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Lecture 16: Fourier transform - MIT OpenCourseWare

: Signals and Systems Fourier transform November 3, 2011 1 Last Time: Fourier Series Representing periodic signals as sums of sinusoids. new representations for systems as filters. Today: generalize for aperiodic signals. 2 Fourier transform An aperiodic signal can be thought of as periodic with infinite period. Let x(t) represent an aperiodic signal. x(t)t S S 0 Periodic extension : xT (t) = x(t + kT ) k= xT(t)t S STThen x(t) = lim xT (t). T 32 sin S 0= 2 /T =k 0=k2 TTakk Fourier transform Represent xT (t) by its Fourier series. xT(t)t S STak = 1 T T/2 T/2 xT (t)e j 2 Tktdt = 1 T S S e j 2 Tktdt = sin 2 kS T k = 2 T sin S 4 2 sin S 0= 2 /T =k 0=k2 TTakk Fourier transform Doubling period doubles # of harmonics in given frequency interval. xT(t)t S STak = 1 T T/2 T/2 xT (t)e j 2 Tktdt = 1 T S S e j 2 Tktdt = sin 2 kS T k = 2 T sin S 5 2 sin S 0= 2 /T =k 0=k2 TTakk Fourier transform As T , discrete harmonic amplitudes a continuum E( ).

Frequency plots provide intuition that is difficult to otherwise obtain. 13. Check Yourself. Find the Fourier transform of the following square pulse. 1 1 x. 1 (t) 1 t 1. X. 1 (jω ...

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Transcription of Lecture 16: Fourier transform - MIT OpenCourseWare

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