EE / EEE SAMPLE STUDY MATERIAL. GATE, IES & PSUs Signal System. Electrical Engineering. Postal Correspondence Course

Size: px

Start display at page:

Download "EE / EEE SAMPLE STUDY MATERIAL. GATE, IES & PSUs Signal System. Electrical Engineering. Postal Correspondence Course"

Baldwin Brown
5 years ago
Views:

1 Sigal-EE Postal Correspodece Course 1 SAMPLE STUDY MATERIAL Electrical Egieerig EE / EEE Postal Correspodece Course GATE, IES & PSUs Sigal System

2 Sigal-EE Postal Correspodece Course CONTENTS 1. SIGNAL & SYSTEM LINEAR TIME-INVARIANT SYSTEMS FOURIER REPRSENTATION OF SIGNALS FOURIER TRANSFORM: CTFT AND DTFT LAPLACE TRANSFORM THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEM SAMPLING AND DISCRETE FOURIER TRANSFORM IES Questios GATE Questio

3 Sigal-EE Postal Correspodece Course 3 CHAPTER-1 SIGNALS AND SYSTEM 1.1 INTRODUCTION The cocept ad theory of sigals ad systems are eeded i almost all electrical egieerig fields ad i may other egieerig ad scietific disciplies as well. I this chapter we itroduce the mathematical descriptio ad represetatio of sigals ad systems ad their classificatios. We also defie several importat basic sigals essetial to our studies. What are Sigals? Sigals are represeted as fuctios of oe or more idepedet variables. For example, a digital image ca be represeted by itesity as a fuctio of time. All sigals carry some kid of iformatio. Classificatio of sigals Sigals ca be classified o the basis of differet parameters: A. Cotiuous time ad discrete time sigals A sigal is said to be cotiuous time sigal if it is defied at all values of time parameter t or we ca say t must be cotiuous variable. x() t t Fig. 1 (a) A discrete-time sigal ca be obtaied from samplig a cotiuous time sigal x(t) at discrete time istats like t =, 1,,. It is represeted by x[]. Various values are give by x[], x[1] Fig. 1 (b) x[], x[1] are called samples of the cotiuous time sigal x(t). For example: 1 x[ ] 3 =.., 1,, 1,,.. B. Aalog ad Digital Sigals If a cotiuous-time sigal x(t) ca take o ay value i the cotiuous iterval (a, b), where a may be - ad b may be +, the the cotiuous-time sigal x(t) is called a aalog sigal. If a discrete-time sigal x[] ca take o oly a fiite umber of distict values, the we call this sigal a digital sigal. Digital sigals are discretized i both time ad value. C. Real ad Complex Sigals

4 Sigal-EE Postal Correspodece Course 4 A sigal x(t) is a real sigal if its value is a real umber, ad a sigal x(t) is a complex sigal if its value is a complex umber. A geeral complex sigal x(t) is a fuctio of the form t x t jx t x 1. (1.1) Where x 1 t ad t x are real sigal ad j 1. Note that i Eq. (1.1) t represet either a cotiuous or a discrete variable. D. Determiistic ad Radom Sigals Determiistic sigals are those sigals whose values are completely specified for ay give time. Thus, a determiistic sigal ca be modeled by a kow fuctio of time t. Radom sigals are those sigals that take radom values at ay give time ad must be characterized statistically. Radom sigals will ot be discussed i this text. For example: y (t) = t + 1 So y(t) is defied ad determiistic for each value of t. Ad suppose temperature of a city is a radom sigal which ca take arbitrary values. E. Eve ad Odd Sigals A sigal x(t) or x[] is referred to as a eve sigal if x(-t) = x(t) x[-] = x[] (1.) A sigal x(t) or x[] is referred to as a odd sigal if x(-t) = -x(t) x[-] = -x[]...(1.3) Examples of eve ad odd sigals are show i Fig. 1.. Ay sigal x(t) or x[] ca be expressed as a sum of two sigals, oe of which is eve ad oe of which is odd. That is, x(t) = x e (t) + x o (t)...(1.4) x[] = x e [] + x o [] where 1 xe t xt xt eve part of xt 1 xe x x eve part of x (1.5) 1 x t x t x t odd part of xt 1 xo x x odd part of x (1.6)

5 Sigal-EE Postal Correspodece Course 5 So, from the above formulae, we ca split ay sigal ito its eve ad odd parts. Ad subsequetly we ca fid the eve ad odd part of ay sigal x(t). Note that the product of two eve sigals or of two odd sigals is a eve sigal ad that the product of a eve sigal ad a odd sigal is a odd sigal (Solved Problem 1.7). F. Periodic ad No-periodic Sigals A cotiuous-time sigal x(t) is said to be periodic with period T if there is a positive ozero value of T for which xt xt T for all t (1.7) A example of such a sigal is give i Fig. 1.3(a). From Eq. (1.7) or Fig. 1.3(a) it follows that xt mt xt (1.8) For all t ad ay iteger m. The fudametal period T of x(t) is the smallest positive value of T for which Eq. (1.7) holds. Ay cotiuous-time sigal which is ot periodic is called a o-periodic (or aperiodic). Periodic discrete-time sigals are defied aalogously. A sequece (discrete -time sigal) x[] is periodic with period N if there is a positive iteger N for which x N x all (1.9) A example of such a sequece is give i Fig. 1.3(b). From Eq. (1.9) ad Fig. 1.3(b) it follows that x mn x (1.1)

6 Sigal-EE Postal Correspodece Course 6 for all ad ay iteger m. The fudametal period N of x[] is the smallest positive iteger N for which Eq. (1.9) holds. Ay sequece which is ot periodic is called a o-periodic (or aperiodic) sequece. Note: (i) Fudametal period of costat sigal is udefied. (ii) Sum of two cotiuous periodic sigals may ot be periodic but sum of two periodic sequeces is always periodic. G. Eergy ad Power Sigals For a arbitrary cotiuous-time sigal x(t), the ormalized eergy cotet E of x(t) is defied as E x t dt (1.11) The ormalized average power P of x(t) is defied as P 1 T / lim x t dt (1.1) T T T / Similarly, for a discrete-time sigal x[], the ormalized eergy cotet E of x[] is defied as E x (1.13) The ormalized average power P of x[] is defied as 1 N P lim x (1.14) N N 1 For Example: N Cosider v(t) to be the voltage across a resistor R producig a curret i(t). The istataeous power p(t) per ohm i defied as v()() t i t (1.15) R p()() t i t Total eergy E ad average power P o a per ohm basis are E = Ad P = i () t dt. joules (1.16) T 1 lim() i t dt T T T watts (1.17) Based o defiitios (1.11) to (1.14), the followig classes of sigals are defied: 1. x(t) (or x[]) is said to be a eergy sigal (or sequece) if ad oly if < E <, ad so P =.. x(t) (or x[]) is said to be a power sigal (or sequece) if ad oly if < P <, thus implyig that E

7 Sigal-EE Postal Correspodece Course 7 =. 3. Sigals that satisfy either property are referred to as either eergy sigals or power sigals. Note that a periodic sigal is a power sigal if its eergy cotet per period is fiite, ad the the average power of this sigal eed oly be calculated over a period. 1.3 BASIC CONTINUOUS-TIME SIGNALS A. The uit step fuctios The uit step fuctio u(t), also kow as the Heaviside uit fuctio, is defied as u t 1 t t Which is show i Fig. 1.4(a) Note that it is discotiuous at t = ad that the value at t = is udefied. Similarly, the shifted uit step fuctio ut t u t t 1 t t t t This is show i fig 1.4(b) is defied as B. The uit impulse fuctio The uit impulse fuctio (t), also kow as the Dirac delta fuctio, plays a cetral role i system aalysis. Traditioally, (t) is ofte defied as the limit of a suitably chose covetioal fuctio havig uity area over a ifiitesimal time iterval as show i Fig. 1.5 ad possesses the followig properties: t t t tdt 1

8 Sigal-EE Postal Correspodece Course 8 But a ordiary fuctio which is everywhere except at a sigle poit must have the itegral (i the Riema itegral sese). Thus (t) caot be a ordiary fuctio ad mathematically it is defied by Where t tdt (1.18) t is ay regular fuctio cotiuous at t =. A alterative defiitio of (t) is give by b a a b t t dt a b or a b udefied a or b (1.19) Note that equatio (1.18) to (1.19 ) is a symbolic expressio ad should ot be cosidered a ordiary Riema itegral. I this sese, (t) is ofte called a geeralized fuctio ad (t) is kow as a testig fuctio. For coveiece, t ad t t are depicted graphically as show i Fig Fig. 1.6 (a) Uit Impulse Fuctio; (b) Shifted Uit Impulse Fuctio Some additioal properties of (t) are 1 (1.) a at t t t (1.1)

9 Sigal-EE Postal Correspodece Course 9 If If x t t x t (1.) xt is cotiuous at t =. x t t t x t t t (1.3) x t is cotiuous at t t. Usig Equatios (1.19) ad (1.1), ay cotiuous-time sigal x t x t d (1.4) xt ca be expressed as Note: Uit step fuctios ad uit impulse fuctio are very much used i etworks, cotrol systems. So better uderstadig is ecessary for these sigals. C. Complex Expoetial Sigals: The complex expoetial sigal j e x t t (1.5) is a importat example of a complex sigal. Usig Euler s formula, this sigal ca be defied as Thus, j t x t e cos t jsi t (1.6) x t is a complex sigal whose real part is cost ad imagiary part issi t. A importat property of the complex expoetial sigal xt period T of xtis give by (Solved Problem 1.9) T (1.7) i Eq. (1.5 ) is that it is periodic. The fudametal Note that xt is periodic for ay value of Geeral Complex Expoetial Sigals Let s j be a complex umber. We defie xt as st j t t x t e e e cost j sit (1.8)

10 Sigal-EE Postal Correspodece Course 1 Figure-1.7: (a) Expoetially Icreasig siusoidal sigal (b) Expoetially decreasig siusoidal Sigal The sigal e t xt i Eq. (1.7 ) is kow as a geeral complex expoetial sigal whose real part t cos t ad imagiary part e sit are expoetially icreasig siusoidal sigals (Fig 1.7). or decreasig Real expoetial sigals Note that if s sigal x t (a real umber), the Eq. (1.8) reduces to a real expoetial t e (1.9) As illustrated i Figure below, if, the xt is a growig expoetial; ad if, the xt is a decayig expoetial. 1.4 BASIC DISCRETE-TIME SIGNALS A. The Uit Step Sequece The uit step sequece u[] is defied as u 1 (1.3) Which is show i Fig. 1.1(a). Note that the value of u[] at = is defied [ulike the cotiuous-time step fuctio u(t) at t = ] ad equals uity. Similarly, the shifted uit step sequece u[ - k] is defied as Which is show i Fig. 1.1(b). u k 1 k (1.31) k

11 Sigal-EE Postal Correspodece Course 11 B. The Uit Impulse Sequece The uit impulse (or uit sample) sequece [] is defied as 1 (1.3) Which is show i Fig. 1.11(a). Similarly, the shifted uit impulse (or sample) sequece k defied as Which is show i Fig. 1.11(b). k 1 k (1.33) k Ulike the cotiuous-time uit impulse fuctio t, complicatio or difficulty. From defiitios (1.3) ad (1.33) it is readily see that x x (1.34) is is defied without mathematical x k x k k (1.35) Which are the discrete-time couterparts of Equatios (1.1) ad (1. ), respectively. Above property is called siftig property. From defiitios (1.3) to (1.31), [] ad u[] are related by [] = u[] - u[ -1] (1.36) k k u (1.37)

12 Sigal-EE Postal Correspodece Course 1 Which are the discrete-time couterparts of Equatios (1.3) ad (1.31), respectively. Usig defiitio (1.33), ay sequece x[] ca be expressed as x xk k (1.38) k Which correspods to Eq. (1.7) i the cotiuous-time sigal case. C. Complex Expoetial Sequeces The complex expoetial sequece is of the form j e x Agai, usig Euler s formula, x[] ca be expressed as (1.39) j x e cos jsi (1.4) Thus x[] is a complex sequece whose real part is cos ad imagiary part issi. j Periodicity of e is order for e j to be periodic with period N (> ), must satisfy the followig coditio, m = positive iteger (1.41) m N j Thus, the sequece e is ot periodic for ay value of. It is periodic oly if / is a ratioal umber. Note that this property is quite differet from the property that the cotiuous- time sigal j t e is periodic for ay value of cos. Thus, if satisfies the periodicity coditio i Eq. (1.41),, N ad m have o factors i commo, the the fudametal period of the sequece x[] i Eq. (1.39) is N give by N m (1.4) Aother very importat distictio betwee the discrete-time ad cotiuous-time complex expoetials is

13 Sigal-EE Postal Correspodece Course 13 j t that the sigals e are all distict for distict values of but that this is ot the case for the j sigalse. Cosider the complex expoetial sequece with frequecy k j k j jk j e e e e (1.43), where k is a iteger: Sice j k e 1, From Eq. (1.56) we see that the complex expoetial sequece at frequecy is the same as that at frequecies, 4, ad so o. Therefore, i dealig with discrete- time expoetials, we oly cosider a iterval of legth ad usually, we will use the iterval or the iterval. So i frequecy domai, these complex expoetial sequeces are periodic with period. D. Siusoidal Sequeces A siusoidal sequece ca be expressed as cos x A (1.44) If is dimesioless, the both ad have uits of radias. Two examples of siusoidal sequeces are show i Fig As before, the siusoidal sequece i Eq. (1.44) ca be expressed as j Acos A Re e (1.45)

14 Sigal-EE Postal Correspodece Course 14 As we observed i the case of the complex expoetial sequece i Eq. (1.39 ), the same observatios [Equatios (1.41) ad (1.43)] also hold for siusoidal sequeces. For istace, the sequece i Fig. 1.13(a) is periodic with fudametal period 1, but the sequece i Fig. 1.13(b) is ot periodic. 1.5 SYSTEMS AND CLASSIFICATION OF SYSTEMS A. System Represetatio A system is a mathematical model of a physical process that relates the iput (or excitatio) sigal to the output (or respose) sigal. Let x ad y be the iput ad output sigals, respectively, of a system. The the system is viewed as a trasformatio (or mappig) of x ito y. This trasformatio is represeted by the mathematical otatio y = T{x} (1.46) Where T is the operator represetig some well-defied rule by which x is trasformed ito y. Relatioship (1.46) is depicted as show i Fig. 1.14(a). Multiple iput ad/or output sigals are possible as show i Fig. 1.14(b). We will restrict our attetio for the most part i this text to the sigle-iput, sigle-output case.

15 Sigal-EE Postal Correspodece Course 15 B. Cotiuous-Time ad Discrete-Time Systems If the iput ad output sigals x ad y are cotiuous-time sigals, the the system is called a system [Fig. 1.15(a)]. If the iput ad output sigals are discrete-time sigals or sequeces, the the system is called a discrete-time system [Fig. 1.15(b)]. For Example: y(t) = x(t) is a example of cotiuous time system. Ad y() = x( k) is a example of discrete time system. To Buy Postal Correspodece Package call at

Mathematical Description of Discrete-Time Signals. 9/10/16 M. J. Roberts - All Rights Reserved 1

Mathematical Description of Discrete-Time Signals. 9/10/16 M. J. Roberts - All Rights Reserved 1 Mathematical Descriptio of Discrete-Time Sigals 9/10/16 M. J. Roberts - All Rights Reserved 1 Samplig ad Discrete Time Samplig is the acquisitio of the values of a cotiuous-time sigal at discrete poits