{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

HW5 Solns

# HW5 Solns - (Problem 4.26 a(a i Given signals x t = te −...

This preview shows pages 1–6. 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 is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: (Problem 4.26 a) (a) i. Given signals x ( t ) = te − 2 t u ( t ) and h ( t ) = e − 4 t u ( t ), find the output y ( t ) using the convolution theorem and the inverse Fourier transform. Using the table, we know that X ( jω ) = 1 (2 + jω ) 2 (11) and also that H ( jω ) = 1 4 + jω (12) Use the convolution theorem. Y ( jω ) = H ( jω ) X ( jω ) = 1 (2 + jω ) 2 (4 + jω ) (13) To find the inverse Fourier transform of Y ( jω ), use a partial fraction expansion. 1 (2 + jω ) 2 (4 + jω ) = A (2 + jω ) 2 + B 2 + jω + C 4 + jω (14) Then, we get that 1 = A (4 + jω ) + B (2 + jω )(4 + jω ) + C (2 + jω ) 2 (15) Substitute ω = j 4 which then leaves the following 1 = C (2 + j ( j 4)) 2 = C ( − 2) 2 = 4 C (16) So C = 1 4 . Substitute ω = j 2 which then leaves the following 1 = A (4 + j ( j 2)) = A (2) = 2 A (17) So A = 1 2 . Then, substitute ω = 0. So, we get 1 = 4 A + 8 B + 4 C (18) Solving this, B = − 1 4 ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 13

HW5 Solns - (Problem 4.26 a(a i Given signals x t = te −...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online