FM Theory

[ukcol]

When I was at college in the 1960s doing my City & Guilds radio & TV servicing course we learnt about bandwidth as applied to FM broadcasting and the reasons that the bandwidth requirements for FM were much higher than those for AM.

Refreshing my memory on this subject I find that there is a rule called the Carson Bandwidth Rule which can be used to calculate FM bandwidth. It states that

Carson bandwidth requirement = 2(deviation frequency + modulation frequency)

For a modern FM stereo broadcast this becomes

Cbr = (75+53)kHz or 256kHz.

The 53kHz is made up of the 15kHz audio and the side bands for the stereo signal either side of the 38kHz pilot tone.


Now we come to the bit that has always puzzled me.

When we came to the end of the 1970s and the VHS home video recorders, I learnt that the 3mHz luminance signal could be FM modulated and handled by an amplifier that did not require anything like the bandwidth that this rule suggests. This situation applies to all the VTR standards of course.

Is it that the distortion caused to the demodulated video signal is tolerated by the eye or is there some other factor I have missed?

[G8HQP Dave]

The bandwidth rule you state does not give the total signal bandwidth (which is infinite) but the bandwidth which accomodates the significant sidebands. You can have less bandwidth and so fewer sidebands at the expense of greater distortion, or more bandwidth and less distortion. The eye is less fussy than the ear. If the deviation is low (i.e. deviation much smaller than baseband frequency) then almost all the signal is in the first set of sidebands. You could have an alternative expression where bandwidth is 2 x MAX( deviation f, modulation f) i.e. just use whichever number is bigger.

[ukcol]

Thanks Dave.

I was aware that the side bands extend infinitely at smaller and smaller amplitudes but you are the first person to tell me that the reason you can get away with low bandwidth for video is the tolerance of the eye.

My expression for Carson's bandwidth requirement should have included MAX deviation.

[Synchrodyne]

The attached charts, taken from the book: “V.H.F. Radio Manual”, by P.R. Keller, Newnes, 1957, plot sideband relative amplitude variation against modulation index. This book provides a good, accessible treatment of the topic.

Narrowband FM, in which the bandwidth corresponds to only the first pair of sidebands for the highest modulating frequency, has also been used for broadcast audio purposes, albeit in subcarrier form.

The Japanese FM-FM system for stereo TV sound used for the S (L-R) channel an FM subcarrier at 2fH (roundly 31.5 kHz) whose maximum deviation was ±10 kHz, and whose nominal bandwidth, with 15 kHz maximum audio frequency, was 30 kHz.

The American MTS system (Zenith-DBX) for TV stereo and secondary audio used an FM subcarrier at 5fH (roundly 78.7 kHz) for the second audio program (SAP) channel. Maximum deviation was also ±10 kHz, but in this case the maximum audio frequency was limited to 10 kHz, so the nominal bandwidth was 20 kHz. DBX companding was applied to achieve acceptable signal-to-noise ratios.

Long before these systems appeared, FM subcarriers were used by American FM broadcasters for SCA (subsidiary communications authorization) audio channels. I am not sure if hard specifications were defined for them, but with the advent of the Zenith-GE stereo system in 1961, stations using this had to confine their SCA channels to the 52 to 75 kHz range, and 67 kHz subcarriers became the norm, with 16 kHz or slightly lower bandwidth, which in turn implied a maximum audio frequency of 8 kHz or less. Later on the FCC rules must have changed, as second SCA subcarriers at 92 kHz were common by the 1980s, if not earlier.

Narrowband FM (or rather a mix of FM and PM) was also used for the carrier channels on CD-4 discrete quadraphonic discs. Here the bandwidth was 20 to 45 kHz about a 30 kHz subcarrier, which with 15 kHz maximum audio frequency implied asymmetric sideband operation, unusual I think in the FM/PM world.

Cheers,

[Radio Wrangler]

In quoting any figure less than infinity for the bandwidth of an FM signal, there has been a decision made on how small sideband components have to be before they don't count. The result says as much about this decision as it does about the signal.

Synchrodyne's posted the standard plots of Bessel functions which show the sideband amplitudes versus modulation index (ratio of deviation and modulating frequency). Negative amplitude values mean the phase is reversed, of course. The points where terms pass through zero are interesting. If the modulation frequency is adjusted to make a sideband term (or even the carrier) hit zero, and the mod frequency is noted, then the deviation can be calculated very accurately.

This is all looking at FM at the wrong end, where the mechanisms and reasons for what is happening aren't very easy to see. It all becomes easier if you try to answer the question "How do all those sidebands add up to make an FM signal".

David

[ukcol]

I have just read, in Radio Communication (Reyner & Reyner 1972), that in practice if you neglect any side frequency of less than 1 % amplitude of the carrier frequency you can use

Carson bandwidth requirement = 2(MAX deviation frequency + modulation frequency).

The same book notes that the modulation index m (also called the Bessel function) in not very useful in design calculations, a more useful parameter being frequency deviation.

BTW the modulation index m =

deviation of carrier from mean value/ modulation frequency.

[Synchrodyne]

If one uses the Bessel functions and the “1% rule”, then the desirable bandwidth for a 15 kHz modulating signal with ±75 kHz deviation comes out at 240 kHz. On the other hand the formula would suggest 180 kHz. Sideband cutting results in amplitude and harmonic distortion.

Some early VHF radiotelephone systems used ±15 kHz deviation. Assuming a maximum modulation frequency of around 3 kHz (the nominal voice channel upper limit is 3.4 kHz), then the Bessel function and the 1% rule indicate a bandwidth of 48 kHz, whereas the formula gives 36 kHz. In practice, 30 kHz bandwidth was typically used (often with purpose-built with 10.7 MHz lumped selectivity receiver crystal IF filters, this being an application where highish distortion (as compared with broadcast reception) was acceptable.

So QED to David R.W.’s comment that the actual bandwidths chosen for FM systems reflect circumstantial trades-off with respect to the effects of sideband cutting. I imagine that in some cases it is likely that the Bessel functions were used as an input to the deliberations.

Regarding the question as to how an FM signal is synthesized from the sidebands and carrier, I suppose one could use the vector approach with the sidebands as vectors contra-rotating about the tip of the carrier vector. Odd sideband pairs are net normal to the carrier and so sum to provide the frequency deviation. Even sidebands are net aligned with the carrier vector in the same way as AM sidebands and so sum to keep resultant overall amplitude constant. Thus the removal of some sidebands would result in both frequency and amplitude errors. Simple to say, but complicated to work out, I think.

Cheers,

[Radio Wrangler]

Yup, the vector approach was what I was hinting at.

The true frequency (or phase) modulated signal has a constant length vector whose tip wags around following the circumference of a circle.

The addition of many modulation vector pairs (with 90 degree phase shifts to the carrier) add to form an approximation to the circular trajectory. An imperfect approximation if less than an infinite number of components are added. Squaring the circle!

David

[Synchrodyne]

I rather like that “squaring the circle” analogy. If one takes the case of an AM system with a single modulating frequency and then shifts the carrier vector phase π/2 relative to the sideband vector phase, then one ends up with a mixed AM-PM system in which the resultant vector swings about and varies in length with its point tracing a straight line. I haven’t worked through the calculations, but I imagine that the vector swing in this case is not sinusoidal. Thus as additional sideband pairs are added, that straight line gets closer to being circular, and the swing motion gets closer to being sinusoidal.

Regarding narrow band systems, VHF R/T went progressively from ±15 kHz deviation with 50 kHz channel spacing and 30 kHz bandwidth through ±5 kHz/25 kHz/15kHz to ±2.5 kHz/12.5 kHz/7.5 kHz as improved techniques, including better carrier frequency stability, allowed. In the last-mentioned case, only the first sideband pair of a 3 kHz modulating signal was transmitted.

The previously mentioned FM SCA system might be thought of as being double NBFM. The subcarrier could be modulated by frequencies up to 8 kHz with a maximum deviation of ±8 kHz, a modulation index of 1. For an 8 kHz modulating signal, only the first sideband pair fitted within the allocated bandwidth. Then the subcarrier, occupying 59 to 75 kHz, was allowed 10% maximum of the main carrier deviation, i.e. ±7.5 kHz, meaning a modulation index of 0.1. For the whole subcarrier bandwidth, only the first sideband pair was transmitted.

Cheers,

[Radio Wrangler]

I once saw amachine in the museum at Liverpool. I had a number of cranks rotating at integer multiples of a fundamental frequency, and it had a chain threaded round pulleys on each crank in turn. The chain added the resolved 1-dimensional position of each crank and added the lot together. The stroke of each crank was adjustable and done before pressing 'go' the whole thing was a Fourier synthesiser and was used to automatically compute tide tables.

What the sidebands of an FM transmission do is two-dimensional, they add up to a vector whose tip (not the whole vector, just the tip) describes a varying length of arc to either side of a vector representing the carrier. The sidebands are rigidly stylised, their frequencies are only the harmonics of the modulation, and their phases are regularly shifted. Only the amplitudes vary. Circling-the-square by means of a mathematical series of terms of constrained frequencies and phases.

Cheers
David