Notes for Signals and Systems Version 1.0

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1 Noes for Signals and Sysems Version.0 Wilson J. Rugh These noes were developed for use in 50.4, Signals and Sysems, Deparmen of Elecrical and Compuer Engineering, Johns Hopkins Universiy, over he period As indicaed by he Table of Conens, he noes cover radiional, inroducory conceps in he ime domain and frequency domain analysis of signals and sysems. No as complee or polished as a book, hough perhaps subjec o furher developmen, hese noes are offered on an as is or use a your own risk basis. Prerequisies for he maerial are he arihmeic of complex numbers, differenial and inegral calculus, and a course in elecrical circuis. (Circuis are used as examples in he maerial, and he las secion reas circuis by Laplace ransform.) Concurren sudy of mulivariable calculus is helpful, for on occasion a double inegral or parial derivaive appears. A course in differenial equaions is no required, hough some very simple differenial equaions appear in he maerial. The maerial includes links o demonsraions of various conceps. These and oher demonsraions can be found a hp:// commens o rugh@jhu.edu are welcome. Copyrigh , Johns Hopkins Universiy and Wilson J. Rugh, all righs reserved. Use of his maerial is permied for personal or non-profi educaional purposes only. Use of his maerial for business or commercial purposes is prohibied.

2 Noes for Signals and Sysems Table of Conens 0. Inroducion Inroducory Commens 0.. Background in Complex Arihmeic 0.3. Analysis Background Exercises. Signals 0.. Mahemaical Definiions of Signals.. Elemenary Operaions on Signals.3. Elemenary Operaions on he Independen Variable.4. Energy and Power Classificaions.5. Symmery-Based Classificaions of Signals.6. Addiional Classificaions of Signals.7. Discree-Time Signals: Definiions, Classificaions, and Operaions Exercises. Coninuous-Time Signal Classes..3.. Coninuous-Time Exponenial Signals.. Coninuous-Time Singulariy Signals.3. Generalized Calculus Exercises 3. Discree-Time Signal Classes Discree-Time Exponenial Signals 3.. Discree-Time Singulariy Signals Exercises 4. Sysems Inroducion o Sysems 4.. Sysem Properies 4.3. Inerconnecions of Sysems Exercises 5. Discree-Time LTI Sysems DT LTI Sysems and Convoluion 5.. Properies of Convoluion - Inerconnecions of DT LTI Sysems 5.3. DT LTI Sysem Properies 5.4. Response o Singulariy Signals 5.5. Response o Exponenials (Eigenfuncion Properies) 5.6. DT LTI Sysems Described by Linear Difference Equaions Exercises 6. Coninuous-Time LTI Sysems CT LTI Sysems and Convoluion

3 6.. Properies of Convoluion - Inerconnecions of DT LTI Sysems 6.3. CT LTI Sysem Properies 6.4. Response o Singulariy Signals 6.5. Response o Exponenials (Eigenfuncion Properies) 6.6. CT LTI Sysems Described by Linear Difference Equaions Exercises 7. Inroducion o Signal Represenaion Inroducion o CT Signal Represenaion 7.. Orhogonaliy and Minimum ISE Represenaion 7.3. Complex Basis Signals 7.4. DT Signal Represenaion Exercises 8. Periodic CT Signal Represenaion (Fourier Series) CT Fourier Series 8.. Real Forms, Specra, and Convergence 8.3. Operaions on Signals 8.4. CT LTI Frequency Response and Filering Exercises 9. Periodic DT Signal Represenaion (Fourier Series) DT Fourier Series 9.. Real Forms, Specra, and Convergence 9.3. Operaions on Signals 9.4. DT LTI Frequency Response and Filering Exercises 0. Fourier Transform Represenaion for CT Signals Inroducion o CT Fourier Transform 0.. Fourier Transform for Periodic Signals 0.3. Properies of Fourier Transform 0.4. Convoluion Propery and LTI Frequency Response 0.5. Addiional Fourier Transform Properies 0.6. Inverse Fourier Transform 0.7. Fourier Transform and LTI Sysems Described by Differenial Equaions 0.8. Fourier Transform and Inerconnecions of LTI Sysems Exercises. Unilaeral Laplace Transform 43.. Inroducion.. Properies of he Laplace Transform.3. Inverse Transform.4. Sysems Described by Differenial Equaions.5. Inroducion o Laplace Transform Analysis of Sysems Exercises. Applicaion o Circuis Circuis wih Zero Iniial Condiions.. Circuis wih Nonzero Iniial Condiions Exercises 3

4 Noes for Signals and Sysems 0. Inroducory Commens Wha is Signals and Sysems? Easy, bu perhaps unhelpful answers, include he α and he ω, he quesion and he answer, he fever and he cure, calculus and complex arihmeic for fun and profi, More seriously, signals are funcions of ime (coninuous-ime signals) or sequences in ime (discree-ime signals) ha presumably represen quaniies of ineres. Sysems are operaors ha accep a given signal (he inpu signal) and produce a new signal (he oupu signal). Of course, his is an absracion of he processing of a signal. From a more general viewpoin, sysems are simply funcions ha have domain and range ha are ses of funcions of ime (or sequences in ime). I is radiional o use a fancier erm such as operaor or mapping in place of funcion, o describe such a siuaion. However we will no be so formal wih our viewpoins or erminologies. Simply remember ha signals are absracions of ime-varying quaniies of ineres, and sysems are absracions of processes ha modify hese quaniies o produce new ime-varying quaniies of ineres. These noes are abou he mahemaical represenaion of signals and sysems. The mos imporan represenaions we inroduce involve he frequency domain a differen way of looking a signals and sysems, and a complemen o he ime-domain viewpoin. Indeed engineers and scieniss ofen hink of signals in erms of frequency conen, and sysems in erms of heir effec on he frequency conen of he inpu signal. Some of he associaed mahemaical conceps and manipulaions involved are challenging, bu he mahemaics leads o a new way of looking a he world! 0. Background in Complex Arihmeic We assume easy familiariy wih he arihmeic of complex numbers. In paricular, he polar form of a complex number c, wrien as j c c= c e is mos convenien for muliplicaion and division, e.g., j c j c j( c+ c) cc = c e c e = c c e The recangular form for c, wrien c= a+ jb where a and b are real numbers, is mos convenien for addiion and subracion, e.g., c + c = a+ jb + a + jb = ( a+ a) + j( b+ b) Of course, connecions beween he wo forms of a complex number c include c = a+ jb = a + b, c= ( a+ jb) = an ( b/ a) and, he oher way round, 4

5 a= Re{ c} = c cos( c), b= Im{ c} = c sin( c) Noe especially ha he quadran ambiguiy of he inverse angen mus be resolved in making hese compuaions. For example, ( j) = an ( /) = π / 4 while ( + j) = an (/( )) = 3 π /4 I is imporan o be able o menally compue he sine, cosine, and angen of angles ha are ineger muliples of π /4, since many problems will be se up his way o avoid he disracion of calculaors. You should also be familiar wih Euler s formula, jθ e = cos( θ ) + jsin( θ ) and he complex exponenial represenaion for rigonomeric funcions: jθ jθ jθ jθ e + e e e cos( θ) =, sin( θ) = j Noions of complex numbers exend o noions of complex-valued funcions (of a real variable) in he obvious way. For example, we can hink of a complex-valued funcion of ime, x(), in he recangular form x() = Re { x() } + jim { x() } In a simpler noaion his can be wrien as x() = xr() + jxi() where xr () and xi () are real-valued funcions of. Or we can consider polar form, j x() x () = x () e where x( ) and x() are real-valued funcions of (wih, of course, x( ) nonnegaive for all ). In erms of hese forms, muliplicaion and addiion of complex funcions can be carried ou in he obvious way, wih polar form mos convenien for muliplicaion and recangular form mos convenien for addiion. In all cases, signals we encouner are funcions of he real variable. Tha is, while signals ha are complex-valued funcions of, or some oher real variable, will arise as mahemaical conveniences, we will no deal wih funcions of a complex variable unil near he end of he course. 0.3 Analysis Background We will use he noaion x[ n ] for a real or complex-valued sequence (discree-ime signal) defined for ineger values of n. This noaion is inended o emphasize he similariy of our reamen of funcions of a coninuous variable (ime) and our reamen of sequences (in ime). Bu use of he square brackes is inended o remind us ha he similariy should no be overdone! Summaion noaion, for example, 5

6 3 x[ k] = x[] + x[] + x[3] k = is exensively used. Of course, addiion is commuaive, and so we conclude ha 3 x[ k] = x[ k] k= k= 3 Care mus be exercised in consuling oher references since some use he convenion ha a summaion is zero if he upper limi is less han he lower limi. And of course his summaion limi reversal is no o be confused wih he inegral limi reversal formula: 3 x() d = x() d 3 I is imporan o manage summaion indices o avoid collisions. For example, 3 zk [ ] xk [ ] k = is no he same hing as 3 zk [ ] xk [ ] k = Bu i is he same hing as 3 zk [ ] x[ j] j= All hese observaions are involved in changes of variables of summaion. A ypical case is 3 x[ n k] k = Le j = n k (relying on conex o disinguish he new index from he imaginary uni j ) o rewrie he sum as n 3 n x[ j] = x[ j] j= n j= n 3 Someimes we will encouner muliple summaions, ofen as a resul of a produc of summaions, for example, x[ k] z[ j] = x[ k] z[ j] = x[ k] z[ j] k= j= 0 k= j= 0 j= 0k= The order of summaions here is immaerial. Bu, again, look ahead o be sure o avoid index collisions by changing index names when needed. For example, wrie x[ k] z[ k] = x[ k] z[ j] k= k= 0 k= j= 0 before proceeding as above. These consideraions also arise, in slighly differen form, when inegral expressions are manipulaed. For example, changing he variable of inegraion in he expression 6

7 x( τ ) dτ 0 o σ = τ gives 0 x( σ )( dσ) = x( σ) dσ 0 We encouner muliple inegrals on rare occasions, usually as a resul of a produc of inegrals, and collisions of inegraion variables mus be avoided by renaming. For example, x() d z() d = x() d z( τ ) dτ = x () z( τ) ddτ 0 The Fundamenal Theorem of Calculus arises frequenly: d x( τ) dτ = x( ) d For finie sums, or inegrals of well-behaved (e.g. coninuous) funcions wih finie inegraion limis, here are no paricular echnical concerns abou exisence of he sum or inegral, or inerchange of order of inegraion or summaion. However, for infinie sums or improper inegrals (over an infinie range) we should be concerned abou convergence and hen abou various manipulaions involving change of order of operaions. However, we will be a bi cavalier abou his. For summaions such as x[ k] k = a raher obvious necessary condiion for convergence is ha xk [ ] 0 as k ±. Typically we will no worry abou general sufficien condiions, raher we leave consideraion of convergence o paricular cases. For inegrals such as x() d an obvious necessary condiion for convergence is ha x ( ) 0 as ±, bu again furher deails will be ignored. We especially will ignore condiions under which he order of a double (infinie) summaion can be inerchanged, or he order of a double (improper) inegral can be inerchanged. Indeed, many of he mahemaical magic ricks ha appear in our subjec are explainable only by aking a very rigorous view of hese issues. Such rigor is beyond our scope. For complex-valued funcions of ime, operaions such as differeniaion and inegraion are carried ou in he usual fashion wih j viewed as a consan. I someimes helps o hink of he funcion in recangular form o jusify his view: for example, if x() = xr() + jxi(), hen 7

8 x( τ ) dτ = x ( τ) dτ + j x ( τ) dτ R I Similar commens apply o complex summaions and sequences. Pahologies ha someimes arise in he calculus, such as everywhere coninuous bu nowhere differeniable funcions (signals), are of no ineres o us! On he oher hand, cerain generalized noions of funcions, paricularly he impulse funcion, will be very useful for represening special ypes of signals and sysems. Because we do no provide a careful mahemaical background for generalized funcions, we will ake a very formulaic approach o working wih hem. Impulse funcions aside, fussy maers such as signals ha have inconvenien values a isolaed poins will be handled informally by simply adjusing values o achieve convenience. Example Consider he funcion, = 0 x () = 0, else Cerainly he inegral of x() beween any wo limis, is zero here being no area under a single poin. The derivaive of x() is zero for any 0, bu he derivaive is undefined a = 0, here being no reasonable noion of slope. How do we deal wih his? The answer is o view x() as equivalen o he idenically-zero funcion. Indeed, we will happily adjus he value of a funcion a isolaed values of for purposes of convenience and simpliciy. In a similar fashion, consider, > 0 u () = 0, < 0 which probably is familiar as he uni-sep funcion. Wha value should we assign o u (0)? Again, he answer is ha we choose u(0) for convenience. For some purposes, seing u (0) = / is mos suiable, for oher purposes u (0) = is bes. Bu in every insance we freely choose he value of u(0) o fi he purpose a hand. The derivaive of u () is zero for all 0, bu is undefined in he usual calculus sense a = 0. However here is an inuiive noion ha a jump upward has infinie slope (and a jump downward has slope ). We will capure his noion using generalized funcions and a noion of generalized calculus in he sequel. By comparison, he signal x() in he example above effecively exhibis wo simulaneous jumps, and here is lile alernaive han o simplify x() o he zero signal. Excep for generalized funcions, o be discussed in he sequel, we ypically work in he conex of piecewise-coninuous funcions, and permi only simple, finie jumps as disconinuiies. Exercises. Compue he polar form of he complex numbers e j( + j) and j π / +. ( je ). Compue he recangular form of he complex numbers j5 /4 e π and e π π j j6 + e. 8

9 3. Evaluae, he easy way, he magniude ( j) 3 and he angle ( j). j 4. Using Euler's relaion, e θ = cosθ + jsinθ, derive he expression j j cos e θ θ = + e θ 5. If z and z are complex numbers, and a sar denoes complex conjugae, express he following quaniies in erms of he real and imaginary pars of z and z : Re[ z z ], Im[ zz ], Re[ z/ z ] 6. Wha is he relaionship among he hree expressions below? x( σ ) dσ, x( σ) dσ, x( σ) dσ 7. Simplify he hree expressions below. 0 0 d ( ), d ( ), d ( ) d x σ dσ d x σ dσ x d dσ σ σ 0 9

10 . Mahemaical Definiions of Signals Noes for Signals and Sysems A coninuous-ime signal is a quaniy of ineres ha depends on an independen variable, where we usually hink of he independen variable as ime. Two examples are he volage a a paricular node in an elecrical circui and he room emperaure a a paricular spo, boh as funcions of ime. A more precise, mahemaical definiion is he following. A coninuous-ime signal is a funcion x() of he real variable defined for < <. A crude represenaion of such a signal is a skech, as shown. On plane earh, physical quaniies ake on real numerical values, hough i urns ou ha someimes i is mahemaically convenien o consider complex-valued funcions of. However, he defaul is real-valued x(), and indeed he ype of skech exhibied above is valid only for real-valued signals. A skech of a complex-valued signal x() requires an addiional dimension or muliple skeches, for example, a skech of he real par, Re{ x( )}, versus and a skech of he imaginary par, Im{ x( )}, versus. Remarks: A coninuous-ime signal is no necessarily a coninuous funcion, in he sense of calculus. Disconinuiies (jumps) in a signal are indicaed by a verical line, as drawn above. The defaul domain of definiion is always he whole real line a convenien absracion ha ignores various big-bang heories. We use ellipses as shown above o indicae ha he signal coninues in a similar fashion, wih he meaning presumably clear from conex. If a signal is of ineres only over a paricular inerval in he real line, hen we usually define i o be zero ouside of his inerval so ha he domain of definiion remains he whole real line. Oher convenions are possible, of course. In some cases a signal defined on a finie inerval is exended o he whole real line by endlessly repeaing he signal (in boh direcions). The independen variable need no be ime, i could be disance, for example. Bu for simpliciy we will always consider i o be ime. An imporan subclass of signals is he class of unilaeral or righ-sided signals ha are zero for negaive argumens. These are used o represen siuaions where here is a definie saring ime, usually designaed = 0 for convenience. A discree-ime signal is a sequence of values of ineres, where he ineger index can be hough of as a ime index, and he values in he sequence represen some physical quaniy of ineres. Because many discree-ime signals arise as equally-spaced samples of a coninuous-ime signal, i is ofen more convenien o hink of he index as he sample number. Examples are he closing Dow-Jones sock average each day and he room emperaure a 6 pm each day. In hese cases, he sample number would be day 0, day, day, and so on. 0

11 We use he following mahemaical definiion. A discree-ime signal is a sequence x[ n] defined for all inegers < n <. We display x[ n ] graphically as a sring of lollypops of appropriae heigh. Of course here is no concep of coninuiy in his seing. However, all he remarks abou domains of definiion exend o he discree-ime case in he obvious way. In addiion, complexvalued discree-ime signals ofen are mahemaically convenien, hough he defaul assumpion is ha x[ n ] is a real sequence. In due course we discuss convering a signal from one domain o he oher sampling and reconsrucion, also called analog-o-digial (A/D) and digial-o-analog (D/A) conversion.. Elemenary Operaions on Signals Several basic operaions by which new signals are formed from given signals are familiar from he algebra and calculus of funcions. Ampliude Scale: y() = a x(), where a is a real (or possibly complex) consan Ampliude Shif: y() = x()+ b, where b is a real (or possibly complex) consan Addiion: y() = x() + z() Muliplicaion: y() = x() z() Wih a change in viewpoin, hese operaions can be viewed as simple examples of sysems, a opic discussed a lengh in he sequel. In paricular, if a and b are assumed real, and z() is assumed o be a fixed, real signal, hen each operaion describes a sysem wih inpu signal x() and oupu signal y(). This viewpoin ofen is no paricularly useful for such simple siuaions, however. The descripion of hese operaions for he case of discree-ime signals is compleely analogous..3 Elemenary Operaions on he Independen Variable Transformaions of he independen variable are addiional, basic operaions by which new signals are formed from a given signal. Because hese involve he independen variable, ha is, he argumen (), he operaions someimes subly involve our cusomary noaion for funcions. These operaions can be viewed as somewha less simple examples of sysems, and someimes such an alernae view is adoped.

12 As is ypical in calculus, we use he noaion x() o denoe boh he enire signal, and he value of he signal a a value of he independen variable called. The inerpreaion depends on conex. This is simpler han adoping a special noaion, such as x() i, o describe he enire signal. Subleies ha arise from our dual view will be discussed in he paricular conex. Time Scale: Suppose y() = x( a) where a is a real consan. By skeching simple examples, i becomes clear ha if a >, he resul is a ime-compressed signal, and if 0 < a <, he resul is ime dilaion. Of course, he case a = 0 is rivial, giving he consan signal y() = x(0) ha is only slighly relaed o x(). For a 0, x() can be recovered from y(). Tha is, he operaion is inverible. If a < 0, hen here is a ime reversal, in addiion o compression or dilaion. The recommended approach o skeching ime-scaled signals is simply o evaluae y() for a selecion of values of unil he resul becomes clear. For example, Noice ha in addiion o compression or dilaion, he `beginning ime or `ending ime of a pulse-ype signal will be changed in he new ime scale. Time Shif: Suppose y() = x( T) where T is a real consan. If T > 0, he shif is a righ shif in ime, or a ime delay. If T is negaive, we have a lef shif, or a ime advance. For example,

13 Combinaion Scale and Shif: Suppose y () = xa ( T). I is emping o hink abou his as wo operaions in sequence -- a scale followed by a shif, or a shif followed by a scale. This is dangerous in ha a wrong choice leads o incorrec answers. The recommended approach is o ignore shorcus, and figure ou he resul by brue-force graphical mehods: subsiue various values of unil y () becomes clear. Coninuing he example, Example: The mos imporan scale and shif combinaion for he sequel is he case where a =, and he sign of T is changed o wrie y () = xt ( ). This is accomplished graphically by reversing ime and hen shifing he reversed signal T unis o he righ if T > 0, or o he lef if T < 0. We refer o his ransformaion as he flip and shif. For example, The flip and shif operaion can be explored in he apple below. However, you should verify he inerpreaion of he flip and shif by hand skeches of a few examples. flip and shif.4 Energy and Power Classificaions The oal energy of a coninuous-ime signal x(), where x() is defined for < <, is T E = x () d = lim x () d T T 3

14 In many siuaions, his quaniy is proporional o a physical noion of energy, for example, if x() is he curren hrough, or volage across, a resisor. If a signal has finie energy, hen he signal values mus approach zero as approaches posiive and negaive infiniy. The ime-average power of a signal is T P = lim x ( ) d T T T For example he consan signal x () = (for all ) has ime-average power of uniy. Wih hese definiions, we can place mos, bu no all, coninuous-ime signals ino one of wo classes: An energy signal is a signal wih finie E. For example, x() = e, and, rivially, x() = 0, for all are energy signals. For an energy signal, P = 0. A power signal is a signal wih finie, nonzero P. An example is x() =, for all, hough more ineresing examples are no obvious and require analysis. For a power signal, E =. Example Mos would suspec ha x() = sin() is no an energy signal, bu in any case we firs compue T T sin ( ) d = ( cos( ) ) d = T sin( T) T T Leing T confirms our suspicions, since he limi doesn exis. The second sep of he power-signal calculaion gives P = lim ( sin( ) T T T ) = T and we conclude ha x() is a power signal. Example The uni-sep funcion, defined by, > 0 u () = 0, < 0 is a power signal, since T / T / T T = T T T / 0 = lim T T = T lim u ( ) d lim d Example There are signals ha belong o neiher of hese classes. For example, x() = e is a signal wih boh E and P infinie. A more unusual example is /, x () = 0, < This signal has infinie energy bu zero average power. 4

15 The RMS (roo-mean-square) value of a power signal x() is defined as P. These energy and power definiions also can be used for complex-valued signals, in which case we replace x () by x( )..5 Symmery-Based Classificaions of Signals A signal x() is called an even signal if x( ) = x( ) for all. If x( ) = x( ), for all, hen x() is called an odd signal. The even par of a signal x() is defined as and he odd par of x() is x ev x() + x( ) () = x() x( ) xod () = The even par of a signal is an even signal, since x( ) + x( ) xev( ) = = xev ( ) and a similar calculaion shows ha he odd par of a signal is an odd signal. Also, for any signal x() we can wrie a decomposiion as x() = x () + x () ev These conceps are mos useful for real signals. For complex-valued signals, a symmery concep ha someimes arises is conjugae symmery, characerized by x() = x ( ) where superscrip sar denoes complex conjugae..6 Addiional Classificaions of Signals Boundedness: A signal x() is called bounded if here is a finie consan K such ha x( ) K, for all. (Here he absolue value is inerpreed as magniude if he signal is complex valued.) Oherwise a signal is called unbounded. Tha is, a signal is unbounded if no such K exiss. For example, x() = sin(3) is a bounded signal, and we can ake K =. Obviously, x() = sin(3) is unbounded. Periodiciy: A signal x() is called periodic if here is a posiive consan T such ha x() = x( + T), for all. Such a T is called a period of he signal, and someimes we say a signal is T-periodic. Of course if a periodic signal has period T, hen i also has period T, 3T, and so on. The smalles value of T for which x() = x( + T), for all, is called he fundamenal period of he signal, and ofen is denoed T o. Noe also ha a consan signal, x() = 3, for example, is periodic wih period any T > 0, and he fundamenal period is no well defined (here is no smalles posiive number). od 5

16 Examples To deermine periodiciy of he signal x() = sin(3), and he fundamenal period T o if periodic, we apply he periodiciy condiion sin(3( + T) = sin(3 ), < < Rewriing his as sin(3+ 3 T) = sin(3 ), < < i is clear ha he condiion holds if and only if 3T is an ineger muliple of π, ha is, T is a posiive ineger muliple of π /3. Thus he signal is periodic, and he fundamenal period is To = π /3. As a second example, we regard x() = u() + u( ) as periodic, by assuming for convenience he value u (0) = /, bu here is no fundamenal period. Periodic signals are an imporan subclass of all signals. Physical examples include he ocean ides, an a-res ECG, and musical ones (bu no unes). Typically we consider he period of a periodic signal o have unis of seconds, and he fundamenal frequency of a periodic signal is defined by π ω o = T o wih unis of radians/second. We will use radian frequency hroughou, hough some oher sources use frequency in Herz, denoed by he symbol f o. The relaion beween radian frequency and Herz is ωo fo = = π To The main difference ha arises beween he use of he wo frequency unis involves he placemen of π facors in various formulas. Given a lieral expression for a signal, solving he equaion x() = x( + T ), for all for he smalles value of T, if one exiss, can be arbirarily difficul. Someimes he bes approach is o plo ou he signal and ry o deermine periodiciy and he fundamenal period by inspecion. Such a conclusion is no definiive, however, since here are signals ha are very close o, bu no, periodic, and his canno be discerned from a skech. Noe ha he average power per period of a T-periodic signal x() becomes, T / PT = () T x d T / or, more generally, PT = x () d T T where we have indicaed ha he inegraion can be performed over any inerval of lengh T. To prove his, for any consan o consider o + T x () d T o 6

17 and perform he variable change τ = o T /. The average power per period is he same as he average power of he periodic signal. Therefore he RMS value of a periodic signal x() is P = x () d T T Example Ordinary household elecrical power is supplied as a 60 Herz sinusoid wih RMS value abou 0 vols. Tha is, x( ) = Acos(0 π) and he fundamenal period is T o = /60 sec. The ampliude A is such ha / cos = A (0 π) d 0 from which we compue A Discree-Time Signals: Definiions, Classificaions, and Operaions For discree-ime signals, x[ n ], we simply need o conver he various noions from he seing of funcions o he seing of sequences. Energy and Power: The oal energy of a discree-ime signal is defined by N E = x [ n] = lim x [ n] n= N n= N The ime-average power is P lim N = N x [ n] N + n= N and discree-ime classificaions of energy signals and power signals are defined exacly as in he coninuous-ime case. Examples The uni pulse signal,, n = 0 δ[ n] = 0, n 0 is an energy signal, wih E =. The uni-sep signal,, n 0 un [ ] = 0, n < 0 is a power signal wih ime-average power 7

18 N P = lim N u [ n] N + n= N N = limn N + n= 0 N + = limn = N + Periodiciy: The signal x[ n ] is periodic if here is a posiive ineger N, called a period, such ha x[ n+ N] = x[ n] for all ineger n. The smalles period of a signal, ha is, he leas value of N such ha he periodiciy condiion is saisfied, is called he fundamenal period of he signal. The fundamenal period is denoed N, hough someimes he subscrip is dropped in paricular conexs. o Example To check periodiciy of he signal x[ n] = sin(3 n), we check if here is a posiive ineger N such ha sin(3( n+ N)) = sin(3 n), n= 0, ±, ±, Tha is sin(3n+ 3 N) = sin(3 n), n= 0, ±, ±, This condiion holds if and only if 3N is an ineger muliple of π, a condiion ha canno be me by ineger N. Thus he signal is no periodic. Elemenary operaions: Elemenary operaions, for example addiion and scalar muliplicaion, on discree-ime signals are obvious conversions from he coninuous-ime case. Elemenary ransformaions of he independen variable also are easy, hough i mus be remembered ha only ineger argumen values are permied in he discree-ime case Time Scale: Suppose y[ n] = x[ an], where a is a posiive or negaive ineger (so ha he produc, an, is an ineger for all ineger n). If a =, his is a ime reversal. Bu for any case beyond a = ±, be aware ha loss of informaion in he signal occurs, unlike he coninuous-ime case. Example For a =, compare he ime scaled signal wih he original: 8

19 Time Shif: Suppose y[ n] = x[ n N], where N is a fixed ineger. If N is posiive, hen his is a righ shif, or delay, and if N is negaive, i is a lef shif or advance. Combinaion Scale and Shif: Suppose yn [ ] = xan [ N], where a is a nonzero ineger and N is an ineger. As in he coninuous-ime case, he safes approach o inerpreing he resul is o simply plo ou he signal y[n]. Example: Suppose yn [ ] = xn [ n]. This is a flip and shif, and occurs sufficienly ofen ha i is worhwhile verifying and remembering he shorcu: y[n] can be ploed by ime-reversing (flipping) x[n] and hen shifing he reversed signal o move he original value a n = 0 o n= N. Tha is, shif N samples o he righ if N > 0, and N samples o he lef if N < 0. Exercises. Given he signal shown below, skech he signal y() = (a) x () x ( ) (b) x( ) (c) x( ) u( ) (d) x( ) + x( 3 ) (e) x(3 ). Deermine if he following signals are power signals or energy signals, and compue he oal energy or ime-average power, as appropriae. (a) x() = sin() u() 9

20 (b) x() = e (c) x() = u() (d) 3 x() = 5 e u() 3. For an energy signal x(), prove ha he oal energy is he sum of he oal energy of he even par of x() and he oal energy of he odd par of x(). 4. If a given signal x() has oal energy E = 5, wha is he oal energy of he signal y () = x(3 4)? α 5. Under wha condiions on he real consan α is he coninuous-ime signal x() = e u( ) an energy signal? When your condiions are saisfied, wha is he energy of he signal? 6. Skech he even and odd pars of he signals below. (a) (b) 7. Suppose ha for a signal x() i is known ha Ev{()} x = Od{()} x = for > 0.Wha is x()? 8. Deermine which of he following signals are periodic, and specify he fundamenal period. jπ (a) x () = e cos( π + π ) (b) x() = sin () (c) x() = u( k) u( k) k = j3π (d) x () = 3e 9. Suppose ha x () and x () are periodic signals wih respecive fundamenal periods T and T. Show ha if here are posiive inegers m and n such ha 0

21 T T m = n (ha is, he raio of fundamenal periods is a raional number), hen x() = x () + x () is periodic. If he condiion holds, wha is he fundamenal period of x()? 0. Deermine which of he following signals are bounded, and specify a smalles bound. 3 (a) x() = e u() 3 (b) x() = e u( ) 6 (c) x () = 4e 3 5 (d) x() = e sin( ) u(). Given he signal xn [ ] = δ[ n] δ[ n ], skech y[ n ] = (a) x[4n ] (b) x[ nu ] [ n] (c) 3 x[ n+ 3] (d) x[ n] x[ n]. Deermine wheher he following signals are periodic, and if so deermine he fundamenal period. (a) x[ n] = u[ n] + u[ n] j3π n (b) xn [ ] = e n j n (c) xn [ ] = ( ) + (d) x[ n] = cos( π n) 4 e π 3. Suppose x[n] is a discree-ime signal, and le y[n]=x[n]. (a) If x[n] is periodic, is y[n] periodic? If so, wha is he fundamenal period of y[n] in erms of he fundamenal period of x[n]? (b) If y[n] is periodic, is x[n] periodic? If so, wha is he fundamenal period of x[n] in erms of he fundamenal period of y[n]? 4. Under wha condiion is he sum of wo periodic discree-ime signals periodic? When he condiion is saisfied, wha is he fundamenal period of he sum, in erms of he fundamenal periods of he summands? n 5. Is he signal x[ n] = 3( ) u[ n] an energy signal, power signal, or neiher? 6. Is he signal jπ n jπ n xn [ ] = e + e periodic? If so, wha is he fundamenal period? 7. Answer he following quesions abou he discree-ime signal (a) Is x[ n ] periodic? If so, wha is is fundamenal period? xn [ ] j( π /) n = e.

22 (b) Is x[ n ] an even signal? Is i an odd signal? (c) Is x[ n] an energy signal? Is i a power signal? 8. Which of he following signals are periodic? For hose ha are periodic, wha is he fundamenal period? (a) (b) (c) xn [ ] = j 4 e π j xn [ ] = e xn [ ] = e 8 n π n 7 8 j π ( n )

23 Noes for Signals and Sysems Much of our discussion will focus on wo broad classes of signals: he class of complex exponenial signals and he class of singulariy signals. Though i is far from obvious, i urns ou ha essenially all signals of ineres can be addressed in erms of hese wo classes.. The Class of CT Exponenial Signals There are several ways o represen he complex-valued signal a x () = ce, < < where boh c and a are complex numbers. A convenien approach is o wrie c in polar form, and a in recangular form, jφ c= c e o, a= σ o + jωo where φ o = c and where we have chosen noaions for he recangular form of a ha are cusomary in he field of signals and sysems. In addiion, he subscrip o s are inended o emphasize ha he quaniies are fixed real numbers. Then jφ ( ) () o σo+ jωo x = ce e σ ( ) o j ωo+ φ = ce e o Using Euler s formula, we can wrie he signal in recangular form as σ () o σ cos( ) o x = ce ωo+ φo + jce sin( ωo+ φo) There are wo special cases ha are of mos ineres. Special Case : Suppose boh c and a are real. Tha is, ω o = 0 and φ o is eiher 0 or π. Then we have he familiar exponenials σ o ce, ifφo = 0 x () = σ o ce, ifφo = π Or, more simply, x() = ce σ o Special Case : Suppose c is complex and a is purely imaginary. Tha is, σ o = 0. Then j( ω ) () o+ φ x = ce o = c cos( ωo+ φo) + j c sin( ωo+ φo) Boh he real and imaginary pars of x() are periodic signals, wih fundamenal period T o = π ωo Since he independen variable,, is viewed as ime, unis of ω o ypically are radians/second and unis of T o naurally are seconds. A signal of his form is ofen called a phasor. 3

24 Lef in exponenial form, we can check direcly ha given anyω o, x() is periodic wih periodto = π / ωo : j[ ω ( ) ] ( ) o + To + φ x+ T o o = ce j( ω ) o+ φo ± j π = ce e j( ω ) o+ φ = ce o = x () Also, i is clear ha T o is he fundamenal period of he signal, simply by aemping o saisfy he periodiciy wih any smaller, posiive value for he period. We can view a phasor signal as a vecor a he origin of lengh c roaing in he complex plane wih angular frequency ω o radians/second, beginning wih he angle φ o a = 0. Ifω o > 0, hen he roaion is couner clockwise. Ifω o < 0, hen he roaion is clockwise. Of course, ifω o = 0, hen he signal is a consan, and i is no surprising ha he noion of a fundamenal period falls apar. The apple in he link below illusraes his roaing-vecor inerpreaion, and also displays he imaginary par of he phasor, ha is, he projecion on he verical axis. One Phasor Sums of phasors ha have differen frequencies are also very imporan. These are bes visualized using he head-o-ail consrucion of vecor addiion. The apple below illusraes. Sum of Two Phasors The quesion of periodiciy becomes much more ineresing for phasor sums, and we firs discuss his for sums of wo phasors. Consider j j () x ce ω ω = + c e The values of c and c are no essenial facors in he periodiciy quesion, bu he values of ω and ω are. I is worhwhile o provide a formal saemen and proof of he resul, wih assumpions designed o rule ou rivialiies and needless complexiies. Theorem The complex valued signal j j () x ce ω ω = + c e wih c, c 0 and ω, ω 0 is periodic if and only if here exiss a posiive frequency ω 0 and inegers k and l such ha ω = kω0, ω = lω0 (.) Furhermore, if ω0 is he larges frequency for which (.) can be saisfied, in which case i is called he fundamenal frequency for x(), hen he fundamenal period of x() isto = π / ω0. 4

25 Proof Firs we assume ha posiive ω 0 and he inegers k, l saisfy (.). ChoosingT = π / ω0, we see ha jkω0( + T ) jlω0( + T ) x ( + T) = ce + ce jkπ jkω0 jlπ jlω0 = e ce + e ce jkω0 jlω0 = ce + ce = x () for all, and x() is periodic. I is easy o see ha if ω o is he larges frequency such ha (.) is saisfied, hen he fundamenal period of x() isto = π / ωo. (Perhaps we are beginning o abuse he subscrip oh s and zeros ) Now suppose ha x() is periodic, and T > 0 is such ha x( + T) = x( ) for all. Tha is, j ( T) j ( T) j j ce ω + ce ω + + = ce ω + ce ω for all. This implies jωt jωt j( ω ω) ( e ) c + ( e ) ce = 0 for all. Picking he paricular imes = 0 and = π /( ω ω) gives he wo algebraic equaions jωt jωt ( e ) c + ( e ) c = 0 jω ( T jω ) T e c ( e ) c = 0 By adding hese wo equaions, and also subracing he second from he firs, we obain j ω T j ω e = e T = Therefore boh frequencies mus be ineger muliples of frequency π /T. Example The signal j j3 x() = 4e 5e is periodic wih fundamenal frequencyω o =, and hus fundamenal period π. The signal () 4 j j x = e 5e π is no periodic, since he frequencies and π canno be ineger muliples of a fixed frequency. The heorem generalizes o he case where x() is a sum of any number of complex exponenial erms: he signal is periodic if and only if here exiss ω 0 such ha every frequency presen in he sum can be wrien as an ineger muliple ofω 0. Such frequency erms are ofen called harmonically relaed. The apple below can be used o visualize sums of several harmonically relaed phasors, and he imaginary par exhibis he corresponding periodic, real signal. Phasor Sums 5

26 . The Class of CT Singulariy Signals The basic singulariy signal is he uni impulse, δ (), a signal we inven in order o have he following sifing propery wih respec o ordinary signals, x() : x() δ () d = x(0) (.) Tha is, δ () causes he inegral o sif ou he value of x (0). Here x() is any coninuousime signal ha is a coninuous funcion a = 0, so ha he value of x() a = 0 is well defined. For example, a uni sep, or he signal x() = /, would no be eligible for use in he sifing propery. (However, some reamens do allow a finie jump in x() a = 0, as occurs in he uni sep signal, and he sifing propery is defined o give he mid-poin of he jump. Tha is, + x(0 ) + x(0 ) x () δ () d= For example, if he signal is he uni sep, hen he sif would yield /.) A lile hough, reviewed in deail below, shows ha δ () canno be a funcion in he ordinary sense. However, we develop furher properies of he uni impulse by focusing on implicaions of he sifing propery, while insising ha in oher respecs δ () behave in a manner consisen wih he usual rules of arihmeic and calculus of ordinary funcions. Area δ () d = (.3) Considering he sifing propery wih he signal x () =, for all, we see he uni impulse mus saisfy (.3). Time values δ () = 0, for 0 (.4) By considering x() o be any signal ha is coninuous a = 0 wih x (0) = 0, for example, he 3 signals x() =,,,, i can be shown ha here is no conribuion o he inegral in (.) for nonzero values of he inegraion variable. This indicaes ha he impulse mus be zero for nonzero argumens. Obviously δ (0) canno be zero, and indeed i mus have, in some sense, infinie value. Tha is, he uni impulse is zero everywhere excep = 0, and ye has uni area. This makes clear he fac ha we are dealing wih somehing ouside he realm of basic calculus. Noice also ha hese firs wo properies imply ha a δ () d = a for any a > 0. 6

27 Scalar muliplicaion We rea he scalar muliplicaion of an impulse he same as he scalar muliplicaion of an ordinary signal. To inerpre he sifing propery for aδ (), where a is a consan, noe ha he properies of inegraion imply x()[ aδ()] d = a x() δ() d = ax(0) The usual erminology is ha aδ() is an impulse of area a, based on choosing x () =, for all, in he sifing expression. Signal Muliplicaion z () δ () = z(0) δ () When a uni impulse is muliplied by a signal z(), which is assumed o be coninuous a = 0, he sifing propery gives x( ) [ z( ) δ( )] d = [ x( ) z( )] δ( ) d = x(0) z(0) This is he same as he resul obained when he uni impulse is muliplied by he consan z (0), x( )[ z(0) δ( )] d = z(0) x( ) δ( ) d = z(0) x(0) Therefore we conclude he signal muliplicaion propery shown above. Time shif We rea he ime shif of an impulse he same as he ime shif of any oher signal. To inerpre he sifing propery for he ime shifed uni impulse, δ ( o ), a change of inegraion variable from o τ = o gives x() δ( o) d = x( τ + o) δ( τ) dτ = x( o) This propery, ogeher wih he funcion muliplicaion propery gives he more general saemen z() δ ( o) = z( o) δ ( o) where o is any real consan and z() is any ordinary signal ha is a coninuous funcion of a = o. Time scale Since an impulse is zero for all nonzero argumens, ime scaling an impulse has impac only wih regard o he sifing propery where, for any nonzero consan a, x () δ ( a) d= x(0), a 0 a 7

28 To jusify his expression, assume firs ha a > 0. Then he sifing propery mus obey, by he principle of consisency wih he usual rules of inegraion, and in paricular wih he change of inegraion variable from o τ = a, x() δ( a) d = x( / a) ( ) d a τ δ τ τ = a x(0), a > 0 A similar calculaion for a < 0, where now he change of inegraion variable yields an inerchange of limis, gives x() δ( a) d = a x( τ / a) δ( τ) dτ = ( / ) ( ) a x τ a δ τ dτ = a x(0), a< 0 These wo cases can be combined ino one expression given above. Thus he sifing propery leads o he definiion: δ( a) = δ( ), a 0 a Symmery Noe ha he case a = in ime scaling gives he resul ha δ( ) acs in he sifing propery exacly as δ(), so we regard he uni impulse as an even funcion. Oher inerpreaions are possible, bu we will no go here. We graphically represen an impulse by an arrow, as shown below. (If he area of he impulse is negaive, a < 0, someimes he arrow is drawn poining souh.) We could coninue his invesigaion of properies of he impulse, for example, using he calculus consisency principle o figure ou how o inerpre δ ( a o ), z ( ) δ ( a), and so on. Bu we only need he properies jusified above, and wo addiional properies ha are simply wondrous. These include an exension of he sifing propery ha violaes he coninuiy condiion: Special Propery δ ( τ) δ( τ) dτ = δ( ) Noe here ha he inegraion variable is τ, and is any real value. Even more remarkable is an expression ha relaes impulses and complex exponenials: Special Propery jω δ () = e dω π 8

29 Noe here ha he inegral simply does no converge in he usual sense of basic calculus, since j e ω = for any (real) values of and ω. Remark Our general approach o hese impulse properies will be don hink abou impulses simply follow he rules. However, o provide a bi of explanaion, wih lile rigor, we briefly discuss one of he mahemaical approaches o he subjec. To arrive a he uni impulse, consider he possibiliy of an infinie sequence of funcions, dn(), n=,,3,, ha have he uni-area propery dn ( ) d =, n=,,3, and also have he propery ha for any oher funcion x() ha is coninuous a = 0, lim n dn( ) x( ) d = x(0) Here he limi involves a sequence of numbers defined by ordinary inegrals, and can be inerpreed in he usual way. However we nex inerchange he order of he limi and he inegraion, wihou proper jusificaion, and view δ() as some sor of limi: δ () = lim n dn() This view is useful for inuiion purposes, bu is dangerous if pursued oo far by elemenary means. In paricular, for he sequences of funcions dn() ypically considered, he limi does no exis in any usual sense. Examples Consider he recangular-pulse signals n, < < d () n n n =, n=,,3, 0, else The pulses ge aller and hinner as n increases, bu clearly every dn() is uni area, and he mean-value heorem can be used o show /( n) x(0) dn() x() d = n x() d n /( n) n wih he approximaion geing beer as n increases. Thus we can casually view a uni impulse as he limi, as n, of hese uni-area recangles. A similar example is o ake dn() o be a riangle of heigh n, widh /n, cenered a he origin. Bu i urns ou ha a more ineresing example is o use he sinc funcion defined by sin( π ) sinc( ) = ( π ) and le dn( ) = nsin c( n), n=,,3, I can be shown, by evaluaing an inegral ha is no quie elemenary, ha hese signals all have area π, and ha he sifing propery dn( ) x() d x(0) 9

30 is a beer and beer approximaion as n grows wihou bound. Therefore we can view an impulse of area π, ha is, πδ ( ), as a limi of hese funcions. This sequence of sinc signals is displayed in he apple below for a range of n, and you can ge a picorial view of how an impulse migh arise from sinc s as n increases, in much he same way as he impulse arises from heigh n, widh /n, recangular pulses as n increases. Family of Sincs Remark Special Propery can be inuiively undersood in erms of our casual view of impulses as follows. Le W () j ω d W = e d π ω W W = [cos( ) jsin( )] d π ω + ω ω W W W j = cos( ω) dω sin( ω) dω π + π W W Using he fac ha a sinusoid is an odd funcion of is argumen, W d () W = cos( ) d π ω ω 0 sin( W) = π = W sinc( W) π π This dw () can be shown o have uni area for every W > 0, again by a non-elemenary inegraion, and again he sifing propery is approximaed when W is large. Therefore he Special Propery migh be expeced. The apple below shows a plo of dw () as W is varied, and provides a picure of how he impulse migh arise as W increases. Addiional Singulariy Signals Anoher Sinc Family From he uni impulse we generae addiional singulariy signals using a generalized form of calculus. Inegraion leads o 0, < 0 δτ ( ) dτ =, > 0 which is he familiar uni-sep funcion, u ().(We leave he value of u(0), where he jump occurs, freely assignable following our general policy.) The running inegral in his expression acually can be inerpreed graphically in very nice way. And a variable change from τ o σ = τ gives he alernae expression 30

31 u () = δ ( σ) dσ 0 Analyically his can be viewed as an applicaion of a sifing propery applied o he case x() = u(): δ( σ) dσ = u( σ) δ( σ) dσ = u( σ) δ( σ) dσ = u( ) 0 This is no, sricly speaking, legal for = 0, because of he disconinuiy here in u(), bu we ignore his issue. By considering he running inegral of he uni-sep funcion, we arrive a he uni-ramp: 0, < 0 u( τ) dτ =, 0 = u() Ofen we wrie his as r(). Noice ha he uni ramp is a coninuous funcion of ime, hough i is unbounded. Coninuing, 0, < 0 r( τ) dτ = /, 0 = u () which migh be called he uni parabola, p(). We sop here, as furher inegraions yield signals lile used in he sequel. We can urn maers around, using differeniaion and he fundamenal heorem of calculus. Clearly, d d p() = r( τ) dτ = r() d d and his is a perfecly legal applicaion of he fundamenal heorem since he inegrand, r(), is a coninuous funcion of ime. However, we go furher, cheaing a bi on he assumpions, since he uni sep is no coninuous, o wrie d d r () = u( τ) dτ = u () d d Tha his cheaing is no unreasonable follows from a plo of he uni ramp, r(), and hen a plo of he slope a each value of. Cheaing more, we also wrie d d u () = δ ( τ) dτ = δ() d d Again, a graphical inerpreaion makes his seem less unreasonable. We can also consider derivaives of he uni impulse. The approach is again o demand consisency wih oher rules of calculus, and use inegraion by pars o inerpre he sifing propery ha should be saisfied. We need go no furher han he firs derivaive, where 3

32 ()[ d x δ()] d x() δ() x() δ() d d = = x (0) This assumes, of course, ha x ( ) is coninuous a = 0, ha is, x() is coninuously differeniable a = 0. This uni-impulse derivaive is usually called he uni double, and denoed δ (). Various properies can be deduced, jus as for he uni impulse. For example, choosing x () =, < <, he sifing propery for he double gives δ () d = 0 In oher words, he double has zero area a rue ghos. I is also easy o verify he propery x() δ ( o) d = x ( o) and, finally, we skech he uni double as shown below. All of he generalized calculus properies can be generalized in various ways. For example, he produc rule gives d [ u () ] = u () + δ () d = u () where we have used he muliplicaion rule o conclude ha δ () = 0for all. As anoher example, he chain rule gives d u ( o) = δ ( o) d Remark These maers can be aken oo far, o a poin where ambiguiies begin o overwhelm and worrisome liberies mus be aken. For example, using he produc rule for differeniaion, and ignoring he fac ha u () is he same signal as u(), d u () uu () () uu () () u () () d = + = δ The muliplicaion rule for impulses does no apply, since u() is no coninuous a = 0, and so we are suck. However if we inerpre u(0) as ½, he midpoin of he jump, we ge a resul consisen wih u () = δ (). We will no need o ake maers his far, since we use hese generalized noions only for raher simple signals..3 Linear Combinaions of Singulariy Signals and Generalized Calculus For simple signals, ha is, signals wih uncomplicaed wave shapes, i is convenien for many purposes o use singulariy signals for represenaion and calculaion. Example The signal shown below can be wrien as a sum of sep funcions, 3

33 x ( ) = 4 u ( + ) 4 u ( ) Anoher represenaion is x() = 4 u( + ) u( ) This uses he uni seps as cuoff funcions, and someimes his usage is advanageous. However for furher calculaion, represenaion as a linear combinaion usually is much simpler. Differeniaion of he firs expression for x() gives, using our generalized noion of differeniaion, This signal is shown below. x ( ) = 4 δ ( + ) 4 δ ( ) The same resul can be obained by differeniaing he sep cuoff represenaion for x(), hough usage of he produc rule and inerpreaion of he final resul makes he derivaive calculaion more difficul. Tha is, x () = 4 δ ( + ) u( ) + 4 u( + ) δ ( ) = 4 δ( + ) 4 δ( ) = 4 δ( + ) 4 δ( ) (The firs sep makes use of he produc rule for differeniaion, he second sep uses he signalmuliplicaion rule for impulses, and he las sep uses he evenness of he impulse.) Of course, regardless of he approach aken, graphical mehods easily verify x ( τ) dτ = x( ) in his example. Noe ha he consan of inegraion is aken o be zero since i is known ha he signal x() is zero for <. Example The signal shown below can be wrien as x ( ) = r ( ) r ( ) u ( 3) 33

34 Again, he derivaive is sraighforward o compue, and skech, x () = u () u ( ) δ ( 3) Graphical inerpreaions of differeniaion suppor hese compuaions. While represenaion in erms of linear combinaions of singulariy signals leads o convenien shorcus for some purposes, cauion should be exercised. In he examples so far, well-behaved energy signals have been represened as linear combinaions of signals ha are power signals, singulariy signals, and unbounded signals. This can inroduce complicaions in some conexs. Someimes we use hese generalized calculus ideas for signals are nonzero for infinie ime inervals. Example A righ-sided cosine signal can be wrien as x() = cos() u() Then differeniaion using he produc rule, followed by he impulse muliplicaion rule, gives x () = sin() u() + cos() δ () = sin( ) u( ) + δ ( ) You should graphically check ha his is a consisen resul, and ha he impulse in x () is crucial in verifying he relaionship x ( τ) dτ = x( ) Example The periodic signal shown below can be wrien as x() = 4 u( + 3 k) 4 u( 3 k) k= [ ] 34