¾ Starting from this lecture to unify the notation we use to denote the Fourier

¾ starting from this lecture to unify the notation

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¾ Starting from this lecture , to unify the notation, we use to denote the Fourier transform ¾ Note that, the Laplace transform is a mapping of a complex variable to complex amplitude. And, when s = j (purely imaginary), it is the Fourier transform. 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 7 Our new notation of Fourier transform:
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Connection with FT ¾ We have seen ¾ A more general connection is as follows. From we have The Laplace transform is the Fourier transform of the function x ( t ) multiplied by a real exponential function. 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 8
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LT May Resolve the Convergence Issue of FT ¾ Why is LT useful? Laplace transform may converge when Fourier transform doesn’t. The convergence issue of FT can be resolved by some choice of in LT. ¾ Example Find the LT of the step function u ( t ) Recall that we cannot apply the analysis equation to compute the FT of u ( t ), because u ( t ) is not absolutely integrable, and thus, FT does not converge. However, in LT: it is integrable under some choice of s . Caution: the algebraic expression is only valid for some s . (This is a key distinction from FT.) 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 9
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Examples ¾ Example 9.1 ¾ Example 9.2 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 10
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Examples – Graphical Illustration 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 11 a > 0 a < 0 0 t 0 t 0 t 0 t FT: FT: doesn’t exist FT: FT: doesn’t exist LT: LT:
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The Key Distinctions from the Previous Examples ¾ In FT, the existence of the FT X ( j ) depends on the signal itself ( e.g. , the parameter a of the signal . Once it exists, the transform is valid for all . ¾ In LT, the transform always converges ( i.e. , the algebraic expression of the transform does not depend on the signal x ( t )). However, the transform X ( s ) may be valid only for a limited range of s . ¾ This observation leads to a very important concept called Region of Convergence (ROC) of Laplace transform. In LT, not only do we need to specify the algebraic expression, but also we have to specify the region of convergence over which the expression is valid. 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 12
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ROC: Region of Convergence ¾ ROC: the set of s values over which Laplace transform exists. ¾ Example 9.3 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 13 s -plane x x o -2 -3/2
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ROC: Region of Convergence ¾ ROC: the set of s values over which the LT expression is valid.
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