Signals and Systems Lecture 6: Spectral Representation

2 Today's lecture −Spectrum of a Sinusoid −Graphical Spectrum −Amplitude Modulation

3 General Form

4 Definition of Spectrum −Can be expresses as set of pairs { (0,X 0 ), (f 1,1/2 X 1 ), (-f 1,1/2 X * 1 ), ……(f k,1/2 X k ), (-f k,1/2 X * k )} −Each pair of (f k,1/2 X k ) indicates the complex amplitude of the sinusoidal component at the frequency f k −Spectrum is the frequency domain representation of a signal −Up-till now we have seen the time-domain representation of signals

5 Graphical Spectrum

6 Spectrum of Sinusoid

7 Gather (A,w,0)Info

8 Add Spectral Components

9

10 Simplify Components

11 Final Answer

12 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

13 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)??

14 Beat Note Waveform −Beat notes are produced by adding two sinusoids with nearly identical frequencies −x(t)= cos(2πf 1 t) + cos(2πf 2 t) where f 1 = f c – f Δ and f 2 = f c + f Δ f c is the center frequency = (f 1 + f 2 )/2 f Δ is the deviation frequency = (f 2 – f 1 )/2 −x(t)= 2cos(2πf Δ t) cos(2πf c t)

15 Amplitude Modulation: x(t)= v(t)cos(2πf c t)

16 Amplitude Modulation Waveform

17 Figure 3.7: Spectrum of AM signal x(t)= cos (2π(20)t) cos (2π(200)t)