6 003 Signals and Systems Sampling November 24 2009 Sampling Conversion of a continuous time signal to discrete time x t x n n t 0 2 4 6 8 10 0 2 4 6 We have used sampling a number of times before Today new insights from Fourier representations 8 10 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 Sampling Sampling is pervasive Example digital cameras record sampled images y I x y n x I m n m Sampling Photographs in newsprint are half tone images black or white and the average conveys brightness Each point is Sampling Zoom in to see the binary pattern Sampling Even high quality photographic paper records discrete images When AgBr crystals 0 04 1 5 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 Sampling Every image that we see is sampled by the retina which contains 100 million rods and 6 million cones average spacing 3 m which act as discrete sensors http webvision med utah edu imageswv sagschem jpeg 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 adequate to resolve individual pixels in this slide Check Yourself The spacing of rods and cones limits the angular resolution of your retina to approximately eye 3 10 6 m rod cone spacing 10 4 radians diameter of eye 3 cm The angle between pixels viewed from the center of the classroom is approximately pixels screen size 1024 3 m 1024 3 10 4 radians distance to screen 10 m Light from a single pixel falls upon multiple rods and cones Sampling How does sampling affect the information contained in a signal Sampling We would like to sample in a way that preserves information which may not seem possible x t t Information between samples is lost Therefore the same samples can represent multiple signals cos 7 3 n cos 3 n t Sampling and Reconstruction To determine the effect of sampling compare the original signal x t to the signal xp t that is reconstructed from the samples x n Uniform sampling sampling interval T x n x nT t n Impulse reconstruction xp t X x n t nT n t n Sampling Impulse reconstuction produces a signal xp t that is equal to the original signal x t multiplied by an impulse train xp t X x n t nT n X n X x nT t nT x t t nT n X x t t nT n z p t xp t is motivated by impulse reconstruction top line can be understood entirely within CT framework bottom line Sampling Multiplication by an impulse train in time is equivalent to convolution by an impulse train in frequency generates multiple copies of original frequency content X j 1 W W P j 2 T s s 1 X j P j Xp j 2 1 T s s 2 T Check Yourself What is the relation between the DTFT of x n x nT P and the CTFT of xp t x n t nT for X j below X j 1 W W 1 Xp j X e j 2 Xp j X e j T 3 Xp j X e j T 4 Xp j X e j 5 none of the above Check Yourself DTFT X ej X x n e j n n CTFT of xp t Z Xp j X x n t nT e j t dt n X Z x n n X t nT e j t dt x n e j nT n X ej T Check Yourself Xp j X e j T X j 1 W W 1 X j P j Xp j 2 1 T s s 2 T X ej Xp j T 1 T 2 2 Check Yourself What is the relation between the DTFT of x n x nT P and the CTFT of xp t x n t nT for X j below P j 2 T s s 1 Xp j X e j 2 Xp j X e j T 3 Xp j X e j T 4 Xp j X e j 5 none of the above Sampling The high frequency copies can be removed with a low pass filter also multiply by T to undo the amplitude scaling 1 X j P j Xp j 2 1 T T s 2 s 2 Impulse reconstruction followed by ideal low pass filtering is called bandlimited reconstruction 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 2 2 m s T The minimum sampling frequency 2 m is called the Nyquist rate Summary Three important ideas Sampling x t x n x nT Bandlimited Reconstruction x n Impulse Reconstruction LPF T xp t P x n t nT Sampling Theorem If X j 0 2s s 2 s then xr t x t 2 xr t Check Yourself We can hear sounds with frequency components between 20 Hz and 20 kHz What is the maximum sampling interval T that can be used to sample a signal without loss of audible information 1 100 s 3 25 s 5 50 s 2 50 s 4 100 s 6 25 s Check Yourself s 2 2 2T 1 1 T 25 s 2fm 2 20 kHz 2 fm m Check Yourself We can hear sounds with frequency components between 20 Hz and 20 kHz What is the maximum sampling interval T that can be used to sample a signal without loss of audible information 1 100 s 3 25 s 5 50 s 2 50 s 4 100 s 6 25 s CT Model of Sampling and Reconstruction Sampling followed by bandlimited reconstruction is equivalent to multiplying by an impulse train and then low pass filtering LPF x t T xp t s 2 2s xr t p t p t sampling function 0 T t Aliasing What happens if X contains frequencies X j T P j 2 T s s 1 X j P j Xp j 2 1 T 2s s 2 Aliasing What happens if X contains frequencies X j T P j 2 T s s 1 X j P j Xp j 2 1 T 2s s 2 Aliasing What happens if X contains frequencies X j T P j 2 T s s 1 X j P j Xp j 2 1 T 2s s 2 Aliasing What happens if X contains frequencies X j T P j 2 T s s 1 X j P j Xp j 2 1 T 2s s 2 Aliasing The effect of aliasing is to wrap frequencies Output frequency s 2 Input …
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