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

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

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 Sinusoidal Signals of Different Frequencies

5 Inverse Euler’s Formula

6 Interpretation of Spectrum

7 Spectrum of a Sine

8 General Form

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 Graphical Spectrum

11 Spectrum of Sinusoid

12 Gather (A,ω,φ)Info

13 Add Spectral Components

14 Add Spectral Components

15 Simplify Components

16 Final Answer

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