Communication Systems · 96 h · Topic 5 of 11

Exponential modulation: FM and PM

A commercial FM station occupies about 180 kHz to transmit 15 kHz audio. That bandwidth, twelve times larger, is the price of its quality.

Modulation index Bessel functions Carson's rule Narrowband PLL

01Modulating the angle

In exponential or angle modulations, the message changes the phase or the frequency of the carrier and the amplitude stays constant. In phase modulation (PM), the phase follows the message; in frequency modulation (FM), the instantaneous frequency does:

\[ x_{PM}(t) = A_c \cos\big(2\pi f_c t + k_p\, m(t)\big) \] \[ x_{FM}(t) = A_c \cos\Big(2\pi f_c t + 2\pi k_f \int_0^t m(\tau)\, d\tau\Big) \] In FM the instantaneous frequency is \( f_c + k_f m(t) \), and its maximum excursion is the frequency deviation \( \Delta f \). PM and FM are the same family: FM is PM of the integrated message.

Since the envelope is constant (as seen in complex envelope), FM transmitters use saturated amplifiers, and receivers clip the amplitude before demodulating, eliminating a good part of the noise and interference that affect the amplitude.

02The FM spectrum and Bessel functions

Unlike AM, FM is not linear: its spectrum is not that of the shifted message. With a tone of frequency \( f_m \), the modulation index is \( \beta = \Delta f / f_m \), and the signal has infinitely many lines spaced \( f_m \) apart around the carrier, with amplitudes given by the Bessel functions:

\[ x(t) = A_c \sum_{n=-\infty}^{\infty} J_n(\beta)\cos\big(2\pi (f_c + n f_m) t\big) \] The total power does not change with modulation: \( \sum J_n^2(\beta) = 1 \). Modulation only distributes it between the carrier and the sidebands; with β = 2.405 the carrier disappears.

The lines become negligible shortly after \( n = \beta + 1 \), and from this comes Carson's rule for the bandwidth, which contains about 98 % of the power:

\[ B \approx 2(\Delta f + W) = 2(\beta + 1)\, W \] \( W \) is the maximum frequency of the message. A commercial FM station, with Δf = 75 kHz and audio up to 15 kHz (β = 5), occupies about 180 kHz: that is why channels are spaced 200 kHz apart.

Lab · the FM spectrum

A carrier frequency-modulated by a 15 kHz tone. The lines are the amplitudes \( |J_n(\beta)| \) of the carrier and the sidebands; the yellow strip, the Carson bandwidth; the gray one, the bandwidth that contains 98 % of the power.

Modulation index β =

03Narrowband and wideband

With β much less than 1 (narrowband FM), only the carrier and the first pair of lines are appreciable, and the bandwidth is \( 2W \), as in AM: it is used in two-way radios, where spectrum is scarce. With large β (wideband), the bandwidth grows almost in proportion to the deviation. In return, as seen in the unit on noise in analog modulations, the output signal-to-noise ratio improves with the square of β: wideband FM trades bandwidth for quality.

04Generation and demodulation

Function	Methods
Direct generation	A voltage-controlled oscillator (VCO), stabilized with a PLL
Indirect generation (Armstrong)	Narrowband PM of the integrated message, and frequency multiplication
Slope demodulation	A resonant circuit converts frequency into amplitude, followed by an envelope detector
Quadrature discriminator	Compares the signal with a copy phase-shifted by an amount that depends on frequency
PLL	The control voltage of the locked VCO follows the frequency of the signal: it is the message

05In the lab

Activity 1 · Nulling the carrier

Frequency-modulate a generator with a 1 kHz tone and increase the deviation until the carrier disappears on the spectrum analyzer. Verify that it happens with Δf = 2.405 kHz: it is the classic method for calibrating the deviation.

Activity 2 · Carson

Measure the bandwidth that contains 98 % of the power for several indices and compare it with Carson's rule.

Activity 3 · Demodulating with a PLL

Demodulate an FM signal with an integrated PLL and observe the VCO control voltage.

06Common mistakes

Calculating the FM bandwidth as 2Δf. The message term is missing: it is 2(Δf + W).
Believing that modulation adds power. In FM the total power is constant.
Confusing frequency deviation with index. The index is the deviation divided by the message frequency.
Forgetting pre-emphasis. FM stations boost the high frequencies before transmitting and the receiver attenuates them, to reduce noise.

07Self-assessment

What modulation index does an FM signal with Δf = 5 kHz and a 1 kHz tone have?

β = 5.

What Carson bandwidth does a two-way radio with Δf = 2.5 kHz and audio up to 3 kHz occupy?

\( 2(2.5 + 3) = 11 \) kHz.

At what index does the carrier disappear?

At β = 2.405, the first zero of \( J_0 \).

What fraction of the power remains in the carrier with β = 1?

\( J_0^2(1) = 0.765^2 \approx 0.59 \): 59 %.

08Further reading

B. P. Lathi and Zhi Ding. Modern Digital and Analog Communication Systems. 5th ed., Oxford University Press, 2019. Angle modulations, Bessel functions and Carson's rule.
Simon Haykin and Michael Moher. Communication Systems. 5th ed., Wiley, 2009. FM with its demodulators and the PLL.
Wayne Tomasi. Electronic Communications Systems: Fundamentals Through Advanced. 4th ed. (Spanish edition, “Sistemas de comunicaciones electrónicas”), Pearson, 2003. FM transmitters and receivers from the circuits side.

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Pulse modulation and PCM