Fourier Transform ,LAPLACE TRANSFORM,ROC and its Properties
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1. The document discusses Fourier transforms, Laplace transforms, and their applications. Fourier transforms represent non-periodic signals as a function of frequency by decomposing them into simpler constituent parts. Laplace transforms transform time domain signals to the complex s-domain. 2. Key aspects covered include the Fourier analysis and synthesis equations, properties of the Fourier transform such as its magnitude and phase spectra. Conditions for the existence of Fourier transforms are explained. The region of convergence where the Laplace transform converges is also defined. 3. Examples are provided to demonstrate the calculation of Fourier and Laplace transforms of simple signals and determining their regions of convergence. Poles and zeros of transforms are also explained.
Fourier Transform ,LAPLACE TRANSFORM,ROC and its Properties
- 1. Dr.K.G.SHANTHI Professor/ECE shanthiece@rmkcet.ac.in RMK College of Engineering and Technology
- 2. 2 Fourier series Fourier Transform Laplace Transforms
- 3. 3 Fourier Series Fourier Transform CTF S DTF S CTF T DTF T Fourier Transform can be used for Periodic signal also If the input signal x(t)
- 4. 4 Mathematical transformation employed to transform signals between time (or spatial) domain and frequency domain. Fourier method of representing non-periodic signals as a function of frequency Fundamental period T tend to infinity F.T Analysis : Break the signal or functions into simpler constituent parts dt e t x j X t x F t j d e j X t x t j 2 1 Analysis Equation: Synthesis Equation F.T Synthesis : Reassemble a signal from its constituent parts Fourier transform Pair : Analysis+ Synthesis
- 5. j X j j X j X i r j X j Xr of part Real j X j Xi of part Imaginary 2 2 j X j X j X i r or j X j X j X * j X of Conjugate j X * The X(jω) is a complex function of ω. Hence it can be expressed as The magnitude of X(jω) is called Magnitude Spectrum.
- 6. 6 The phase of X(jω) is called Phase Spectrum The phase spectrum can be written as The magnitude and phase spectrum together is called frequency spectrum j X j X j X r i 1 tan
- 7. 7 Fourier Transform does not exist for some signals. For example t u e t x t 2 Fourier Transform for x(t)does not exists because it is not absolutely integrable Existence of Fourier Transform-The Dirichlet Conditions should be satisfied Signal should have finite number of maxima and minima Signal should have finite number of discontinuities Signal should be absolutely integrable dt t x
- 8. 8 It is used to transform a time domain to complex frequency domain signal (s-domain) Two Sided Laplace transform (or) Bilateral Laplace transform Let 𝑥(𝑡) be a continuous time signal defined for all values of 𝑡. Let 𝑋(𝑆) be Laplace transform of 𝑥(𝑡)(non-causal signal ). One sided Laplace transform (or) Unilateral Laplace transform Let 𝑥(𝑡) be a continuous time signal defined for 𝑡≥0 (ie If 𝑥(𝑡) is causal) then, dt e t x s X t x L t s dt e t x s X t x L t s 0 Complex variable, S= σ+ jω
- 9. 9 Inverse Laplace transform (S-domain signal 𝑋(𝑆) Time domain signal x(t) ) s X t x Laplace transformX(s) and Inverse Laplace transform x(t) are called Laplace Transform Pair and can be expressed as ds s X j t x s X L j s j s 2 1 1
- 10. 10 Not absolutely integrable Absolutely integrable for σ>2 t u e t x t 2 t u e e t x t u e e e t x t t t t t ) 2 ( 2 converges
- 11. 11 The Laplace transform of a signal is given by The range of ‘s’ (σ) for which the Laplace transform converges (Finite) is called region of convergence dt e t x t s Complex variable, S= σ+ jω Re(s) - ∞ 0 ∞ jω σ LHS RHS Img(s) S plane
- 12. 12 The zeros are found by setting the numerator polynomial to Zero The zeros of the transform X(s)are the values of s for which the Transform is Zero The Poles are found by setting the Denominator polynomial to Zero. The Poles of the transform X(s)are the values of s for which the Transform is infinite. ) ( ) ( s D s N s X
- 13. 0 a where t u e t x Let t a Now Laplace transform of x(t) is given by, dt e t x s X t x L t s dt e t u e s X t s t a ) ( dt e e st at 0 dt e t a s 0 0 a s e t a s a s a s e e s X 1 ) ( 0 Since the given signal is right sided signal or causal signal then, a ROC : s-Plane Case i: Causal Signal or Right sided Signal a a s a s o a s ) Re(
- 14. Now Laplace transform of x(t) is given by, dt e t x s X t x L t s a : ROC Case ii: Non causal Signal or Left sided Signal 0 Let a where t u e t x t a dt e t u e s X t s t a ) ( dt e e st at 0 dt e t s a 0 0 a s e t s a a s s X 1 s a s a e e a s 0 . 1 b a cx b a cx e e dt e t s a 0 a a s a s a s ) Re( 0
- 15. 15 a : ROC Non causal Signal or Left sided Signal
- 16. 16 Case iii: Two sided Signal Let t u e t u e t x t b t a t x t x t x 2 1 t u e t x t a 1 t u e t x t b 2 and s X Find 1 dt e t u e s X t s t a ) ( dt e e st at 0 dt e t a s 0
- 17. 17 dt e e st bt 0 dt e t s b 0 s b s b t s b e e b s b s e 0 . 0 1 b a cx b a cx e e s X Find 2 0 a s e t a s a s a s e e s X 1 ) ( 0 dt e t u e s X t s t b ) ( 2 b s s X 1 2 a is ROC b is ROC dt e t b s 0 ) (
- 18. 18 Therefore ROC of X(s) is the region between two lines passing through poles –a and –b that is b a s-Plane ROC of a Two sided Signal
- 19. 19 Property 1 The ROC of X(s) consists of parallel strips to the imaginary axis. Property 2 The ROC of Laplace transform does not include any pole of X(s)
- 20. 20 Property 3 If x(t) is right sided or causal signal ,the ROC of X(s) extends to the right of the right most poles and no pole is located inside the ROC. Property 4 If x(t) is left sided or non causal signal ,the ROC of X(s) extends to the left of the left most poles and no pole is located inside the ROC. t u e t x E t a g a s s X 1 ) ( a ROC : t u e t x t a Eg a s s X 1 a : ROC
- 21. 21 Property 5 If x(t) is two sided signal the ROC of X(s) is a strip in the s-plane bounded by poles and no pole is located inside the ROC. s-Plane Property 7 Impulse function is the only function for which the ROC is the entire plane. Property 6 The ROC of the sum of two or more signals is equal to the intersection of the ROCs of those signals.