ID
|
Function
|
Time domain
|
Laplace domain
|
Region of convergence for causal systems
|
1
|
Ideal delay
|
|
|
|
1a
|
Unit impulse
|
|
|
|
2
|
Delayed nth power with frequency shift
|
|
|
|
2a
|
nth Power
|
|
|
|
2a.1
|
qth Power
|
|
|
|
2a.2
|
Unit step
|
|
|
|
2b
|
Delayed unit step
|
|
|
|
2c
|
Ramp
|
|
|
|
2d
|
nth Power with frequency shift
|
|
|
|
2d.1
|
Exponential decay
|
|
|
|
3
|
Exponential approach
|
|
|
|
4
|
Sine
|
|
|
|
5
|
Cosine
|
|
|
|
6
|
Hyperbolic sine
|
|
|
|
7
|
Hyperbolic cosine
|
|
|
|
8
|
Exponentially-decaying sine
|
|
|
|
9
|
Exponentially-decaying cosine
|
|
|
|
10
|
nth Root
|
|
|
|
11
|
Natural logarithm
|
|
|
|
12
|
Bessel function of the first kind, of order n
|
|
|
|
13
|
Modified Bessel function of the first kind, of order n
|
|
|
|
14
|
Bessel function of the second kind, of order 0
|
|
|
|
15
|
Modified Bessel function of the second kind, of order 0
|
|
|
|
16
|
Error function
|
|
|
|
17
|
Constant
|
|
|
|
Explanatory notes:
- represents the Heaviside step function.
- represents the Dirac delta function.
- represents the Gamma function.
- is the Euler-Mascheroni constant.
|
- , a real number, typically represents time,
although it can represent any independent dimension.
- is the complex angular frequency.
- , , , and are real numbers.
- is an integer.
|
- A causal system is a system where the impulse response h(t) is zero for all time t prior to t = 0. In general, the ROC for causal systems is not the same as the ROC for anticausal systems. See also causality.
|