FM Part 5. How to Calculate Bandwidth Using Carson’s Rule (Approximate) & Bessel Function Tables.
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Knowing the bandwidth of a frequency modulated (FM) signal is important for several reasons:
1. Efficient Spectrum Utilization
Communication systems allocate specific frequency bands for different applications (FM radio, TV broadcasting, wireless communication).
Knowing the bandwidth ensures that FM signals do not interfere with adjacent channels.
2. Regulatory Compliance
Governments and regulatory bodies (e.g., FCC, ITU) set bandwidth limits to manage spectrum allocation efficiently.
Ensuring compliance prevents unauthorized spectrum usage and interference.
3. Receiver Design
FM receivers need filters that match the bandwidth of the signal.
A filter that is too narrow may cut off important signal components, leading to distortion.
A filter that is too wide may allow unwanted noise and interference.
4. Signal Quality & Noise Performance
A wider bandwidth improves signal fidelity but may require more spectrum.
A narrower bandwidth may lead to signal degradation and increased noise.
5. Trade-off Between Bandwidth and Power
FM signals with higher frequency deviation require more bandwidth but offer better noise immunity.
Knowing the bandwidth helps in optimizing the trade-off between power efficiency and spectrum usage.
6. Multiplexing and System Design
In multi-user communication systems (e.g., cellular networks, satellite communication), bandwidth knowledge ensures multiple signals coexist without interference.
Theoretically, FM signal has an infinite number of side frequencies and hence would have an infinite bandwidth. However, for practical purposes, side frequencies less than 1% of Ec are considered as insignificant and hence can be ignored.
The bandwidth for FM based on the Bessel Function table is
BW = Highest frequency - lowest frequency = 2 x nmax x fm (Hz)
where nmax is the order of the highest significant side frequency pairs
Another expression, known as Carson’s rule, can be used to approximate the bandwidth of the FM signal:
BW = 2(f + fm) Hz
Where:
Frequency Deviation (Δf): This is determined by the amplitude of the modulating signal and the modulation index. It represents how much the carrier frequency varies from its center frequency.
Modulating Signal Frequency (fm): This is the highest frequency component present in the modulating signal (e.g., audio or data signal).
Carson's rule is an empirical formula used to estimate the bandwidth of a frequency-modulated (FM) signal. It provides a straightforward method based on the modulation index, encompassing 98% of the signal power, to calculate the necessary bandwidth for transmission without significant distortion. The rule is particularly useful in communication systems where efficient utilization of the frequency spectrum is critical.
BW = 2(f + fm) Hz
Limitations:
While Carson's rule is a useful approximation, it may not be accurate for all scenarios, especially when the modulation index is very high or very low. In such cases, more precise methods may be required to determine the bandwidth
The total power in a FM signal is equal to the power of the unmodulated carrier as the peak amplitude of the modulated carrier remains at the value of the unmodulated carrier.
Total average FM power PT = Average power in the unmodulated carrier
The power of the FM signal is distributed across the carrier & side frequencies as shown in power spectrum.
An FM signal e(t) = 20 cos[2 108t + 0.5 sin(2500 x 103t)] volts is applied to a 50 antenna.
Determine the following:
(a) the modulating signal, fm
(b) the modulation index, mf
(c) the peak frequency deviation, f
(d) the total power, PT
(e) the bandwidth using the Bessel function method and the
Carson’s rule
(f) draw the power spectrum
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