Lesson_310

# Lesson_310 - Lesson 31 Lesson 31 Challenge 30 Lesson 31 –...

This preview shows pages 1–8. 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 is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lesson 31 Lesson 31 Challenge 30 Lesson 31 – Continuous-time Fourier transform Challenge 31 Lesson 31 Challenge 30 A full-wave rectified signal is shown below. ϖ =2 π /T T T /2 Lesson 31 Challenge 30 The Fourier transform is: What is the value of a 0 (DC level)? ( 29 ( 29 ( 29 ∑ ∞ ≠ =-- = 2 2 1 cos 4 i even i i i t i V a t v ϖ π ( 29 ( 29 π ϖ ϖ π π ϖ 2 / T /2 T T 2 cos 2 / T 2 ) sin( 2 / T 1 x(t)dt 2 / T 1 / 2 2 V t V dt t V a T f =- = = = = = ∫ ∫ Lesson 31 Lesson 31 The continuous-time Fourier transform (CTFT) is defined in terms of the analysis equation : and the synthesis equation ∫ ∞ ∞-- = dt x(t)e ) X(j t j ϖ ϖ ∫ ∞ ∞- Ω = d )e X(j x(t) j t ϖ ϖ Lesson 31 Lesson 31 The Fourier transform of x(t) exists if x(t) is absolutely integrable or square integrable (finite energy), plus some exceptions. The so-called Dirichlet conditions state that: The inverse Fourier transform of X(j ϖ ) equals x(t) at points where x(t) is continuous. The inverse Fourier transform of X(j ϖ ) converges to the mid-point of x(t) at points where x(t) is discontinuous. ∫ ∞ ∞-- = dt x(t)e ) X(j t j ϖ ϖ Lesson 31 Lesson 31 Example: Single sideband ϖ ϖ ( 29 ( 29 ( 29 t j t j t j e d e d e 2 1 2 1 2 1 1 ϖ ϖ ϖ π ϖ π ϖ ϖ ϖ δ π ϖ ϖ δ = =- =- ℑ ∫ ∫ ∞ ∞- ∞ ∞-- ( 29 ( 29 2 / 1 ϖ ϖ δ π ϖ- → ← ℑ t j e ( 29 ( 29 2 ϖ ϖ δ π ϖ- → ← ℑ t j e Lesson 31 Lesson 31 Example: x(t)=sgn(t)={1 if t>0, -1 if t<0, 0 if t=0} ( 29 [ ] ) ( ) ( sgn →--- = a at at t u e t u e t ( 29 ( 29 [ ] ( 29 ϖ ϖ ϖ ϖ ϖ j a j j a j a t u e t u e t a a a at at 2 2 1 1 ) ( ) ( sgn 2 2 = +- = -- + =-- ℑ = ℑ → → →-...
View Full Document

{[ snackBarMessage ]}

### Page1 / 26

Lesson_310 - Lesson 31 Lesson 31 Challenge 30 Lesson 31 –...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online