ELEC584_Signals&Systems_lecture_notes_week_1 - ELEC 584 Signals and Systems Chapter 1 Signals and Systems A Signals and Systems Signals A signal

ELEC584_Signals&Systems_lecture_notes_week_1 - ELEC...

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Unformatted text preview: ELEC 584: Signals and Systems Chapter 1: Signals and Systems A. Signals and Systems . Signals: A signal is a set of data or information. a Systems: Signals may be processed further by systems, which may modify them or extract additional information from them. B. Energy and Power - Size of a Signal The size of any entity is a number that indicates the largeness or stregth of the entity. How can a number indicate the size (or strength) of a signal? 0 Signal Energy: The signal energy E3, is defined as: E, = f z2(t)dt, (la) 7;, a? E. = / |m(t)|2dt. (1b) '10] for real-valued and complex-valued signals, respectively. A signal can have finite or infinite energy. The necessary con- dition for the energy to be finite is that the signal amplitude —> D as t —; 00. A signal with finite energy is shown to the right. - Signal Power: When the signal energy is infinite, a more meaningful measure of the signal size would be the time everage of the energy, if it exists. This measure is called power of the signaland is defined as: 1 772 2 P, = 1' — , 2 T3201, _T/2.’E (t) dt ( a) 1 772 P, = lim — ]x(t)l2dt, (2b) T—roo T —T/2 for real-valued and complex-valued signals, respectively. Gen- erally speaking, power exists if the signal is either periodic or has a statistical regularity. If such a condition is not satisfied, the averaage may not exist. For instance, for a ramp signal :c[t) = t, either the energy nor power exists. However, the unit step signal :c(t) = u(t), which is not periodic nor has statistical regularky, does have a finite power. A signal with finite power is shown to the right. uare root of ower is the root-mean-s uare (ms) of arm. Units: The above equations are not correct dimensionaly. We are use these terms not in the conventional sense, but to indicate the signal size. The units of energy and power depend on the actual signal :J:(t). For example, if :r(t) is a voltage signal, then the units for energy and power are V2.5- and volts squared. respectively. . Example: Determine the suitable measures of the signals shown below. E?’ #1536, {Rafi-WW??? +‘3i‘? 501%? 0+ "bk: ‘fCi DUNN? 383% \u xtH, C ozé MW) [h :3) = cl uztw‘fifitgrj + c; mum-m Mags»; 3:) _ = 136.3%": ' 30km? r ox 65 v6 3m 1 ”99% :95 gm (mum .1. :2 x <3 “mm 3 Egg Ear £2 «E s .33 «“3 .v a 5A 3115. x, 5 L > 320 Inc: {J 1. AWE é. .2w1w . A 6; q (a; . so , 0 2w 1 + E é um fr 5 m 8\ 1M 0 min ‘oms . IN 1 SW“: - combim‘DN ‘fiLHfi ”Femfiw‘ ‘b E} @Nen why-.49, sketév‘w KC’W'M W Kev-6) \c) XL‘bJré) : xfflwfl b!) )CHv—b) : xE—hwwfl‘ x ha?) L993 Vex/"W Lu W W LC) fer WT) ; XH’rb)‘ mar-— XHH) = Km): 171% :0' Xc‘fio): )CUH-fi) =X{b) =+7(JF6 ‘6‘ w] {a flaw =X(-Jv-6) XokE~6)-'-= NH): NJ: *(vo ‘ ' lg Classification of Signals: 02° Continuous-time vs. discrete-time: ? o A signal that is specified for a continuum of value ‘ is a continuous-time signal. 0 A single that is only specified at discrete values of time is a discrete-time signal. 0:. Analog vs. discrete: o The concept of continuous-time is oftern confused with that of analog. These two are not the same. The same is true of the concept of discrete signal with digital. o A signal whose amplitude can take on any value in a continuous range is an analog signal. This means that analog signals’ amplitude can take on an infinite number of values. 0 A digital signal, on the other hand, is one whose amplitude can take on only a finite number of values. 0 The terms continuous and discrete qualify the nature of a signal along the time (horizontal) axis. The terms analog and digital qualify the nature of a signal along the am litude vertical axis. xm x0) _ r —|- r —h- (a) (b) .110) t—h' (C) (d) Figure 1.11 Exainples of signals: (a) analog. continuous time, (1}) digital, continuous time, (c) analog, discrete time, and (d) digital, discrete time. ‘2’ Periodic vs. aperiodic: o A signal x(t) is said to be periodic if for some positive constant To, x(t) = x(t + To), for all I o The smallest value of To that satisfies the above condition is called the fundamental period of x(t). A signal is aperiodic if it is not periodic. 4* Energy vs. power signals. '2‘ Deterministic vs. random: 0 A signal, whose physical description is known completely, either in a mathematical form or a graphic form, is a deterministic signal. ) o A signal whose values cannot be predicted precisely but are known only in terms of probabilistic description, such as mean value or mean-squared value, is a random signal. .4 E _ N33 U%fui 52:15: fit UM: Ste? mem um. ={ I, 17,0 ‘ 1 ' o 1720. _. 7.0, “was, {7 Lu ubH Um Ex 6AM *1) mama named W befiiw wé its FDV' Qfib‘xmflfi Hm} 363::5‘6 "M i5 an exPowoM-Emi mfiém out fmwm E3 NWT W“ W by??? viz!)- we aim}; “figmfiw ‘E‘i-fi‘rfiw {an J 1 8—D: Mb” 0 t U); W) Es also very mu! :m sgecfifr ‘3 0&1;waer arm ot‘flfmw Wfihew "am WWW ever firm Wm“? sz'fj) a {7mm «act-‘5) MW) +2Lt-33ui,+ '3) =Wfi) —— SCJcQNH-é) + 2&3) was?) ‘ 'pxmnw wvfie fie wwwwrw dawfim wffie qtof’wM 9mm WEN“) xxw=gucwfi . ‘ . wwkmnqéwa 6"”‘1' w 16) ;:«am pew: (fiafmec—D -M(:t—2)] + ¥ - EAR—2) - uwfi = w) N467!) — (fiW-fi) manor) ...
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