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

# T he m ean of an e ntity averaged over a large t ime

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Unformatted text preview: esistor (or if a current f (t) were t o b e passed t hrough t he I -ohm resistor). T he m easure of &quot;energy&quot; is, therefore indicative of t he e nergy capability of thE signal a nd n ot t he a ctual energy. For t his reason t he c oncepts of conservation oj energy should not be applied t o t his &quot;signal energy&quot;. P arallel observation applieE t o &quot;signal power&quot; defined in Eq. (1.3) o r (1.4). These measures are b ut convenient indicators of t he signal size, which prove useful in many applications. For instance, il we a pproximate a signal f (t) b y a nother signal g(t), t he e rror in t he approximation is e(t) = f (t) - g(t). T he energy (or power) of e(t) is a convenient indicator 01 t he goodness of t he a pproximation. I t provides us with a quantitative measure oj d etermining t he closeness of t he a pproximation. I n c ommunication systems, during transmission over a channel, message signals a re c orrupted by unwanted signal! (noise). T he q uality of t he received signal is j udged by t he relative sizes of thE 54 I ntroduction t o S ignals a nd S ystems 1.1 55 S ignals • o -\ 2 4 1- E xample 1 .2 D etermine t he power and the rms value of ( a) f (t) = 0 cos (Wot + II) ( b) f (t) = 01 cos ( Wit + I IIl + 02 cos (w2t+1I2) ( WI ;.!oW2). ( c) f (t) = D eiwo '. ( a) T his is a periodic signal with period To = 271' /wo. T he s uitable measure of this signal is its power. Because i t is a periodic signal, we may compute its power by averaging its energy over one period To = 271' /wo. However, for t he sake of demonstration, we shall solve this problem by averaging over a n infinitely large time interval using E q (1.3). 1 l TI2 2 P I = lim 0 cos 2 (wot + II) d t T~oo T - T12 0 21TI2 = lim 2 T dt T~oo - T12 Fig. 1 .2 Signals for Example 1.1. d esired s ignal a nd t he u nwanted s ignal ( noise). I n t his c ase t he r atio o f t he m essage s ignal a nd n oise s ignal p owers ( signal t o n oise p ower r atio) is a g ood i ndication o f t he r eceived s ignal q uality. Units of Energy and Power: E quations (1.1) a nd (1.2) a re n ot c orrect d imensionally. T his is b ecause h ere we a re u sing t he t erm e nergy n ot i n i ts c onventional s ense, b ut t o i ndicate t he s ignal size. T he s ame o bservation a pplies t o E qs. (1.3) a nd (1.4) f or p ower. T he u nits o f e nergy a nd p ower, a s d efined h ere, d epend o n t he n ature o f t he s ignal f (t). I f f (t) is a v oltage s ignal, i ts e nergy E f h as u nits o f V 2s ( volts s quared-seconds) a nd i ts p ower P , h as u nits o f V 2 ( volts s quared). I f f (t) is a c urrent s ignal, t hese u nits will b e A 2s ( amperes s quared-seconds) a nd A 2 ( amperes s quared), r espectively. + = 0 21TI2 lim - T T~oo 2 - T12 0 21TI2 lim 2 T T~oo - T12 [I + cos (2wot + 2(1)] dt cos (2 wot + 2(1) d t T he first t erm on t he r ight-hand side is e qual t o 0 2 /2. Moreover, t he second t erm is zero because t he integral appearing in this t erm represents t he a rea u nder a sinusoi...
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