Frequency modulation in video tape recorders

3916 IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998

Frequency Modulation in Video Tape RecordersJohn C. Mallinson,Fellow, IEEE

Abstract—Essentially all consumer video tape recorders usefrequency modulation for the luminance component of the videosignal and demodulation is performed by zero-crossing detection.Distortion-free video requires that the zero crossings of thefrequency modulation waveforms be preserved exactly. In thispaper, the complete set of mathematical conditions for zero-crossing preservation is derived from first principles. While someof these conditions have been discussed previously, the completeset derived here is believed to be novel.

Index Terms—Equalizers, linear distortion, magnetic tape re-cording, nonlinear distortion, video recording.

I. INTRODUCTION

OVER 50 10 consumer video tape recorders aremanufactured annually and virtually all use frequency

modulation (FM) for at least some part of the compositevideo signal. The frequency demodulation is achieved by hardlimiting, zero-crossing detection, pulse forming, and low passfiltering. It is, therefore, of cardinal importance that the posi-tions of the zero crossings of the FM waveform be preservedor maintained after the magnetic writing, reading, and post-equalization processes have occurred, otherwise distortionsmay appear in the demodulated video signal.

Notwithstanding the widespread application of FM in con-sumer video tape recorders, there is to be found in theliterature no mathematically correct formulation of the preciseconditions necessary to preserve the zero crossings. For exam-ple, the frequency domain transfer function of all physicallyrealizable, linear systems must have an amplitude frequencyresponse which is an even function of frequency, sothat . Despite this, the two principal publisheddiscussions on zero-crossing preservation in FM consider onlynonrealizable amplitude responses with odd symmetry. Thus inCherry and Rivlin [1], [2], the impossible amplitude response

is studied. Similarly in Felix and Walsh[3], the nonphysical amplitude response isconsidered.

Perhaps because these analyzes consider only nonphysicallyrealizable amplitude spectra, there has arisen a certain confu-sion about the role of negative frequency components of theFM waveform. Some of these components, which appear asfolded sidebands on a spectrum analyzer, can cause the zerocrossings positions to be moved and thus cause distortion inthe demodulated video.

In this paper, the problem is treated anew with proper regardto the properties of physically realizable systems. Descriptionsof FM signals in both the time and frequency domains are

Manuscript received March 24, 1998; revised July 20, 1998.The author is with Mallinson Magnetics, Inc., Belmont, CA 94002 USA.Publisher Item Identifier S 0018-9464(98)08251-X.

given. The origin of the folded sidebands is discussed and itis pointed out that, because magnetic recording systems areband-pass systems which lack both low and high frequencyresponse, folded sidebands must inevitably introduce distortionin video tape recorders. Furthermore, since a band-pass systemcannot have high frequency response above some limit, itfollows that missing upper sidebands must also inevitablycause video distortion.

The question of preserving the zero crossings is consid-ered here for several linear systems of interest in magneticrecorders. It is shown that a differentiator does not preservezero crossings. On the other hand, a differentiator followed bya phase corrector does preserve zero-crossing providedthere are negligible folded sidebands and that the instanta-neous frequency remains always positive. Finally, the so-calledstraight line equalization used in all FM video recorders istreated. It is shown that now there are three separate criteriawhich must be met in order to preserve zero crossings. Theyare: that there be negligible folded lower sidebands, that theupper sidebands above the cut-off frequency be negligible,and that the instantaneous frequency always remains belowthe cut-off frequency.

In this paper, extensive use will be made of several typesof spectra. They are the complex spectrum, the amplitudespectrum, and the phase spectrum. If the complex spectrum is

, then the amplitude spectrum isand the phase spectrum is . Although andmay be positive or negative, is always positive. All thespectra are two sided, that is they extend to both positive andnegative frequencies. Because and must be evenand odd functions of frequency, respectively, for physicallyrealizable systems, it follows that the amplitude and phasespectra must also be even and odd functions of frequency,respectively. A spectrum analyzer, of course, shows poweronly, versus positive frequency only.

II. FREQUENCY-MODULATED SIGNALS

When a signal, for example a video signal, is made tochange the phase of a constant amplitude carrier, the processis called angle modulation. Two cases may be distinguished

(1)

and

(2)

where

a phase modulated wave;a frequency modulated wave;

0018–9464/98$10.00 1998 IEEE

MALLINSON: FREQUENCY MODULATION IN VIDEO TAPE RECORDERS 3917

Fig. 1. The amplitude spectrumA(w) of an FM wave.

the amplitude of the angle modulated wave;the carrier frequency;

and proportionality constants;the modulating video signal.

Clearly there is no important difference between phase andfrequency modulation; any difference is simply a matter of thedefinition of the modulating signal. Since in phase modulation,the video signal changes the carrier phase angle directly, thenomenclature is self-evident. The name frequency modulationapparently arises from the mathematical definition that thetime differential of the phase angle represents a frequency.Upon differentiating the argument of (2), we have the so-calledinstantaneous frequency

(3)

However, since frequency can only be accurately defined byobservations over long periods of time, it is understood thatcan hardly be considered a true frequency. Indeed, as is shownbelow, the instantaneous frequency does not even appear inthe spectrum of an FM signal!

Suppose, in the interests of simplicity, that the modulatingsignal is itself a sinusoid

(4)

Equation (2) now becomes

(5)

where the term is called the modulation index. ByFourier analysis, it may be shown that an alternative expressionfor (5) is

(6)

where is a Bessel function of the first kind andth order.Extensive tabulations of this Bessel function exist [4].

Clearly (6) represents the superposition of a large numberof (co)sinusoids. Since the Fourier transform of a cosinehas both positive and negative frequency components, it isobvious that the complex spectrum of an FM wave or signalmust properly include both positive and negative frequencycomponents. Moreover, since when is oddand when is even, the complex spectrumsidebands have both polarities.

In Fig. 1 the amplitude spectrum, which of course does notdisplay the sideband polarities, of an FM wave with smallmodulation index is sketched. It shows the existence of bothpositive and negative frequency side bands. The side bands arespaced at integer multiples of from the carriers at .

As was mentioned above, the instantaneous frequency doesnot appear in this spectrum. For small modulation indices, theamplitude of the sidebands decreases rapidly as the orderincreases. It can be seen that even though the higher ordersidebands become extremely small, strictly speaking the FMwave has an infinitely large bandwidth because the sidebandamplitudes never vanish. Moreover, the lower sidebands ofthe positive frequency carrier must go through dc into negativefrequencies and vice versa for the upper sidebands of the lowercarrier. This phenomenon, which always exists to some degree,is called folding of the sidebands.

It should be noted, at this point, that these folded sidebandsare in no sense spurious or detrimental to the FM signal. Onthe contrary, they are an absolutely necessary and correct partof the spectrum of the FM wave. For the zero crossings ofthis FM wave to be in exactly their correct positions, each andevery folded sideband must be present without attenuation.

On the other hand, when such an FM wave is passed througha channel, such as a magnetic recording channel, which hasno low frequency response, then the zero-crossing positionsgenerally cannot be preserved because some low frequencysidebands, folded or otherwise, will be missing. In ac coupledchannels lacking low frequency and dc response, preservationof zero crossings can only be guaranteed when the amplitudesof both the direct and the folded sidebands at frequencies closeto dc are negligibly small.

As a practical matter, negligibly small might mean thatare less than 1% (40 dB) of the FM wave amplitude . Itshould be noted that the amplitudeis not the amplitude ofthe spectral lines appearing at the carrier frequency in thespectrum. The amplitude is the amplitude of the FM wavein the time domain, as is indicated in (2).

III. T HE MAGNETIC RECORDING CHANNEL

It will be assumed in this paper that when a currentis applied to the writing head, the magnetization

waveform written on the tape preserves the zero-crossingpositions of the current waveform. This implies that a) there isnegligible nonlinear distortion in the writing process and b) thesources of both “avoidable” and “unavoidable” FM distortionare made negligible in the writing process.


All of these distortions are well understood. Non-lineardistortion occurs in the writing process a) at high frequencies,because the magnetic flux rise time in the writing head is tooslow and b) at high linear densities, because the demagnetizingfields from the previous magnetization zero crossing interferewith the writing of the current transition [5], [6]. “Avoidable”distortion is, by definition, due to the less than perfect oper-ation of components in the system, for example, lack of dcbalance of the writing amplifier [3]. “Unavoidable” distortionarises most fundamentally because the writing process acts asa low pass band amplitude limiter. An amplitude limiter ofsufficient bandwidth, of course, passes all higher order oddharmonics and would produce no “unavoidable” distortion.With a pass band amplitude limiter, however, the higher orderharmonics are attenuated or missing, causing “unavoidable”distortion. A major source of “unavoidable” distortion in videotape recorders comes from those lower sidebands of the thirdharmonic of the FM carrier which fall in the pass band of therecorder [3].

Assuming then that the amplitude limited magnetizationwaveform written on tape has the same zero crossings as

, the fundamental of the reading headoutput voltage is, disregarding mere proportionalityfactors

kgkg

(7)

where

the wave-number ;the head-tape relative velocity;the tape magnetic coating thickness;the head-tape spacing;the reading head gap-length.

Equation (7) shows the well-known product of the thicknessloss, the spacing loss, and the gap loss and the fact that the out-put voltage leads in phase, by , the written magnetization[6]. The amplitude frequency response of the magneticrecording channel is that of a band-pass filter which has zeroresponse at dc and at the gap-null frequencies,and the phase response is for and for

; see Fig. 2. At low frequencies, (7) also shows that thesimplest possible model of the magnetic recording channel isjust a differentiator.

In this paper, the behavior of frequency modulated signalspassing through a magnetic recording channel is analyzedfor three cases. First, the recording channel is modeled asa differentiator. Second, it is modeled as a differentiatorfollowed by a phase corrector. Finally, the channel ismodeled with the straight line amplitude response and zerophase response which corresponds to that actually used invideo tape recorders.

IV. CASE 1: A DIFFERENTIATOR

This model of the reading channel is of interest becauseinductive heads produce their output voltage by temporaldifferentiation of the reading head flux . It shouldbe noted that since shielded magnetoresistive heads are spatial

Fig. 2. Amplitude and phase transfer function of a magnetic recordingchannel.

Fig. 3. Amplitude and phase transfer function of a differentiator.

differentiators , their output voltage is similar because, where is the

head-tape relative velocity [7].The amplitude and phase frequency transfer functions of a

differentiator are shown in Fig. 3.Suppose the FM input wave is

(8)


The output of the differentiator is

(9)

and zero crossings evidently occur whenever

(10)

or

(11)

It is clear that, even with the restriction that the instanta-neous frequency, never falls to zero, the zero crossings of

cannot be at the same positions as those of. The zerocrossings of sines and cosines are not identical.

It should be noted, however, that since, the zero crossings of

correspond to those of a different FM input wave,where the phase of the FM carrier

is now lagging in phase by . This different set of zerocrossings would, of course, also be a perfectly satisfactoryset for FM demodulation. In fact, the zero crossings of

where is an arbitrary FM carrier phaseshift, also form a satisfactory set.

V. CASE 2: A DIFFERENTIATOR WITH PHASE CORRECTION

In this case a differentiator, with a complex transfer function, is followed by a phase corrector. A phase

corrector, which is often called a Hilbert transformer, has acomplex transfer function , where when

and when .Again, suppose the FM input waveform is

(12)

The output of the differentiator, Hilbert transformer cascadeis

(13)

where denotes Hilbert transformation.Now, providing the FM input wave has negligible folded

sidebands

(14)

The proof of this statement is shown in Figs. 4 and 5, wherethe upper and lower sidebands are shown as continua ratherthan as discrete spectral lines.

Fig. 4(a) and (b) show the complex spectra of a cosineFM wave with no folded sidebands before and after Hilberttransformation, respectively. Clearly Fig. 4(b) is the complexspectrum of a sine FM wave without folded sidebands.

Fig. 5(a) shows the complex spectrum of a cosine FMwave with folded sidebands. Upon Hilbert transformation thecomplex spectrum shown in Fig. 5(b) appears. Note carefullythat the spectral shape at low frequencies in Fig. 5(b) is notidentical to that of a sine FM wave with folded sidebands,shown in Fig. 5(c).

(a)

(b)

Fig. 4. Complex spectrum of a cosine FM wave (a) before and (b) afterHilbert transformation.

Combining (13) and (14) yields

(15)

and zero crossings now occur whenever

(16)

or

(17)

Providing the instantaneous frequencynever falls to zero,it is clear that and have identical zero crossings.

When there are no folded sidebands and the instantaneousfrequency remains positive, a phase corrected differentia-tor preserves the zero crossings of an FM wave.

VI. CASE 3: A STRAIGHT LINE EQUALIZER WITH ZERO PHASE

The amplitude frequency response of a straight line equal-izer used in all FM video recorders is shown in Fig. 6. Notethat it is an even function of frequency and is, therefore, phys-ically realizable. Moreover, note that the amplitude responseis zero above the cut-off frequency.

Let the amplitude spectrum, above the low frequency null,be

(18a)

(18b)

and the phase response be zero. In a video tape recorder,zero phase is usually achieved by integrating the output, thusintroducing a phase shift to correct the Faraday’s lawdifferentiation phase shift.


(a)

(b)

(c)

Fig. 5. Complex spectrum of (a) a cosine FM wave (with folded sidebands),(b) a cosine FM wave (with folded sidebands) after Hilbert transformation,and (c) a sine FM wave with folded sidebands.

Fig. 6. Amplitude spectrum of the straight line equalization used in all FMvideo recorders.

For an input FM wave , the output,following the preceding analysis, is

(19)

If there are no folded sidebands, so that

(20)

and

(21)

then

(22)

It is now seen that zero crossings of the output occurwhenever

(23)

or

(24)

It follows, therefore, that provided the instantaneous frequencyremains less than the cut-off frequency , the input and

output FM waves have identical zero crossings.Zero crossing preservation with a straight line, zero phase

filter occurs only when two independent conditions are met:a) the lower sidebands must be negligible at dc, and b) theinstantaneous frequency must remain always less than thecut-off frequency.

A practical difficulty arises with the second requirement,that the instantaneous frequency remains below the cut-offfrequency, because the instantaneous frequency itself is notobservable. Of course, mathematically, it is just the derivativeof the phase angle. The practical difficulty, however, is that itdoes not appear in any spectrum.

Accordingly, an alternative way of understanding the behav-ior of the straight-line filter is outlined below. Examination of(6) reveals that the FM wave can be considered to be madeupon of a) a component of amplitude at frequency , b)two components of amplitude at frequencies , c)two components of amplitude at frequencies , andso on. These components can be added as vectors. In such aphasor diagram, the spatial direction of each component vectorcorresponds to its phase angle. It was first shown by Felixand Walsh, that, at the moment of maximum phase deviation[when , see (5)], the resultant total vector, shownin a coordinate system rotating at the carrier frequencyin Fig. 7, is unaffected by the straight-line equalization. Thereason for this is that the straight-line equalization changesthe amplitudes of each pair of upper and lower sidebands byequal and opposite amounts, so that all the sumsremain unchanged. Fig. 7 shows this effect of straight-lineequalization on the first, second, third, and higher ordersidebands.

It is clear that the sums can only remainunchanged as long as a) the upper sidebands remain belowthe cut-off frequency and again b) the lower sidebands remainunfolded. If an upper sideband is above the cut-off frequencyand is, therefore, not passed by the straight-line filter, zero-crossing distortion inevitably occurs.

The set of conditions necessary for zero-crossing preser-vation with the straight-line equalization may, therefore, berestated: a) the lower sidebands must not be folded, b) theupper sidebands must not exceed the cut-off frequency, andc) the instantaneous frequency must remain below the cut-offfrequency.


Fig. 7. Vector, or phasor, diagram of sidebands, shown in a coordinatesystem rotating at the carrier frequency, at the moment of maximum phasedeviation.

VII. D ISCUSSION

The principal work on FM in video recorders is, of course,the landmark, seminal paper by Felix and Walsh. It is, giventhe clarity and wide scope of their studies, curious that theirdiscussion of negative frequencies and folded sidebands leavesquestions unanswered.

In the present paper, the roles of negative frequenciesand folded sidebands emerge naturally as a necessary partof the spectrum of any FM wave. Distortions of the zerocrossings can be caused by appreciable folded sidebands fortwo independent reasons: a) a magnetic recorder is a band-pass channel which lacks low frequency response and b) themagnetic recorder’s amplitude frequency response falls offrapidly at higher frequencies thus forcing the use of a straightline equalization, which only works in the absence of foldedsidebands.

When Felix and Walsh discuss the effect of limiters incausing “unavoidable” distortion on the other hand, the anal-ysis is both succinct and clear. To quote one sentence (p.1661): “Although odd-order distortion of itself does not affectthe cross overs, it does introduce carrier harmonics and theirsidebandswhich must pass undistorted through the system.”Having realized that missing sidebands cause distortion in thiscase, one wonders why they did not simply say that missingfolded sidebands also cause distortions?

Felix and Walsh did not deduce the requirement that theupper sidebands must not exceed the cut-off frequency. Indeed,they state just the contrary (p. 1662), “... there is no restrictionon the position of the sidebands which will be found on bothsides of (the cut-off frequency).” Presumably this error isrelated to the fact that the amplitude response they studied

continues with negative sign at frequenciesabove the cut-off frequency ! However, as hasbeen shown above, by definition, cannot be a negativequantity.

Any missing sidebands, be they lower sidebands passingthrough zero frequency or upper sidebands exceeding the cut-off frequency must inevitably introduce distortion in the zero-crossing positions. Whether these distortions pass the videodemodulation filters and thus cause video signal distortion,however, is beyond the scope of this paper.

It will be realized that almost all of the analysis anddiscussion in this paper on preserving zero crossings couldequally well be applied to binary digital recording. Indeed,after hard limiting the output signal of an FM video recorder,the resultant waveform is indistinguishable from that of aproperly equalized digital recorder’s waveform. Ultimately,both analogue FM video recorders and digital recorders aresimply “timing machines” which mark the times (or positions)of the zero crossings of the written magnetization. In bothcases zero crossing preservation is mandatory.

ACKNOWLEDGMENT

The author is deeply indebted to both J. Miller and R. Woodfor their many helpful suggestions concerning this paper.

REFERENCES

[1] E. C. Cherry and R. S. Rivlin, “Non-linear distortion with particularreference to frequency-modulated waves—Part I,”Philosophical Mag.,vol. 32, no. 213, pp. 265–281, Oct. 1941.

[2] , “Non-linear distortion with particular reference to frequency-modulated waves—Part II,”Philosophical Mag., vol. 33, no. 219, pp.272–293, Apr. 1942.

[3] M. O. Felix and H. Walsh, “F. M. systems of exceptional bandwidth,”in Proc. Inst. Elect. Eng., vol. 112, no. 9, pp. 1659–1668, Sept. 1965.

[4] M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functionswith Formulas, Graphs and Mathematical Tables, 10th ed. New York:Wiley, 1972.

[5] H. Neal Bertram,Theory of Magnetic Recording, Cambridge UniversityPress, 1994.

[6] J. C. Mallinson,Foundations of Magnetic Recording, 2nd ed. NewYork: Academic, 1993.

[7] , Magneto-Resistive Heads: Fundamentals and Applications.New York: Academic, 1995.

John C. Mallinson (SM’74–F’82) was born in Bradford, England, on January30, 1932. He received the M.A. degree in natural philosophy (physics) fromUniversity College, Oxford, England, in 1953 and the D.Sc. degree in 1997.

He joined Amp., Inc., Harrisburg, PA, in 1954 to work on the theory anddesign of all-magnetic logic elements. In 1962 he joined Ampex Corporation,Redwood City, CA, where he held many positions concerned with theunderstanding and development of magnetic recording systems. From 1976 to1978, as Manager of High Bit Rate Recording in the Data Systems Division,he was concerned with the initial design of a 750 Mb/s digital recorder. From1978 to 1984 he supervised the Magnetic Recording Technology Department,a multidisciplinary group working in magnetic recording theory, high-densityhead fabrication, coding and communications theory, and the exploration ofadvanced concepts in various areas of recording. In 1984 he was appointedFounding Director of the Center for Magnetic Recording Research at theUniversity of California, San Diego. In 1990 he resigned his universityposition in order to pursue his research and consulting business interests. He isnow President of Mallinson Magnetics, Inc., Belmont, CA. He has publishedover 70 papers on a wide variety of theoretical topics in magnetic recording.He is the author of two textbooks,The Foundations of Magnetic Recording(New York: Academic, 1993) andMagneto-Resistive Heads: Fundamentalsand Applications(New York: Academic, 1995).

Dr. Mallinson was an IEEE Magnetics Society Distinguished Lecturerin 1983. In 1984 he was awarded the Alexander M. Poniatoff Award for“leadership in the theory and practice of magnetic recording.”

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Frequency modulation in video tape recorders