The Laplace and Fourier transforms are continuous (integral) transforms of continuous functions.
The Laplace transform maps a function f(t)
to a function F(s) of the complex variable s, where s=σ+jω.
Since the derivative f˙(t)=df(t)dt
maps to sF(s), the Laplace transform of a linear differential equation is an algebraic equation. Thus, the Laplace transform is useful for, among other things, solving linear differential equations.
If we set the real part of the complex variable s to zero, σ=0
, the result is the Fourier transform F(jω) which is essentially the frequency domain representation of f(t).
The Z transform is essentially a discrete version of the Laplace transform and, thus, can be useful in solving difference equations, the discrete version of differential equations. The Z transform maps a sequence f[n]
to a continuous function F(z) of the complex variable z=rejΩ.
If we set the magnitude of z to unity, r=1
, the result is the Discrete Time Fourier Transform (DTFT) F(jΩ) which is essentially the frequency domain representation of f[n].
In the following links, you can find more answers and comments on your questions
Best Luck.
http://electronics.stackexchange.com/questions/86489/relation-and-difference-between-fourier-laplace-and-z-transforms
https://www.quora.com/Signal-Processing/What-are-the-differences-between-Laplace-and-Fourier-Transform
http://forum.allaboutcircuits.com/threads/comparison-of-fourier-z-and-laplace-transform.30658/
http://functionspace.com/topic/257/Differences-between-Fourier---Laplace-transforms-
The Beladel answer is okay and very satisfactory, but i want to stress some important points concerning this question and its answer.
The question is a very interesting one and is useful for the researcher. It is about the the understanding of most important tools for signal and systems analysis and synthesis.
The systems operate in time domain and therefore its performance is analysed and described in time domain by means of differential equations. The system of equations describing the system is called the mathematical model of the system. It is challenging to solve the system of equations in the time domain. La Place invented a tool to transform the differential equations into algebraic equations provided that equations are linear. It is his La place transformation. Where a function f(t) is transformed into F(S) a function in the complex frequency domain S where the differentiation of f(t) is just a multiplication of F(S) by S where the integration is a division by S in the S domain. So, the invention of the S-domain has simplifies the system and signal analysis very much.
The signals and systems can be described also in the frequency domain. The frequency domain is a special domain of the la Place domain by formally making S= jw where j is the imaginary and w is the frequency. Now, with this substitution, one gets the Fourier transform. As a co sequence of this transform one can can get the frequency content of any function f(t). This means that such functions can be analysed to its frequency components or tones. Conversely functions cab be constructed from their frequency components.The concept of the bandwidth is the consequence of such signal description. Concerning the system description one obtained the amplitude frequency response and phase response. Which is equivalent to the time domain description but easy to measure and analyse suing harmonic signals inform of sine and cosines.
This concerns the continuous time descriptions. For discrete time functions and systems one has the Z-domain. The z domain is the discrete S domain where by definition Z= exp S Ts with Ts is the sampling time. It is also a special domain of the S-domain. Also the discrete time functions and systems can be easily mathematically described and synthesized in the Z-domain exactly like the S-domain for continuous time systems and signals.
In my post i wished to underline the usefulness of such tools for the function ans systems descriptions. The analysis of linear systems became much easier. Thank you La Place. Than you Fourier.
