{"id":2719,"date":"2024-08-08T18:19:51","date_gmt":"2024-08-08T12:49:51","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=2719"},"modified":"2024-08-13T17:37:48","modified_gmt":"2024-08-13T12:07:48","slug":"laplace-transform","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/ec\/signals-systems\/laplace-transform","title":{"rendered":"Laplace Transform"},"content":{"rendered":"

In Fourier transform, one problem we faced that Fourier transform is not applicable for signals that are not absolutely integrable. This problem could be resolved by generalizing Fourier transform which leads to development
\nof Laplace transform. The Laplace transform can be applied to broader class of signals compared to Fourier transform and analysis of many unstable systems thus play an important role in study of stability of systems. There are two varieties of Laplace transform: bilateral and unilateral. The bilateral one, also known as two- sided Laplace transform is defined for \u2013 \u221e \u2264 t \u2264 \u221e can handle all causal and non causal signals. It provides insights about system\u2019s characteristics such as stability, causality and frequency response.
\nThe unilateral one, also known as one-sided Laplace transform can handle only causal signals and mainly used to solve differential equations with initial conditions.<\/p>\n

The Definition of Laplace Transform<\/h2>\n

Consider a continuous time signal x (t ). Its Laplace transform (bilateral)\"Laplace\"Convergence\"<\/p>\n

where \u2018c\u2019 is constant and responsible for convergence of integral.<\/p>\n

Relationship between Laplace Transform and Fourier Transform<\/h3>\n

For a signal x(t), its Laplace transform X(s) is given by\"Laplace<\/p>\n

Region of Convergence (ROC) for Laplace Transform<\/h3>\n

Region of Convergence (ROC) is the set of values of \u2018s\u2019 for which the Laplace transform of the signal converges. The Laplace transform is guaranteed to converge if x(t)e\u2013\u03c3t<\/sup> is absolutely integrable<\/p>\n

\"Convergence\"<\/p>\n

Thus ROC consist of those values of \u201c\u03c3\u201d [Re {s}] for which the Fourier transform of x(t) e\u2013\u03c3t<\/sup> converges.<\/p>\n

Properties of ROC<\/h4>\n
    \n
  1. ROC along with algebraic expression of X(s) provides complete specification of Laplace transform.<\/li>\n
  2. In case of identical algebraic expression for X(s) the Laplace transforms are distinguishable only by ROC.<\/li>\n
  3. The ROC of X(s) consists of strips parallel to the j\u03c9-axis in s-plane.<\/li>\n
  4. If signal x(t) is of finite duration then the ROC is entire s-plane excluding possibly s = 0, \u221e, \u2013\u221e.<\/li>\n
  5. For rational Laplace transform, the ROC does not contain any poles.<\/li>\n
  6. For rational Laplace transform, if x(t) is left sided, the ROC is the region in the s-plane to the left of the left most pole and if x(t) is right sided, the ROC is the region to the right of the right most pole.<\/li>\n
  7. If ROC for X(s) includes j\u03c9 axis, the Fourier transform to x(t) is possible.<\/li>\n<\/ol>\n

    The s-plane and poles and zeros<\/h4>\n

    The graphical representation of complex frequency \u2018s\u2019 in terms of complex plane is called s-plane. with s = \u03c3 + j\u03c9, horizontal axis represents the real part (\u03c3), and vertical axis represents imaginary part (j\u03c9) Consider a ratio of two polynomials, say F(s)
    \n\"S-Plane\"<\/p>\n

     <\/p>\n

     <\/p>\n","protected":false},"excerpt":{"rendered":"

    In Fourier transform, one problem we faced that Fourier transform is not applicable for signals that are not absolutely integrable.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[729,6],"tags":[738,739],"class_list":["post-2719","post","type-post","status-publish","format-standard","hentry","category-signals-systems","category-ec","tag-bilateral","tag-properties-of-roc"],"_links":{"self":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/2719","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/comments?post=2719"}],"version-history":[{"count":0,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/2719\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/media?parent=2719"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/categories?post=2719"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/tags?post=2719"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}