FOURIER TRANSFORMS. JELMAAN FOURIER: Definition of the Fourier transforms Relationship between Laplace Transforms and Fourier Transforms Fourier transforms

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  • FOURIER TRANSFORMS
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  • JELMAAN FOURIER: Definition of the Fourier transforms Relationship between Laplace Transforms and Fourier Transforms Fourier transforms in the limit Properties of the Fourier Transforms Circuit applications using Fourier Transforms Parsevals theorem Energy calculation in magnitude spectrum
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  • CIRCUIT APPLICATION USING FOURIER TRANSFORMS Circuit element in frequency domain:
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  • Example 1: Obtain v o (t) if v i (t)=2e -3t u(t)
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  • Solution: Fourier Transforms for v i
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  • Transfer function:
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  • Thus,
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  • From partial fraction: Inverse Fourier Transforms:
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  • Example 2: Determine v o (t) if v i (t)=2sgn(t)=-2+4u(t)
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  • Solution:
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  • JELMAAN FOURIER: Definition of the Fourier transforms Relationship between Laplace Transforms and Fourier Transforms Fourier transforms in the limit Properties of the Fourier Transforms Circuit applications using Fourier Transforms Parsevals theorem Energy calculation in magnitude spectrum
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  • PARSEVALS THEOREM Energy absorbed by a function f(t)
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  • Parsevals theorem stated that energy also can be calculate using,
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  • Parsevals theorem also can be written as:
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  • PARSEVALS THEOREM DEMONSTRATION If a function,
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  • Integral left-hand side:
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  • Integral right-hand side:
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  • JELMAAN FOURIER: Definition of the Fourier transforms Relationship between Laplace Transforms and Fourier Transforms Fourier transforms in the limit Properties of the Fourier Transforms Circuit applications using Fourier Transforms Parsevals theorem Energy calculation in magnitude spectrum
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  • ENERGY CALCULATION IN MAGNITUDE SPECTRUM Magnitude of the Fourier Transforms squared is an energy density (J/Hz)
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  • Energy in the frequency band from 1 and 2 :
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  • Example 1: The current in a 40 resistor is: What is the percentage of the total energy dissipated in the resistor can be associated with the frequency band 0 23 rad/s?
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  • Solution: Total energy dissipated in the resistor:
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  • Check the answer with parsevals theorem: Fourier Transform of the current:
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  • Magnitude of the current:
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  • Energy associated with the frequency band:
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  • Percentage of the total energy associated:
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  • Example 2: Calculate the percentage of output energy to input energy for the filter below:
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  • Energy at the input filter:
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  • Fourier Transforms for the output voltage:
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  • Thus,
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  • Energy at the output filter:
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  • Thus the percentage: