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## Q. 1.1.13

In a double side-band (DSB) full carrier AM transmission system, if the modulation index is doubled, then the ratio of total sideband power to the carrier power increases by a factor of _____.

## Verified Solution

Ratio of $P_{S B}$ to PC

$g_{+}(t)=e^{-a t} e^{+j\left(w_{c} t+\Delta w\right) t}$

$\left(\frac{P_{S B}}{P}\right)_{1}=\frac{\mu^{2} T}{2}$

and $\left(\frac{P_{S B}}{P_{C}}\right)_{2}=\frac{\mu_{0}^{2}}{2}$

where  $\mu_{2}=2 \mu$

Hence,  $\left(\frac{\frac{P_{S B}}{P_{C}}}{\frac{P_{SB}}{P_{C}}}\right)=\frac{1}{4}$

Four times.
Therefore,

The Fourier transform of output y(t) of the high pass filter centroid at fc is

Now, the output of second multiplier is y(t).w(t) = z(t)

After passing through the low pass filter centred at $f_{ c }$ the Fourier transform of output S(t) is

Thus, bandwidth = (A B) – 0
= 100 – 40
= 60 Hz