What are the similarities and differences between the Laplace transform and Fourier transform?

Fourier transform is defined only for functions defined for all the real numbers, whereas Laplace transform does not require the function to be defined on set the negative real numbers. Every function that has a Fourier transform will have a Laplace transform but not vice-versa.

What is the relation between FT & ZT justify your answer?

There is a close relationship between Z transform and Fourier transform. If we replace the complex variable z by e –jω, then z transform is reduced to Fourier transform. The frequency ω=0 is along the positive Re(z) axis and the frequency ∏/2 is along the positive Im(z) axis.

What is the relationship between Fourier transform and Fourier series?

Fourier series is an expansion of periodic signal as a linear combination of sines and cosines while Fourier transform is the process or function used to convert signals from time domain in to frequency domain.

What is the relationship between Fourier transform and FFT?

The fast Fourier transform (FFT) is an efficient implementation of the discrete Fourier Transform (DFT). There is also the discrete-time Fourier transform (DTFT) which under some stimulus conditions is identical to the DFT. There are also continuous time Fourier transforms.

Why do we use Fourier Transform and Laplace transform?

Fourier transform helps us to study anything in the frequency domain whereas laplace transform is usually done for complex analysis (when anything is not easier to analyse in time domain, we convert it into s domain and then take the inverse laplace transform to complete the analysis).

What is Fourier Transform in signals and systems?

The Fourier Transform is a mathematical technique that transforms a function of time, x(t), to a function of frequency, X(ω). Likewise, we can derive the Inverse Fourier Transform (i.e., the synthesis equation) by starting with the synthesis equation for the Fourier Series (and multiply and divide by T).

What is the difference between the Fourier Laplace transform?

Fourier transform is defined only for functions defined for all the real numbers, whereas Laplace transform does not require the function to be defined on set the negative real numbers . Fourier transform is a special case of the Laplace transform. It can be seen that both coincide for non-negative real numbers.

What is a Fourier transform and how is it used?

The Fourier transform is a mathematical function that can be used to show the different parts of a continuous signal. It is most used to convert from time domain to frequency domain. Fourier transforms are often used to calculate the frequency spectrum of a signal that changes over time.

What exactly is Laplace transform?

Laplace transform. In mathematics, the Laplace transform is an integral transform named after its discoverer Pierre-Simon Laplace (/ləˈplɑːs/). It takes a function of a real variable t (often time) to a function of a complex variable s (complex frequency).

What is the Laplace transform in its simplified form?

Laplace Transform Laplace Transform of Differential Equation. The Laplace transform is a well established mathematical technique for solving a differential equation. Step Functions. The step function can take the values of 0 or 1. Bilateral Laplace Transform. Inverse Laplace Transform. Laplace Transform in Probability Theory. Applications of Laplace Transform.