Calculation of frequencies and voltages of carrier and modulating signals in AM wave

If an AM signal is represented by
v = ( 15 + 3 Sin( 2Π * 5 * 10³ t) ) * Sin( 2Π * 0.5 * 10⁶ t) volts

i.) Calculate the values of the frequencies of carrier and modulating signals.
ii.) Calculate the value of modulation index.
iii.) Calculate the value of bandwidth of this signal.

a.) 1.6MHz and 8KHz, 0.6, 16 MHz
b.) 1.9MHz and 18KHz, 0.2, 16 KHz
c.) 2.4 MHz and 18KHz, 0.2, 16 KHz
d.) 1.6MHz and 8KHz, 0.2, 16 KHz

Correct Answer: d) 1.6MHz and 8KHz, 0.2, 16 KHz

Explanation:

The amplitude modulated wave equation is
v = ( 10 + 2 Sin( 2Π * 8 * 10³ t) ) * Sin( 2Π * 1.6 * 10⁶ t) volts

Instantaneous value of AM signal is represented by the equation
v = {V_c + V_m Sin ( ω_m t )} * Sin ( ω_c t )
comparing it with the given equation,

V_c = 10 V
V_m = 2V
ω_c (= 2Π f_c) = 2Π * 1.6 * 10⁶
ω_m (= 2Π f_m) = 2Π * 8 * 10³

(i) The carrier frequency f_c is = 1.6 * 10⁶ = 1.6 MHz
The modulating frequency f_m is = 8* 10³ = 8 kHz

(ii) The modulation index m = V_m/V_c = 2/10 = 0.2

(iii) The bandwidth BW = 2 f_m = 16 kHz