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FM modulation

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FM modulation
Prelab 3
1:
The operations involved in the modulation of DSB-SC are:
A message signal and carrier signal are fed in to a mixer which effectively modulates the message signal. The figure below demonstrates this step:

Mathematically, this is expressed as the following:

The resultant waveform can be demodulated by multiplying the modulated signal by the carrier signal of the same frequency and then using the low pass filter to filter out the remaining frequencies to get the frequency required. In terms of the frequency domain, the following figure shows the effect of DSB-SC modulation

The effect of the demodulation process in frequency domain:

b)
AsAccos2πfstcos2πfct=AsAc2(cosfs+fc2πt+cos⁡(fs-fc2πt))
xt=AsAc2cos(ωs+ωct+cos⁡(ωs-ωct))Xω=AsAcπ(δω+ωs+ωc+δω-ωs-ωc)There are two distinct frequencies are (ωs+ωc) ,-ωs+ωcc)
AdAsAccos2πfct+ϕcos2πfstcos2πfct=AsAcAd2(cos2πfct+ϕcos(fs+fc2πt)+cos2πfct+ϕcos⁡(fs-fc2πt))
AsAcAd4(cos2ωc+ωst+ϕ+cosωst-ϕ+cosωst+ϕ+cos2ωc-ωst+ϕ)After low pass filter the two 2ωc term sinusoids are filtered so:
AsAcAd4(cosωst-ϕ+cos⁡(ωst+ϕ))=AsAcAd2cos⁡(ωst)cos⁡(ϕ)There is only one distinct frequency that remains after demodulating and low pass filtering: ωs. The effect of the phase shift is the multiplication of the output with a constant cos (Φ). Therefore one needs to know the phase shift of the demodulation signal in order to demodulate and get the original signal.
Q2:
Advantages of DSB-LC when compared to DSB-SC
Demodulations requires simple circuits while DSB-SC requires complicated circuits
Does not suffer from phase shift and frequency errors
The receiver does not need to be in phase as required in the case of DSB-SC Disadvantages of DSB-LC when compared to DSB-SC
Inefficient in terms of power when compared to DSB-SC
Can suffer from over modulation resulting from incorrect modulation index Mathematical representation of a modulated signal in DSB-LC: xt=Ac1+ftcos2πfct where μ=AmAc is the modulation
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