6 003 Signals and Systems Lecture 17 November 10 2009 6 003 Signals and Systems Mid term Examination 3 Fourier Transform Wednesday November 18 7 30 9 30pm Walker Memorial this exam is after drop date No recitations on the day of the exam Coverage cumulative with more emphasis on recent material lectures 1 18 homeworks 1 10 Homework 10 will not collected or graded Solutions will be posted Closed book 3 page of notes 8 12 11 inches front and back Designed as 1 hour exam two hours to complete Review sessions during open office hours November 10 2009 Conflict Contact freeman mit edu by Friday November 13 2009 Last Week Fourier Series Fourier Transform Representing periodic signals as sums of sinusoids An aperiodic signal can be thought of as periodic with infinite period new representations for systems as filters Let x t represent an aperiodic signal x t This week generalize for aperiodic signals S Periodic extension xT t t S X x t kT k xT t S S t T Then x t lim xT t T Fourier Transform Fourier Transform Represent xT t by its Fourier series Doubling period doubles of harmonics in given frequency interval xT t S ak xT t S t T S Z Z 2 sin 2 kS 1 T 2 1 S j 2 kt 2 sin S 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 ak S Z Z 2 sin 2 kS 1 T 2 1 S j 2 kt 2 sin S T xT t e j T kt dt e T dt T T 2 T S k T T ak 2 T t T 2 sin S k k 0 k 2 T k 0 2 T 0 2 T 1 6 003 Signals and Systems Lecture 17 November 10 2009 Fourier Transform Fourier Transform As T discrete harmonic amplitudes a continuum E As T synthesis sum integral xT t S ak xT t S t T S T ak 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 lim T ak lim T T T 2 2 sin S k 0 k t 2 T k 2 k 0 k T Z T 2 0 2 T 2 sin S E Z X X 2 1 0 1 x t E ej T kt E ej t E ej t d 2 2 T z lim T ak lim T 0 2 T x t e j t dt T k Z T 2 S 2 sin S E T T 2 k ak x t e j t dt k Fourier Transform Relation between Fourier and Laplace Transforms Replacing E by X j yields the Fourier transform relations 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 E X s s j X j Fourier transform Z X j x t e j t dt Laplace transform Z x t e st dt X s analysis equation Z 1 x t X j ej t d 2 Fourier transform Z X j x t e j t dt H s s j synthesis equation Relation between Fourier and Laplace Transforms Relation between Fourier and Laplace Transforms Fourier transform inherits properties of Laplace transform There are also important differences 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 Differentiation dx t dt 1 s X a a e j t0 X j 1 j X a a sX s j X j Multiply by t tx t Convolution x1 t x2 t d X s ds X1 s X2 s Compare Fourier and Laplace transforms of x t e t u t x t t Laplace transform Z Z X s e t u t e st dt e s 1 t dt 1 d X j j d 0 1 Re s 1 1 s a complex valued function of complex domain X1 j X2 j Fourier transform Z Z X j e t u t e j t dt e j 1 t dt 0 a complex valued function of real domain 2 1 1 j 6 003 Signals and Systems Lecture 17 November 10 2009 Laplace Transform Fourier Transform The Laplace transform maps a function of time t to a complex valued function of complex valued domain s The Fourier transform maps a function of time t to a complex valued function of real valued domain x t 1 1 s Magnitude X s t 10 1 X j 1 j 0 Ima 1 0 gin 1 ary s 1 1 0eal s R 0 1 Frequency plots promote intuition in ways not possible with s Check Yourself Laplace Transform Laplace transform complex valued function of complex domain Find the Fourier transform of the following square pulse X1 s x1 t 1 s e e s s 1 30 1 t 1 1 e e 2 3 X1 j e e 1 X1 j 20 2 X1 j 1 sin 4 X1 j 2 sin 10 0 5 5 0 0 5 5 5 none of the above Fourier Transform Fourier Transform The Fourier transform is a function of real domain frequency One of the most useful features of the Fourier transform and Fourier series is the simple inverse Fourier transform Z X j x t e j t dt Fourier transform Time representation X j 0 Frequency representation 1 1 1 j Z 1 x t X j ej t d 2 X1 j 2 3 inverse Fourier transform 6 003 Signals and Systems Lecture 17 November 10 2009 Inverse Fourier Transform Fourier Transform Find the impulse reponse of an ideal low pass filter The Fourier transform and its inverse have very similar forms Z x t e j t dt Fourier transform X j H j 1 h t 0 0 Z 1 x t X j ej t d 2 Z Z 0 sin 0 t 1 1 1 ej t 0 H j ej t d ej t d 2 2 0 2 jt 0 t inverse Fourier transform Convert one to the other by t t scale by 2 h t 0 t 0 This result is not so easily obtained without inverse relation Duality Duality The Fourier transform and its inverse have very similar forms X j x t Z Using duality to find new transform pairs x1 t f t X1 j g x t e j t dt Z 1 X j ej t d 2 x2 t g t X2 j 2 f Two differences t flip 2 t f t t minus sign flips time axis …
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