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

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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 —&gt; 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+ &quot;bk: ‘fCi DUNN? 383% \u xtH, C ozé MW) [h :3) = cl uztw‘ﬁﬁtgrj + c; mum-m Mags»; 3:) _ = 136.3%&quot;: ' 30km? r ox 65 v6 3m 1 ”99% :95 gm (mum .1. :2 x &lt;3 “mm 3 Egg Ear £2 «E s .33 «“3 .v a 5A 3115. x, 5 L &gt; 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/&quot;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 &quot;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&amp;1;waer arm ot‘ﬂfmw Wﬁhew &quot;am WWW ever ﬁrm Wm“? sz'fj) a {7mm «act-‘5) MW) +2Lt-33ui,+ '3) =Wﬁ) —— SCJcQNH-é) + 2&amp;3) was?) ‘ 'pxmnw wvﬁe ﬁe wwwwrw dawﬁm wfﬁe qtof’wM 9mm WEN“) xxw=gucwﬁ . ‘ . wwkmnqéwa 6&quot;”‘1' w 16) ;:«am pew: (ﬁafmec—D -M(:t—2)] + ¥ - EAR—2) - uwﬁ = w) N467!) — (ﬁW-ﬁ) manor) ...
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