d) The message m(t) = 2sin2000nt – 3cos4000nt is used in an AM system with a modulation index of 70 % and carrier frequency of 580 kHz. i. What is the power efficiency? ii. If the net transmitted power is 10 watts, find the magnitude spectrum of the transmitted signal. (Note: Use Fourier transform is required)

The question is asking for two things: the power efficiency of an Amplitude Modulation (AM) system and the magnitude spectrum of the transmitted signal.

i. Power Efficiency: In an AM system, the power efficiency (η) is given by the formula:

η = (m^2) / (2 + m^2)

where m is the modulation index. In this case, m is given as 70% or 0.7. Substituting this into the formula gives:

η = (0.7^2) / (2 + 0.7^2) = 0.33 or 33%

So, the power efficiency of the system is 33%.

ii. Magnitude Spectrum: The magnitude spectrum of a signal can be found using the Fourier Transform. However, without the specific form of the signal, it's impossible to calculate the exact Fourier Transform.

The given message signal m(t) = 2sin2000nt – 3cos4000nt is a combination of two sinusoidal signals. The Fourier Transform will give two delta functions at the frequencies of the sinusoids (2000n and 4000n).

The amplitude of the delta functions will be proportional to the amplitude of the sinusoids (2 and -3 respectively).

The net transmitted power is given as 10 watts. This power is divided between the carrier and the sidebands in an AM signal. Without knowing the power distribution or the exact form of the signal, it's impossible to give the exact magnitude spectrum.

In general, the magnitude spectrum of an AM signal will have a large peak at the carrier frequency (580 kHz in this case) and smaller peaks at the sideband frequencies (580 kHz ± the frequencies of the message signal). The exact heights of these peaks would depend on the power distribution and the form of the message signal.