102_1_lecture12_student

# 102_1_lecture12_student - UCLA Fall 2011 Systems and...

This preview shows pages 1–11. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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”....
View Full Document

## 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.

### Page1 / 34

102_1_lecture12_student - UCLA Fall 2011 Systems and...

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online