3. fourier series and transform

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Find the response of an LTI system by expressing the

excitation as a linear combination of impulses and

expressing the response as a linear combination of 

impulse responses.

We called that technique convolution.

In earlier lectures…. 

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Sinusoidal Signal 

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Representing Real Signals using Complex

Phasors 

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Advantages1. Simplifies mathematical derivations and

analyses.

2. Addition of signals is easy

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Fourier series/Transform

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Time property of signal and itsFourier representations

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• Fourier Series is an alternate method of signal

representation.

• Signal is represented as a function of 

frequency.

• Every Periodic Signal (real world signal) can

be expressed in terms of its Fourier

coefficients.

• If the Fourier coefficients are known it is

possible to synthesize the periodic signal

Fourier series

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Periodic Signal 

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Trigonometric Form

Exponential Form

Fourier series representation

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Trigonometric Form 

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Compact Trigonometric Form 

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Example

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Example 2

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Exponential Form

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Example

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Periodic symmetries & Fouriercoefficients

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Fourier coefficients

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Line Spectrum

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2-Sided Line Spectrum

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P S f i di

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Power Spectrum of a periodicsignal

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• It is the difference between the highest and lowestfrequencies of the spectral component of a signal.

Bandwidth of the signal

D t i ti f f d t l

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Determination of fundamentalfrequency and period

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Transforms a signal (i.e., function) from the spatialdomain to the frequency domain.

If the signal is aperiodic signal, Fourier Transform isused.

Fourier Transform

An aperiodic function is a function that never repeats itself 

Never Repeating itself is equivalent to Repeating at Infinity. 

=> Aperiodic Function spans a Continuum of Frequencies of 

the Fourier Series Envelope

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In Fourier series,

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Examples

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Dirichlet’s conditions

Fourier Transform of basic

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Fourier Transform of basicfunctions

Impulse function ----- Fourier Transform

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Inverse Fourier Transform

The Fourier transform and its inverse are referred

to as a Fourier transform pair

Inverse Fourier tranform of

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Inverse Fourier tranform ofImpulse function

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R f th t

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Response of the system

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P i f F i T f

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Properties of Fourier Transform

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