Technology
Amplitude Modulation and Radiated Power Calculation in FM Signals
Amplitude Modulation and Radiated Power Calculation in FM Signals
Amplitude modulation (AM) is a fundamental technique in communications that alters the amplitude of a carrier wave in response to an input signal. One common question is whether a 100% amplitude modulation (AM) increases the radiated power by 50%. This article will explore this concept, delving into the mathematical derivations and practical implications.
Amplitude Modulation Fundamentals
When a carrier wave is amplitude modulated at a frequency of 100%, the carrier's amplitude changes between 0 and its peak value. This means that the upper and lower side bands each carry 50% of the carrier's amplitude. Importantly, the power in a signal is proportional to the square of the amplitude. Therefore, each side band will have 25% of the carrier's power.
Calculating the Combined Power
To determine the combined power of the side bands, consider the formula for the sum of sines and cosines derived from the product of two sines or cosines. This allows us to represent the modulated signal in terms of the side bands. The combined power of the upper and lower side bands will be the sum of the individual powers of these side bands, which together account for 50% of the carrier's power.
Integrating the Modulated Signal
To understand the radiated power further, we can integrate the square of the amplitude of the modulated signal. Let's take a simplified case where the modulated signal is described by a function like (12 sin^2(x)) between the limits of 0 and (2pi).
Mathematical Integration
The integral of (12 sin^2(x)) can be found using the identity for the square of a sine function:
[int sin^2(x) dx frac{1}{2} (3x - 4 cos(x) - sin(x) cos(x))]When evaluating this definite integral between 0 and (2pi), we get:
[int_{0}^{2pi} 12 sin^2(x) dx 12 cdot left[frac{1}{2} (3x - 4 cos(x) - sin(x) cos(x))right]_0^{2pi}]The terms involving (cos(x)) and (sin(x) cos(x)) will be zero at both limits, leaving us with:
[int_{0}^{2pi} 12 sin^2(x) dx 12 cdot left[frac{1}{2} cdot 6piright] 36pi / 2 18pi]This integral represents the radiated power over one period, indicating that the modulated signal has a 50% increase in radiated power compared to the unmodulated carrier.
Conclusion
In summary, a 100% amplitude modulation does indeed increase the radiated power by 50% due to the increased amplitude of the side bands. This relationship is a fundamental aspect of amplitude modulation and is crucial for understanding the efficiency and impact of different modulation techniques in communication systems.