1. Trang chủ
  2. » Thể loại khác

Wiley signals and systems e book TLFe BO 214

1 0 0

Đang tải... (xem toàn văn)

THÔNG TIN TÀI LIỆU

Thông tin cơ bản

Định dạng
Số trang 1
Dung lượng 53,27 KB

Nội dung

9.3 Sirriilarities arid r.>ifF(irenc*t.sbc+wecn Fourier a~tdLaplace Transforms 199 Example 9.1 The functioii (t) = &(t)e-? LI >0 (9.10) has tfre Laplace trarisfoum (9.11) Tlie region of ronvcqcncc lies i o tii(2 right of the negative rinmhcr - a, and tlius contains thc imaginary axis of the s-plane with Re{s} = The FoLrrirr transform is obtained by substituting (9.10) into the clefintion of thrx Pomicr txansforni (9.1) arid evaluating the iritcgial yL+n X(jw) = , (9.12) This clearly sl.iows the coiinection (13.9) betwreri the Foiirier and Laplace traiisforms If, however, U < irr (9.10) the Fomier transforin tlors riot exist, hccauie the Fourier integral does r i d conveige The Laplace tritnsforni, 011 the othcr hand i s unch;t.rigd (9.11), although the region of coiivergcnc*clriow lies to the right of the positive mimtter -a arid no longer corit;tiiis the imaginary axis If the rcgiori of c onvergence of the Lapldce tr arisforni cwiitains the irnaginai y axis fir Fourier transkorrn (9.9) can thcrr be differentiated any riiirrihcr of imes The Lttplace tiarisforni can tkierciorr easily be obtained from tlie Fourirr tr;uisform using analytic tontinuation (C3-1aptt.r 5.4.3) In practice, Lhis mclarrs that ?U: is replaced by s and the rcgiori of convergcuc dcfiiics the Laplace t iansform Example 9.2 Wiat is tlie Laplace transform of the signal wil 11 Fourier traiisforrn X(,jU)= I ? (9.13) l+W‘L Expression (9.13) has rio singularities for rcal values of w aid c m be diffcrent i a l d at all points iFTe can porform analytir c-onlimitition 0x1 X ( j w ) wl-ieie we write x (.jw)= I - (,jw)2 I -1 X(s) = -= 1- s ( s - l)(s (9.14) + 1) (9.15)

Ngày đăng: 25/10/2022, 09:20

TÀI LIỆU CÙNG NGƯỜI DÙNG

  • Đang cập nhật ...

TÀI LIỆU LIÊN QUAN