# Lect-03Part2_07 - Fourier transform of an Impulse train An...

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Fourier transform of an Impulse train An impulse train in time domain is transformed into another impulse train in the frequency domain.

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Sampling Theorem(contd.) Impulse train and its spectrum
Rayleigh’s Energy Theorem: - - = ϖ d X X dt t x t x ) ( * ) ( ) ( * ) ( 2 | ) ( | ) ( X xx S = Energy Spectral Density(ESD) - = π d xx S E ) ( ) 2 / 1 ( = 0 2 | ) ( | / 1 d X

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If input ESD is S xx (ω), for a LTI system, output ESD will be x(t) S xx (ω) h(t) LTI system y(t) ) ( 2 | ) ( | ) ( ϖ xx S H yy S = Where = ) ( ) ( t h F H
Need for going to frequency domain: There are three primary reasons 1. To estimate the spectral occupancy of a given signal [ i.e. ., the frequency extent of the signal for subsequently estimating Bandwidth ] 2. Convolution in time domain reduces to the multiplication operation in frequency domain 3. Differential equation become algebraic expressions in the frequency domain

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Lect-03Part2_07 - Fourier transform of an Impulse train An...

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