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Unformatted text preview: 6.003: Signals and Systems Sampling and Quantization December 1, 2009 Last Time: Sampling and Reconstruction Uniform sampling (sampling interval T ): t n x [ n ] = x ( nT ) Impulse reconstruction: t n x p ( t ) = n = x [ n ] ( t nT ) Relation: x p ( t ) = x ( t ) p ( t ) where p ( t ) = n = ( t nT ) Fourier Representation Sampling followed by impulse reconstruction multiplication by an impulse train in time convolving by an impulse train in frequency X ( j ) W W 1 P ( j ) s s 2 T X p ( j ) = 1 2 ( X ( j ) P ( j ))( ) s s = 2 T 1 T Fourier Representation Bandlimited reconstruction: lowpass filter, remove high frequencies. X ( j ) W W 1 P ( j ) s s 2 T X p ( j ) = 1 2 ( X ( j ) P ( j ))( ) 1 T s 2 s 2 T The Sampling Theorem If signal is bandlimited sample without loosing information. If x ( t ) is bandlimited so that X ( j ) = 0 for   > m then x ( t ) is uniquely determined by its samples x ( nT ) if s = 2 T > 2 m . The minimum sampling frequency, 2 m , is called the Nyquist rate. Aliasing Aliasing results if there are frequency components with > s 2 . X p ( j ) = 1 2 ( X ( j ) P ( j ))( ) s 2 s 2 1 T P ( j ) s s 2 T X ( j ) 1 Aliasing Aliasing results if there are frequency components with > s 2 . X p ( j ) = 1 2 ( X ( j ) P ( j ))( ) s 2 s 2 1 T P ( j ) s s 2 T X ( j ) 1 Aliasing Aliasing results if there are frequency components with > s 2 . X p ( j ) = 1 2 ( X ( j ) P ( j ))( ) s 2 s 2 1 T P ( j ) s s 2 T X ( j ) 1 Aliasing Aliasing results if there are frequency components with > s 2 . X p ( j ) = 1 2 ( X ( j ) P ( j ))( ) s 2 s 2 1 T P ( j ) s s 2 T X ( j ) 1 AntiAliasing Filter To avoid aliasing, remove frequency components that alias before sampling. s 2 s 2 1 s 2 s 2 T x ( t ) p ( t ) x r ( t ) x p ( t ) Reconstruction Filter Antialiasing Filter Aliasing Aliasing increases as the sampling rate decreases. X p ( j ) = 1 2 ( X ( j ) P ( j ))( ) s 2 s 2 1 T P ( j ) s s 2 T X ( j ) 1 Aliasing Aliasing increases as the sampling rate decreases. X p ( j ) = 1 2 ( X ( j ) P ( j ))( ) s 2 s 2 1 T P ( j ) s s 2 T Antialiased X ( j ) Aliasing Aliasing increases as the sampling rate decreases....
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This note was uploaded on 08/24/2011 for the course EECS 6.003 taught by Professor Dennism.freeman during the Spring '11 term at MIT.
 Spring '11
 DennisM.Freeman

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