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Unformatted text preview: 6.003: Signals and Systems Lecture 22 December 1, 2009 1 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 ) = ∞ Ø n = −∞ x [ n ] δ ( t − nT ) Relation: x p ( t ) = x ( t ) p ( t ) where p ( t ) = ∞ Ø n = −∞ δ ( t − nT ) Fourier Representation Bandlimited reconstruction: lowpass filter, remove high frequencies. ω X ( jω ) − W W 1 ω P ( jω ) − ω s ω s 2 π T ω X p ( jω ) = 1 2 π ( X ( j · ) ∗ P ( j · ))( ω ) 1 T − ω s 2 ω s 2 T The Sampling Theorem If signal is bandlimited → sample without loosing information. If x ( t ) is bandlimited so that X ( jω ) = 0 for | ω | > ω m then x ( t ) is uniquely determined by its samples x ( nT ) if ω s = 2 π T > 2 ω m . The minimum sampling frequency, 2 ω m , is called the “Nyquist rate.” Aliasing Aliasing results if there are frequency components with ω > ω s 2 . ω X p ( jω ) = 1 2 π ( X ( j · ) ∗ P ( j · ))( ω ) − ω s 2 ω s 2 1 T ω P ( jω ) − ω s ω s 2 π T ω X ( jω ) 1 Anti-Aliasing Filter To avoid aliasing, remove frequency components that alias before sampling. − ω s 2 ω s 2 ω 1 × − ω s 2 ω s 2 ω T x ( t ) p ( t ) x r ( t ) x p ( t ) Reconstruction Filter Anti-aliasing Filter 6.003: Signals and Systems Lecture 22 December 1, 2009 2 Aliasing Aliasing increases as the sampling rate decreases. ω X p ( jω ) = 1 2 π ( X ( j · ) ∗ P ( j · ))( ω ) − ω s 2 ω s 2 1 T ω P ( jω ) − ω s ω s 2 π T ω Anti-aliased X ( jω ) Anti-Aliasing Demonstration Sampling Music ω s = 2 π T = 2 πf s • f s = 11 kHz without anti-aliasing • f s = 11 kHz with anti-aliasing • f s = 5 . 5 kHz without anti-aliasing • f s = 5 . 5 kHz with anti-aliasing • f s = 2 . 8 kHz without anti-aliasing • f s = 2 . 8 kHz with anti-aliasing J.S. Bach, Sonata No. 1 in G minor Mvmt. IV. Presto Nathan Milstein, violin Quantization Analog to digital conversion requires not only sampling in time but also quantization in amplitude....
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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