Frequency Modulation Average Transmitted Power

To compute the average transmitted power of a frequency modulated (FM) wave, we first need to understand the fundamental characteristic of FM signals. In frequency modulation, the frequency of the carrier signal is varied in accordance with the modulating signal, while its amplitude remains constant. This is a key difference from amplitude modulation (AM), where the amplitude changes.

Carrier Signal Analysis

The given carrier signal is described as \( \cos(2\pi 1000t) \). A standard form for an unmodulated carrier signal is \( V_c(t) = A_c \cos(\omega_c t) \), where:

• \( A_c \) is the amplitude of the carrier signal.
• \( \omega_c \) is the angular frequency of the carrier signal.

By comparing the given signal \( \cos(2\pi 1000t) \) with the standard form, we can identify the carrier amplitude:

• The amplitude \( A_c \) is the coefficient of the cosine term. In this case, there is no explicit coefficient, which implies \( A_c = 1 \).
• The angular frequency \( \omega_c = 2\pi \times 1000 \) radians per second.

Therefore, the carrier amplitude \( A_c \) for the given frequency modulated wave is \( 1 \).

Average Transmitted Power Calculation

For any sinusoidal signal with an amplitude \( A \) (like our carrier amplitude \( A_c \)), the average power dissipated across a 1-ohm resistive load is given by the formula:

\\[ P_{avg} = \frac{A^2}{2} \\]

Since the amplitude of an FM wave remains constant at the carrier amplitude \( A_c \) throughout the modulation process, the average transmitted power of the frequency modulated wave is determined solely by this constant carrier amplitude.

Using the identified carrier amplitude \( A_c = 1 \):

\\[ P_{avg} = \frac{A_c^2}{2} \\]

\\[ P_{avg} = \frac{1^2}{2} \\]

\\[ P_{avg} = \frac{1}{2} \\]

\\[ P_{avg} = 0.5 \\]

Thus, the average transmitted power of the frequency modulated wave is 0.5 units (e.g., Watts, if the amplitude is in Volts and load is 1 Ohm).

Summary of Power in FM Waves

It is crucial to remember that unlike Amplitude Modulation (AM), where the total power transmitted changes with the modulating signal because the amplitude changes, in Frequency Modulation (FM), the total power (average transmitted power) remains constant and is entirely contained within the carrier and its sidebands. This constant average transmitted power is a significant characteristic of FM systems.