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

Wtilw 211 wo 211 n oo lj 1 00 f weiwtdw 415

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: - 00 where Y (w), t he Fourier transform of y(t), is given b yt h (t)e-iwtdt (4.17) T he r ight-hand side is t he Fourier transform of h (t), and according t o o ur n otation introduced i n Eq. (4.3) this is H (w), whereas t he same entity is d enoted by H (jw) t The r elationship (4.19) applies only t o t he a symptotically s table s ystems. T he r eason is t hat when s = jw, t he i ntegral o n t he r ight-hand side o f Eq. (2.48) does n ot converge for u nstable systems. Moreover, even for marginally s table s ystems, t hat i ntegral does n ot converges in t he o rdinary sense, a nd H(jw) [or H(w)) c annot b e o btained b y r eplacing s i n H(s) w ith jw. As shown in Eq. (4.44b), Eq. (4.19) c an b e applied t o m arginally s table s ystems p rovided H(w) is i nterpreted as t he F ourier t ransform o f h(t) r ather t han a s H(s) w ith s r eplaced b y jw. 244 4 Continuous-Time Signal Analysis: T he Fourier Transform Y (w) = F (w)H(w) 4.2 Transforms of Some Useful Functions 1 (4.19) 245 r eet( x ) I n c onclusion, we showed t hat for a n LTIC s ystem with transfer function H (s), if t he i nput a nd t he o utput a re f it) a nd y(t), respectively, a nd if f it) = F(w) y(t) = -I Y (w) 1: 1 "2 "1 2" F ig. 4 .7 A gate pulse. t hen for asymptotically stable systems Y (w) = F(w)H(w) We shall derive this result again l ater in a more formal way. T he p rocedure o f t he frequency-domain m ethod is identical t o t hat o f t he timedomain m ethod. I n t he t ime-domain case we express t he i nput f (t) a s a s um o f its impulse components; in t he frequency-domain case, t he i nput is expressed as a s um o f everlasting exponentials (or sinusoids). I n t he former case, t he response y(t) o btained by summing t he s ystem's responses t o impulse components results in t he c onvolution integral; in t he l atter case, t he response o btained b y summing t he s ystem's response to everlasting exponential components results in t he Fourier integral. T hese ideas can be expressed mathematically as follows: 4 .2 Transforms o f S ome Useful Functions F or convenience, we now introduce a compact n otation for some useful functions such a s g ate, triangle, a nd i nterpolation functions. U nit Gate Function We define a u nit g ate f unction reet (x) as a gate pulse of u nit height a nd u nit w idth, centered a t t he origin, a s i llustrated in Fig. 4.7a:t Ixl > ~ Ixl = ~ Ixl < ~ 1 For t he t ime-domain case tilt) ===> h it) f it) = 1: the impulse response of the system is h (t) f (x)ti(t - x)dx T he g ate pulse in Fig. 4.7b is t he u nit g ate p ulse rect (x) e xpanded by a factor a nd t herefore can be expressed as reet(~) (see Sec. 1.3-2). Observe t hat T , t he d enominator o f t he a rgument o f rect(~), i ndicates t he w idth o f t he pulse. expresses J (t) as a sum of impulse components U nit Triangle Function expresses yet) as a sum of responses to impulse components y(t) = [ : f (x)h(t - x)dx We define a u nit t riangle function t .(x) as a t riangular pulse of unit height a nd u nit width, cent...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online