MIT16_36s09_lec02

MIT16_36s09_lec02 - MIT OpenCourseWare http://ocw.mit.edu...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 16.36 Communication Systems Engineering Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Lecture 2: The Sampling Theorem Eytan Modiano Eytan Modiano Slide 1 Sampling Given a continuous time waveform, can we represent it using discrete samples? How often should we sample? Can we reproduce the original waveform? Eytan Modiano Slide 2 The Fourier Transform Frequency representation of signals Definition: X ( f ) = x ( t ) e ! j 2 " ft !# # $ dt x ( t ) = X ( f ) e j 2 " ft !# # $ df Notation: X(f) = F[x(t)] X(t) = F-1 [X(f)] x(t) X(f) Eytan Modiano Slide 3 Unit impulse (t) ! ( t ) = 0, " t # ! ( t ) = 1 $% % & ! ( t ) x ( t ) = x (0) "# # $ ! ( t " % ) x ( % ) = x ( t ) "# # $ F [ ! ( t )] = ! ( t ) e " j 2 # ft dt = e = 1 "$ $ % ! ( t ) & 1 ! ( t ) 1 F [ ! ( t )] 0 Eytan Modiano Slide 4 Rectangle pulse ! ( t ) = 1 | t | < 1/2 1/2 | t | = 1/2 otherwise " # $ % $ F [ ! ( t )] = ! ( t ) e & j 2 ft dt = &( ( ) e & j 2 ft dt & 1/2 1/2 ) = e & j f & e j f & j 2 f = Sin ( f ) f = Sinc ( f ) ! ( t ) 1/2 1/2 Eytan Modiano Slide 5 1 Properties of the Fourier transform Linearity x1(t) <=> X1(f), x2(t) <=>X2(f) => x1(t) + x2(t) <=> X1(f) + X2(f) Duality X(f) <=> x(t) => x(f) <=> X(-t) and x(-f)<=> X(t)...
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MIT16_36s09_lec02 - MIT OpenCourseWare http://ocw.mit.edu...

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