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MIT16_36s09_lec02

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

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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 .

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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? λ λ λ λ λ λ λ λ λ λ Eytan Modiano Slide 2

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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 δ (t) ! ( t ) = 0, " t # 0 ! ( t ) = 1 \$% % & ! ( t ) x ( t ) = x (0) "# # \$ ! ( t " % ) x ( % ) = x ( t ) "# # \$ F [ ! ( t )] = ! ( t ) e " j 2 # ft dt = e 0 = 1 "\$ \$ % ! ( t ) & 1 ! ( t ) 1 F [ ! ( t )] 0 Eytan Modiano Slide 4

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Rectangle pulse ! ( t ) = 1 | t | < 1/ 2 1/ 2 | t | = 1/ 2 0 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) => α x1(t) + β x2(t) <=> α X1(f) + β X2(f) Duality X(f) <=> x(t) => x(f) <=> X(-t) and x(-f)<=> X(t) Time-shifting: x(t- τ ) <=> X(f)e -j2 π f τ Scaling: F[(x(at)] = 1/|a| X(f/a) Convolution: x(t) <=> X(f), y(t) <=> Y(f) then, F[x(t)*y(t)] = X(f)Y(f)

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