6 003 Signals and Systems Sampling and Quantization April 29 2010 What to do with a billion transistors Gene Frantz Texas Instruments Seminar today 32 155 4pm We are getting closer to a time when we will be able to cost effectively integrate billions of transistors on an integrated circuit In fact we are seeing the beginning of this era with the broad adoption of multi processing system on chips which has both advantages and disadvantages that should be considered This talk will discuss the options we have the issues we must face and the future we can look forward to Last Time 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 Last Time Sampling Theory 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 Practice Aliasing anti aliasing filter 2s s 2 s then xr t x t 2 xr t Today Digital recording transmission storage and retrieval requires discrete representations of both time e g sampling and amplitude audio MP3 CD cell phone pictures digital camera printer video DVD everything on the web Quantization discrete representations for amplitudes Quantization Output voltage We measure discrete amplitudes in bits 1 2 bits 3 bits 10 4 bits 01 0 1 00 1 0 Input voltage 1 1 0 Input voltage 1 1 0 Input voltage 1 0 5 Time second 1 0 0 5 Time second 1 0 0 5 Time second 1 1 0 1 0 Bit rate bits sample samples sec Check Yourself We hear sounds that range in amplitude from 1 000 000 to 1 How many bits are needed to represent this range 1 2 3 4 5 5 bits 10 bits 20 bits 30 bits 40 bits Check Yourself How many bits are needed to represent 1 000 000 1 bits range 1 2 2 4 3 8 4 16 5 32 6 64 7 128 8 256 9 512 10 1 024 11 2 048 12 4 096 13 8 192 14 16 384 15 32 768 16 65 536 17 131 072 18 262 144 19 524 288 20 1 048 576 Check Yourself We hear sounds that range in amplitude from 1 000 000 to 1 How many bits are needed to represent this range 1 2 3 4 5 5 bits 10 bits 20 bits 30 bits 40 bits 3 Quantization Demonstration Quantizing Music 16 bits sample 8 bits sample 6 bits sample 4 bits sample 3 bits sample 2 bit sample J S Bach Sonata No 1 in G minor Mvmt IV Presto Nathan Milstein violin Quantization Output voltage We measure discrete amplitudes in bits 1 2 bits 3 bits 10 4 bits 01 0 1 00 1 0 Input voltage 1 1 0 Input voltage 1 1 0 Input voltage 1 0 5 Time second 1 0 0 5 Time second 1 0 0 5 Time second 1 1 0 1 0 Example audio CD bits samples sec 2 channels 16 44 100 60 74 min 6 3 G bits sample sec min 0 78 G bytes Quantizing Images Converting an image from a continuous representation to a discrete representation involves the same sort of issues This image has 280 280 pixels with brightness quantized to 8 bits Quantizing Images 8 bit image 7 bit image Quantizing Images 8 bit image 6 bit image Quantizing Images 8 bit image 5 bit image Quantizing Images 8 bit image 4 bit image Quantizing Images 8 bit image 3 bit image Quantizing Images 8 bit image 2 bit image Quantizing Images 8 bit image 1 bit image Check Yourself What is the most objectionable artifact of coarse quantization 8 bit image 4 bit image Dithering One very annoying artifact is banding caused by clustering of pixels that quantize to the same level Banding can be reduced by dithering Dithering adding a small amount 21 quantum of random noise to the image before quantizing Since the noise is different for each pixel in the band the noise causes some of the pixels to quantize to a higher value and some to a lower But the average value of the brightness is preserved Quantizing Images with Dither 7 bit image 7 bits with dither Quantizing Images with Dither 6 bit image 6 bits with dither Quantizing Images with Dither 5 bit image 5 bits with dither Quantizing Images with Dither 4 bit image 4 bits with dither Quantizing Images with Dither 3 bit image 3 bits with dither Quantizing Images with Dither 2 bit image 2 bits with dither Quantizing Images with Dither 1 bit image 1 bit with dither Check Yourself What is the most objectionable artifact of dithering 3 bit image 3 bit dithered image Robert s Technique One annoying feature of dithering is that it adds noise The noise can be reduced using Robert s technique Robert s technique add a small amount 12 quantum of random noise before quantizing then subtract that same amount of random noise Quantizing Images with Robert s Method 7 bits with dither 7 bits with Robert s method Quantizing Images with Robert s Method 6 bits with dither 6 bits with Robert s method Quantizing Images with Robert s Method 5 bits with dither 5 bits with Robert s method Quantizing Images with Robert s Method 4 bits with dither 4 bits with Robert s method Quantizing Images with Robert s Method 3 bits with dither 3 bits with Robert s method Quantizing Images with Robert s Method 2 bits with dither 2 bits with Robert s method Quantizing Images with Robert s Method 1 bits with dither 1 bit with Robert s method Quantizing Images 3 bits 8 bits 3 bits dither Robert s Quantizing Images 2 bits 8 bits 2 bits dither Robert s Quantizing Images 1 bit 8 bits 1 bit dither Robert s Progressive Refinement Trading precision for speed Start by sending a crude representation then progressively update with increasing higher fidelity versions Discrete Time Sampling Resampling DT sampling is much like CT sampling x n p n xp n P k n kN x n 0 p n 0 xp n 0 n n n Discrete Time Sampling As in CT sampling introduces additional copies of X ej x n xp n p n P k n kN X ej 1 2 0 P ej 2 2 3 2 4 3 2 3 2 0 3 j Xp e 4 3 2 4 3 2 1 3 2 4 3 2 3 0 2 3 Discrete Time Sampling Sampling a finite sequence gives rise to a shorter sequence x n n 0 xp n n 0 xb n n 0 Xb ej X n xb n e j n X n xp 3n e j n X k xp k e j k 3 Xp ej 3 Discrete Time Sampling But the shorter sequence has a wider frequency representation X ej 1 2 0 2 Xp ej …
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