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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