The Fourier Transform

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The Fourier Transform
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Fourier Transform

• Relates t and f domains for energy signals
• Pair is unique

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Fourier Transform (contd)

• For real x(t),
• X(f) X(-f) Even Symmetric
• ?(f) - ?(- f) Odd Symmetric

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Example for Amplitude and Phase Spectra of a
pulse signal

• (a) Time Function

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Example for Amplitude and Phase Spectra of a
pulse signal (contd)

• (b) Amplitude Spectrum

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Exercise

• Does Fourier Series give continuous spectrum?

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Parsevals theorem for energy signals
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Some Fourier transforms involving impulse
functions

• The impulse function
• The eternal complex exponential

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• Sinusoidal signals
• Signum function and unit step

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• Periodic functions

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Properties of Fourier Transform

• Linearity
• Complex conjugate

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Properties of Fourier Transform (cont.)

• Symmetry
• Duality

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Properties of Fourier Transform (cont.)

• Scaling
• Time shifting

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Properties of Fourier Transform

• Frequency Shifting

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Properties of Fourier Transform

• Differentiation and Integration

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Properties of Fourier Transform

• Time convolution
• Frequency convolution