2 sin 0 sin y = cos(0 – 4) – cos(0+4) Equation 1 - Trigonometric Identity - Product of Sines Equation 2 represents the mathematical function for Amplitude Modulation (AM). With this AM equation and the use of , we are able to derive the resulting DSB-AM signals observed in AM demonstration seen in class using signal generators. Equation 3,5, and 6 below for c(t), m(t), and y(t) represents the resulting AM waveform after performing the multiplication (mixing) of the carrier waveform c(t), and the information bearing signal, m(t). m(t) y(t) = |1+ c(t) A = [1+m cos(27fmt+ ¢)] A sin(2n fct) Equation 2 - Amplitude Modulation General Equation In class we saw how the product of sines trigonometric identity was applied to the product of two waveforms, c(t) and m(t) to provide the resulting AM modulated waveform shown in Equation 5. The resulting AM waveform consists of the original carrier signal with the resulting lower and upper sideband products. A quick inspection shows that the resulting sinusoidal frequency components are simply the sum and difference of the information- bearing signal relative to the carrier. You can see that the information bearing signal is frequency translated by fc ± fm- c(t) = A sin(27 fct) m(t) = M cos(2n fmt + ¢) = Am cos(27 fmt + 4) Equation 3 - Carrier Wave General Sine Wave Equation Equation 4 - Message Signal General Equation 1 y(t) = A sin(27 fct) + Am [sin(27 [fc + fm]t+ ¢) + sin(27 [fc – fm]t – ø)] 2 Equation 5 - Resultant Expanded AM Equation