6 003 Signals and Systems Lecture 22 April 29 2010 6 003 Signals and Systems What to do with a billion transistors Sampling and Quantization 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 April 29 2010 Last Time Sampling Last Time Sampling Sampling allows the use of modern digital electronics to process record transmit store and retrieve CT signals Theory Sampling x t x n x nT audio MP3 CD cell phone pictures digital camera printer video DVD everything on the web Bandlimited Reconstruction Impulse Reconstruction x n LPF T xp t P 2s x n t nT Sampling Theorem If X j 0 s 2 xr t s then xr t x t 2 Practice Aliasing anti aliasing filter Quantization Digital recording transmission storage and retrieval requires discrete representations of both time e g sampling and amplitude We measure discrete amplitudes in bits Output voltage Today audio MP3 CD cell phone pictures digital camera printer video DVD everything on the web 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 Quantization discrete representations for amplitudes 0 1 0 Bit rate bits sample samples sec 1 6 003 Signals and Systems Lecture 22 Check Yourself April 29 2010 Quantization Demonstration Quantizing Music We hear sounds that range in amplitude from 1 000 000 to 1 16 bits sample How many bits are needed to represent this range 8 bits sample 6 bits sample 1 2 3 4 5 5 bits 10 bits 20 bits 30 bits 40 bits 4 bits sample 3 bits sample 2 bit sample J S Bach Sonata No 1 in G minor Mvmt IV Presto Nathan Milstein violin Quantizing Images We measure discrete amplitudes in bits Converting an image from a continuous representation to a discrete representation involves the same sort of issues Output voltage Quantization 1 2 bits 3 bits 10 4 bits This image has 280 280 pixels with brightness quantized to 8 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 Check Yourself Dithering What is the most objectionable artifact of coarse quantization Dithering adding a small amount 12 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 8 bit image 4 bit image 2 6 003 Signals and Systems Lecture 22 April 29 2010 Check Yourself Robert s Technique What is the most objectionable artifact of dithering Robert s technique add a small amount 12 quantum of random noise before quantizing then subtract that same amount of random noise 3 bit image 3 bit dithered image Quantizing Images with Robert s Method 3 bits with dither Quantizing Images 3 bits 8 bits 3 bits dither Robert s 3 bits with Robert s method Progressive Refinement Discrete Time Sampling Resampling Trading precision for speed DT sampling is much like CT sampling Start by sending a crude representation then progressively update with increasing higher fidelity versions x n xp n p n k n kN P x n 0 p n 0 xp n 0 3 n n n 6 003 Signals and Systems Lecture 22 April 29 2010 Discrete Time Sampling Discrete Time Sampling As in CT sampling introduces additional copies of X ej Sampling a finite sequence gives rise to a shorter sequence x n xp n p n x n k n kN P 2 0 P ej 2 2 3 2 4 2 3 3 n 0 xp n X ej 1 2 0 3 Xp ej 4 3 2 n 0 xb n n 0 1 3 2 2 3 4 3 0 2 3 4 3 2 Xb ej Discrete Time Sampling X n xb n e j n X n xp 3n e j n X xp k e j k 3 Xp ej 3 k Discrete Time Sampling But the shorter sequence has a wider frequency representation X ej 1 0 2 2 Xp ej 1 3 2 0 2 Xb ej Xp ej 3 1 3 2 4 3 2 3 0 2 3 4 3 2 Discrete Time Sampling Progressive Refinement JPEG Example JPEG Joint Photographic Experts Group encodes images by a sequence of transformations color encoding DCT discrete cosine transform a kind of Fourier series quantization to achieve perceptual compression lossy Huffman encoding lossless information theoretic coding We will focus on the DCT and quantization of its components the image is broken into 8 8 pixel blocks each block is represented by its 8 8 DCT coefficients each DCT coefficient is quantized using higher resolutions for coefficients with greater perceptual importance 4 6 003 Signals and Systems Lecture 22 April 29 2010 JPEG JPEG Discrete cosine transform DCT is similar to a Fourier series but high frequency artifacts are typically smaller Periodically extend a row and represent it with a Fourier series x n x n 8 Example imagine coding the following 8 8 block 0 n 8 There are 8 distinct Fourier series coefficients 2 1 X ak x n e jk 0 n 0 8 8 For a two dimensional transform take the transforms of all of the rows assemble those results into an image and then take the transforms of all of the columns of that image n 8 JPEG Check Yourself DCT is based on a different periodic representation shown below Which signal has greater high frequency content y n y n 16 x n x n 8 0 16 n 0 n 8 y n y n 16 0 16 n JPEG JPEG Periodic extension of an 8 8 pixel block can lead to a discontinuous function even when the block was taken from a smooth image Periodic extension of the type done for JPEG generates a continuous function from a smoothly varying image original row original row n 0 8 pixel block 8 pixel block n 0 8 n 0 x n x n 8 0 n 0 y n y n 16 n 0 5 16 n 6 003 Signals and Systems Lecture 22 April 29 2010 JPEG JPEG Although …
View Full Document
Unlocking...