f--

2n/0

]/2zo

- - (z i + z~)

]/

(11)

z2

These relationships can form the basis of a method of frequency of period d e t e r m i n a t i o n for sine waves. Figure 1 shows the block diagram of a d i g i t a l frequency m e t e r . The frequency is determined by measuring the instantaneous values of the input signal by means of ADC. The measured values pass to a computing unit, which produces the quantity (z z + z3)/2zz, which is stored in the counter CL In the i n i t i a l states, the value of ~t0 is set arbitrarily in a reversible counter. The unit that reproduces cos wt0 works into a counter C2, whose content is compared via the comparator device CD with the content of C1. If C2 > C1, a command is produced to reduce wt0; in the contrary case, wt0 is increased until the contents of C1 and C2 are equal. The value of oJt0 is then the desired value. The result is presented on the indicator unit I, whose presentation may correspond to the frequency or period of the signals. If the random quantity 6n has a normal distribution [7], we can round-off the d i g i t a l eqnivalents of the e q u i distant ordinates in accordance with the following principles: in the case Y~< V (t~) < g ~ + (1/2) A,

gn = K~;

and in the case W + (1/2) A < !/(t~) < Y~+t,

y ~ = ( ~ + 1) A.

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We assume that the interval between equidistant ordinates is 1 ~ We use the round-off rules and calculate zl, ze, and z3 as follows: m--I

zl = ~Ko ) asin ~ (15 A q-

m~l

m--1

,

z_, : ~ko

)a sin ~ (35 A+

'

-Z a sin (55 -@to)~ z3 - ~

A

We use the values of these quantities for different m with (10) to determine the period of frequency of the sine wave Equation (10) can also be used to determine the frequency or period of a harmonic signal masked by random noise of zero mean [8]. Consider a signal containing a random stationary ergodic process g(t) with zero mean and a deterministic signal y(t) = a sin (cot + ~0); then z (t) = y (t) + ~ ( t ) .

Then following the above we write m--I

g~0

m--1

~r

m--1

(12)

g~0

In the case of a representative sample m--I

"~ ~ (t,d ~ o, it follows from (12) that we get (6), which results in (10). Conclusions. A new method and an algorithm are described for determining the frequency or period of a harmonic signal no matter what the period of the signal. The principle can be used in digital measuring systems, particularly at low and infralow frequencies. LITERATURE CITED 1. 2. 3. 4. 5. 6. 7. 8.

1712

~. K. Shakhov, Avtomet~iya, No. 2 (1966). M . V . Leitman and E. L. Bar'yudin, Izmeritel'. Tekh., No. 8 (1968). I . G . Shafir, Author's certificate No. 237262, byuL Izobr., No. 8 (1969). A . A . Muchiauri, Author's certificate No. 263744, Byul. Izobr., No. 8 (1970). A . S . Nemirovskii, Probability Methods on Metrology [in Russian], Izd. Standattov, Moscow (1964). A . A . Muchiauri, Proceedings of the First Republic Conference on Metrology [in Russian], Vol. 1, Tbilisi (1970). ~. I. Gitis and E. G. Men'shikh, Avtometriya, No. 2 (1966). B.R. Levin, Theoretical Principles of Statistical Radio Engineering [in Russian], Sov. Radio, Moscow (1968).