Carson's Rule FM Bandwidth

BW = 2 × (Δf + fm)

Quick answer

This calculator solves BW ≈ 2 × (Δf + f_m) for Occupied Bandwidth. With Peak Frequency Deviation 0.22 MHz, Max Modulating Frequency 0.47 MHz, the result is 1.38 MHz. It solves in 3 directions, so you can enter the values you have and get the one you need.

Calculator

Occupied Bandwidth (kHz)
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Formula

BW ≈ 2 × (Δf + f_m)

Description

Carson's rule estimates the approximate bandwidth occupied by a frequency-modulated (FM) signal. The bandwidth depends on both the peak frequency deviation (Δf) and the highest modulating frequency (fm). For narrowband FM (Δf << fm), the bandwidth approaches 2×fm (similar to AM). For wideband FM (Δf >> fm), the bandwidth approaches 2×Δf. This rule captures about 98% of the signal power. It is the standard method for estimating FM channel bandwidth in communications system design.

Variables

  • BW — Approximate occupied bandwidth (Hz)
  • Δf — Peak frequency deviation (Hz)
  • f_m — Maximum modulating frequency (Hz)

Practical Notes

Commercial FM radio: Δf = ±75 kHz, fm = 15 kHz, BW = 2(75+15) = 180 kHz, rounded to 200 kHz channel spacing. Narrowband FM (amateur radio, public safety): Δf = ±5 kHz, fm = 3 kHz, BW = 2(5+3) = 16 kHz. The modulation index β = Δf/fm determines the spectral shape: β < 0.5 is narrowband FM, β > 1 is wideband FM.

Symbols and units

Symbol Meaning Default unit Accepted units
BW Occupied Bandwidth kHz Hz, kHz, MHz, GHz
df Peak Frequency Deviation kHz Hz, kHz, MHz, GHz
fm Max Modulating Frequency kHz Hz, kHz, MHz, GHz

Step by step

  1. 1. What you know: Peak Frequency Deviation 0.22 MHz, Max Modulating Frequency 0.47 MHz
  2. 2. The relation: BW ≈ 2 × (Δf + f_m)
  3. 3. Substitute: df = 0.22 MHz, fm = 0.47 MHz
  4. 4. Result: Occupied Bandwidth = 1.38 MHz

Worked examples

Minimum range

Given: Peak Frequency Deviation 0.22 kHz, Max Modulating Frequency 0.47 kHz

Result: Occupied Bandwidth = 1.38 kHz

Low range

Given: Peak Frequency Deviation 2.2 kHz, Max Modulating Frequency 4.7 kHz

Result: Occupied Bandwidth = 13.8 kHz

Typical

Given: Peak Frequency Deviation 0.22 MHz, Max Modulating Frequency 0.47 MHz

Result: Occupied Bandwidth = 1.38 MHz

High range

Given: Peak Frequency Deviation 1.1 MHz, Max Modulating Frequency 2.35 MHz

Result: Occupied Bandwidth = 6.9 MHz

Maximum range

Given: Peak Frequency Deviation 11 MHz, Max Modulating Frequency 23.5 MHz

Result: Occupied Bandwidth = 69 MHz

Solving for each variable

Solve for Occupied Bandwidth

Peak Frequency Deviation 0.22 kHz, Max Modulating Frequency 0.47 kHz → 1.38 kHz

Solve for Peak Frequency Deviation

Occupied Bandwidth 100 Hz, Max Modulating Frequency 0.47 kHz → -0.42 kHz

Solve for Max Modulating Frequency

Occupied Bandwidth 100 Hz, Peak Frequency Deviation 0.22 kHz → -0.17 kHz

How Occupied Bandwidth changes with Peak Frequency Deviation

Peak Frequency Deviation (input) Occupied Bandwidth (result)
48.4 kHz 2.157 MHz
121 kHz 2.302 MHz
0.242 MHz 2.544 MHz
0.363 MHz 2.786 MHz
0.484 MHz 3.028 MHz
0.726 MHz 3.512 MHz
0.968 MHz 3.996 MHz
1.936 MHz 5.932 MHz
3.388 MHz 8.836 MHz
4.84 MHz 11.74 MHz

The same result in other units

Unit Value
Hz 1380000
kHz 1380
MHz 1.38
GHz 0.00138

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Frequently asked questions

How do you calculate Carson's Rule FM Bandwidth?

Use BW ≈ 2 × (Δf + f_m). With Peak Frequency Deviation 0.22 MHz, Max Modulating Frequency 0.47 MHz, Occupied Bandwidth works out to 1.38 MHz.

How do you calculate carson bandwidth rule?

Rearrange for Occupied Bandwidth: with Peak Frequency Deviation 2.2 kHz, Max Modulating Frequency 4.7 kHz, Occupied Bandwidth is 13.8 kHz.

What is carson's rule?

Rearrange for Occupied Bandwidth: with Peak Frequency Deviation 0.22 MHz, Max Modulating Frequency 0.47 MHz, Occupied Bandwidth is 1.38 MHz.

What is the formula for Carson's Rule FM Bandwidth?

The formula is BW ≈ 2 × (Δf + f_m). Carson's rule estimates the approximate bandwidth occupied by a frequency-modulated (FM) signal.

How do you find Peak Frequency Deviation when you know the other values?

Rearrange for Peak Frequency Deviation: with Occupied Bandwidth 100 Hz, Max Modulating Frequency 0.47 kHz, Peak Frequency Deviation is -0.42 kHz.

How do you find Max Modulating Frequency when you know the other values?

Rearrange for Max Modulating Frequency: with Occupied Bandwidth 100 Hz, Peak Frequency Deviation 0.22 kHz, Max Modulating Frequency is -0.17 kHz.

Related Concepts

All Signal & RF formulas →
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