102_1_F08-102-Week-3-Recap

# 102_1_F08-102-Week-3-Recap - s = ± i ω and single zero at...

This preview shows pages 1–2. Sign up to view the full content.

14 WEEK-3 RECAP 10. Laplace Transforms: Definition (10.1) L s { f ( t ) } := ± 0 e - st f ( t ) dt := F ( s ) where f ( t ) is 0 for t < 0 — hence we could have written f ( t ) U ( t ) — and s is a complex variable, which can be written as s = α + i ω . L s { f ( t ) } if it exists is a function of s . The part of the s -plane over which F ( s ) exists is called the DOC – Domain Of Convergence — of F ( s ). FACT . L s is a Linear Transformation: L s { af 1 ( t ) + bf 2 ( t ) } = a L s { f 1 ( t ) } + b L s { f 2 ( t ) } In the next page we list Laplace Transforms of simple signals and some basic properties of L s {·} which will be needed later.

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

View Full Document
15 11. Laplace Transforms of Simple Signals In the following table you have time-functions on the left and their Laplace Transforms on the right. f ( t ) | L s { f ( t ) } U ( t ) | 1 s , Re s > 0 , single (‘order 1’) pole at s = 0 δ ( t ) | 1 e - at | 1 s + a , Re s > - Re a, single pole at s = - a e at | 1 s - a , Re s > Re a, single pole at s = a a : real or complex t | 1 s 2 , Re s > 0 , double (‘order 2’) pole at s = 0 t n , n > 0 , | n ! s n +1 , Re s > 0 , pole of order n + 1 at s = 0 sin ω 0 t | ω 0 s 2 + ω 2 0 , Re s > 0 , single poles at s = ± i ω 0 cos ω 0 t | s s 2 + ω 2 0 , Re s > 0 , single poles at
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: s = ± i ω and single zero at s = 0 sinh at | a s 2-a 2 , Re s > a, single poles at s = ± a cosh at | s s 2-a 2 , Re s > a, single poles at s = ± a and single zero at s = 0 More Properties : e ∓ β t f ( t ) | F ( s ± β ) , F ( s ) := L s { f ( t ) } f ( t-t d ) , t d > | e-st d F ( s ) , F ( s ) := L s { f ( t ) } ± t f ( τ ) d τ | 1 s F ( s ) , F ( s ) := L s { f ( t ) } t f ( t ) |-d ds F ( s ) , F ( s ) := L s { f ( t ) } t n f ( t ) , n ≥ | (for you to ﬁll in) d dt f ( t ) | sF ( s )-f (0) , F ( s ) := L s { f ( t ) } d 2 dt 2 f ( t ) | s 2 F ( s )-sf (0)-˙ f (0) , d n dt n f ( t ) | s n F ( s )-s n-1 f (0)-s n-2 ˙ f (0) · · ·-f n-1 (0) , n ≥ 1...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

102_1_F08-102-Week-3-Recap - s = ± i ω and single zero at...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online