Bessel Functions in FM

When an FM signal is generated with a sinusoidal message, its spectrum is not simply a carrier with two sidebands as in AM. Instead, the FM spectrum consists of the carrier plus an infinite number of sideband pairs, each spaced by the message frequency fm. The amplitudes of these spectral components are determined by Bessel functions of the first kind. Understanding Bessel functions is essential for FM spectrum analysis and for predicting which sidebands carry significant power.

Core Concept Explanation

The FM signal for a single-tone message can be expressed as s(t) = Ac cos[2pi fc t + mf sin(2pi fm t)]. Expanding this using the Jacobi-Anger expansion yields s(t) = Ac * sum over n from -infinity to +infinity of Jn(mf) * cos[2pi(fc + n*fm)t]. Here Jn(mf) is the Bessel function of the first kind of order n evaluated at mf. Each term n corresponds to a spectral component at frequency fc + n*fm.

The Bessel functions Jn(x) satisfy the property J(-n)(x) = (-1)^n Jn(x), which means the spectrum is symmetric about the carrier. The component at fc has amplitude J0(mf), the first sideband pair at fc plus/minus fm has amplitude J1(mf), and generally the nth sideband pair has amplitude Jn(mf). Since the Bessel functions decrease in magnitude with increasing order for small n, the sidebands farther from the carrier have smaller amplitudes.

A particularly important phenomenon is carrier null. At specific values of mf, the zeroth-order Bessel function J0(mf) becomes exactly zero, meaning the carrier component vanishes entirely. The first carrier null occurs at mf approximately equal to 2.405. This is a unique feature of FM and has no equivalent in AM. GATE problems frequently exploit this property by asking for the value of mf that eliminates the carrier.

Another key property is power conservation. The total power in an FM signal is always Ac^2 / 2, regardless of the modulation index. This means as mf increases and power redistributes into more sidebands, the carrier and lower sidebands lose power, but the total sum remains constant. This is expressed as sum over n from -infinity to +infinity of Jn^2(mf) = 1.

Mathematical Expression

For practical bandwidth calculation, only sidebands with significant amplitude matter. The significant sideband rule states that sidebands where Jn(mf) is greater than 0.01 (1% of unmodulated carrier) are considered significant. The number of significant sideband pairs N can be estimated from Bessel function tables. For mf = 1, approximately 3 significant pairs; for mf = 2, approximately 5 pairs; for mf = 5, approximately 8 pairs; for mf = 10, approximately 13 pairs.

The FM bandwidth based on the significant sideband criterion is BW = 2N * fm, where N is the number of significant sideband pairs. This is the exact bandwidth formulation. Carson's rule provides the approximation BW = 2(delta_f + fm) = 2fm(mf + 1), which contains approximately 98% of the total power and is used as the standard approximation in most engineering calculations.

Practical Understanding

Bessel function tables are standard tools in FM system design. Engineers look up the Jn(mf) values to determine how many sidebands carry significant power, and from this determine the required channel bandwidth. A channel that is too narrow will cut off significant sidebands, introducing distortion. A channel that is too wide wastes spectrum.

In FM stereo broadcasting, the baseband signal has components up to 53 kHz, and the total FM deviation is 75 kHz. The Bessel-based analysis would require tables with many orders. In practice, Carson's rule is always applied for such complex signals, and the Bessel analysis is reserved for single-tone theoretical analysis and examination problems.

Given:
FM signal with mf = 1
Carrier amplitude Ac = 10 V
Carrier frequency fc = 100 MHz, fm = 10 kHz

Why this formula applies:
Spectral component amplitudes = Jn(mf) * Ac

Bessel values for mf = 1 (from standard table):
J0(1) = 0.765, J1(1) = 0.440, J2(1) = 0.115, J3(1) = 0.020

Formula:
Amplitude of nth component = Jn(mf) * Ac

Substitution:
Carrier (n=0): 0.765 * 10 = 7.65 V at fc
First sidebands (n=1): 0.440 * 10 = 4.40 V at fc +/- 10 kHz
Second sidebands (n=2): 0.115 * 10 = 1.15 V at fc +/- 20 kHz
Third sidebands (n=3): 0.020 * 10 = 0.20 V at fc +/- 30 kHz

Calculation:
Power check: J0^2 + 2*J1^2 + 2*J2^2 + 2*J3^2
= 0.585 + 2*0.194 + 2*0.013 + 2*0.0004 = 0.585+0.388+0.026+0.0008 ≈ 1.0

Final Answer:
3 significant sideband pairs, BW = 2*3*10 kHz = 60 kHz. Total power preserved.

Exam Tip: The first carrier null in FM occurs at mf = 2.405 (first zero of J0). Memorize this value. GATE also tests the power conservation property: sum of squares of all Bessel coefficients equals 1, regardless of mf.

Bessel Function Properties Relevant to FM

• s(t) = Ac * sum Jn(mf) cos[2pi(fc + n fm)t] -- the FM signal is a sum of infinite spectral components.
• The component at frequency fc + n fm has amplitude Jn(mf) * Ac. Symmetry: J(-n) = (-1)^n Jn.
• Power conservation: sum from -inf to +inf of Jn^2(mf) = 1, regardless of mf value.
• Carrier null: J0(mf) = 0 at mf = 2.405, 5.52, 8.65 (first three zeros of J0).
• For large mf, significant sideband count N is approximately mf + 1 (from Carson's rule with one extra pair).
• The 1% significant sideband criterion defines the practical FM bandwidth for system design.

Quick Revision

• FM spectral component at fc + n fm has amplitude Jn(mf) * Ac where Jn is Bessel function order n.
• Carrier (n=0) amplitude: J0(mf) * Ac. First carrier null at mf = 2.405.
• Power conservation: sum of Jn^2(mf) for all n = 1, meaning total FM power = Ac^2/2 always.
• GATE trap: At mf = 2.405, the carrier disappears but the signal still exists; power shifts into sidebands.
• Significant sidebands: those with |Jn(mf)| greater than 0.01. Number increases with mf.
• Memorize: J0(1)=0.77, J1(1)=0.44, J0(2)=0.22, J1(2)=0.58, J2(2)=0.35 for GATE problems.
• FM bandwidth from Bessel: BW = 2N fm where N = significant sideband pairs.