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# An AM signal is represented by the equation $25+5\sin (2\pi\times 8\times 10^3\;t) \times \sin (2\pi\times 0.25\times10^6\;t)$. Calculate the bandwidth, the modulating factor and the amplitude of each side frequency?

Given the signal: $25+5\sin (2\pi\times 8\times 10^3\;t) \times \sin (2\pi\times 0.25\times10^6\;t)$
This is of the form: $v = V_c \big [ $$\sin (\omega_c t) + \large\frac{1}{2}$$m (\cos ((\omega_c - \omega_m)t) + \cos ((\omega_c+\omega_m)t)$$\big ] Comparing this to the give signal, we get the following: (1) \; \omega_c = 2\pi f_c = 2\pi \times 0.25 \times 10^6 (2) \; \omega_m = 2\pi f_m = 2\pi \times 8 \times 10^3 (3)\; V_c = 25\;V (4)\; V_m = 5\;V \Rightarrow Carrier Frequence f_c = 0.25 \times 10^6 = 0.25\;MHz \Rightarrow Modulating Frequency f_m = 8 \times 10^3 = 8\;kHz \Rightarrow Bandwidth = 2f_m = 16\;kHz \Rightarrow Modulation index m = \large\frac{V_m}{V_c}$$ = \large\frac{5}{25} $$= 0.2 Therefore, the amplitude of each side frequency can be calculated as \large\frac{m \times V_c}{2}$$ = 0.2 \times \large\frac{25}{2}$$= 2.5\;V$

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