Signals & Systems

Lecture 11:

Chapter 3

Spectrum Representation

(Book: Signal Processing First)

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Today's lecture

Definition of Spectrum Spectrum of a Sinusoid Graphical Spectrum Interpretation of Spectrum General Form

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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.

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Sinusoidal Signals of Different Frequencies

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Inverse Euler’s Formula

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Interpretation of Spectrum

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Spectrum of a Sine

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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,ω,φ)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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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 f1 has complex amplitude 1/2 X1 then –f1 will have complex amplitude 1/2 X*

1 if x(t) is real