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)