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Common Z-Transforms

Module by: Dan Calderon. E-mail the author

Table 1: Common Continuous Time Fourier Series
Signal Z-transform Region of Convergence
δ ( t ) δ ( t ) 1 all ss
δ ( t - T ) δ ( t - T ) e - ( s T ) e - ( s T ) all ss
u ( t ) u ( t ) 1 s 1 s R > 0 R > 0
- u ( - t ) - u ( - t ) 1 s 1 s R < 0 R < 0
t u ( t ) t u ( t ) 1 s 2 1 s 2 R > 0 R > 0
t n u ( t ) t n u ( t ) n ! s n + 1 n ! s n + 1 R > 0 R > 0
- ( t n u ( - t ) ) - ( t n u ( - t ) ) n ! s n + 1 n ! s n + 1 R < 0 R < 0
e - ( λ t ) u ( t ) e - ( λ t ) u ( t ) 1 s + λ 1 s + λ R > - λ R > - λ
( - ( e - ( λ t ) ) u ( - t ) ( - ( e - ( λ t ) ) u ( - t ) 1 s + λ 1 s + λ R < - λ R < - λ
t e - ( λ t ) u ( t ) t e - ( λ t ) u ( t ) 1 ( s - λ ) 2 1 ( s - λ ) 2 R > - λ R > - λ
t n e - ( λ t ) u ( t ) t n e - ( λ t ) u ( t ) n ! ( s + λ ) n + 1 n ! ( s + λ ) n + 1 R > - λ R > - λ
- ( t n e - ( λ t ) u ( - t ) ) - ( t n e - ( λ t ) u ( - t ) ) n ! ( s + λ ) n + 1 n ! ( s + λ ) n + 1 R < - λ R < - λ
cos ( b t ) u ( t ) cos ( b t ) u ( t ) s s 2 + b 2 s s 2 + b 2 R > 0 R > 0
sin ( b t ) u ( t ) sin ( b t ) u ( t ) b s 2 + b 2 b s 2 + b 2 R > 0 R > 0
e - ( a t ) cos ( b t ) u ( t ) e - ( a t ) cos ( b t ) u ( t ) s + a ( s + a ) 2 + b 2 s + a ( s + a ) 2 + b 2 R > - a R > - a
e - ( a t ) sin ( b t ) u ( t ) e - ( a t ) sin ( b t ) u ( t ) b ( s + a ) 2 + b 2 b ( s + a ) 2 + b 2 R > - a R > - a
d n d t n δ ( t ) d n d t n δ ( t ) s n s n all ss

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

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