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

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 "energy" 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 "signal energy". P arallel observation applieE t o "signal power" 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...
View Full Document

Ask a homework question - tutors are online