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# 303 Class Note for ECE 43800 at Purdue

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Date Created: 02/06/15
Chapter 1 Analysis of DiscreteTime Linear TimeInvariant Systems I 11 Signals I 111 Definitions and Notation A signal is a function signal and function are synonymous The two notions are the same and we will be using them interchangeably The historical reason for the existence of these two terms to denote the same thing is that function is the standard term from mathematics whereas signal is an engineering term which originally was used to denote measurable physical quantities like a voltage signal Continuous time CT or analog signals are 0 de ned for every value of time on an interval possibly an in nite interval AND 0 take on values in an interval A graph of a continuous time function is shown in Fig 11a Discrete Time DT signals or sequences are de ned only at integer values of time A graph of a discretetime function is shown in Fig 11b To emphasize the difference between continuous time and discrete time we will use n instead oft for discrete time A digital signal or digital sequence is a DT signal which can take on only integer values Fig 11c is a digital signal which takes on only two different values sometimes such signals are called binary signals Sometimes notation such as f Z a R is used to indicate that f is a discretetime signal Here R is the set of all real numbers ie the real line Z is the set of all integers 7271012 In order to completely understand this notation it is important to recall that a function is a mapping from one set to another WHAT IS A FUNCTION A function is a RULE for producing a number in its range given a number from its domain It is helpful to think of a function as a block diagram shown in Fig 12a 10 CHAPTER 1 ANALYSIS OF DISCRETEiTIME LINEAR TIMEilNVARIANT SYSTEMS 5 2 1 a 31 0 5 1 0 50 100 0 5 0 0 2 4 t n n a b f1ltn Figure 11 a A continuoustime function b A discretetime function c A digital function a single number n a single number taken from the in the range argument TL Value nS domain of f D of f R an mteger Function a real number gt function f f ndjvide by 37 gt a A generic function b Function nS Figure 12 Functions as block diagrams Example 11 The concept of function or signal has a straightforward programming analogy you can think of a signal as a program that takes a single number as its input and produces another number as the output for example float divideby3n int n float x x n30 returnx The function which performs diuision by three can be thought of as this module of code or a rule or an algorithm Then you can call this subroutine from elsewhere and eualuate it for a particular argument for example main x divideby35 Sec 11 Signals 11 When you eualuate the function you will be assigning to m a particular number in this case 53 or approximately 53 modulo computer precision So a function is a procedure which takes in one number and produces another number I When we write R a R to describe continuous time functions we mean that continuous time functions can take in any number on the real line and produce another number anywhere on the real line1 Discrete time functions on the other hand can only take in an integer number but can produce a real number Z a R Digital functions take in an integer and produce an integer Z a Z Note the important distinction between a discrete time signal f and its n th sample fn which is a single number Sometimes it is convenient to abuse this notation and refer to signal fn In this case it is implied that we are referring to a signal f de ned for integer n I 112 Specifying 3 Signal There are many different ways to specify or represent a function a formula eg fn n3 for n O 1 2 34 b graphical representations note that for 2 D functions surface plots and intensity 2 8 1 images can be very useful c a list of all values for all arguments n 012 34 fn013 23143 d A vector in an N dimensional space see Fig 13 which will be used for o N point signals 1More generally a continuoustime signal is described by 1 gt 2 where 1 and 2 are two intervals on the real line Similarly for DT signals and digital signals 12 CHAPTER 1 ANALYSIS OF DISCRETEiTIME LINEAR TIMEilNVARIANT SYSTEMS o periodic signals with period N This is done by recording the N values of the signal fn as a column vector We will typically denote vectors by boldface letters thus the vector corresponding to an N point or an N periodic signal 1 is f m 1 This approach is very important and will be emphasized throughout the course It provides geometric intuition into many key theoretical results and helps turn complicated formulas and proofs into very natural intuitive statements For example when signals are viewed as vectors in an N dimensional Euclidean space it turns out that the Discrete Fourier Transform is essentially a rotation in this space see Fig 13b Parseval s theorem therefore simply says that if you rotate a vector you do not change its length We will also occasionally treat random variables as vectors to gain geometric insight into linear prediction and recursive estimation I 113 Properties of Signals Different types of functions require different processing tools It will be important for us to know is a function periodic or not ls it nite duration ls it bounded ls its energy nite a Periodicity lf fn fnN for some xed N and all n we say that f is periodic with period N For example the function given by the formula f1 71 for all integer n is periodic with period 2 as shown in Fig 14 left we assume here that the signal 1711 g 2 i i i i i i i 72 extends 1n n1tely in both directions The function f2n 7 Otherwise 07 not periodic as shown in Fig 14 right b Finiteinfinite duration If fn 0 outside of a nite interval 1 is a signal of nite duration otherwise 1 is a signal of in nite duration For example the signal f1n de ned above is in nite duration f2 is nite duration c The energy of a signal 1 will be denoted f A more standard notation which you will nd in mathematics literature is The energy is de ned as follows 00 51 Z lleZ 11 n7oo Sec 11 Signals 13 R3 FT RN X2 b Figure 13 a A vector space representation for Npoint or Nperiodic signals With N 3 b In this framework the Fourier transform is Very similar to a rotation it preserves distances and angles 1 5 A A T 50 50000P oo 1 N 9 1 9 1 1 5 0 5 4 2 0 2 4 11 11 Figure 14 A periodic signal left and a nonperiodic signal right The absolute value needs to be in the de nition for the case when fn is complex valued For example the energy of fl is 1 1 1 which is in nite The energy of f2 isU 241 2127 24221 An important remark here is that since we will often be dealing with sums of the type 1 q q 14 CHAPTER 1 ANALYSIS OF DISCRETEiTIME LINEAR TIMEilNVARIANT SYSTEMS it is useful to remember the formulas for summing the geometric series m m1 N71 7 qmin lfmN Zand0 mltNthenq q q 7 1 q 00 gm d f0lt lt1 th an1 in en q liq nm To verify the rst formula multiply both sides by 1 7 q and cancel some terms on the lefthand side To verify the second formula take the limit of both sides of the rst formula as N a 00 Why would the second formula not work for lql 2 17 d The magnitude of a signal 1 is the maximum of its absolute value MU max WON 12 ieoltnlteo Another notation for the magnitude of f is For example the magnitude of fl is 1 the magnitude of f2 is 4 If a signal has a nite magnitude we say that it is bounded I 114 Special Signals 1 1 Q A 0305 gas 0 4 2 0 2 4 4 2 0 2 4 11 11 Figure 15 Unit sample left and unit step right There are several special signals which we will encounter very often a Unit sample or unlt ImPUIse7 6W i 07 n 7g 0 1 n 2 0 In Unit step 7 07 n lt 0 c Sinusoids sinum gt or cosum gt Sec 11 Signals 15 cosTct cos3Tct and cosnncos37cn l cosZTcn cos0n 1 a 0 05 0 l 4 2 0 2 4 0 11 Figure 16 Left the DT frequency 27r is the same as the DT frequency zero Right adding 27r t0 the frequency does not Change the DT signal 0 Apparent motionk J Actual motion Figure 17 I 115 Peculiar Properties of DT Sinusoids a The highest rate of oscillation in a discrete time sinusoid is attained when w 7139 or w 77139 For example the DT frequency 27139 is actually smaller than the DT frequency 7139 Indeed since 71 is integer we have cos27m 1 cosO 71 for all 71 SO the DT frequency 27139 is the same as the DT frequency 0 16 CHAPTER 1 ANALYSIS OF DISCRETEeTIME LINEAR TIMEelNVARIANT SYSTEMS b Discrete time sinusoids whose frequencies differ by an integer multiple of 27139 are iden tical cosw 27139n gt cosum gt This is illustrated in Fig 16 right Notice that the continuous time signals cos7rt and cos37rt are the same at integer points So if we sample either of these signals at integer points we will get the same signal cos7r0 cos37r0 cos7r1 cos37r1 cos7r2 cos37r2 cos7m cos37m for any integer n More generally cosw 27139n gt cosum gt 1 27m coswn gtcos27rn 7 sinwn gtsin27rn cosum gt 17 sinum gt 0 cosum gt Even though the two continuous time signals in Fig 16 right are different their sampling at integer points is the same The dashed continuous time signal oscillates faster but it all happens in between sampling instants The sampling points so not see this activity This is why two different continuous time frequencies can appear to be the same discretetime frequency This phenomenon is called aliasing You have all encountered aliasing when watching a movie You must have noticed that sometimes a car moves in one direction but its wheels seem to be rotating in the opposite direction A simplistic picture of this is shown in Fig 17 Between each pair of consecutive movie frames the wheels rotate 270 degrees threequarters of one full revolution which looks like a 90 degree rotation backwards c DT sinusoids are not necessarily periodic Suppose you are sampling a CT sinusoid t at integer points 71 0i1i2 If z0 1 as in Fig 18left then in order for the DT sinusoid to be periodic it has to have a value of one again some time in the future In Fig 18left this happens at n 5 From then on the DT signal will start repeating For this particular example 5 4quot and so an 43quot Even in the case when the DT sinusoid is periodic its period may be different from the period of Note that while the fundamental period of cos 437 is 25 the fundamental period of cos is 5 But suppose now that w 1 as in Fig 18b Then there is no value of n besides n 0 for which 1 So the value of the sample z0 will never repeat As you can see cos6 is pretty close to 1 it is in fact approximately 096 however there is no integer 71 except 0 for which cosn is exactly equal zero2 2It is possible to show however that cosn can be arbitrarily Close to zero In other words for any E gt 0 no matter how small there exists a positive integer n for which cosn gt 1 7 E Sec 11 Signals 17 cos4Tct5 cos4nn5 cost cosn it A m it ill NJ tn tn Figure 18 Left DT sinusoid Whose frequency 0 47r5 is a rational multiple of 7r is periodic Right DT sinusoid Whose frequency 0 1 is not a rational multiple of 7r is not periodic What should happen for the sampled signal to be periodic An integer number of the continuous time periods has to eventually become an integer In other words7 there must exist two integers k and m such that k 2quot m 17 ie the continuous time period 3 must be a rational number If this happens7 then the sample at n m will have the same value as the sample at n 07 and the resulting sinusoid will be periodic If this never happens7 then no sample will ever have the same value as the sample at n O

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