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# Difference between Z-Transform and Laplace Transform

## Z-Transform

The Z-transform (ZT) is a mathematical tool which is used to convert the difference equations in time domain into the algebraic equations in z-domain.

Mathematically, if $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete-time signal or sequence, then its *bilateral* or *two-sided Z-transform* is defined as −

$$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\sum_{\mathit{n=-\infty }}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}\:\:\:\:\:\:...(1)}$$

Where, z is a complex variable.

Also, the *unilateral* or *one-sided z-transform* is defined as −

$$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\sum_{\mathit{n=\mathrm{0} }}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}\:\:\:\:\:\:...(2)}$$

## Laplace Transform

The Laplace transform is a mathematical tool which is used to convert the differential equation in time domain into the algebraic equations in the frequency domain or *s*-domain.

Mathematically, if $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ is a time domain function, then its Laplace transform is defined as −

$$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{-\infty }^{\infty }\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-\mathit{st}}\:\mathit{dt}}\:\:\:\:\:\:...(3)}$$

Equation (1) gives the bilateral Laplace transform of the function $\mathit{x}\mathrm{\left(\mathit{t}\right)}$. But for the causal signals, the unilateral Laplace transform is applied, which is defined as −

$$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{\mathrm{0} }^{\infty }\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-\mathit{st}}\:\mathit{dt}}\:\:\:\:\:\:...(4)}$$

## Difference between Z-Transform and Laplace Transform

The following table highlights some of the major points that differentiate Z-Transform and Laplace Transform −

Z-Transform | Laplace Transform |
---|---|

The Z-transform is used to analyse the discrete-time LTI (also called LSI - Linear Shift Invariant) systems. | The Laplace transform is used to analyse the continuous-time LTI systems. |

The ZT converts the time-domain difference equations into the algebraic equations in z-domain. | The LT converts the time domain differential equations into the algebraic equations in s-domain. |

ZT may be of two types viz. onesided (or unilateral) and two-sided (or bilateral). | LT may also be of two types viz. one-sided (or unilateral) and twosided (or bilateral). |

The ZT is a simple and systematic method and the complete solution can be obtained in one step. Also, the initial conditions can be introduced in the beginning of the process. | The LT is also a simple and systematic method and the complete solution can be obtained in one step. Also, the initial conditions can be introduced in the beginning of the process. |

The set of points in z-plane for which the function $\mathit{X}\mathrm{\left(\mathit{z}\right)}$ converges is called the ROC of $\mathit{X}\mathrm{\left(\mathit{z}\right)}$. | The set of points in s-plane for which the function X(s) converges is called the ROC of X(s). |

The ROC of the Z-transform $\mathit{X}\mathrm{\left(\mathit{z}\right)}$ consists of a ring in z-plane centred at the origin. | The ROC of LT X(s) consists of strip parallel to jω-axis in s-plane. |

When the magnitude of z is unity, i.e.,$\left|\mathit{z} \right|$ = 1, then the ZT becomes discrete-time Fourier transform (DTFT). | When the real part of the variable 's'
is equal to zero, i.e., σ = 0, then the
LT becomes the continuous-time
Fourier transform (CTFT). |

Convolution in time-domain is equal to multiplication in z-domain. | Convolution in time-domain is also equal to multiplication in s-domain. |

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