Signals and Systems
Lecture 6:
Spectral Representation
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Today's lecture −Spectrum of a Sinusoid−Graphical Spectrum−Amplitude Modulation
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General Form
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Definition of Spectrum−Can be expresses as set of pairs
{ (0,X0), (f1,1/2 X1), (-f1,1/2 X*1),
……(fk,1/2 Xk), (-fk,1/2 X*k)}
−Each pair of (fk,1/2 Xk) indicates the complex amplitude of the sinusoidal component at the frequency fk
−Spectrum is the frequency domain representation of a signal
−Up-till now we have seen the time-domain representation of signals
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Graphical Spectrum
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Spectrum of Sinusoid
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Gather (A,w,0)Info
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Add Spectral Components
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Add Spectral Components
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Simplify Components
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Final Answer
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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
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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)??
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Beat Note Waveform−Beat notes are produced by adding two
sinusoids with nearly identical frequencies
−x(t)= cos(2πf1t) + cos(2πf2t)
where f1 = fc – fΔ and f2 = fc + fΔ
fc is the center frequency = (f1 + f2)/2
fΔ is the deviation frequency = (f2 – f1)/2
−x(t)= 2cos(2πfΔt) cos(2πfct)
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Amplitude Modulation: x(t)= v(t)cos(2πfct)
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Amplitude Modulation Waveform
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Figure 3.7: Spectrum of AM signalx(t)= cos (2π(20)t) cos (2π(200)t)