lec21 - 6.003: Signals and Systems Sampling November 24,...

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6.003: Signals and Systems Sampling November 24, 2009
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Sampling Conversion of a continuous-time signal to discrete time. t x ( t ) 0 2 4 6 8 10 n x [ n ] 0 2 4 6 8 10 We have used sampling a number of times before. Today: new insights from Fourier representations.
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Sampling Sampling allows the use of modern digital electronics to process, record, transmit, store, and retrieve CT signals. audio: MP3, CD, cell phone pictures: digital camera, printer video: DVD everything on the web
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Sampling Sampling is pervasive. Example: digital cameras record sampled images. x y I ( x,y ) m n I [ m,n ]
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Sampling Photographs in newsprint are “half-tone” images. Each point is black or white and the average conveys brightness.
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Sampling Zoom in to see the binary pattern.
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Sampling Even high-quality photographic paper records discrete images. When AgBr crystals ( 0 . 04 1 . 5 µ m) are exposed to light, some of the Ag is reduced to metal. During “development” the exposed grains are completely reduced to metal and unexposed grains are removed.
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Sampling Every image that we see is sampled by the retina, which contains 100 million rods and 6 million cones (average spacing 3 µ m) which act as discrete sensors. http://webvision.med.utah.edu/imageswv/sagschem.jpeg
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Check Yourself Your retina is sampling this slide, which is composed of 1024 × 768 pixels. Is the spatial sampling done by your rods and cones ade- quate to resolve individual pixels in this slide?
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Check Yourself The spacing of rods and cones limits the angular resolution of your retina to approximately θ eye = rod/cone spacing diameter of eye 3 × 10 6 m 3 cm 10 4 radians The angle between pixels viewed from the center of the classroom is approximately θ pixels = screen size / 1024 distance to screen 3 m / 1024 10 m 3 × 10 4 radians Light from a single pixel falls upon multiple rods and cones.
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Sampling How does sampling affect the information contained in a signal?
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Sampling We would like to sample in a way that preserves information, which may not seem possible. t x ( t ) Information between samples is lost. Therefore, the same samples can represent multiple signals. t cos 7 π 3 n ? cos π 3 n ?
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Sampling and Reconstruction To determine the effect of sampling, compare the original signal x ( t ) to the signal x p ( t ) that is reconstructed from the samples x [ n ] . 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 )
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Sampling Impulse reconstuction produces a signal x p ( t ) that is equal to the original signal x ( t ) multiplied by an impulse train. x
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lec21 - 6.003: Signals and Systems Sampling November 24,...

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