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The parameter s is in general complex:

Table of common Laplace transform pairs Time domain ID

Function

1

ideal delay

1a

unit impulse

2

delayed nth power with frequency shift

2a

nth power

Métodos Estocásticos – Prof. Emilio Wille – UTFPR

Frequency domain

Region of convergence for causal systems

1

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

Métodos Estocásticos – Prof. Emilio Wille – UTFPR

2

8

Exponentiallydecaying sine wave

9

Exponentiallydecaying cosine wave

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

Bessel function 14 of the second kind, of order 0

Modified Bessel function 15 of the second kind, of order 0

Métodos Estocásticos – Prof. Emilio Wille – UTFPR

3

16 Error function

Explanatory notes: •

•

•

represents the Dirac delta function.

• • •

•

represents the Heaviside step 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 is an integer.

are real numbers.

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.

Unilateral Z-Transform Alternatively, in cases where x[n] is defined only for n ≥ 0, the single-sided or unilateral Ztransform is defined as

In signal processing, this definition is used when the signal is causal.

Table of common Z-transform pairs Signal, x[n]

Z-transform, X(z)

ROC

1

Métodos Estocásticos – Prof. Emilio Wille – UTFPR

4

2

3

4

5

6

7

8

9

10

11

Métodos Estocásticos – Prof. Emilio Wille – UTFPR

5

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