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Understanding Beat Notes and Amplitude Modulation in Signal Processing

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In this lecture, we explore the multiplication of sinusoids and their impact on beat notes and amplitude modulation (AM) in signal processing. When two sinusoids with slightly different frequencies combine, they produce a beat note, a phenomenon commonly heard in musical instruments. We will examine the spectrum representation of the product of two sinusoids, and derive the formula for beat notes, illustrating with examples. Additionally, we’ll look at amplitude modulation, a key concept in radio broadcasting, and plot the spectral components involved.

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Understanding Beat Notes and Amplitude Modulation in Signal Processing

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Presentation Transcript

  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)

  7. 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)

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

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

  10. Amplitude Modulation Waveform

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