Signals introduction Signals and Systems ECE, Lecture notes of Signals and Systems Theory

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Signals and Systems i) Signals and Systems - Simon Haykin ir) Signals avd, Systems ~ Alan V O ppenheim ti) Linear Systems and Sigrels - B-P- Lathi Signals : A signal is Aefined asa function of one or ore variables , which con vey s intovmatim on the nature of a physical phenomenon. When the functino Aeperds on a sin le Variable, the signal is soid to be one- dimensionalen pees When the function depends on Lwo or More vaviables the signal jis said to be malti- Aimersioral (ex: image video Fignald) Systems: A sgstem is defined as an entity that manipulates One sr move Signals to acesmplish a tu netio . Input Signal Output Signal —___»] _—$$_—_» System Block Diagram Representation ot a System Estimate Message Transmitted Received of message signal i signal signal ae Transmitter |_signal Channel _Se Receiver a Elements of a communication System Classification of Signals (1) Continuous- time and _Discrete~time signals A signal x(t) is said to be a continuous- time signal if it is detined for all time t. : x —> Aependert veriable x(f) t> indeperdent Ex. aN “— a : Continuous-time signal. A discrete -time signal is defined only at Asscrete instants of time. the independent Variable (time) Aas values only. A discrete-time sigra| is often discrete derived from a conti nucuwy - time signal by sampling ‘t ot a unitom rate. Let T denote the sampling juaterval (sampling pertod) . Sampling of a continuo ~ time Signa x(t) at time t=anT gields a sample of value 2(n7). A set of such samples Generates a discrek- time Signal , devoted by Hn] = X(nT) ; N=O,41 42,43, .-. xf o} x(t) ez) ae) 7 “nl woenet wa Me a7) a : salle a (a) (b) (a) Continuous-time signal x(t). (b) Representation of x(t) as a discrete-time signal x[7]. A complex sipnal x Tele 5 i i ; tntegvate over integrate over one time period one time peried For a discrete-time signal x(n), the total energy 4 is pe lim Sheol}? = SIC]? N00 2=-N n=-be The Average Power is gn by p_ jim t= peg) ne 2N4+l n=-N The average power in a periodic signal 2 [nf with fundamental period N, is ge by 1 Nev! 2 Pe + & |=fa)/ N, n-0 summation overt one Period A signal is pefewed to a5 anh energy signal if the total energy 4 of the sig nal is Finite. fe, O 0; (b)a< 0. (@) Prove that x(t) = A cos (Wt +f) “s periodic. Ans: z(é+D= oc (t) fo all & oc (44+Te)= Acos (,(t+%) +4) = Acos (ret +Wy Ty + 4) —3 W,* T= Wyx 2h an = Ahem (ust + art db) oT = Aca Cont +p) cos[8 +21] =em B = x(t) Example 1 Sinusoids with Aifferent Frequencies = cos « xy(0) = cos ot alt) = 0s wat x30) ost VAM AAAL 4 NI AN AAP VARY, CW, > Ww, > o,) The discrete-time sinusoidal signal is ow” 4y x[n] = Acos(n +h) xn} is periodic with period N= (2 \m [” is the smollest intes er which makes ml an in teger]. (1) Z mpulse Signal The discrete-time version of the unit impulse is defined by 8[r1] 1 0 ” > a= 3[x] = {3 n#0° . 432-10 123 4 $f] > Kronecker delta The continuous-time version of the unit impulse is defined by the following pair of relations: a(t)=0 for +40 Slt) > Dirac Delta function and [x0 =1. Bantinuous= eer time / Ny Empalee ~/ a(n) \ Wad SIGne } \ =» itn => —=» \ Le} Ara =1 Area =1 Area =1 \ Area = } t t t \ ti Width =1 Width = 7 Width =1 ar) f Pre perties of Continuous. time Vonpulse signal Reet () SCt)= 6) @ bep= 4 6 (ii) COQ) 6G-t) = % He) S(t -F) — Froduct Property ) feo Slt“) = 2 (tq) — Sifting property Properties of oliscrete-time impulse signal Y SE-n) = $67) (Gp) Pr] (1-} = [ne] [n- No] Reoduct Prope vty (CD) = ac[n)} Sfr- ) = x{M] «— Sifting property Basic Operations on Signals OPERATIONS PERFORMED ON DEPENDENT VARIABLES (><) Amplitude scaling: Let x(t) denote a continuous-time signal. Then the signal y(t) result- ing from amplitude scaling applied to x(t) is defined by y(t) = ex(t), where c is the scaling factor. the value of y(t) is obtained by multiplying the corresponding value of x(t) by the scalar c for each instant of time t. For discrete-time signals, we write yn] = cx[n}. Addition: Let x,(t) and x,(t) denote a pair of continuous-time signals. Then the sig: nal y(t) obtained by the addition of x,(t) and x,(t) is defined by peed x, dt) +x, 4) 1 y(t) = x,(t) + x(t). Ex; 7 a le SA xu) ye For discrete-time signals, we write ' ar23 2 olr23 t yn] = xi] + x2[7)- Multiplication: Let x,(t) and x(t) denote a pair of continuous-time signals. Then the signal y(t) resulting from the multiplication of x,(t) by x(t) is defined by y(t) = x,(#)x2(t). That is, for each prescribed time t, the value of y(t) is given by the product of the corre- sponding values of x,(t) and x(t). For discrete-time signals, we write yn] = xi[2]x2[7]. Differentiation: Let x(t) denote a continuous-time signal. Then the derivative of x(t) with respect to time is defined by d y(t) = F(t). Integration: Let x(t) denote a continuous-time signal. Then the integral of x(t) with respect to time ¢ is defined by xe) = [ * a(n) de, 00 OPERATIONS PERFORMED ON THE INDEPENDENT VARIABLE (¢/r) Time scaling. Let x(t) denote a continuous-time signal. Then the signal y(t) obtained by scaling the independent variable, time ¢, by a factor a is defined by To tincl the \ y(t) = (at). new Lime indice s:! | Ifa > 1, the signal y(t) is a compressed version of x(t). If 0 < a < 1, the signal y(t) is an } expanded (stretched) version of x(t). These two operations are illustrated in Fig. 1.20. — oe 2 1 < 2 Example : MH=x0) ELE wo=a(he) bo t Lop 2 1 Oh) ~ 1 a aa t ay vk ooN t Li ofa (23 (07 (24 =I = i) le SL “1 mee We, 10 No Sa —— (a) a o- > = oO & i In the discrete-time case, we write fe finch the ' Le 1 Compression when kal yee time haclices tr] 7 al , Om Expansion when fo} Odd signals, for which we have x(—t) = —x(t) for all ¢; that is, an odd signal is the negative of its reflected version. Exar ple : x(t) y(t) = x(t) | (Q@) For the signal x(+ illustrated in Fig., sketch x(—f), which is time-reversed x(1). x(t) 2 elf? 7 -5 -3 -1 |? i @) A triangular pulse signal x(¢) is depicted in Fig, Sketch each of the following signals derived from x(t}: (a) x(3¢) (b) x(3t + 2) {c} x(~2t — 1} {d) x(2(t + 2)) (e} x(2(t — 2)) (f} x(3t) + x(3r + 2) att) Aas: (0) Gt) : a=3(<1) Pemprsion 1 ¢ aot 3 3 (b) 2(at+2) + (1) SHiPt by 2 to left (0) Seale by 3 @ x (t+2) (ir) x(3t+2) VAN /\ “3-2 -1 0 “lea LO a 3 ©) x(-2t-1) - @) Shift by 1 to right (ii) Scale by 2 followed by reflection () 2e(t-I) Gi) x(2t-1) VAN ° t 2 aa o + | e x€2t-1) 4 > t “a to 2 GQ) x( 20642) = x (244) 1D SHife by 4 fo lett @1) Scale by 2 @ x(tt4) @ x (2 +4) Al. At "“S 27320 oO z 2 (2) X (2(¢-2)) = x(2¥-4) 2 (i) shift by & to gE (i) sole by 2 wW (t-4) ii) EMH) : | /\ ¢ At ° 3045 ° Z2f 2 Zz (#) x(3+) 4 x(t) x(3t +2) aot * 3 3 = 1 + X Gt+2) t /\ ra “lI 2tol id 3 3 3 -1 et (Q) Determine the values of Ps. and £. for each of the following signals: (a) x((t) = &° “u(?) (b) x(t) = ef@rer (e) xx(t ; ) = cos(t @) ifn) = Gyula] @) ela] = efP) ein]: _ sn) -2t 3 Ans: @) “ x- 6 uw fl O >t &) uct) 1 ° t 1 oN : °o ? so gt = fers cfd = fe tar = @ | = : joules =r], m= Li, a fie*uerae = bin de ser o = T lim 5 [eo] Foe or 4 © -4T 677] ae ZF [ine i a-e') = O watts &(~) 3’ +e) jet _2 Je +h) _, To= Bo (9) 2¢,(t) = Note: All continuous-time com plex exponential signals ave periodic. co All periodic signals ave power Signals. [le Jorenayy, ‘dt = Los = 00 joules (at + p= lim tof ‘Plat = tim ‘hat < to geo aw T2027, Teoo 271, (©) %l= cost [eos at] Note: Continucus-Lime sinusoidal signals are always power signals. = flee at = [eteae = [3 coszt oe —¢0 = [2 ej? + + [zee)” = 90 joules ofa E..= =+ fre t/7dt = od L [cos tdt sine “Tops a olintegred 28) A 26 ° -+ Pie: ot = ["ajees L (a) EIS (A) £0) =(£)"ale] = L 0 be]! + watts e- =, \G@) "a6r]] + ay | i aed - —— a= ey 21 O ta by “ — ° L >n i} Ll, Ool23 ao n = = (4)’= +4 + +f+ ere & jotles 3 = Pe fim 2 = jeoul? No 2N+1 x)= O watts cade aC = wari iene Blew IOP ja. 0, malt) me 2X ().L] = -e? 7/2 2) a" 2, samples Binge I Periedic Fx, ee = le oe DI = 20: 41414... te © = 00 joules nak Bowne RIM gp Bene gteveas nzo ro —- (f) x3[r] = cos(En) [oo No)" = 25-)m=8 samples| Periodic Ex, -_= eos (En)|* =-@. + cos(Z(-i))+ os E(6) +5'(E ()) 4.89 ne =n oo joules ZL - fay th Ente Bette FREE = 4 =4 es r£(0] —it watts