Frequency Modulation Fundamentals
August 1939 QST Article
modulation (FM) was, is, and shall always be: x(t) = Xc·cos [Ωct + β·sin (Ωmt)],
where the carrier is xc(t) = Xc·cos (Ωct), and the modulating signal is xm(t)
= β·sin (Ωmt). Yea verily, thus sayeth
Edwin H. Armstrong. Amen. The
methods for generating and degenerating[sic] FM might vary, but the fundamentals do not vary. Mr. Armstrong developed
and patented his system of frequency modulation in the late 1920s and early 1930s, so when this article appeared
in QST in 1939, FM was still fairly new - or even unknown - to most people. Household radio dials still had only
markings for the commercial AM band (520 - 1720 kHz) and, in a few cases, a couple shortwave bands (also AM).
The information presented here is suitable for study by anyone at any time.
of Contents]These articles are scanned and OCRed from old editions of the
ARRL's QST magazine. Here is a list of the
QST articles I have already posted. As time permits, I will
be glad to scan articles for you. All copyrights (if any) are hereby acknowledged.
Frequency Modulation Fundamentals
Modulation Works; Its Advantages in Overcoming Noise and Interference
By Daniel E. Noble, W1CAS
Two 50,000-watt experimental transmitters and several lower-powered transmitters will be placed in regular operation
in the Fall using the Armstrong frequency-modulation system. The marked noise suppression which is the important
characteristic of the system will make possible a new standard of high-fidelity reception. The writer has been asked
to explain the action of this frequency-modulation system without too much technical terminology. With all qualifications
aside, the picture looks something like this:
Fig. 1 - Illustrating amplitude and frequency modulation. A, the motor-driven alternator used as an example;
B, output with constant field and constant speed (sine wave); C, output with constant speed and variable field
(amplitude modulation); D, constant amplitude and variable speed (frequency modulation).
Fig. 2 - Vector Diagram of phase modulation. The modulator vector reverses, producing a resultant Θ degrees
ahead or behind the carrier vector. This is equivalent to a sudden change in the time axis, with the result
that the frequency changes. The vector will oscillate back and forth between A and B at the modulating frequency.
The more rapid the oscillation the faster the change in the time axis, therefore the greater the frequency deviation
Every amateur knows what frequency modulation is - it's something
in his transmitter operation that he doesn't want! To make the picture a little more exact, we shall make use of
a pure sine wave alternator. A pure sine wave is a single-frequency wave; that is, no side bands and no harmonics
will be associated with it. A perfect frequency meter could locate only one frequency with such a wave. If our alternator
is the usual motor-driven type with an external field supply, we can vary the voltage output of the alternator by
varying the field current. Let's vary the field current slowly up and down and observe the result. First, the output
voltage of the alternator will increase and decrease, and we have a condition commonly referred to as amplitude
modulation. See Fig. 1 (A, B, and C). Second, the output wave is no longer a pure sine wave, and if we examine the
wave with our perfect frequency meter we shall find several frequencies present, because only the pure sine wave
will be limited to a single frequency. So much for amplitude modulation.
Now regulate the field supply so that the amplitude of the alternator output will not change while the driving
motor is made to speed up and slow down. The frequency of the alternator will be determined by the speed of the
motor; if we speed up the motor the output frequency will increase, and it will decrease when the motor slows down.
Assuming that the amplitude of the output remains constant, we have produced a frequency-modulated wave by the simple
process of speeding up and slowing down the motor. What has happened to the wave? First, obviously the wave is no
longer a pure sine wave, since the frequency is changing. Second, since the wave is not a pure sine wave, several
frequencies will be present (theoretically, an infinite number). When we neglect inertia and speed up and slow down
the motor in such a way that the change in speed is at the rate of ten cycles per second, and the cycles are perfect
sine-wave cycles, we will produce a frequency series for a 1000-cycle generator something like this: . . . 1000
- 30, 1000 - 20, 1000 - 10, 1000, 1000 + 10, 1000 + 20, 1000 + 30 . . . and so on to an infinite number of side
bands. Although frequency modulation will produce a composite wave made up of the carrier, plus and minus a regular
harmonic series of the modulating-signal frequency and the carrier, we are fortunate in the fact that the amplitudes
of the side bands decrease rapidly as the signal harmonic number increases.
To go back to our motor-generator
again, the motor was speeded up and slowed down to produce our frequency modulation but we didn't say how much we
speeded it up or how much we slowed it down. We can change the motor speed so that the frequency will vary instantaneously
as follows: 1000 --> 1025 --> 1000 --> 975 --> 1000 cycles, and make the entire excursion in one-tenth
of a second for a modulating frequency of ten cycles per second. Or we can go 1000 -->1050 --> 1000 -->
950 --> 1000 in one-tenth of a second for a 10-cycle modulation frequency. The difference is found in the more
extended change in frequency in the second case. This change is called the "deviation." For the first case the deviation
is 25 cycles and for the second, 50 cycles. Deviation is then the maximum instantaneous change in frequency. Just
to increase the confusion, we might add that we can't find the deviation with the frequency meter since no continuous
spectrum is produced but, rather, we produce discrete side bands which may be detected and their physical existence
made evident by means of our frequency meter. These side bands may be found far beyond the limits of the deviation.
We might define the maximum instantaneous frequency for our special case as the frequency we would get from our
alternator if we held the speed constant when the maximum speed was reached. We do not actually produce such a maximum
frequency because the speed does not remain constant. All this leads to conclusion that we can expect the band-width
of the frequency modulated wave to be greater than twice the deviation.
Fig. 3 - A practicable frequency-modulator circuit, after Weir. The oscillator is frequency-modulated
by the a.f.c. tube (modulator) which causes a frequency deviation in proportion to the amplitude of the audio voltage.
A small part of the output signal is fed to the converter tube, which is heterodyned by a stable crystal oscillator
to give a beat frequency at 1500 kc. The i.f. output operates the rectifier (discriminator) and by providing the
modulator with a d.c. bias which varies when the mean oscillator frequency tends to change (a.f.c. action) maintains
the carrier frequency constant. Deviations of approximately 30 to 40 kc. may be obtained in the region of 20 Mc.
using a 6L6 modulator and 6F6 oscillator. The stability of the system will be determined by the discriminator circuit
Producing Frequency Modulation
A frequency-modulated wave may
be produced much more readily with vacuum tube equipment than with rotating machinery. Rotating a condenser back
and forth to change the capacity in an oscillator circuit will produce a frequency-modulated wave. Placing a condenser
microphone in an oscillator circuit in such a way that changes in the microphone capacity will influence the frequency
of the oscillator is an obvious means of producing a modulated wave. The circuit used in automatic frequency control
systems is an excellent frequency-modulation system.
The modulation method invented by Major Edwin Armstrong is very stable since the carrier is controlled by a quartz
crystal oscillator. A 200-kc. oscillator supplies voltage to a phase-shift network from which two components of
the carrier are extracted, differing only in phase. One component is 90° out of phase with the other. Mathematically,
the difference between the amplitude-modulated wave and the frequency-modulated wave is the difference in the phase
relations between side bands and carrier. If the side bands of an amplitude-modulated wave could be extracted from
the carrier, shifted in phase 90°, and then recombined with the carrier, a frequency- modulated wave would result.
Major Armstrong did not extract the side bands but he did arrange to produce side bands without a carrier by means
of a balanced modulator working with one of the 200-kc. components mentioned above, and then to combine the side
bands with the second component in such a way that the side bands were 90° out of phase with the normal arrangement
for carrier and side bands in the amplitude modulated wave. His result was a frequency-modulated wave of the special
type sometimes referred to as a "phase-modulated" wave. Another way to describe the action of Major Armstrong's
modulator is to say that he combined a carrier voltage with a side-band voltage which had been rotated through 90°.
This gives us the simple picture of two vectors 90° out of phase combining to give the resultant voltage. Fig. 2
will assist the reader to visualize the process. As the side-band voltage is increased and decreased, the resultant
of the two vectors is caused to shift phase. The shift in phase corresponds to a frequency change, and the amount
of frequency change produced will depend upon the magnitude of the phase shift and upon how rapidly the phase shift
is taking place. Since the magnitude and the speed of the phase shift is determined by the side-band vector, the
deviation produced will be determined by the magnitude and the frequency of the modulating signal. The only difference
between pure frequency modulation1 and phase modulation is the fact that the deviation is a function
of the amplitude only of the modulating signal for pure frequency modulation while the frequency of the signal also
determines the deviation for phase modulation. A network placed in the audio input amplifier making the output signal
voltage inversely proportional to frequency will make the overall response independent of signal frequency, and
thus the phase modulator will produce a pure frequency-modulated wave. The actual deviation produced at 200 kc.
is small, something of the order of 15 to 20 cycles. Therefore, a series of doublers must be introduced to increase
the maximum deviation to 100 kc. A total of twelve or thirteen doubler stages is used to reach the required deviation.
Fig. 4 - Essentials of a superheat receiver suitable for frequency-modulated signals.
Fig 5. - Elementary detector circuit for frequency-modulated waves. CD is tuned to the lower extremity of useful
side bands, AB to the upper extremity. The voltage appearing across either circuit is determined by the amplitude
of the audio modulating voltage. A hybrid wave appears across each circuit. Rectification recovers the audio
A system of modulation suggested by Murray C. Crosby and developed by Irving R. Weir makes use of the automatic
frequency control variable oscillator for modulating the frequency, and of the a.f.c. discriminator circuit for
stabilizing the oscillator carrier. Fig. 3 illustrates the type of circuit used. The modulator tube injects 90°
out-of-phase current into the oscillator tank circuit. The effect of changing the modulator injector current is
comparable to changing the tank capacity in the oscillator circuit. The stabilizing circuit functions in the usual
The receiver requirements are not so complicated as one might suspect. The usual super-heterodyne is used with a
few additions and changes. The pass band must be greater than twice the transmitter deviation. A limiter precedes
the detector, and this limiter has the very important function of "wiping off" any amplitude modulation which may
have been introduced by noise voltages. The limiter passes on the frequency-modulated wave with constant amplitude
to the frequency detector, which changes the frequency-modulated wave into a hybrid wave with both amplitude and
frequency modulation components. An ordinary detector then recovers the signal from the amplitude component. Fig.
4 illustrates the line-up. Figs. 5 and 6 show two frequency detectors.
Fig. 6 - The discriminator circuit combines the functions of frequency detector and rectifier to recover the
Fig. 7 - A parallel-resonant circuit will act as a frequency detector when tuned to carrier at points A or B.
With the circuit tuned so that the carrier is at A. The voltage across the circuit will rise and fall in step
with the deviation produced by the modulating voltage; the result is an amplitude- modulated wave which is also
frequency-modulated. A rectifier will recover the amplitude audio component.
Transmitter house and six-bay turnstile antenna at W1XPW, a 1000-watt experimental frequency-modulated transmitter
located on top of West Peak, Meriden Mountain, near Meriden, Conn. The transmitter operates on 43.4 megacycles.
WDRC, Inc., is the owner.
A simple circuit of the type shown
in Fig. 7 also will act as a frequency detector. The carrier is tuned in on one side of the resonance curve. A steady-state
r.f. voltage will result from the unmodulated carrier, and modulation will produce instantaneous frequency changes.
Taking A as the operating point, any change corresponding to an increase in frequency of the signal will increase
the amplitude of the voltage across the parallel-resonant circuit, and an equivalent decrease in frequency will
decrease the voltage across the circuit. Therefore, since the modulation produces magnitudes of frequency change
or deviation corresponding to the amplitude of the audio modulating signal, and since the rate at which the changes
or deviations take place corresponds to the frequency of the audio signal, the voltage appearing across the parallel-resonant
circuit will be amplitude-modulated. Frequency modulation will also be present but we are no longer interested in
that. Rectification will recover the audio signal. Any receiver of the usual type can be made to receive frequency-
modulated signals after a fashion by detuning slightly, but the reader is assured that the "fashion" is not very
Remarkable results in the suppression of noise and
interference are possible with the frequency modulation system. Since the limiter wipes off all amplitude variations,
noise of this type must appear as frequency modulation produced by the phase shift resulting from the combination
of the signal and noise voltages. For the case where the peak noise amplitude is half the signal amplitude and the
phase relation between signal and noise is ninety degrees, the maximum phase shift would be approximately 26.5°.
Very little frequency modulation will be produced if this phase shift is the result of noise modulated by a low-frequency
audio component, but the frequency modulation will increase directly with the frequency of the audio noise component.
The receiver will display greatest susceptibility to noise frequencies above audibility. Logical design of the receiver
would call for a sharp cut-off of the audio amplifier response or, better still, a falling high-frequency characteristic
which will reduce the hiss response. A simple predistortion network at the transmitter will present a compensating
rising high-frequency response so that the overall response of the system is flat. This is the arrangement used
in the stations now on the air.
The very remarkable effect of the limiter action upon the suppression of
interference has been demonstrated by Weir.2 He reports that with two stations operating on the same
channel the stronger station would prevail 100 per cent at the receiver whenever the stronger station's signal was
more than twice the strength of the weaker signal. He also reports in the same paper that no interference area of
the usual kind existed where the signals were of nearly the same amplitude. In this area the movement of the antenna
a few inches would throw one program out and bring in the other one. The presence of standing waves accounts for
the phenomenon since the nodes would permit the selection of the required voltage radio.
action of the limiter is rather complicated, but the results of the limiter action are an overall effect of cutting
the amplitude of the received voltage in such a way that the strong signal component dominates while the weak signal
is suppressed. In other words, the strong signal will always take control of the receiver. The frequency-modulation
system permits2 as much as 25 dB gain in signal-plus-noise- to-noise ratio over that possible with an
amplitude system of equal carrier strength.
While this gain in equivalent power is due in part to the limiter
action it is also the result of the very interesting effect which makes the magnitude of the recovered power at
the receiver a function of the modulation deviation. If a deviation of 50 kc. produces voltage A at the receiver,
then a deviation of 100 kc. will produce a voltage two times A at the receiver. Here the received voltage has been
doubled without changing the carrier power at the transmitter. In the amplitude case the peak carrier power must
increase four times when the modulation changes from zero to 100 per cent. Since without the power change the received
voltage increases for the frequency-modulated system with an increase in deviation, it follows that the advantage
of the system over the amplitude system will increase as the deviation is increased. The practicable limits must
be determined by available channel width. The Federal Communications Commission has assigned 200-kc. channels for
the broadcast stations now in operation. For this channel width the deviation will probably be restricted to 80
kc. or less. Present standards seem to point to a modulation index (that is: ratio of deviation to audio frequency)
of 80,000/15,000' or approximately 5.3.
Necessarily this is a very sketchy account of Major Armstrong's invention.
The writer hopes that it may serve as an introduction to the subject, and for those who are interested in the more
detailed and technical aspects a carefully selected bibliography is appended.
1: The term "phase modulation" is something of a pain. Actually there are as many types of
as there may be functions of X. Phase modulation is one type. The type
referred to as "pure" frequency modulation
is the unadulterated, holy, sweet, etc. variety
in which the deviation produced is a linear function of the modulating
signal amplitude only.
"Phase modulation" is still "frequency modulation."
2: I.R. Weir, "Field Tests and Amplitude Modulation
with Ultra-High-Frequency Waves," Part I. General Electric Review,
Selected Frequency Modulation Bibliography
Carson, John R. "Notes on the Theory of Modulation," Proceedings IRE, Vol. 10, No.1, February, 1922. First paper
in which the frequency-modulated wave is analyzed mathematically and the required "band width" determined.
2. Armstrong, Edwin H. "A Method of Reducing Disturbances in Radio Signaling by a System of Frequency Modulation."
Proc. I.R.E., Vol. 24, No.5, May, 1936. Undoubtedly the classical paper in the field. The first published account
of the wide-band frequency modulation system.
3. Crosby, Murray C. "Frequency Modulation Noise Characteristics."
Proc. I.R.E., Vol. 25, No.4, April, 1937. Mathematical treatment and experimental verification of wide-band frequency
modulation vs. amplitude noise suppression.
4. Roder, Hans. "Frequency Modulation." Electronics, Vol. 10,
No.5, May, 1937. Mathematical analysis of validity of noise-suppression effect in wide-band frequency modulation.
5. Carson, John R. and Fry, Thornton C. "Variable Frequency Electric Circuit Theory with Application to the
Theory of Frequency Modulation." Bell System Technical Journal, Vol. 16, No.4. Fundamental formulas for variable
frequency electric circuit theory are developed. Transmission, reception and detection of frequency modulated waves
are studied analytically.
6. Roder, Hans. "Tuned Circuits and a Frequency Modulated Signal." Proc. I.R.E.,
Vol. 25, No. 12, Mathematical treatment of tuned circuits.
7. Weir, I. R. "Field Tests of Frequency-and-Amplitude-Modulation
with Ultra-High-Frequency Waves." General Electric Review, May, 1939, Part I; June, 1939, Part II. A very important
paper of interest to both the technical and non-technical readers. Describes a simplified transmitter.
Day, John R . "A Receiver for Frequency Modulation." Electronics, June, 1939. The first published constructional
data for a seven-tube frequency-modulation receiver.