Signal Processing and Linear Systems-B.P.Lathi copy

51 52 11 1 i ntroduction t o signals a nd systems 53

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Unformatted text preview: pe of a person is a cylinder of variable radius r (which varies w ith t he height h) t hen a reasonable measure of the size of a person of height H is t he p erson's volume V , given by V= 7r lH r 2(h) dh Signal Energy Arguing in this manner, we may consider t he a rea under a signal f (t) as a possible measure of its size, because i t takes account of not only t he a mplitude, b ut also t he d uration. However, this will b e a defective measure because f (t) could be a large signal, y et i ts positive and negative areas could cancel each other, indicating a signal of s mall size. This difficulty can be corrected by defining the signal size as the area u nder f 2(t), which is always positive. We call this measure t he s ignal e nergy E f, defined (for a real signal) as Ef = i: f2(t) dt (1.1) T his definition c an b e generalized t o a complex valued signal f (t) as (1.2) There are also o ther possible measures of signal size, such as the area under I f(t)l. T he energy measure, however, is n ot only more tractable mathematically, b ut is also more meaningful (as shown later) in t he sense t hat i t is indicative of the energy t hat c an be e xtracted from t he signal. Signal Power T he signal energy must be finite for i t t o be a meaningful measure of t he signal size. A necessary condition for the energy t o b e finite is t hat t he signal amplitude - > 0 as It I ...... 0 0 (Fig. 1.1a). Otherwise t he i ntegral in Eq. (1.1) will not converge. In some cases, for instance, when the amplitude of f (t) does not ...... 0 as It I ...... 0 0 (Fig. L Ib), t hen, t he signal energy is infinite. A more meaningful measure of t he signal size in s uch a case would be the time average of the energy, if it exists. This measure is c alled the p ower of the signal. For a signal f (t), we define its power P f as 1 j T/2 f2(t) dt T -oo T - T/2 P f = lim - (1.3) F ig. 1.1 power. Examples of Signals: (a) a signal with finite energy (b) a signal with finite We c an generalize t his definition for a complex signal f (t) as 1 j T/2 P f = lim -T T~oo - T/2 If(t)12 d t (1.4) Observe t hat t he signal power P f is t he t ime average (mean) of t he signal amplitude squared, t hat is, t he m ean-squared value of f (t). I ndeed, t he s quare root of P f is t he familiar r ms ( root mean square) value of f (t). T he m ean of an e ntity averaged over a large t ime interval approaching infinity exists if t he e ntity is e ither periodic or h as a s tatistical regularity. I f such a condition is n ot satisfied, t he average may n ot exist. For instance, a ramp signal f (t) = t increases indefinitely as It I --+ 0 0, a nd n either t he energy nor t he power exists for t his signal. C omments T he signal energy as defined in Eq. (1.1) or Eq. (1.2) does not indicate the actual energy of t he signal because t he signal energy depends not only on t he signal, b ut also on t he load. I t can, however, be interpreted as t he energy dissipated in a normalized load of a I -ohm resistor if a voltage f (t) were t o b e applied across the I -ohm r...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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