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102_1_lecture12_student - UCLA Fall 2011 Systems and...

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Unformatted text preview: UCLA Fall 2011 Systems and Signals Lecture 12: Fourier Transform and Frequency Response of Systems November 7, 2011 EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 1 Today’s topics : • Review • Fourier Transforms of periodic signals • Limiting transforms EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 2 Review Properties of Fourier Transform • Shift x ( t- t ) ↔ X ( jω ) e- jωt • Scaling x ( at ) ↔ 1 | a | X ( j ω a ) • Derivative dx ( t ) dt ↔ jωX ( jω ) • Modulation x ( t ) e jω t ↔ X ( j ( ω- ω )) • Duality X ( t ) ↔ 2 πx (- ω ) EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 3 • Parseval R ∞-∞ | x ( t ) | 2 dt = 1 2 π R ∞-∞ | X ( jω ) | 2 dω • ? The convolution theorem ? x ( t ) * y ( t ) ↔ X ( jω ) Y ( jω ) EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 4 Example. x(t) y(t) System H System G *h(t) H(jw) *g(t) G(jw) Output is given by: y ( t ) = x ( t ) * h ( t ) * g ( t ) This calculation is tedious. In frequency domain, output is given by: Y ( jω ) = X ( jω ) H ( jω ) G ( jω ) Analysis is often simpler EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 5 Example (Oppenheim & Willsky 4.14). Consider a signal x ( t ) with Fourier transform X ( jω ) . Suppose we are given the following facts. • x ( t ) is real and nonnegative. • F- 1 { (1 + jω ) X ( jω ) } = Ae- 2 t u ( t ) , where A is independent of t . • R ∞-∞ | X ( jω ) | 2 dω = 2 π Determine a closed-form expression for x ( t ) . EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 6 Solution: EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 7 Solution cont.: EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 8 Another version of convolution theorem We’ve studied: x ( t ) * y ( t ) ↔ X ( jω ) Y ( jω ) Convolution in time is equivalent to multiplication in frequency domain Another version: 1 2 π X ( jω ) * Y ( jω ) ↔ x ( t ) y ( t ) Convolution in frequency domain is equivalent to multiplication in time domain EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 9 Suppose Z ( jω ) = X ( jω ) * Y ( jω ) : z ( t ) = 1 2 π Z ∞-∞ Z ( jω ) e jωt dω Z ( jω ) = X ( jω ) * Y ( jω ) = Z ∞-∞ X ( jθ ) Y ( j ( ω- θ )) dθ z ( t ) = 1 2 π Z ∞-∞ Z ∞-∞ X ( jθ ) Y ( j ( ω- θ )) dθ e jωt dω Rearrange the integrals: z ( t ) = 1 2 π Z ∞-∞ X ( jθ ) Z ∞-∞ Y ( j ( ω- θ )) e jωt dω dθ Do ”u-substitution”. LetDo ”u-substitution”....
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This note was uploaded on 03/30/2012 for the course ELEC ENGR 102 taught by Professor Lee during the Fall '11 term at UCLA.

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102_1_lecture12_student - UCLA Fall 2011 Systems and...

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