1.1 What is a Signal. 1.2 What is signal processing
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1 1.1 What is a Sigal We are all immersed i a sea of sigals. All of us from the smallest livig uit, a cell, to the most complex livig orgaism(humas) are all time time receivig sigals ad are processig them. Survival of ay livig orgaism depeds upo processig the sigals appropriately. What is sigal? To defie this precisely is a difficult task. Aythig which carries iformatio is a sigal. I this course we will lear some of the mathematical represetatios of the sigals, which has bee foud very useful i makig iformatio processig systems. Examples of sigals are huma voice, chirpig of birds, smoke sigals, gestures (sig laguage), fragraces of the flowers. May of our body fuctios are regulated by chemical sigals, blid people use sese of touch. Bees commuicate by their dacig patter.some examples of moder high speed sigals are the voltage charger i a telephoe wire, the electromagetic field emaatig from a trasmittig atea,variatio of light itesity i a optical fiber. Thus we see that there is a almost edless variety of sigals ad a large umber of ways i which sigals are carried from o place to aother place. I this course we will adopt the followig defiitio for the sigal: A sigal is a real (or complex) valued fuctio of oe or more real variable(s). Whe the fuctio depeds o a sigle variable, the sigal is said to be oedimesioal. A speech sigal, daily maximum temperature, aual raifall at a place, are all examples of a oe dimesioal sigal.whe the fuctio depeds o two or more variables, the sigal is said to be multidimesioal. A image is represetig the two dimesioal sigal,vertical ad horizotal coordiates represetig the two dimesios. Our physical world is four dimesioal(three spatial ad oe temporal). 1.2 What is sigal processig By processig we mea operatig i some fashio o a sigal to extract some useful iformatio. For example whe we hear same thig we use our ears ad auditory path ways i the brai to extract the iformatio. The sigal is processed by a system. I the example metioed above the system is biological i ature. We ca use a electroic system to try to mimic this behavior. The sigal processor may be a electroic system, a mechaical system or eve it might be a computer program. The word digital i digital sigal processig meas that the processig is doe either by a digital hardware or by a digital computer. 1
2 1.3 Aalog versus digital sigal processig The sigal processig operatios ivolved i may applicatios like commuicatio systems, cotrol systems, istrumetatio, biomedical sigal processig etc ca be implemeted i two differet ways (1) Aalog or cotiuous time method ad (2) Digital or discrete time method. The aalog approach to sigal processig was domiat for may years. The aalog sigal processig uses aalog circuit elemets such as resistors, capacitors, trasistors, diodes etc. With the advet of digital computer ad later microprocessor, the digital sigal processig has become domiat ow adays. The aalog sigal processig is based o atural ability of the aalog system to solve differetial equatios the describe a physical system. The solutio are obtaied i real time. I cotrast digital sigal processig relies o umerical calculatios. The method may or may ot give results i real time. The digital approach has two mai advatages over aalog approach (1) Flexibility: Same hardware ca be used to do various kid of sigal processig operatio,while i the core of aalog sigal processig oe has to desig a system for each kid of operatio. (2) Repeatability: The same sigal processig operatio ca be repeated agai ad agai givig same results, while i aalog systems there may be parameter variatio due to chage i temperature or supply voltage. The choice betwee aalog or digital sigal processig depeds o applicatio. Oe has to compare desig time,size ad cost of the implemetatio. 1.4 Classificatio of sigals As metioed earlier, we will use the term sigal to mea a real or complex valued fuctio of real variable(s). Let us deote the sigal by x(t). The variable t is called idepedet variable ad the value x of t as depedet variable. We say a sigal is cotiuous time sigal if the idepedet variable t takesvaluesiaiterval. For example tɛ(, ),ortɛ[0, ] or t ɛ [T 0,T 1 ] The idepedet variable t is referred to as time,eve though it may ot be actually time. For example i variatio if pressure with height t refers above mea sea level. Whe t takes a vales i a coutable set the sigal is called a discrete time sigal. For example tɛ{0,t,2t,3t,4t,...} or tɛ{... 1, 0, 1,...} or tɛ{1/2, 3/2, 5/2, 7/2,...} etc. 2
3 For coveiece of presetatio we use the otatio x[] to deote discrete time sigal. Let us pause here ad clarify the otatio a bit. Whe we write x(t) ithas two meaigs. Oe is value of x at time t ad the other is the pairs(x(t),t) allowable value of t. By sigal we mea the secod iterpretatio. To keep this distictio we will use the followig otatio: {x(t)} to deote the cotiuous time sigal. Here {x(t)} is short otatio for {x(t),tɛi} where I is the set i which t takes the value. Similarly for discrete time sigal we will use the otatio {x[]}, where{x[]} is short for {x[],ɛi}. Note that i {x(t)} ad {x[]} are dummy variables ie. {x[]} ad {x[t]} refer to the same sigal. Some books use the otatio x[ ] to deote{x[]} ad x[] to deote value of x at time x[] refers to the whole waveform,while x[] refers to a particular value. Most of the books do ot make this distictio clea ad use x[] todeotesigaladx[ 0 ]todeoteaparticularvalue. As with idepedet variable t, the depedet variable x ca take values i a cotiues set or i a coutable set. Whe both the depedet ad idepedet variable take value i itervals, the sigal is called a aalog sigal. Whe both the depedet ad idepedet variables take values i coutable sets(two sets ca be quite differet) the sigal is called Digital sigal.whe we use digital computers to do processig we are doig digital sigal processig. But most of the theory is for discrete time sigal processig where default variable is cotiuous. This is because of the mathematical simplicity of discrete time sigal processig. Also digital sigal processig tries to implemet this as closely as possible. Thus what we study is mostly discrete time sigal processig ad what is really implemeted is digital sigal processig. Exercise: 1.GIve examples of cotiues time sigals. 2.Give examples of discrete time sigals. 3.Give examples of sigal where the idepedet variable is ot time(oedimesioal). 4.Give examples of sigal where we have oe idepedet variable but depedet variable has more tha oe dimesio.(this is sometimes called vector valued sigal or multichael sigal). 5.Give examples of sigals where depedet variable is discrete but idepedet variable are cotiues. 3
4 1.5 Elemetary sigals There are several elemetary sigals that feature promietly i the study of digital sigals ad digital sigal processig. (a)uit sample sequece δ[]: Uit sample sequece is defied by { 1, =0 δ[] = 0, 0 Graphically this is as show below. δ[] Uit sample sequece is also kow as impulse sequece. This plays role aki to the impulse fuctio δ(t) of cotiues time. The cotiues time impulse δ(t) is purely a mathematical costruct while i discrete time we ca actually geerate the impulse sequece. (b)uit step sequece u[]: Uit step sequece is defied by { 1, 0 u[] = 0, < 0 Graphically this is as show below u[] (c) Expoetial sequece: The complex expoetial sigal or sequece x[] is defied by x[] =Cα where C ad α are, i geeral, complex umbers. Note that by writig α = e β, we ca write the expoetial sequece as x[] =ce β. 4
5 Real expoetial sigals: If C ad α are real, we ca have oe of the several type of behavior illustrated below {x[] =α,α>1} {x[] =α, 0 <α<1} {x[] =α, 1 <α<0} {x[] =α,α< 1} if α > 1 the magitude of the sigals grows expoetially, whlie if α < 1, we have decayig expoetial. If α is positive all terms of {x[]} have same sig, but if α is egative the sig of terms i {x[]} alterates. (d)siusoidal Sigal: The siusoidal sigal {x[]} is defied by x[] =A cos(w 0 + φ) Euler s relatio allows us to relate complex expoetials ad siusoids. ad e jw 0 =cosw 0 + j si w 0 A cos(w 0 + φ) = A 2 ejφ e jw 0 + A 2 e jφ e jw 0 The geeral discrete time complex expoetial ca be writte i terms of real expoetial ad siusiodal sigals.specifically if we write c ad α i polar for C = C e jθ ad α = α e jw 0 the Cα = C α cos(w 0 + θ)+j C α si(w 0 + θ) 5
6 Thus for α = 1, the real ad imagiary parts of a cmplex expoetial sequece are siusoidal. For α < 1, they correspod to siusoidal sequece multiplied by a decayig expoetial, ad for α > 1 they correspod to siusiodal sequece multiplied by a growig expoetial. 1.6 Geeratig Sigals with MATLAB MATLAB, acroym for MATrix LABoratory has become a very porplar software eviromet for complex based study of sigals ad systems. Here we give some sample programmes to geerate the elemetary sigals discussed above. For details oe should cosider MATLAB maual or read help files. I MATLAB, oes(m,n) is a M-by-N matrix of oes, ad zeros(m,n) is a M-by-N matrix of zeros. We may use those two matrices to geerate impulse ad step sequece. The followig is a program to geerate ad display impulse sequece. >> % Program to geerate ad display impulse respose sequece >> = 49 : 49; >> delta =[zeros(1, 49), 1,zeros(1, 49)]; >> stem(, delta) Here >> idicates the MATLAB prompt to type i a commad, stem(,x) depicts the data cotaied i vector x as a discrete time sigal at time values defied by. Oe ca add title ad lable the axes by suitable commads. To geerate step sequece we ca use the followig program >> % Program to geerate ad display uit step fuctio >> = 49 : 49; >> u =[zeros(1, 49),oes(1, 50)]; >> stem(, u); We ca use the followig program to geerate real expoetial sequece >> % Program to geerate real expoetial sequece >> C =1; >> alpha =0.8; >> = 10 : 10; >> x = C alpha. >> stem(, x) Note that, i their program, the base alpha is a scalar but the expoet is a vector, hece use of the operator. to deote elemet-by-elemet power. Exercise: Experimet with this program by chagig differet values of alpha (real). Values of alpha greater the 1 will give growig expoetial ad less tha 1 will give decayig expoetials. 6
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