Contains reports on three completed research projects and reports on two current research projects.Joint Services Electronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DA 36-039-AMC-03200(E)National Aeronautics and Space Administration (Grant NsG-496)National Science Foundation (Grant GK-835)National Aeronautics and Space Administration Grant (NsG-334

Bauer, R. F.

Bose, A. G.

Bruce, J. D.

Nelsen, D. E.

Oppenheim, Alan V.

DSpace@MIT

COMMUNICATION SCIENCESANDENGINEERINGXI. STATISTICAL COMMUNICATION THEORYAcademic and Research StaffProf. Y. W. LeeProf. A. G. BoseProf. H. L. Van TreestM. E. Austin tA. B. BaggeroertR. F. BauerL. D. CollinstD. A. FeldmanProf. J. D. BruceProf. A. V. OppenheimGraduate StudentsT. HuangJ. F. KososkiA. PitegoffL. R. PouloA. E. RollandR. W. SchaferProf. D. E. NelsenProf. D. L. SnyderJ. J. WawzonekJ. E. SchindallJ. C. StaffordJ. L. WalkerC. J. WeinsteinP. D. WolfeA. WORK COMPLETED1. CLASS-D AMPLIFICATION OF RADIOFREQUENCY AMPLITUDE-MODULATEDSINE WAVESThis study has been completed by D. A. Feldman. It was submitted as a thesis inpartial fulfillment of the requirements for the Degree of Master of Science, Departmentof Electrical Engineering, M. I. T. , August, 1966.A. G. Bose2. SPECTRUM ANALYSIS OF THE LOGARITHM OF A FUNCTIONThis study has been completed by J. F. Kososki. It was submitted as a thesis inpartial fulfillment of the requirements for the Degree of Master of Science, Departmentof Electrical Engineering, M. I. T. , August, 1966.A. V. Oppenheim3. AN INVESTIGATION OF TIME JITTER IN NEON BULB THRESHOLD-CROSSING DETECTORSThis study has been completed by J. C. Stafford. It was submitted as a thesis inpartial fulfillment of the requirements for the Degree of Master of Science, Departmentof Electrical Engineering, M. I. T. , August, 1966.D. E. NelsenThis work was supported by the Joint Services Electronics Programs (U. S. ArmyU. S. Navy, and U. S. Air Force) under Contract DA 36-039-AMC-03200(E), the NationalAeronautics and Space Administration (Grant NsG-496), and the National Science Foun-dation (Grant GK-835).tThis work is supported by the National Aeronautics and Space AdministrationGrant (NsG-334).QPR No. 83(XI. STATISTICAL COMMUNICATION THEORY)B. OPTIMUM QUANTIZATION OF A TWO-LEVEL SIGNAL AFTERCONTAMINATION BY NOISEOne of the important problems of statistical decision theory is the binary detectionproblem; that is, the problem of deciding after reception which of two possible signalshas been transmitted over a noisy channel. In this paper we shall approach this prob-lem from a point of view different from that traditionally taken.We begin by considering the message signal s(t) which takes on values s I and s2 withprobabilities q and (1-q), respectively. For convenience we assume throughout the paperthat s I < s 2 . s(t) is contaminated by an additive, independent noise signal n(t). Thisresults in a received signal x(t),x(t) = s(t) + n(t). (1)The amplitude probability density of the noise signal n(t) is pn(p ) . Our problem is todetermine the two-level quantizer which, when operating on x(t), yields a signal y(t),y(t) = Q[x(t)], (2)that minimizes an appropriate mean value of the quantizer error signal, subject to theconstraint that the quantizer output signal can assume only the values s 1 and s 2 . Thatis, an appropriate mean value of the quantizer error signal will be minimized subjectto the constraint that the quantizer's representation values are constrained to be s 1 ands2 'Mathematically, this problem can be formulated in the following manner. We desirethat the quantized signal Q[x(t)] be an instantaneous replica of the message portion s(t)of the quantizer-input signal. Generally, we demand more than the quantizer can accom-plish. There will be an error,e(t) = s(t) - Q[x(t)]. (3)We shall take an appropriate mean value of e(t) as a measure of how well the quantizerperforms with respect to the demands. This measure of the error is given by= dY dq {g[q-Q( )] px s(, 1) } .  (4)Px, s( , r) is the joint amplitude probability density of the quantizer-input signal x andthe message signal s (which is also the desired output signal). From our statement ofthe problem px, s (  l ) isPx, s( , ) = [q P n(-sl)] uoh(-s 1)+ [(1-q) " p n(-s2)] uo(1-s2). (5)QPR No. 83(XI. STATISTICAL COMMUNICATION THEORY)uo(t) is the unit-impulse function. g[Ti-Q(g)] is a function of the error that we call theerror-weighting function. Q( ) is the quantizer input-output characteristic and is definedin the following way:s I  ( < xlQ() = (6)s 2  ( > x I .Substituting Eqs. (5) and (6) into (4) we have for the measure of the error= d drn {g(i-s 1 )[q P(sl ) u (-s1) + (-q) * pn(-s 2 ) uo(-s2)]}co oo+ di dn {g(ln-s2)[ qpn(_) uo(l-sl) + (1-q) Pn(-2) Uo(n-sZ)]} (7)x 1  - aor, upon integration with respect to rl,6= q g(0) 0' Pn(a-sl) d + (1-q) • g(s 2 -s 1  Pn(a-s2) d+ q . g(s 1 -s 2 ) pn(-sl) d + (1-q) - g(o) pn( -sz) d . (8)-o o x 1Referring to Eq. (8) we observe that the error 6' is a function of only one variable, x1 .We find the value of s 1 which minimizes g by taking the first partial derivative of Swith respect to x 1 and equating it to zero. Doing this we have0 = q g(0) Pn(X1-Sl) + (1-q) - g(s 2 -s 1 ) Pn(X-S) - q. g(sl-s 2) Pn(x1-S) - (1-q) - g(O) Pn(X1-S 2 ) (9)or, after algebraic manipulation,q g(sl-s 2 ) - g(O) Pn(Xl-S 2) (10)q g(sZsl ) - g(0) Pn(X1-1)The value of xl which satisfies (10) minimizes the error 6. It should be observed thatin most cases of interest the error-weighting function will satisfy the conditiong(e) = g(-e). (11)Applying this condition to (10) we haveQPR No. 83(XI. STATISTICAL COMMUNICATION THEORY)q Pn(x 1 -s 2 )1 -q (12)p (x1-s)which is independent of the error-weighting function.We now want to compare this result to that obtained using one of the traditional detec-tion theory approaches. The approach which we select for comparison is that of theideal observer. This point of view maximizes the probability of a correct decision. Thex1 which is optimum from the ideal observer point of view satisfies the equation (see,for example, Wainstein and Zubakov I)px(_ s2) Pr(s 2)= (13)Px(I sl )  Pr(s1)=x 1For this particular problem (13) becomesPn(x1-s 2) qPn(xlSz2) - 1 - qwhich is identical to (12). Therefore, in this case the detection theory point of viewyields a result which is identical to the result obtained by applying optimum quantizationtechniques.Consider the following example which indicates how (12) might be solved for xl. Weassume that the noise signal is Gaussian with mean n and mean-- square valuen" Then,p (p ) = exp (15)(2rr n) 1/2 nUpon substitution of (15) in (12) we have11/2 [(X1-is]2 2nq (2r n ) 1/2(16)(1-q) 1 exp_1 _exp -(27r a n) 1/2 2 nAfter algebraic manipulation this becomesQPR No. 83(XI. STATISTICAL COMMUNICATION THEORY)[q s1 +s2x- n +- + n. (17)Ss 2 - s1 -q 2which is the desired result.J. D. BruceReferences1. L. A. Wainstein and V. D. Zubakov, Extraction of Signals From Noise (Prentice-Hall, Inc. , New York, 1962), pp. 150-153.C. NOISE IN MAGNETIC RECORDING SYSTEMS CAUSED BY THERECORDING TAPE1. IntroductionThis report summarizes the results of a thesis in which we studied the cause ofnoise in magnetic recording systems that is due to the recording tape.Previous investigations have indicated that there are two types of noise which aredue to the tape: background noise which limits the smallest signal that can be recorded,and noise that amplitude-modulates the recorded signal and thus broadens its spectrum.This modulation noise, as it is called, is also evidenced by an increase in the noise fromthe recording system when the tape is uniformly magnetized. It has been felt that thesetwo noise terms are both due to randomness in the magnetic properties of the tape, butno generally accepted and verified theory exists for relating the noise to the physicalproperties of the tape.2. Statement of the ProblemThe purpose of this study was to relate the observed noise to the physical propertiesof the recording tape in order to determine its cause.The procedure was to derive a power density spectrum for the noise flux in thereproduce head in terms of a model incorporating all possible random properties of thetape, which would account for the experimentally observed increase in the noise withuniform tape magnetization. This power density spectrum was then checked throughfurther experiments. These experiments verified the model and made it possible todetermine the cause of the noise.3. Power Density Spectrum of the Flux through the Reproduce Head Because of theMagnetization of the TapeThe relation between the flux through the reproduce head and the magnetization of thetape l , 2 is given byQPR No. 83(XI. STATISTICAL COMMUNICATION THEORY)= H. M dv, (1)where M is the magnetic moment density of the tape, H is the field set up by the repro-duce head for a unit current in the head coil, and v is the volume of the tape.The magnetic coating of the tape is not continuous, however, but is an assembly ofsmall magnetic particles dispersed in a binder that is coated on a plastic backing. 3The flux relation can thus be written4(x) = W M i * H(x-x i , yi )  (2)iunder the assumptions that the field function can be adequately represented in just twodimensions,2 and the variation of the field over the volume of a particle is negligible.W is the width of the tape.The autocorrelation function of the head flux in terms of the random properties ofthe magnetic particles1 is2 T k TRkksinh 2 (k) sinh (k)2 b k2 F(k) 2 e-2k(d+T/2) E[m 2](k0 T Tk--) cosh (k2) - 100+ 2NE [cos 0] E[imli] Z E[s si+n] Re MD. (k) . cos (kvT), (3)n= 1 i, 1+nwhere F(k) is the two-dimensional transform of the magnetic scalar potential of the head,d is the spacing between the head and tape, T is the thickness of the magnetic coatingof the tape, N is the average number of particles per unit length of the tape, m is theparticle magnetic dipole moment, 0 is the angle from the x axis of the projection of theparticle on the x-y plane, s is the sign of the particle, MD. (k) is the characteristic.th i, i+nfunction of the random distance between the i t h and the (i+n)t h particles, and vis the tape velocity.The power density spectrum of the flux is2 2 -2 (d+T/2) sinh 2 (L E[s2]( T )inhS (W) = W F() e v2 Emcosh -+ 2NE 2 [cos ] E 2 [Iml] E[sisi+n] Re MD. . (4)n= 1QPR No. 83 100(XI. STATISTICAL COMMUNICATION THEORY)Correcting this equation for the transfer characteristic of the head gives the powerdensity spectrum of the magnetization:sinh E2[ Iml ]SM( ) = NE[m 2 ]  + 2E2[cos O] 2M EOT E [mcosh ( ) - E[m1E[sis i+n] Re MD., .i( (5)n: s i, i+nn=1This power density spectrum can be rewrittensinh _H T 2 00SM(W) = E[m 2 ]  ) + 2E2[cos 0] (2 - Re MD.v4T E[m I D.I, i ,+ncosh - 1 n=c os+ 8E[cos] E 2[Im ]  -P (n) Re MD , (6)E[m ] n=, i+nwhere P = p(si=+l), P_ = P(si=-l), and P(n) = p(s =+1 s =-1). Note also that+ i 1 i- i +nP+ - P_ is proportional to the magnetization of the tape.Examining Eq. 6, we see that the first term is a noise term independent of the mag-netization, the second term contains the signal and can also contain a noise term that isdependent on the magnetization, and the third term is also a noise term dependent onmagnetization. These two noise terms, which change with magnetization, are causedin two different ways: the first is due to the distribution of the magnetic particles, andthe second to interaction between the particles.4. Experimental ResultsWe shall now present the experimental results that allow us to determine the causeof the modulation noise. In Fig. XI-1 we show the power density spectra of the flux forvarious levels of magnetization from the erased state to the saturated state. Thesecurves show the manner in which the power density increases with magnetization.To determine the cause of the noise that increases with magnetization, we correctthese spectra for the transfer function of the head and compare them with the theoreticalspectrum from Eq. 6. We note that if the modulation noise is caused by the particledistribution alone, the shape of the spectrum in the erased state isQPR No. 83 1011 .4 amp,0.675 amp8 12 16 20 24 28 32FREQUENCY (kc/sec)Power density spectrum of noise through the reproduce headas a function of recording current. Tape speed: 7. 5 ips.DOTS : NOISE FROM THE SATURATED TAPE COMPENSATEDFOR THE TRANSFER FUNCTION OF THE HEADLINE : THEORETICAL SPECTRUM(a)DOTS : NOISE FROM THE ERASED TAPE COMPENSATEDFOR THE TRANSFER FUNCTION OF THE HEADLINE : THEORETICAL SPECTRUM0 4 8 12 16 20 24 28 32(b)FREQUENCY ( kc/sec)Comparison of measured power density spectra in the(a) saturated and (b) erased states with f(w).Fig. XI-1.30201020100Fig. XI-2.QPR No. 83 102(XI. STATISTICAL COMMUNICATION THEORY)sinh (V T )f(w) ( ) h 1' (7)cosh - 1(v 2while if the modulation noise is caused by particle interaction alone the shape of the spec-trum in the saturated state will be f(w).In Fig. XI-2 we compare the measured power density spectra in the erased andsaturated states with f(w). This shows that the measured noise spectrum from the erasedtape most closely matches f(w). This leads us to the conclusion that the modulation noiseis caused by the manner in which the particles are distributed on the tape.The spectral shape of the noise, which increases with magnetization, is determinedby subtracting the noise that is due to the erased tape from the other spectra. Theresulting power density spectra for various values of magnetization, corrected for the1.4 amp20200Fig. XI-3.DOTS : MEASURED MODULATION NOISE0 00.675 amp0.45 amp4 8 12 16 20 24 28 32FREQUENCY (kc/sec )Power density spectra for various values of magnetizationcorrected for the transfer function of the head.transfer function of the head, are shown in Fig. XI-3. These plots indicate that themodulation-noise term has a relatively flat spectrum in the frequency range 0-30 kc.5. ConclusionsWe have derived the power density spectrum for the flux through the reproduce headof a magnetic recorder in terms of the random properties of the magnetic particles inQPR No. 83- *103(XI. STATISTICAL COMMUNICATION THEORY)the recording tape. This power density spectrum has two noise terms that are a functionof tape magnetization and one noise term independent of the magnetic state of the tape.Comparison of the theoretical spectrum with measured spectra shows that the noisewhich increases with magnetization is caused by the distribution of particles, not byinteraction between the particles. This comparison also indicates that the spectrum ofthe noise which increases with magnetization is flat in the frequency range 0-30 kc.R. F. BauerReferences1. R. F. Bauer, "A Theoretical and Experimental Study of Noise from MagneticRecording Tape," Ph. D. Thesis, Department of Electrical Engineering, M. I. T. ,September 1966.2. G. J. Fan, "A Study of the Playback Process of a Magnetic Ring Head," IBM J. Res.Develop., Vol. 5, pp. 321-325, October 1961.3. C. D. Mee, The Physics of Magnetic Recording, John Wiley and Sons, Inc.,New York, 1964).QPR No. 83 104

Statistical Communication Theory

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Statistical Communication Theory

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