1. Signals & Systems Lecture 11: Chapter 3 Spectrum Representation (Book: Signal Processing First)

2. 2 Today's lecture  Definition of Spectrum  Spectrum of a Sinusoid  Graphical Spectrum  Interpretation of Spectrum  General Form

3. 3 Concept of the Spectrum  Complicated waveforms can be constructed as sums of sinusoidal signals of different amplitudes, phases and frequencies,  Spectrum simply shows the relationship between frequency,phase and amplitude.

4. 4 Sinusoidal Signals of Different Frequencies

5. 5 Inverse Euler’s Formula

6. 6 Interpretation of Spectrum

7. 7 Spectrum of a Sine

8. 8 General Form

9. 9 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

10. 10 Graphical Spectrum

11. 11 Spectrum of Sinusoid

12. 12 Gather (A,ω,φ)Info

13. 13 Add Spectral Components

14. 14 Add Spectral Components

15. 15 Simplify Components

16. 16 Final Answer

17. Important: Property of Real Signals x(t)  The complex amplitude of each negative frequency component is the complex conjugate of the complex amplitude at the corresponding positive frequency component  Frequency f 1 has complex amplitude 1/2 X 1 then –f 1 will have complex amplitude 1/2 X * 1 if x(t) is real 17