IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

Analysis of a Detection Technique for a Spin-

ning-Goniometer Direction-Finding System*

C. E. LINDAHLt, MEMBER, IEEE, AND B. F. BARTONt, MEMBER, IEEE

Summary-Fourier analysis is used to show that the frequencyspectrum of the output voltage of the linear detector of a spinning-goniometer direction-finding receiver varies as a function of timefor a CW received signal. A filter for the second detector is consideredhaving a bandwidth which varies with time in a similar way, thuseffecting a "match" between the signal and the filter. A simplesignal-to-noise ratio analysis shows that an improvement in signal-to-noise ratio is possible, compared to performance of a time-invariant system. The improvement is described by the equationI = 10 log,0 TIT,, where T is the observation time and T, is the basicperiod of the envelope of the input signal to the second detectorproduced by goniometer modulation.

INTRODUCTION

7lf HE CHARACTERISTICS of the receiver sectionof a spinning-goniometer direction-finding systemstrongly influence system performance. Previous

material',2 has dealt with the design of the receiver.Here, certain results are presented bearing on the designof the second detector. Apart from other possible ap-plications, these results are contributory to an over-allreceiver design specification.The detection problem is onie of extractinig a par-

ticular signal from corrupting noise after linear detec-tioIn of the IF signal. A complication is that while themodulation resulting from goniometer rotation is knowna great variety of superposed modulations of the re-ceived signal may be encountered. There is, further, therestriction that only a finite amount of time is availablefor acquisition aild processing. Under these conditions,one wishes to specify a filter which would separate thesignal from the noise in some "optimumn" way. Thisbrings up the very difficult question as to what criterionof excellence should be chosen in such a filter op-timization. Even the formulation of an answer on thebasis of a statistical theory is complicated because thestatistics of the noise and of a specific set of target sig-nals which may be encountered at a given installationare not likely to be known a priori. Further, the effectof preceding receiver sections on a variety of receivedsignals makes it difficult, if not impossible, to specifyusefully the statistics of the detector input signal.

* Received March 23, 1963. This work was supported by the U. S.Army Materiel Command and by the U. S. Army Signal Corps.

t Cooley Electronics Laboratory, The University of Michigan,Ann Arbor, Mich.

I S. F. George, "Direction Finder Bandwidth Requirements,"Naval Research Laboratory, Washington, D. C., Rept. No. R-3182;October, 1947.

2 H. Busignies and M. Dishal, "Some relations between speed ofindication, bandwidth, and signal-to-random-noise ratio in radionavigation and direction finding," PROC. IRE, vol. 37, pp. 478-488;May, 1949.

As an initial step towards a solution, we uindertakehere a simplified problem under the assumption that thereceived signal is a CW signal. The output of the lineardetector, in the absence of any corrupting noise, thenhas the form shown in Fig. 1 as a result of goniometeraction. Note the assumption of a finite number k ofperiods of signal cycles which may result either from alimit on the duration of the monitored signal or of theobservation time in automatic operation. If the outputwere not truncated, it would consist of a wave whichhad the same frequency components as a full-wave-rectified sinusoidal voltage of period T1.3 In the follow-ing, the effect of the finite observation time T=kTj onthe frequency spectrum of the idealized detector outputsignal is conisidered. A commendable filter operationfor this signal in the presence of noise and possible filterrealizations are then discussed.

k T

-Vm |SIN (Wrt)|-vmI|SIN ( 2 t)lFig. 1 Output of linear detector with a received CWV signal.

DETERMINATION OF THE FREQUENCY SPECTRUM OF THEABOVE "IDEALIZED" OUTPUT SIGNAL4

A Fourier integral analysis is used here to determinethe frequency spectrum of the signal V(t) of Fig. 1.For convenience, time is referenced from the midpointof signal duration. The frequency spectrum is given by

(1)

3 Note that T, = Tr/2, where Tr is the rotation period of thegoniometer.

4 The analysis closely follows that given in S. J. Mason and H. J.Zimmerman, "Electronic Circuits, Signals, and Systems," JohnWiley and Sons, Inc., New York, N. Y.; 1960. See especially Section6.19.

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00

v (w) == V(t),e-jxldt.00

kTi2

t

Lindahl and Barton: Detection for Spinning-Goniometer Direction Finder

By assumption, V(t) =0 for It| > (kT1/2), and thelimits of (1) may be replaced to yield

(2)

where k = the odd number of periods of the voltagewave.5 Note that for a V(t) corresponding to k = 1, oneobtains a function

where w,=r/T,=w,1/2. Substituting into (3) and eval-uating the integral, one obtains

4Vm 2 "Ti= T1Vc = 01 + 2o sin-2

4w 2 CO 12 2 (9)

Hence, the frequency spectrum of k periods of the out-put voltage wave is given by

T1/2'v 1(c) = V(t),E-iwldt.-TI/2

(3)

This expression for the frequency spectrum of oneperiod of the wave looks very much like the integralobtained when one finds the coefficients for the Fourierseries pertaining to a function of period T1. This ob-servation suggests the convenience of introducing afunction Ve, in the analysis defined as

1 rT1/2v = (3 V(t)E-Ttdi. (4)

T1J T1/2

Comparing (3) and (4), one obtains the useful relation

Vi(co) = T1V,,.

1 +2!-{sin i(r

Vk(QO) = 4(-)

isin[ r-.~~ --W-

sInL C- I01

(10)

It is of interest now to examine the character of (10) asa function of the number of periods k of the outputwave. As k becomes large the frequency spectrum be-comes concentrated around the frequencies no.i, wheren=0, + 1, ±2, - * * as shown in Fig. 2 for k=3, 5, and

(5)

Now, note that the wave shown in Fig. 1 may be re-garded as a superposition of a number of waves iden-tical to V1, except that their time origins are shifted tothe right or left by appropriate time intervals. (Theeffect, in the frequency domain, of translating the timeorigin of V(t) in (3) to the right or left by an amount+to is simply to multiply Vj(co) by EJiwto.) Superposi-tion may also be used directly in the frequency domaindue to the linearity of the Fourier integral transforma-tions being considered. Hence, VIk(O) in (2) can bewritten in the form

m

Vk(Wt) = TIV. E E-jiiw Tii=-m

(6)

where m = (k-1)/2. Further, as in Mason and Zimmer-man4 the sum in (6) can be put in the form

sin [k xT]

Vk(c) = T1VW TI

sin2

(7)

T1V, = V1(co) given by (3) is readily evaluated by as-suming the maximum value of V(t) is Vm, so that

V(t) = Vm, sin co.t,

= - Vm sin cw,t,

T1--< t < O

2

T1O< t<-

(a)

(b)Fig. 2 Frequency dependence of (a) Vi(w).

sin k 7r-)(b)

sin )1CO1I

(8)

6 The analysis is facilitated by specification of odd k.

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k T112

Vk (CIO) = V (t) E-'w Idt 2

T112

IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

7.6 In the limit, as k-- , it can be shown that "theFourier transform of the periodic signal consists of im-pulses located at the harmonic frequencies ncol of thesignal and that the area of each impulse is the same asthe value of the corresponding coefficient in the Fourierseries of the periodic signal."5From the above analysis, it is seen that when there is

only a finite observation time T available (as will alwaysbe the case), the frequency spectrum is continuallychanging. Hence, in order to detect the signal in an"optimum" manner, the characteristics of the filter fol-lowing the linear detector should be varied throughoutthe duration of the observation interval in such a waythat its pass bands become concentrated at the har-monic frequencies of the signal. In the followiing section,the accomplishment of this with a liniear network isconsidered.

FILTER CHARACTERISTICS AND METHODOF IMPLEMENTATION

The filter which has the time-varying properties de-scribed above consists merely of a linear device whichadds together samples of the output of the linear de-tector, each added sample having duration T1. (Recallthat T1 is the basic period of the truncated periodicwave of interest.) Specifically, the unit impulse responseof such a filter is shown in Fig. 3. Mathematically, then,it can be described by

is exactly equal to the magnitude of the second factorof (10).7 As k increases the width of the pass bands ofthe proposed filter decreases in exactly the same fashionas the frequency spectrum given by (10) tends to con-centrate around the harmonic frequencies of the inputvoltage waveform. This is precisely the type of behaviorthat was desired.The realization of such a filter is relatively straight-

forward. One such realization is a delay line with tapsat t = 0, T1, 2 T1, , (k-1) T1. The taps are con-nected in parallel, so that the output is the sum of allvoltages appearing at the taps. This is shown schemat-ically in Fig. 4. The length of the delay line will be de-termined by the greatest desired observation time T.Another realization involves the use of a magnetic

recording drum with one record head and (k -1) pickupheads suitably positioned along the periphery of thedrum to give the proper time delay. The output voltagesfrom the pickup heads are added to form the output.The drum must be synchronized with the rotation rateof the signal goniometer; in fact, a simple solutionwould be to drive it directly with the goniometer driveshaft. Such a realization is shown in Fig. 5.

h(t) = (t) + (t - T1) + a(t - 2T1) + *

+ 6[t - (k - 1)TJ].Fig. 3 Unit impulse response of filter,

(1 1)

Conisider the Fourier transform of (11)0x

H(w) = j h(t)e-witd= 1 + E--T1 + E-12w Ti+

+ e- j(k-1)w T

l - e-ikco Ti

1 e-iwTi

sin k T2

sin2

(k- i)T,

Fig. 4-Delay line realization of filter.

wr, GONIOMETER SPIN RATE (RAD/SEC)

This equation is extremely interestinig in the light of theprevious results. In particular, the magnitude of (12)

sin [k --TlH(w) = L 2

(13)

sin

2

6 In terms of the goniometer modulation frequency, as k becomeslarge the frequency spectrum becomes concentrated around the fre-quencies 2nw,.

Fig. 5-Magnetic drum realization of filter.

I Note again that T1= Tr/2, in terms of the goniometer rotationperiod.

RECORDHEAD

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Lindahl and Barton: Detection for Spinning-Goniometer Direction Finder

At this point, the commutated filter analyzed inanother paper8'9 should be mentioned. It, too, has theproperty that the width of its pass bands decreases asa function of time. However, its losses must be strin-gently limited if the required integration time is to berealized. Further, it has the undesirable characteristicof introducing extraneous frequency components. Thisresults from the fact that it is a time-varying network.For this reason, the above-mentioned filters prospec-tively offer performance superior to that of the com-mutated filter.

Thus V(t) contributes nothing to the filter outputsignal variance. The variance oS, of the sum of k sam-ples is therefore given by

,S2 = kowN2. (14)

The rms value of the filter output noise after the kthcontribution is then V\k-2. Let (SIN)o be the outputsignal-to-noise ratio in db and Vrms be the rms value ofthe input signal. The rms value at the filter output dueto the presence of the input component V(t) is k rmsafter k contributions as a result of the linear com-bination. Therefore,

SIGNAL-TO-NOISE RATIO CONSIDERATIONS10

Assume that the input signalfi(t) to the filter consistsof the sum of the idealized signal V(t), having k periodseach of duration T1 seconds, and a random noise com-ponent N(t) with zero mean and variance 0vN2= N2(t).The filters described above are nothing more thanperiodic samplers which add together samples of thesignal every T1 seconds. The initerval T1 is long enoughto ensure that the noise samples are statistically inde-pendent. One recalls that the individual (envelope)samples of V(t) are identical; the succession of con-tributions are thus completely dependent and additive.

8 R. Fischl, "Analysis of a Commutated Network for a Spinning-Goniometer Radio Direction Finder," Cooley Electronics Lab., TheUniversity of Michigan, Ann Arbor, Tech. Rept. No. 131; March,1962.

9 R. Fischl, "Analysis of a Commutated Network," this issue,p. 114.

10 This analysis follows that of Y. XV. Lee, "Statistical Theory ofCommunication," John Wiley and Sons, Inc., New York, N. Y., pp.312-319; 1960.

(SIN)0 = 10 logio [,0N1 Vr2].

The input signal-to-noise ratio (SIN) is given by

Vrms2 -

(SIN)i = l0logio 2L _

(15)

(16)

An improvement I in signal-to-noise ratio (in db) canthen be defined by

I = (SIN)o- (S/N)= 10 logio k. (17)

Eq. (17) states that the improvement in signal-to-noiseratio under the assumed conditions is directly propor-tional to the logarithm of the number of samples taken.Since T=kT,=the length of the observation interval,(17) can be put in the form

TI = 10 login - (18)

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