1394 PROCEEDINGS OF THE IEEE, SEPTEMBER 1970

2) The second term in (2) shows that the admittance looking into the bidirectional port is linear and equal to G.

3) The first term in (3) shows that the output signal u, in response to an input signal ii can be made arbitrarily small by increasing the gain of the balancing amplifier (decreasing UT); criterion b) is satisfied. This is achieved independent of the actual admittance Y connected to the bidirectional port, although it does help to have Y close to its nominal value Yo.

4) The second term in (3) shows that, if there is no input signal i i , a signal i, applied to the bidirectional port appears in undistorted form at the output port; criterion c) is satisfied.

5 ) If both ii and i, are present simultaneously, u., is an intermodulation between them. However, the zero crossings of u, are those off so the output signal is an intelligible but distorted version of the signal applied to the bidirectional port. Double-talking is possible. The degree of intermodulation falls as the balancing gain is reduced.

Thus, the value of U, must be a compromise between echo balance and intermodulation under double-talk conditions. If commonly occurring signal peaks at the bidirectional port are of amplitude Vo (with occasional peaks much larger), the optimum balancing gam appears to be

U, - 0.1 v,. Echo suppression is then about 20 dB relative to a conventional hybrid with the same degree of unbalance, yet the intermodulation under double- talk conditions is difficult to detect; not a single layman in the author’s very limited subjective tests has commented on it.

ACKNOWLEDGMENT The author wishes to thank his colleagues in the Department of Elec-

trical Engineering, Monash University, Australia, for a number of helpful discussions, and particularly the Chairman, Prof. D. G. Lampard.

EDWARD M. CHERRY Dept. of Elec. Engrg. Monash University Clayton, Victoria, Australia 3168

On Implementing Sequential Circuits with Shift Registers Abstract-A theorem, which characterizes two-block partitions

that lead to the formation of shift register, is presented. It can be used to simplify existing approaches to realize sequential machines with shift registers. Furthermore, this register-forming characteristic can be treated as a contributing element in selecting a secondary assign- ment for the implementation of sequential circuits.

INTRODUCTION The advent of integrated circuit technology has prompted the investi-

gations of realizing sequential machines with shift registers. In attempting to implement a given sequential machine with the mini-

mum number of shift registers, the available approaches call for the gen- eration of shift-register partitions or partitions with certain characteristics; and then the selection of an appropriate set of such partitions to achieve a one-to-one or many-to-one assignment.

In this letter. a novel viewpoint is presented. A “minimum-number’’ shift-register realization of a sequential machine is achieved by seeking a secondary assignment in such a way that results in the formation of shift registers with binary storage devices (or simply flip-flops).

ELEMENTARY ASSIGKMENTS Let q denote the number of binary storage devices (BSD), such as flip-

flops, that are to be used in the implementation of a given sequential ma- chine. The secondary assignment is to assign to each state at least one unique code of q bits, each representing the value of a state variable. State variable y i is stored in BSD i, where i = 1,2. . . , q. The assignment of any

TABLE I STATE TRANSITION TABLE FOR MACHIKE A

Next State

0 1 Present State

one state variable constitutes an elementary assignment (EA). An EA is characterized by a two-block partition on the states of the given machine.

Consider Machine A of Table I, for which an arbitrary assignment is given in (1).

state 1 2 3 4 5 6 7

Y l 0 1 0 0 0 1 1

Y2 0 1 0 1 1 0 0

Y3 0 0 1 1 0 0 1 . ( 1 )

The assignment in (1) consists of three EAs, each of which is specified by a two-block partition :

nCvJ = (1, 3,4, 5 ; 2,6, 7)

7d~2) = (1, 3, 6, 7 ; 2,4, 5 )

d ~ 3 ) = ( I , & 5 , 6; 3 ,4 , 7). ( 2 )

rc(yl) specifies the assignment of state variable y,. States in the first block are assigned 0 and those in the second block 1. Similarly, lTcy2) and rc(y3) specify the EAs of y2 and y,, respectively.

The next state functions corresponding to the assignment in (1) or equivalently in (2) are routinely identified :

~-

-~

SHIFT REGISTER FORMATION By the functions given in (3), we see that the output of BSD 1 , y l , serves

as the input to BSD 2, Y2. The same can be said concerning BSD 2 and BSD 3. They form a shift register of length 3.

Definition 1 A BSD i is shift-regster (SR)-related to BSD j if and only if

r, = yi. (4)

Note for the assignment in (l), BSD 1 is SR-related to BSD 2 and BSD 2 is SR-related to BSD 3.

Let x(yi) and x(y j ) be two-block partitions that specify the EAs of BSDs i and j , respectively. Let Bi , (B,o) and Bi , (B j , ) be the two blocks of r c ( y i ) , (n(yj)), and I be the input alphabet of M.

Definition 2 x(yi) has the comapping property with x ( y j ) if and only if

where

S(Bi, I ) = :si1si = 6(s j , Ik); V sjeBi, and V such that 6(sj, I,.) is specified} (6)

Manuscript received June 15, 1970. with r = 0, 1.

PROCEEDINGS LETTERS 1395

6(Blo, I ) = (1, 3, 6, 7) = B,,, 6(Bll, I) = (2, 4, 5 ) = BIl.

Now we are ready to present the main theorem. Theorem: BSD i is SR-related to BSD j if and only if .(-vi) has the co-

mapping property with d ~ ~ ) . Proof: Assume that BSD i is SR-related to BSD j . Let s,EB~o, yi=o,

By (4), y j = O , which implies that 6(s,, Ik ) sBj0 , for any I, . Similarly, if s,€Bil,then6(s,I,.)€Bjl.

To reverse the situation, let S(s, Ik)eBjo, q = O . By (4), yi=O, which impliesthat s ,€BiO Similarly,if6(s,I,.)EBjl, thens,EBil.

The above argument establishes that x ( y J has the comapping property with xbj).

To establish the second part of the theorem. we assume that n(y,) has the comapping property with x(y j ) . Following the same reasoning, we establish that Yj=yi. BSD i is SR-related to BSD j . The theorem is thus established.

This theorem states that pairs of two-blocks partitions and only those pairs of two-block partitions, where one has the comapping property with the other, lead to the formation of shift registers of length 2 or higher.

The identification of such pairs can be easily carried out. See for ex- ample Nichols.’

CONCLUSION

The result reported simplifies the existing in eliminat- ing the need to identify all shift-register partitions of a given machine. Furthermore, we can use the shift-regster-forming characteristic as a contributing element in evaluating the figures of merits of two-block par- titions to obtain an optimum secondary assignment.

H. C. TORNG J. ZALEWSKI School of Elec. Engrg. Cornell University Ithaca. N. Y. 14850

Trans. Electron. Compur., vol. EC-14, pp. 688-700, October 1%5. I A. J. Nichols, “Minimum shift-register realizations of sequential machines,” I€€€

I€€€ Trans. Comput., vol. C-17, pp. 312-324, April 1968. C. C. Su and S. S. Yau, “Unitary shift-register realizations of sequential machines,”

Cornput., vol. C-17, pp. 421431, May 1968. ’ W. A. Davis, “Single shift-register realizations for sequential machines,” I€€€ Trans.

state assignment: Part I and Part II.,” I€€€ Trans. Compur., vol. C-17, pp. 954977, October ’ D. L. Johnson and K. H. O’Keefe, “The application of shift registers to secondary

1968.

Pulse Compression with Periodic Gratings and Zone Plate Gratings

Abstract-Lord Rayleigh noted that a short noise pulse, upon- reflection from a periodic grating, is transformed into a chirped pulse. Conversely, an ascending frequency signal is, upon reflection, compressed into one of shorter length. This letter extends the concept to the compression of pulses reflected from zone plate gratings, and contrasts this with the recently described use of zone plate envelopes for pulse compression. The possibility of compressing laser pulses into shorter, more powerful pulses is discussed.

PERIODIC GRATINGS The phenomenon discussed by Lord Rayleigh [l], whereby a short

noise pulse is transformed into a descending chirp upon reflection from a sinusoidal reflection grating, is shown in Fig. I . In this figure the grating is placed off-axis so as to enhance the first-order diffracted component rela- tive to the zero-order reflections. and the sinusoidal corrugations tend to suppress the higher order diffracted components [4]. As shown in Fig. l(a), the shorter wavelength components of an on-axis noise pulse construc-

Manuscript received June 8, 1970

- ibl

- I C 1

Fig. 1. (a) A periodic grating re0ecting short-wavelength (high-frequency) waves ( i , / 2 )

tion of a handclap heard as a descending chirp. (c) The re0ection of an ascending chirp from points near the axis, longer waves (iJ2) from more distant points. (b) The re0ec-

heard as a handclap.

tively interfere nearer the axis than do the longer wavelength components. Because the more distant reflections return to an on-axis receiving point later. a descending frequency chirp results. as shown in Fig. l(b). If an ascending frequency chirp is used as the outgoing pulse, the reflected signal undergoes a compression in length [2], as shown in Fig. l(c). Had the outgoing pulse been merely a single-frequency signal, the periodic nature of the grating would have caused reflection to occur at only one angle, and the returning pulse would have closely resembled the outgoing pulse.

Electronic implementation of the reflection grating pulse compres- sion procedure for radar use might conceivably be accomplished by aiming two hghly directional on-axis ultrasonic horns (one behind the other) at the sinusoidal grating. The received radar signal, after conversion to an ultrasonic signal, would be introduced into one horn, and the chirped target echoes, after reflection from the grating into the second horn. would be compressed in time. In this process the horn that is nearer the grating would be required to have a high front-to-back ratio in its directivity pattern.

ZONE PLATE GRATINGS Recently, a pulse compression procedure was described in which a zone

plate envelope was imparted to a constant-frequency outgoing pulse [3]. Photographic records of returning signals then consist of many super- imposed zone plates (one for each target), which, upon illumination with coherent light, generate fully resolved small focal spots. the equivalent of highly compressed pulses.

We now consider a grating reflection procedure in whch the grating has a zone plate outline [Fig. 2(a)] rather than the sinusoidal outline of Fig. l(a). For single-frequency waves, a zone plate acts much like a lens in that all portions of it diffract energy toward a focal point [5]. Thus, if a short burst of single-frequency energy is directed toward the reflector of Fig. 2(a). all areas of it would reflect energy to the focal point. Because of the greater time of travel to the more distant areas. however. a short single-frequency pulse would be converted, on reflection, into a long single-frequency pulse [Fig. 2(b)]. This is analogous to the stretching in time of the handclap pulse of Lord Rayleigh [Fig. l(a) and (b)].

One can also run this proces? backward so as to achieve a pulse com- pression effect [Fig. 2(a) and (c)]. Here, a iotating directional horn first illuminates the more off-axis (more distant) portions of the zone plate; later. through its rotational motion, it illuminates the nearer portions. Since all areas reflect the single-frequency waves back toward the focal point, all parts of the long reflected pulse’can be made to amve at this focal area at approximately the same time by making the rotational motion of the horn match the travel times of the various portions of the outgoing pulse [Fig. 2(c)].

It should be possible to shorten the length of a pulsed (single-frequency) laser emission by this procedure. A rapidly rotating plane mirror located on