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# 652 Class Note for ECE 43800 at Purdue

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This 8 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 24 views.

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Date Created: 02/06/15

ECE 438 CLASS NOTES FALL 2004 Digital Signal Processing with Applications Ilya Pollak Purdue University 2004 Purdue University All Rights Reserved Chapter 0 Introduction Why does an Electrical Engineering major need a course on digital signal processing Simply because it is everywhere Whether you become a practicing engineer or go to graduate school you are bound to use the material we cover in this course both if you stay in Electrical and Computer Engineering and if you go into many other branches of science and engineering Some application areas are 0 Consumer Electronics watermarking for copyright protection speech recognition image enhancement 0 Military target tracking detection 0 Finance risk minimization pricing 0 Medicine computer guided procedures medical image analysis for diagnostic pur poses 0 Law Enforcement superresolution of low quality video from surveillance cameras signal enhancement The synthetic example of Fig 1 is prototypical of a situation often encountered in many applications a binary signal top was sent but its noisy version middle was received A signal processing algorithm was used to extract from the received signal a close approximation bottom of the transmitted one The next example involves processing of images A digital grayscale image is simply a rectangular array of numbers typically integers corresponding to image intensities usually 0 for black 255 for white and the numbers in between represent various shades of gray see Fig 2 One bin of a digital image is called a pixel There are various types of pictures which result from medical imaging procedures It is often important to design an computer algorithm for automatic extraction of objects from such imagesiobjects corresponding to for instance internal organs or tumors For example the task in Fig 3 is to extract the outline of a thyroid from an ultrasound image While a humaniespecially a trained professionalimay do this quite successfully writing a computer program that would do this is rather tricky because the image quality is quite poor there is largeamplitude noise and blurring 4 CHAPTER 0 INTRODUCTION T40 200 400 600 800 1000 4 72 T40 200 400 600 800 1000 OJ HJ Il lI I 72 T40 200 400 600 800 1000 Figure 1 Noise removal in 1D original signal top received noisy data middle an estimate recovered from the noisy data using a signal processing algorithm bottom 0 black 255 white 100 gray Figure 2 A digital grayscale image is a rectangular array of numbers which typically range from from 0 black to 255 white The underlying algorithm here is nothing more than a difference equation the algorithm produces a movie whose frame at time n 1 is a certain transformation applied to the frame at time n The initial frame is the input image shown in Fig 3a the nal frame is the two region segmentation of 3d This underlying difference equation is very complicated because it is nonlinear multidimensional and also because its coef cients are discontinuous However by the end of this course we will develop a Sec 0 1 Basic Problems 5 c 170 regions d 2 regions Figure 3 Multiscale segmentation of an ultrasound image basic understanding regarding such algorithms Another image processing example that we will touch upon in the later stages of the course is image compression using transform coding The FBI has approximately 30 million sets of ngerprints 300 million ngers which need to be stored and 40000 new sets arrive every day These papers stored in le cabinets used to occupy a whole oor in the FBI building Digitized at 500 dots per inch the eight bit grayscale image of a single ngerprint can be as large as 10Mb Electronic storage of this data base would therefore require roughly three million Gb An effective image compression algorithm is needed A key requirement for any such algorithm is that it must preserve all the image features which are required by law enforcement experts in order to identify a ngerprint The FBI ngerprint compression standard uses wavelets resulting in acceptable image quality at compression ratio of about 15 JPEG is another compression standard which is widely used for images for example many pictures you see on the Web are JPEG images JPEG uses the Discrete Cosine Transform DCT I 01 Basic Problems The signal processing techniques illustrated above all t neatly into the basic diagram shown in Fig 4a Most applications covered in this course will fall into one of four broad categories of problems that one can pose regarding this diagram CHAPTER 0 INTRODUCTION Input x Output y Sx data ontput gt System S gt gt computer program gt a A generic system b A software system tumor medical image license plate t 011 booth video medical imaging 53 5th surveillance camera c An example of a system d An example of a system Figure 4 A generic system and several examples 1 Filter Design lnput z is known we want to change it according to some objective eg it may be desired to remove some unwanted frequencies from x or to change relative amplitudes of frequency components The question is How to design S This problem has a straightforward programming analogy How to design a com puter program knowing the possible inputs as well as the output speci cations 2 System IDmodeling Several pairs of m y are known What is a plausible S A programming analogy is to investigate the inner workings of some patented software for which you do not have the source code What you may try to do is to look at how it behaves for many kinds of input data 3 Signal recovery reconstruction enhancement The output y and the sys tem S are known or partially known What is a plausible z Both examples of Fig 4cd fall into this category Eg in Fig 4c an object of interest is imaged by an imperfect device to yield a picture such as that of Fig 3 Given this picture the objective is to reconstruct or partially reconstruct the original object 4 When we start our discussion of systems we will begin with the most basic prob lem in systems analysis How to nd y when x and S are known Later in the course you will be exposed to these problems in the context of several applications Before we plunge into that however we need to carefully study some general techniques for approaching these kinds of problems The framework of Fig 4a is too general to make any progress We will therefore concentrate mostly on one class of systems namely linear timeinvariant LTl systems Some of the reasons for studying LTl systems are c it is still a very rich class of systems capable of modeling many phenomena 0 it is tractable o non linear systems can sometimes be approximated by linear ones Sec 0 2 Approximate Syllabus 7 I 02 Approximate Syllabus 1 Analysis of DiscreteTime DT Linear TimeInvariant LTI Systems 11 Signals We will start by considering one of the two basic ingredients of Fig 4a namely signals 12 Systems We will continue with the other basic ingredientisystems through which signals ow 13 Fourier Series and Transforms We will then proceed to frequency analysis and look at what it means that one frequency is lower than another and how they can appear to be the same frequency if sampled incorrectly 14 FFT Our next topic will be the Fast Fourier Transform which is not a new transformiit is just an algorithm to ef ciently compute the Fourier transform 15 Sampling The importance of this topic is due to the fact that while most signals in the physical world are continuous time or continuous space signals our most convenient and powerful signal processing tools deal with discretetime and discrete space signals Sampling is how DT signals are obtained from CT signals 16 Ztransform Next we will cover Z transformsia very useful technique for analyzing DT systems and in particular for talking about system stability 2 Random sequences Since many real world signals are too complex to be mod eled exactly we will next consider random signals Instead of asking what is the exact value of a signal at a point we will be analyzing the average behavior of classes of signals We will be interested in the probability that a signal from a certain class attains a particular value at a particular point This framework is necessary in order to effectively address many of the problems considered above If you have a noisy communications channel there may be no way to exactly model the distortion However it could be possible to devise a good probability model of the average behavior of the noise and from that to obtain a good algorithm for combating the noise Similarly it may not be possible to exactly model the distortions that a medical imaging device introduces into an image But there may be a way to say something useful about this distortion on average and then use that to enhance the image quality or extract some information from the images 21 Introduction to Random Sequences Detection and Estimation 22 Speech processing and linear prediction 23 Geometric interpretation of linear prediction and recursive estimation 3 Image processing noise removal enhancement multiscale ltering tomogra phy compression CHAPTER 0 It would be incorrect to think that Topics 22 and 3 are the only ones worth studying7 and the rest is just something we have to do in order to get to them There is really a lot of beautiful mathematics herekwhich is what makes this subject really interesting It would also be incorrect to think that the mathematical theory is decoupled from When you study the fundamentals7 you should keep in mind the underlying goaliwhich is to understand and analyze Fig 4a7 for a particular class of systems And the reason why we are interested in this picture is to be able to address the types of problems illustrated above These examples constitute the basic motivation for what we will be studying in this course and are the answer to the question that many students have when learning fundamental theory Wait a second7 why are we studying the applications all this77 The ow of events you should have in mind is shown in Fig 5 Motivating applications Topics 22 and 3 Input X The basic framework gt System S Analysis Topics 1 21 and 23 Output y SX a Figure 5 The ow diagram for the course INTRODUCTION

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