# Difference between Z-Transform and Laplace Transform

Signals and SystemsElectronics & ElectricalDigital Electronics

## 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-TransformLaplace 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.