Frequency Domain

# Frequency Domain - Circuits and Systems in the Frequency...

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Circuits and Systems in the Frequency Domain Fall 2010 EE 230: Electronic Circuits and Systems Timothy A. Bigelow 1

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Continuous-Time Periodic Signals: Review of Fourier Series   o jk t xt X ke k  1 j kt k x t e d t 2     o o o T Xk T o o T  T o = fundamental period = fundamental frequency 2 o fundamental frequency k = integer values
Fourier Series Example 1 4 c o s 2 5 s i n 3 t t t       34 xt  3

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Fourier Series Example 1 4 c o s 2 5 s i n 3 t t t       34 xt  4
  1 o o jk t o T X kx t e d t T Fourier Series Example 2 xt   s 1 t -T o -2T o T o 2T o T o /s o /s 5

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MATLAB Plot of Solution, s=5 6
MATLAB Plot of Solution, s=10 7

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MATLAB Plot of Solution, s=25 8
Question: What happens to the Fourier Representation of the signal as the idth in time decreases? width in time decreases? Narrow in Time Broad in Frequency Broad in Time Narrow in Frequency 9

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Continuous-Time Nonperiodic Signals: Fourier Transform ecall for Periodic Signals o j kt t Xke 1 o jk t k x te d t Recall for Periodic Signals     k xt      o o T Xk T (t) x(t) 2 o o T  t -T o -2T o T o 2T o 10
Fourier Transform et T Let T o x(t) t 2 Small Continuous Value oo o k T   11       j t X kX j x t e d t  

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Example for Increasing T o , T o = 1 12
Example for Increasing T o , T o = 2 13

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Example for Increasing T o , T o = 3 14
Example for Increasing T o , T o = 100 15

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Fourier Transform In order to represent aperiodic signals, we need a continuum of equencies frequencies.  1 2 jt xt X j e d  Called Inverse Fourier Transform of X(j ) 16
 jt X jx t e d t  Review of Laplace Transform st X sx t e d

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Frequency Domain - Circuits and Systems in the Frequency...

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