6 003 Signals and Systems Fourier Transform April 6 2010 Mid term Examination 2 Tomorrow April 7 7 30 9 30pm 34 101 No recitations tomorrow Coverage Lectures 1 15 Recitations 1 15 Homeworks 1 8 Homework 8 will not collected or graded Solutions are posted Closed book 2 pages of notes 8 21 11 inches front and back Designed as 1 hour exam two hours to complete Last Week Fourier Series Representing periodic signals as sums of sinusoids new representations for systems as filters This week generalize for aperiodic signals Fourier Transform An aperiodic signal can be thought of as periodic with infinite period Let x t represent an aperiodic signal x t S Periodic extension xT t t S X x t kT k xT t S Then x t lim xT t T S t T Fourier Transform Represent xT t by its Fourier series xT t S ak t S T Z Z 2 sin 2 kS 1 S j 2 kt 2 sin S 1 T 2 T xT t e j T kt dt e T dt T T 2 T S k T T ak 2 sin S k 0 k 2 T k 0 2 T Fourier Transform Doubling period doubles of harmonics in given frequency interval xT t S ak t S T Z Z 2 sin 2 kS 1 S j 2 kt 2 sin S 1 T 2 T xT t e j T kt dt e T dt T T 2 T S k T T ak 2 sin S k 0 k 2 T k 0 2 T Fourier Transform As T discrete harmonic amplitudes a continuum E xT t S ak t S T Z Z 2 sin 2 kS 1 S j 2 kt 2 sin S 1 T 2 T xT t e j T kt dt e T dt T T 2 T S k T T ak 2 sin S k 0 k 2 T k 0 2 T Z T 2 lim T ak lim T T T 2 x t e j t dt 2 sin S E Fourier Transform As T synthesis sum integral xT t S t S T ak T 2 sin S k 0 k 2 T k 0 2 T Z T 2 2 sin S E T T T 2 Z X X 1 0 1 j 2 kt j t x t E e T E e E ej t d T 2 2 z lim T ak lim k ak x t e j t dt k Fourier Transform Replacing E by X j yields the Fourier transform relations E X s s j X j Fourier transform Z x t e j t dt X j analysis equation x t Z 1 X j ej t d 2 synthesis equation Fourier Transform Replacing E by X j yields the Fourier transform relations E X s s j X j Fourier transform Z x t e j t dt X j analysis equation x t Z 1 X j ej t d 2 synthesis equation Form is similar to that of Fourier series provides alternate view of signal Relation between Fourier and Laplace Transforms If the Laplace transform of a signal exists and if the ROC includes the j axis then the Fourier transform is equal to the Laplace transform evaluated on the j axis Laplace transform Z x t e st dt X s Fourier transform Z X j x t e j t dt H s s j Relation between Fourier and Laplace Transforms Fourier transform inherits properties of Laplace transform Property x t X s X j Linearity ax1 t bx2 t aX1 s bX2 s aX1 j bX2 j Time shift x t t0 e st0 X s Time scale x at 1 s X a a e j t0 X j 1 j X a a Differentiation dx t dt sX s j X j Multiply by t tx t Convolution x1 t x2 t d X s ds X1 s X2 s 1 d X j j d X1 j X2 j Relation between Fourier and Laplace Transforms There are also important differences Compare Fourier and Laplace transforms of x t e t u t x t t Laplace transform Z Z t st X s e u t e dt e s 1 t dt 0 1 Re s 1 1 s a complex valued function of complex domain Fourier transform Z Z t j t X j e u t e dt e j 1 t dt 0 a complex valued function of real domain 1 1 j Laplace Transform The Laplace transform maps a function of time t to a complex valued function of complex valued domain s x t t Magnitude X s 1 1 s 10 0 Ima 1 0 gin 1 ary s 1 1 0eal s R Fourier Transform The Fourier transform maps a function of time t to a complex valued function of real valued domain x t t X j 1 1 j 0 1 Fourier Transform The Fourier transform maps a function of time t to a complex valued function of real valued domain x t t X j 1 1 j 0 1 Frequency plots provide intuition that is difficult to otherwise obtain Check Yourself Find the Fourier transform of the following square pulse x1 t 1 1 1 e e 2 3 X1 j e e 1 X1 j 1 t 2 X1 j 1 sin 4 X1 j 2 sin 5 none of the above Fourier Transform Compare the Laplace and Fourier transforms of a square pulse x1 t 1 1 1 Laplace transform Z 1 1 1 s e st X1 s e st dt e e s s 1 s 1 Fourier transform Z 1 1 2 sin e j t X1 j e j t dt j 1 1 t function of s j function of Check Yourself Find the Fourier transform of the following square pulse 4 x1 t 1 1 1 e e 2 3 X1 j e e 1 X1 j 1 t 2 X1 j 1 sin 4 X1 j 2 sin 5 none of the above Laplace Transform Laplace transform complex valued function of complex domain x1 t 1 t 1 1 X s 1 s e e s s 30 20 10 0 5 5 0 0 5 5 Fourier Transform The Fourier transform is a function of real domain frequency Time representation x1 t 1 1 1 t Frequency representation X1 j 2 sin 2 Check Yourself Signal x2 t and its Fourier transform X2 j are shown below x2 t X2 j 1 2 2 b t 0 Which is true 1 2 3 4 5 b 2 and 0 b 2 and 0 b 4 and 0 b 4 and 0 none of the 2 2 2 2 above Check Yourself Find the Fourier transform 2 Z 2 X2 j e 2 j t 2 sin 2 e j t 4 sin 2 dt j 2 2 4 2 Check Yourself …
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