Question

An audio frequency signal 10Sin(2π500)t is used to amplitude modulate a carrier of 50Sin(2π105)t. Calculate.1.Modulation index 2.Sideband frequencies 3.Amplitude of each sideband frequency 4. Bandwidth required 5. Transmission efficiency.

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Answer 1

An audio frequency signal 10sin(2π500t) is used to amplitude modulate a carrier of 50sin(2π × 10⁵t).

We have to calculate..

1. Modulation Index.
2. sideband frequencies.
3. Amplitude of each sideband frequency.
4. Bandwidth required.
5. Transmission efficiency.

1. Here, m(t) = 10sin(2π500t) and c(t) = 50sin(2π10⁵t)

∴ maximum amplitude of modulating wave = 10

and maximum amplitude of carrier wave = 50

$\implies A_m=10\:\:,A_c=50$

We know, the modulation index is given by,

$\text{modulation index}=\frac{A_m}{A_c}=\frac{10}{50}=0.2$

Therefore the modulation Index is 0.2

2. Sideband frequencies are given by,

$(f_c+f_m),(f_c-f_m),(-f_c+f_m),(-f_c-f_m)$

Here,

$f_m=500Hz\\\\f_c=100,000Hz$

Therefore the sideband frequencies are 100.5 Hz, 99.5 Hz, -99.5 Hz, -100.5 Hz.

3. Amplitude of each sideband frequency is given by,

$A=\frac{A_c\times\text{modulation Index}}{2}\\\\=\frac{50\times0.2}{2}=5$

Therefore the amplitude of each sideband frequency is 5.

4. bandwidth required = 2 × maximum frequency of modulating signals.

= 2 × 500 Hz = 1000 Hz

Therefore the bandwidth required is 1000 Hz.

5. Efficiency is given by,

$\eta=\frac{m^2}{2+m^2}$

here m is modulation index = 0.2

$\eta=\frac{0.2^2}{2+0.2^2}=\frac{0.04}{2+0.04}=\frac{0.04}{2.04}\approx0.019$

In percent, the transmission efficiency = 100 × 0.019 = 1.9%.

Therefore the transmission efficiency is 1.9%.

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