Signals & Systems

1. Signals & Systems Lecture 12: Chapter 3 Spectrum Representation

2. Today's lecture • Multiplication sinusoids • Beat notes • Amplitude Modulation

3. Multiplication of Sinusoids • When two sinusoids having different frequencies are multiplied, we get an interesting effect called a ‘Beat note’ • Some musical instruments naturally produce beating tones • Multiplying sinusoids is used for amplitude modulation (AM) in radio broadcasting

4. Example 3.2: Spectrum of a Product x(t)= cos(πt) sin(10πt) x(t)= 1/2cos(11πt- π/2) + 1/2cos(9πt- π/2)?? Plot the spectrum.

5. Beat Note Waveform • Beat notes are produced by adding two sinusoids with nearly identical frequencies • x(t)= cos(2πf1t) + cos(2πf2t) where f1 = fc–fΔandf2 = fc+fΔ fcis the center frequency = (f1 +f2)/2 fΔis the deviation frequency = (f2 –f1)/2 • x(t)= 2cos(2πfΔt) cos(2πfct) • Plot spectrum of the beat signal

6. Plot beat notes x(t)= 2cos(2π(20)t)cos(2π(200)t)

8. Amplitude Modulation: x(t)= v(t)cos(2πfct) Example 3.5 Find the spectral components of x(t) Given v(t)= 5 + 4cos(40πt) and fc = 200 Hz Then x(t) = [5 + 4cos(40πt)] cos(400πt) x(t) = 5cos(400πt) + 4cos(40πt)cos(400πt)

9. Figure 3.7: Spectrum of AM signal x(t)= 5cos(400πt) + 4cos(40πt)cos(400πt)

10. Amplitude Modulation: x(t)= v(t)cos(2πfct)

11. Amplitude Modulation Waveform