PROCEEDINGS OF THE I-R-E

The Response of a Panoramic Receiver

to CW and Pulse Signals*H. W. BATTENt, ASSOCIATE MEMBER, iRE, R. A. JORGENSEN$, A. B. MACNEEt, M-EMBER, IRE,

AND W. W. PETERSONt, ASSOCIATE MEMBER, IRE

Summary-An analysis of the response of a panoramic receiverto cw and pulse signals is presented. The receiver's response isstudied quantitatively as a function of the parameters: signal-pulselength and frequency, receiver bandwidth, sweep-rate, and type ofIF amplifier. The effect of these parameters on the relative outputamplitude, output pulse width, and apparent bandwidth is empha-sized. Two specific cases are considered. An electronic differentialanalyzer is used to study the response of a receiver with a single-tuned IF amplifier to pulses having rectangular envelopes. Theo-retically the response of a receiver with a Gaussian shaped IF pass-band to pulses having Gaussian envelopes is derived. This answer isgiven in closed form. The agreement between these two cases justi-fies the application of the Gaussian case to most practical designproblems.

I. INTRODUCTION

7ff HE RESPONSE of a linear-resonant system to asinusoidal-driving function, having a linear vari-ation of frequency with time, is of importance in

many fields of engineering. This problem is encounteredwhen an engine is accelerated uniformly through a criti-cal frequency.' The same situation occurs in the analysisof records of ocean waves by means of vibration gal-vanometers.2 A panoramic superheterodyne receiver,which also presents this problem, is the subject of thispaper.

SIGNAL INPUTTO RECEIVER FM SIGNAL

The receivers considered in this paper are idealiza-tions of conventional superheterodyne receivers. A blockdiagram is shown in Fig. 1. The function of the mixer isto convert an incoming signal of fixed instantaneousfrequency to one with an instantaneous frequencychanging linearly with time. It is assumed that the en-velope of the incoming signal is not distorted by themixer. The filter of Fig. 1 merely selects the desiredfrequencies, and the detector operates on the output ofthe filter to obtain the envelope. These assumptions re-duce the problem to that of obtaining the response of afilter to a particular FM signal. This is illustrated graph-ically by the time-frequency diagrams of Fig. 2 opposite.The figure also indicates the parameters used to de-scribe the receiver and the incoming signals.The response of a filter with a Gaussian-amplitude re-

sponse and a linear phase-response curve to a cw signal,and to sinusoidal pulses with Gaussian envelopes, hasbeen examined theoretically. These cases are importantbecause answers in closed form can be obtained. TheGaussian filter is not physically realizable; however, ifthe time delay is neglected, the transfer function of nsingle-tuned circuits all at the same frequency ap-proaches the Gaussian function as n becomes large.9 A

FILTER OUTPUT

TRAN3ITTER MIEFLTRW

Fig. 1-Block diagram of idealized superheterodyne receiver.

An analogous second problem is the response of a

system whose resonant frequency varies linearly withtime to a fixed-frequency sinusoidal signal. This problemis encountered in various types of spectrum analyzers

and in panoramic-radio receivers.3-A For the high-Q or

very much underdamped system, the two problemsprove to be essentially equivalent.2' 4

* Decimal classification: R361.121 XR161. Original manuscriptreceived by the IRE, September 22, 1953. This research was sup-ported by the Signal Corps under Contract No. DA-36-039.

t Dept. of Elec. Eng., Univ. of Michigan, Ann Arbor, Mich.t Dept. of Mathematics, Univ. of Minnesota, Minneapolis, Minn.I F. M. Lewis, "Vibration during acceleration through a critical

speed," Trans. Am. Soc. Mech. Eng., APM-54-24, pp. 253-261; 1932.2 N. F. Barber and F. Ursell, "The response of a resonant system

to a gliding tone," London Phil. Mag., vol. 39, pp. 345-361; May,1948.

3 E. M. Williams, "Radio-frequency spectrum analyzers," PROC.I. R.E., vol. 34, pp. 18-22; January, 1946.

4 G. Hok, "Response of linear resonant systems to excitation ofa frequency varying linearly with time," Jour. Appl. Phys., vol. 19,pp. 242-250; March, 1948.

study of the response of 1, 2, and 4 synchronous single-tuned circuits to cw signals and pulses having rectangu-lar envelopes was made with an electronic differentialanalyzer. Over six hundred solutions were obtained.Over-all agreement with the Gaussian case was good.'0

5 H. M. Barlow and A. L. Cullen, "Microwave Measurements,"Constable & Co., Ltd., London, Eng., pp. 320-332; 1950.

6 C. G. Montgomery, "Technique of Microwave Measurements,"M.I.T. Radiation Lab. Series, McGraw-Hill Book Co., Inc., NewYork, N.Y., vol. II, pp. 408-455; 1947.

7 J. Marique, "The response of rlc resonant circuits to emf ofsawtooth varying frequency," PROC. I.R.E., vol. 40, pp. 945-950;August, 1952.

8 H. Salinger, "On the theory of frequency analysis by means ofa searching tone," Eleckt. Nach. Tech., vol. 6, pp. 293-302; August,1929.

9 H. Wallman, and G. E. Valley, Jr., "Vacuum Tube Amplifiers,"M.I.T. Radiation Lab. Series, McGraw-Hill Book Co., Inc., NewYork, N. Y., vol. 18, pp. 723-724; 1948.

'0 H. W. Batten, R. A. Jorgensen, A. B. Macnee, W. W. Peterson,"The Response of a Panoramic Receiver to CW and Pulse Signals,"Tech. Report No. 3, Elec. Defense Group, Univ. of Michigan (un-classified).

DETECTOR OUTPUT

948 Jutne

1954 Batten, Jorgensen, Macnee, and Peterson: Response of a Panoramic Receiver to CW and Pulse Signals 949

II. THE GAUSSIAN CASEThe transfer function assumed is

H(c. = exp+/27r{ ~b2}

The center frequency of the filter is a radians per secoand the bandwidth between e"1/4 points is b radianssecond. Note that the phase delay is completely nlected here. The introduction of a linear-phase dewould not significantly change the answers.

t -c

t - i

(b)

Fig. 2-(a) Time-frequency diagram before mixer.(b) Time-frequency diagram after mixer.

The signal assumed for the cw case is,

f(t) = cos [at + -1] (2)

and for the pulse case is,- (t - C) 2- S t2-

f(t) = exp d2 cos at + 2. (3)

The center-time of the pulse is c, the pulse width be-tween e-lJ4 points is d, and the sweep-rate is s radians persecond per second. The answer is derived first for thepulse case, and the cw case is obtained from it by lettingd approach infinity.The analysis is given in the Appendix. The signal

function is transformed to the w-plane, multiplied by

the transfer function H(c), and transformed back to thet-plane. The envelope of the output for the pulse case is,

(1[ (t-4bm)] 1 2(SC)(1) Ig(t)I =o exp (4)

where Ao, B, W, and tm/c are functions of s, b, and dasfollows:

bAo=,

4 \2 1l/4A [ + b2 + 4S2

1 /4B=b d + b2+ s2d2^

2+ b2 + 4S2B=-17 + 2

b2 ( )4V + b2+ s2d2

d2

sd 1= ,~ and

b A02B4m + b2

tm ~~d2

c 4-+ b2 + s2d2

The envelope response in the cw case reduces to

g(t)| Aoexp {- W2[b }'

where

bAo = + ) and

(b4 + 4s2) 114

(5)

(6)

(7)

(8)

(9)

(10)

and

1 1w = 2 =b+4s2=-2bI AO2

(11)

The quantities Ao, W, and B describe important fea-tures of the response: (1) its peak amplitude, (2) thewidth of the response in time, and (3) the width of thepeak-amplitude curve plotted as a function of sc, thedifference between the filter frequency and the signalfrequency at the center of the pulse. Ao, W, and B areall expressed in dimensionless form; Ao is the peak am-plitude of the response relative to the response to a cwsignal of fixed frequency a; W is the width of the re-sponse in time relative to the time necessary for thereceiver to sweep through its IF bandwidth, b; and B isthe apparent bandwidth of the receiver when sweeping,relative to its steady-stage bandwidth b.1"

11 It can be shown for the case of pulse signals that Ao2B(Wb/sd)=1 (7) regardless of the type of IF filter or the pulse shape, and forthe case of the cw input signal AO2W= 1 (11) regardless of the type ofIF filter. See reference 10.

PROCEEDINGS OF THE I-R-E

s

Fig. 3- The relative amplitude of the response for the Gaussian case.

wIn.

1.0

.01

| - - - - 0 00 - 0

000000oroz~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_________. r____I,/l.̂ ._ A _ _ f _ 7 I Y . .i---t-h1_ - I

bd 27r

////

-

//bd =0

d ir bd

0. 1.0 10s

Fig. 4-The output puilse width for the Gaussian case as a function of sweep-rate.

-'

June950

,..,oo

7

1954 Batten, Jorgensen, Macnee, and Peterson: Response of a Panoramic Receiver to CW and Pulse Signals 951

100

B '0 |

IE

AII,YI/, /,>A>--~~~~~~I0l~~~~~~~~~~~~~~~~~~~~~~~ 000 C ;72/I00L ~~~~~~~~~~~~~~~~~~~~~~~I iooT|1

.01 .1 1.0 10S

Fig. 5-The apparent band width of a Gaussian filter as a function of sweep-rate.

For the Gaussian case, graphs of Ao, W, and B as 00.0

functions of s/b2 are given in Figs. 3, 4, and 5. Ao is de-fined so that it has the value one for an infinite pulselength and zero sweep-rate. Ao is affected little by sweep-ing until s/b2 is of the order 1+ [4/(bd)2], and drops off __rapidly for higher sweep-rates.12By definition, the output pulse width, W, approaches

unity for a cw signal as the sweep-rate approaches zero.The curves in Fig. 4 never fall below W= 2s/b2, whichis the output pulse width corresponding to the impulse 10.0 ||I_ 1response of the filter (bd=O). For low sweep-rates the _||output pulse width is between the value for the cw sig- - _ _I_nal and that for the impulse response. For high sweep- _ _ 1_Irates the output pulse is essentially the impulse re- b

sponse of the filter and is independent of bd. II !The apparent bandwidth, B, is defined so that it is _ -_ - - ,

unity when the sweep-rate is very low and the pulsesvery long. For short pulses, B is greater than one evenfor zero sweep-rate. As the sweep-rate increases above L10

bdVbd2

the curve rises sharply and approaches B=s/b2. bdasymptotically. IIr.The results of (5) and (7) can be plotted in a variety

of ways which may be convenient for the engineer facedwith a particular problem. Two such plots are shown in _ ______Figs. 6 and 7. The question of what IF bandwidth 0°1 byields the optimum resolution for a radio-frequency s

12 s/b2 =1 corresponds to a sweep-rate of 2irmc/, sec. for an IFbandwidth of 1 mc.

Fig. 6 Output pulse width as a function of filterbandwidth for the Gaussian case.

I

'A

PROCEEDINGS OF THE I-R-E

0.1 1.0 10.0

Fnad

Fig. 7-Apparent bandwidth as a function of input pulse width for the Gaussian case.

Fig. 8-Responses for various bandwidths.

952

100.0

b B

10.0

June

1954 Batten, Jorgensen, Macnee, and Peterson: Response of a Panoramic Receiver to CW and Pulse Signals 953

spectrum analyzer arises frequently. From Fig. 6 it isapparent that for cw signals the minimum output pulsewidth occurs for b = V2s radians per second.'3 Fig. 7illustrates the fact that the minimum apparent band-width always occurs for a pulse length d = V* second.14

III. SOLUTIONS BY DIFFERENTIAL ANALYZER

An electronic differential analyzer was used to studythe response of a panoramic receiver, employing a syn-chronously tuned IF amplifier of one, two, and fourstages, to cw signals and pulses having rectangular en-velopes. Over six hundred solutions were run; typicaloutputs taken from the analyzer are shown in Figs. 8and 9. The effect of varying the IF bandwidth of apanoramic receiver, sweeping at a fixed rate on the re-sponse to a cw signal (d = oo) is shown in Fig. 8. Clearly,there is an IF bandwidth which gives the minimum out-put pulse width. This is the same effect illustrated byFig. 6 for the Gaussian case.

IV. COMPARISON OF SOLUTIONSFigs. 10, 11, and 12 give a comparison between the

Gaussian case and some of the data taken from the dif-ferential analyzer for the response of 1, 2, and 4 circuitfilters to rectangular-envelope pulses. In general theagreement between the Gaussian case and filters usingtwo or more circuits is good. Some features of the re-ceiver response in the differential-analyzer solutions dif-fer considerably from the solution of the Gaussian case.For example, the time of maximum response can hardlybe expected to agree since the Gaussian filter is assumedto introduce no phase delay.The solution of the Gaussian case gives an under-

standing of the nature of the response of a panoramicreceiver. Moreover, the Gaussian case is quantitativelyconsistent enough with the other cases studied to beused in many design problems involving peak ampli-tude, output pulse width, apparent bandwidth, andresolution.

APPENDIXDerivation of the Response of a Gaussian Filter

In this appendix the formulas are derived for theresponse of a filter with a Gaussian-shaped tranfer func-tion to a signal which is changing linearly in frequencyand has either a constant amplitude or a Gaussian-shaped envelope.Assume the filter transfer function is

H(W) =I

exp [(w a)]'--,/f7r~~~'I

and the input signal is (the real part of)

J(t) = exp [ + at) d2 ]

(12)

(13)

Fig. 9-Responses for 1, 2, and 4 circuits.

Fig. 9 shows the response of 1, 2, and 4 circuit filters,all having the same bandwidth, to an input pulse havinga center frequency somewhat above the resonant fre-quency of the filters. There is an increase in the delayof the output pulse relative to the input pulse as thenumber of circuits is increased; but otherwise, increasingthe number of circuits has little effect on the relativeamplitude and output pulse width. Note the envelopeof the output pulse tends towards a Gaussian shape asthe number of circuits increases.

13 This corresponds to a bandwidth of 2V/iF mc for a sweep-rateof 1 mc/sAsec.

14 This corresponds to a 1/ViFpsec. long pulse for a sweep-rate of1 mc/ILsec.

The procedure is to find the Fourier transform f(w) off(t), multiply it by H(4w), and transform back to the t-plane. The filter response is the real part of the result-ing function g(t). The calculation is simplified by usingthe convolution formula:

1 XJ-J H(w) .F(.)ei-1dw

,\/27r _.C

1 ( -= _,I-Tf-(X) h(t - )dX (14)

where h(t) is the Fourier transform of H(w)1516Two preliminary remarks will make the derivation go

smoothly. In the first place, the envelope of the realpart of a complex function of time is just the absolutevalue of the function. This can be seen as follows: LetZ(t) be any complex function of t. It can be written,

15 E. C. Titchmarsh, 'Introduction to the Theory of FourierIntegrals," Oxford University Press, New York, N. Y., p. 51; 1937.

16 The use of complex functions for the signal f(t) and the impulseresponse h(t) is justified if the response of the filter is negligible atzero frequency.

PROCEEDINGS OF THE IPR-E

80L0

GAUSSIAN

- CIRCUIT bdE 2 CIRCUITS

0~----' 4 CIRCUITS

10 - -B _L_

.0 10.1 170 s 100 70

Fig. 10-Apparent bandwidth for the Gaussian case and the differential-analyzer solutions.

bd:27r

PULSE CENrEREDON PASSBAND

1.0 - -

s

GAUSSIAN =- 2 CIRCUITS

CIRCUIT a 4 CIRCUITS

Fig. 11-Output pulse width for the Gaussian case anddifferential-analyzer solutions.

Z(t) = IZ(t) I exp (jO(t)), where 0(t) is the argument ofZ(t). The real part of Z(t) is then Z(t) cos 0(t), and theenvelope of this is Z(t) |.

Secondly, in computing the Fourierwill be made of the following formula:

transforms, use

ex t+ v2Jexp (-Ut2 + vt)dt-= / exp--00 u 4u

(15)

The integration is along the real axis in the t-plane,and u and v are complex numbers, with the real part ofu positive. This formula can be derived as follows:

fexp [- ut2 + vt Jdl

-exp [L]" exp[ u[ d]]

Letting Z=u(t-v/2u),

f:exp (-ut2 + vt)dt

1 11rXr2 V2exP exp(Z2)dZ = -1-exp-

\/u 4u _00 u 4u

Note that the path of integration in the Z-plane isnot along the real axis, but along a line which may beoblique to the real axis. From the requirement that thereal part of u be positive, it can be shown that the pathof integration in the Z-plane makes no more than a 45-

954 J-lne

I

D.0

025 J .IU

1954 Batten, Jorgensen, Macnee, and Peterson: Response of a Panoramic Receiiver to CW and Pulse Signals 955

i-r1r IIIrt Ir IDIFFERENTIALANALYZER DATA |

THEORETICAL {

I CIRCUIT2 CIRCUITS4 CIRCUITS

I CIRCUIT

GAUSSIAN

0 -4

0.2 - _0.1 10.0 70.0

degree angle with the real axis. With this restriction theintegral - r+

exp ( Z2)dZ00

is independent of the angle of the path and thus equalto -/7r, which is given by integration along the real axis.Now the calculation of g(t) can be carried out. Before

use is made of (14), h(t) must be calculated.1

h(t) = -JHC(v) exp (jwt)dw-\//27r _0

f0 (L a)2=2 exp wlo - b dw1 r- a2] r [22Cexp exp L + 'tt+ 2) dw

which yields, after application of (15) and simplification,b r b2t2-

h(t) = exp jat - (16)

The -expressions for f(t) and h(t) can now be substi-tuted in (14):

rtg(t) = f(X) h(t-X)dX

_00

1 rX~~ rSX2 . XCs)2-=--_fJ exp [1 2+jaX (2]

b r-b2-7 exp L (t- X)2+ja(t-X)j dX

b c2 b2t2 1- exp -----+jatI

2V\Ir d2 4

. Vsexp[_ /l22 +b _js\s /2c b\1Iexp IX21-+---I+(+-I dX

\d 4 2 \d 2

and using (15) again;

b9(t)=

b I

4

As has been remarked, the required answer is the abso-lute value of g(t), which can be obtained by taking theabsolute value of the first factor and keeping only thereal part of the exponent.

g(t) I= b

[(1 b2 2 s2- 1/4

.exp

r2c b21 2r 1 b2-

c2 b2t2 {d2 2 d}242 4

1.2

A

1.0S

Fig. 12-Relative amplitude for a cw input signal.

Lu -

J --

0.8

-0.02

PROCEEDINGS OF THE I-R-E

The exponent of (18) can be put into the followingform:

b2 E ++b2±+s2d2] [c -+b21 1Fd2(+b +s1d2+

d2 [ +b 2 +4S2 b[+ 2 + s2d2] j

s2c2

4+ +s2d2J

(19)

Referring to the definitions of A, W, and B, and re-calling that in the Gaussian case the width of a curveis taken to the e-14 points, it is clear that

bA0 = , (20)

4 G 1/ i- +b2 + 4s2

1 /4B = b +b2+ s2d2,

and

sd d2sdd/(12+6b2)2+ 4s2 -W =-/=_b2 4- 2 + s2 2 b Ao2B

d2

(21)

(22)

The time of maximum response is given by

4

d2 1tm = C | -

L- + b2 + s2d2

Now g(t) can be written

g(t) I=AOexp { [ b[]B[E]}

(23)

(24)

For a cw input the signal is

exp j -+ at],

and the output can be obtained from (18) by taking thelimit as d approaches infinity.

6 b2s2Tlim Ig\,j = -exp 4s2fd- Ico ( ) I [b4 + 4S2]1[4 exp {-b-4 + 4] 2}

In the notation of (20) to (24),

b

[b4 + 4s2]1/41 1

W = b4+ 4S2 =--62 o

and

lim g(t) = Ao exp {- -2[]s}

(18a)

(20a)

(22a)

(24a)

Mutual Coupling Considerations in

Linear-Slot Array Design*M. J. EHRLICH,t SENIOR MEMBER, I.R.E., AND JOANN SHORTt

Summary--A study was made of the mutual coupling between tworesonant waveguide fed slots on a finite ground plane. The size of theground plane and the relative spacing of the slots were such that thegeometry corresponded to that of a pair of adjacent longitudinal shuntslots on the broad face of a rectangular waveguide. A null bridgemethod of measurement was used to determine accurately the rela-tive field strength excited in the second waveguide by mutualcoupling between the slots when the generator was applied to thefirst waveguide. The change in input slot admittance of the drivenslot arising from the presence of the parasitic slot was also measuredfor matched terminations of the ends of the parasitic slot waveguide.The changes in slot input admittance and excitation arising frommutual coupling were determined. It is shown that the changes maybe neglected in the design of the great majority of linear-slot arrays.

* Decimal classification: R1 18.l X>R142. Original maniuscript re-ceived by the IRE, May 18, 1953; revised manuscript received,October 15, 1953.

This technical memorandum is one of a series of reports which willprovide final information available on Project 532-A.

t Hughes Aircraft Co., Research & Development Labs., CulverCity, Calif.

INTRODUCTIONTVHE DESIGN of linear-slot arrays whose radiat-

ing elements are shunt slots cut on the broad faceof a rectangular waveguide is usually executed

without consideration of the effects of mutual couplingbetween the elements.' The mutual coupling is definedas that coupling which occurs in the free space exteriorto the waveguide. The magnitude of this coupling andsubsequently the evaluation of the validity of its neglectin design have been determined in this study.

NATURE OF THE EXPERIMENTThe particular external slot geometry that is con-

sidered in this experiment corresponds to two adjacentelements of a longitudinal shunt-slot array on the broad

I W. H. Watson, "Waveguide Transmission alid Antenna Sys-tems," Clarendon Press, Oxford, England, chap. 8; 1947.

956 June