Final answer:
The frequency-modulated signal S_FM(t) is calculated by integrating the given message signal, then using the result to modulate the provided carrier signal with specified frequency sensitivity. The resulting modulated signal S_FM(t) is 10cos(2π(20000)t + 5sin(2π×1000t)).
Step-by-step explanation:
The question involves calculating the frequency modulated signal SFM(t) given a message signal m(t) and a carrier signal c(t) with specified frequency sensitivity kf. The student is learning about the principles of frequency modulation (FM) as it applies to communications engineering.
Frequency Modulation Principles
Frequency modulation is a process in which the frequency of the carrier wave is varied in accordance with the amplitude of the modulating audio signal, while the amplitude remains constant. The frequency-modulated signal SFM(t) can be expressed mathematically as:
SFM(t) = Acos[2πfct + 2πkf∫ m(t)dt]
where A is the amplitude of the carrier, fc is the carrier frequency, kf is the frequency sensitivity in Hz/V, and m(t) is the modulating signal.
Since m(t) = 5cos(2π×1000t), we can integrate m(t) with respect to time t to get the phase modulation term. For this signal, the integration yields ∫ m(t)dt = ∫ 5cos(2π×1000t)dt = ∗5/(2π×1000)sin(2π×1000t) + C, where C is the constant of integration which can be ignored if we consider C = 0.
Plugging the integrated m(t) and given values into the formula provides the modulated signal:
SFM(t) = 10cos[2π(20000)t + 2π(600)∗5/(2π×1000)sin(2π×1000t)]
Final Expression
The modulated signal SFM(t) after replacing the values and simplifying is:
SFM(t) = 10cos[2π(20000)t + 5sin(2π×1000t)]