They are similar. In many problems, both methods can be used. You can view Fourier transform is the Laplace transform on the circle, that is |z|=1. When you do Fourier transform, you don't need to worry about the convergence region. However, you need to find the convergence region for each Laplace transform. The discrete version of Fourier transform is discrete Fourier transform, and the discrete version of Laplace transform is Z-transform.
The region of convergence (ROC) of x(z) is the set of all values of z for which x(z) attains a finite value.
Fourier transform and Laplace transform are similar. Laplace transforms map a function to a new function on the complex plane, while Fourier maps a function to a new function on the real line. You can view Fourier as the Laplace transform on the circle, that is |z|=1. z transform is the discrete version of Laplace transform.
.....
Laplace equation: in 3D U_xx+U_yy+U_zz=0 Or in 2D U_xx+U_yy=0 where U is a function of the spatial variables x,y,z in 3D and x,y in 2D.Also, U_xx is the second order partial derivative of u with respect to x, same for y and z. Laplace transform: L(f(t))=integral of (e^(-s*t))*f(t) dt as t goes from 0 to infinity. Laplace transform is more like an operator rather than an equation.
They are similar. In many problems, both methods can be used. You can view Fourier transform is the Laplace transform on the circle, that is |z|=1. When you do Fourier transform, you don't need to worry about the convergence region. However, you need to find the convergence region for each Laplace transform. The discrete version of Fourier transform is discrete Fourier transform, and the discrete version of Laplace transform is Z-transform.
The region of convergence (ROC) of x(z) is the set of all values of z for which x(z) attains a finite value.
Fourier transform and Laplace transform are similar. Laplace transforms map a function to a new function on the complex plane, while Fourier maps a function to a new function on the real line. You can view Fourier as the Laplace transform on the circle, that is |z|=1. z transform is the discrete version of Laplace transform.
The Laplace transform is used for analyzing continuous-time signals and systems, while the Z-transform is used for discrete-time signals and systems. The Laplace transform utilizes the complex s-plane, whereas the Z-transform operates in the complex z-plane. Essentially, the Laplace transform is suited for continuous signals and systems, while the Z-transform is more appropriate for discrete signals and systems.
.....
Laplace equation: in 3D U_xx+U_yy+U_zz=0 Or in 2D U_xx+U_yy=0 where U is a function of the spatial variables x,y,z in 3D and x,y in 2D.Also, U_xx is the second order partial derivative of u with respect to x, same for y and z. Laplace transform: L(f(t))=integral of (e^(-s*t))*f(t) dt as t goes from 0 to infinity. Laplace transform is more like an operator rather than an equation.
the difference is the "S" and "Z" parameters. S used for analog computation while Z for digital processing. basically Z is the digital approximation of the analog frequency domain signal. Z=exp(sT) where T is the sampling time.
The Fourier transform is used to analyze signals in the frequency domain, transforming a signal from the time domain to the frequency domain. The z-transform is used in the analysis of discrete-time systems and signals, transforming sequences in the z-domain. While the Fourier transform is typically applied to continuous signals, the z-transform is used with discrete signals represented as sequences.
z transform perhaps? It's basically a laplace transform for discrete values rather than continuous (although this probably makes no sense to you if you're in algebra. This is stuff used in digital signal processing for Electrical Engineering).
Eginhard J. Muth has written: 'Transform methods' -- subject(s): Engineering, Laplace transformation, Operations research, Z transformation
1.z transform can not aaply in continious signal. 2.z transform can not analyse analog filter
t is the significance of Z-transform