6 003 Signals and Systems Lecture 21 6 003 Signals and Systems April 27 2010 Mid term Examination 3 Tomorrow Wednesday April 28 7 30 9 30pm 34 101 Sampling No recitations tomorrow Coverage Lectures 1 20 Recitations 1 20 Homeworks 1 11 Homework 11 will not collected or graded Solutions are posted Closed book 3 pages of notes 8 12 11 inches front and back Designed as 1 hour exam two hours to complete April 27 2010 Sampling Sampling Conversion of a continuous time signal to discrete time Sampling allows the use of modern digital electronics to process record transmit store and retrieve CT signals x t 0 x n 2 4 6 8 10 t 0 2 4 6 8 10 audio MP3 CD cell phone pictures digital camera printer video DVD everything on the web n We have used sampling a number of times before Today new insights from Fourier representations Sampling Sampling Sampling is pervasive Photographs in newsprint are half tone images black or white and the average conveys brightness Example digital cameras record sampled images y I x y n x I m n m 1 Each point is 6 003 Signals and Systems Lecture 21 April 27 2010 Sampling Sampling Zoom in to see the binary pattern 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 Check Yourself 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 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 http webvision med utah edu imageswv sagschem jpeg Sampling Sampling How does sampling affect the information contained in a signal 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 2 6 003 Signals and Systems Lecture 21 April 27 2010 Sampling and Reconstruction 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 Impulse reconstuction produces a signal xp t that is equal to the original signal x t multiplied by an impulse train Uniform sampling sampling interval T xp t x n x nT t n X x n t nT n X n X x nT t nT x t t nT n Impulse reconstruction xp t X x t X t nT n x n t nT n z p t xp t is motivated by impulse reconstruction top line can be understood entirely within CT framework bottom line t n Sampling Check Yourself Multiplication by an impulse train in time is equivalent to convolution by an impulse train in frequency 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 generates multiple copies of original frequency content X j 1 X j 1 W W W W P j 2 T s Xp j 1 2 1 Xp j X e j s 2 Xp j X e j T 3 Xp j X e j T X j P j 1 T s 4 Xp j X e j 5 none of the above s 2 T Sampling The Sampling Theorem The high frequency copies can be removed with a low pass filter also multiply by T to undo the amplitude scaling If signal is bandlimited sample without loosing information If x t is bandlimited so that 1 X j P j Xp j 2 1 T X j 0 for m T s 2 s 2 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 Impulse reconstruction followed by ideal low pass filtering is called bandlimited reconstruction 3 6 003 Signals and Systems Lecture 21 Summary Check Yourself Three important ideas We can hear sounds with frequency components between 20 Hz and 20 kHz Sampling What is the maximum sampling interval T that can be used to sample a signal without loss of audible information x t x n x nT Bandlimited Reconstruction x n April 27 2010 Impulse Reconstruction LPF xp t P T 2s x n t nT s 2 1 100 s 3 25 s 5 50 s xr t 2 50 s 4 100 s 6 25 s s then xr t x t 2 Sampling Theorem If X j 0 CT Model of Sampling and Reconstruction Aliasing Sampling followed by bandlimited reconstruction is equivalent to multiplying by an impulse train and then low pass filtering What happens if X contains frequencies X j T LPF x t xp t T s 2 2s P j xr t 2 T p t s 1 X j P j Xp j 2 p t sampling function 1 T t 0 T s 2s Aliasing s 2 Check Yourself The effect of aliasing is to wrap frequencies A periodic signal with a period of 0 1 ms is sampled at 44 kHz Output frequency s 2 s 2 To what frequency does the eighth harmonic alias Input frequency 1 18 kHz 3 14 kHz 5 6 kHz X j 1 T 2s s 2 4 2 16 kHz 4 8 kHz 6 none of the above 6 003 Signals and Systems Lecture 21 April 27 2010 Aliasing Aliasing High frequency components of complex signals also wrap Aliasing increases as the sampling rate decreases X j 1 X j 1 P j P j 2 T s 2 T s s 1 X j P j Xp j 2 1 X j P j Xp j 2 1 T 2s s 2 s 1 T 2s 2s Aliasing Demonstration Anti Aliasing Filter Sampling Music To avoid aliasing remove frequency components that alias before sampling s 2 2 fs T fs 44 1 kHz fs 22 kHz fs 11 kHz x t fs 5 5 kHz Anti aliasing Filter 1 s s 2 2 Reconstruction Filter T xp t s s 2 2 p t fs 2 8 kHz J S Bach Sonata No 1 in G minor Mvmt IV Presto Nathan Milstein violin Aliasing Anti Aliasing Demonstration Aliasing increases as the sampling rate decreases Sampling Music Anti aliased X j s fs 11 kHz without anti aliasing P j fs 11 kHz with anti aliasing 2 T s Xp j 1 2 fs 5 5 kHz without anti aliasing s fs 5 5 kHz with anti aliasing fs 2 8 kHz without anti aliasing X j P j 1 T 2s 2s 2 2 fs T fs 2 8 kHz with anti …
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