These three transformations are very important! Any science and engineering disciplines inevitably need these transformations.
The essence of these three transformations is to convert the signal from the time domain to the frequency domain. The emergence of the Fourier transform subverts human perception of the world: the world can not only be seen as changes over time, but also as a combination of various frequencies and different weights. To give an unsuitable example: the sound waveform of a piano music is expressed in the time domain, while his piano score is expressed in the frequency domain.
Since the three transformations can convert differential equations or differential equations into polynomial equations, the calculation cost of differential (differential) equations is greatly reduced.
In addition, in the communication field, without frequency domain analysis of the signal, it will be difficult to understand a signal in the time domain. Because it is often necessary to divide channels by frequency in the communication field, the frequency domain characteristics of a signal are more important than the time domain characteristics.
The analysis of the specific three transformations (should be four) is as follows:
Fourier analysis includes Fourier series and Fourier transform. The Fourier series is used to transform periodic signals, and the Fourier transform is used to transform non-periodic signals.
But for non-convergent signals, the Fourier transform is powerless and can only rely on the Laplace transform. (Mainly used to calculate differential equations)
The z transform can be counted as a discrete Laplace transform. (Mainly used to calculate the difference equation)
Why change?
The meaning of all transformations is to express mathematically what the shape of a wave is . At the beginning, we can arrange an impulse function in a row in the order of time, and then multiply each by its own coefficient (linear combination) to get the shape of a wave on paper. Later, the great Fourier students discovered that not only the impulse function, multiplied by the respective coefficients after superimposing the complex exponential signal, can also express almost all wave waveforms. and! The output calculation method expressed by the complex exponential signal is much more regular than convolution, and this law can be seen from the frequency domain. This discovery made the signal transformation a big step forward.
Periodic signals can be represented by Fourier series, and non-periodic signals can be represented by Fourier transform. This will be off topic if we expand it further. Here is a note of the previous Fourier formula (*^__^*) (from Zhihu user Niu Baa)
Laplace transform : Fourier transform has higher requirements on the signal and is suitable for signals that decay fast. In order to expand the application range of the Fourier transform and enable it to be used in the analysis of more unstable systems, people artificially add a negative exponential function as a coefficient in the calculation process, so that some non-decayed signals decay faster, which is convenient for conversion. . This is the origin of the Labrador transformation. Laplace transform is used for continuous signals.
Labrador transform :
among them
Take S back to the formula to get:
Compared with the formula of Fourier transform, is there only one coefficient difference? Because the transformation has to converge to make sense, the discussion about the region of convergence is to make the 
integration meaningful. This involves a little knowledge of calculus. The final answer is viewed in a rectangular coordinate system, with the dividing line parallel to the Y axis.
Z transform : Similar to the purpose of the Laplace transform, the discrete-time Fourier transform formula is 
replaced by z, and then multiplied by a weighting coefficient to represent the modulus of z (usually equal to 1), and then the z transform is evolved. The z-transform is used for discrete signals.
z Transformation:, 
which 
can be restored by bringing it in.
Similarly, the convergence range of the Z transformation is to make the calculated value meaningful. After the expansion of the geometric formula, it can be seen that z needs to be less than or greater than a certain value. In polar coordinates, it is a circle.
From the complex plane, Fourier analysis pays attention to the imaginary part, Laplace transform pays attention to all complex planes, and z transform projects the Laplace complex plane to the z plane, changing the imaginary axis into one Ring. (An inappropriate analogy is the feeling that a picture can only be seen clearly by placing a metal rod in a fixed position and reflecting the light from the metal rod. )
How to understand?
I assume that now everyone has some understanding of these transformations, at least know how to calculate these transformations. Well, I will explain these changes from several different perspectives. A signal is usually expressed as a function of time 
, which is simple and intuitive, because its function image can be regarded as the waveform of the signal, such as sound waves and water waves. Many times, the signal processing is very special. For example, after a linear circuit processes the input sinusoidal signal, the output is still a sinusoidal signal, but there is a change in amplitude and phase (in fact, mathematically because the exponential function is The characteristic function of a linear differential equation is like the characteristic vector of a matrix, and this complex amplitude corresponds to the characteristic value). Therefore, if we decompose all the signals into linear combinations of sinusoidal signals (Fourier transform), 
then a transfer function can be used 
to describe this linear system. If this signal is very special, for example:, 
Fourier transform does not exist in mathematics, at this time Laplace transform is introduced to solve this problem. 
Such a linear system can use a transfer function
To represent. Therefore, it can be seen from here that decomposing the signal into a sine function (Fourier transform) or a complex exponential function (Laplace transform) is very important for analyzing linear systems.
If you only care about the signal itself and not the system, the relationship between these transformations can be connected through such a process. First of all, we need to make a clear point. No matter whether you use time domain or frequency domain (or s domain) to represent a signal, they all represent the same signal! You can understand this from the perspective of linear space . For the same signal, if different coordinate frames (or basis vectors) are used, their coordinates will be different. For example, if it is used 
as a coordinate, the signal can be expressed as 
, and the signal can be expressed as 
a Fourier transform.
. As mentioned in linear algebra, under two different coordinate frames, the coordinates of the same vector can be connected by a linear transformation. If it is a finite-dimensional space, it can be expressed as a matrix. In this case, it is an infinite dimension. This linear transformation is Fourier transform.
If we draw the Laplace 
domain, it is a complex plane, and the Laplace transform 
is a complex variable function on this complex plane. And this function along the imaginary axis 
value 
is the Fourier transform. Up to now, there are not many assumptions about the form of the signal. If the signal is a bandwidth-limited signal, that 
is , it is only in a small range (for example 
) not zero. According to the sampling theorem, the time domain can be sampled. As long as the sampling frequency is high enough, the signal can be restored without distortion. So what is the effect of sampling on the signal? From the s plane, the sampling in the time domain will be 
extended periodically along the imaginary axis! This property can be easily verified mathematically.
The z-transform can be regarded as a special form of the Laplace transform, that is, a substitution is made 
, and T is the sampling period. This transformation transforms the signal from the s domain to the z domain. Please remember the point mentioned earlier, the s-domain and the z-domain represent the same signal, that is, the signal after sampling. Only sampling will change the signal itself! From the perspective of the complex plane, this transformation transforms a strip parallel to the axis to a single-leaf branch in the z-plane.
You will see that the strips generated by the period extension caused by the previous sampling overlap. Because of the periodicity, The function values 
of different branches in the z-domain are the same. In other words, if there is no sampling and the z-transform is directly performed, a multi-valued complex function will be obtained! So generally only do z-transform on the signal after sampling!
Here we talked about time-domain sampling. After time-domain sampling, the signal has only the 
inter-frequency spectrum, that is, the highest frequency is only half of the sampling frequency. However, to record such a signal, infinite storage space is still needed, and further sampling in the frequency domain is possible. If the time is limited (which contradicts the frequency limitation), the original signal can be recovered from the sampled signal without distortion through frequency domain sampling (period extension in the time domain). And the signal length is limited, this is Discrete Fourier Transform (DFT), it has the famous fast algorithm Fast Fourier Transform (FFT). Why do I say DFT? Because computers need to effectively perform Fourier transform on ordinary signals, which are all realized by DFT. Unless the signal has a simple analytical expression!
In summary, for a linear system, the input and output are linear, whether it is a linear circuit or an optical circuit, as long as it can be described by a linear equation or linear differential equation (such as Laplace equation, Poisson equation, etc.) The system can analyze the characteristics of the system from the frequency domain through Fourier analysis, which is much more powerful than pure time domain analysis! Two well-known application examples are linear circuits and Fourier optics (information optics). Even nonlinear systems use linear systems in many cases! So the Fourier transform is so important! You see, the earliest Fourier was also to solve the heat conduction equation (it can also be regarded as a linear system)!
The idea of Fourier transform is still evolving in different fields. For example, wavelet transform in signal processing. It also uses a set of basis functions to express signals, but it overcomes the problem that Fourier transform cannot be used for time-frequency analysis at the same time. .
Finally, from my angle of pure mathematics say what degree of change in the end Fourier Yes.
Remember the algebraic equations in linear algebra 
? If A is a symmetric square matrix, you can find all the mutually orthogonal eigenvectors 
and eigenvalues of matrix A 
, and then express the vectors x and b as a combination of eigenvectors. 
Due to the orthogonal relationship of eigenvectors, the algebraic equation of the matrix can be transformed into N scalar algebraic equations 
, is it amazing! ! You would ask that this has something to do with the Fourier transform? Don't worry, looking at the non-homogeneous linear ordinary differential equations 
can verify that the exponential function 
is his characteristic function. If the equation is rewritten as an operator 
, then there is 
, is this similar to the characteristic vector eigenvalue of a linear equation? Express both y and z as a linear combination of exponential functions, then after this transformation, the ordinary differential equation becomes a scalar algebraic equation! ! The process of expressing y and z as a linear combination of exponential functions is the Fourier transform (or Laplace transform). There are similar conclusions in partial differential equations such as wave equations! This is what I experienced when I took the course of mathematical equations.
In summary, it means that the Fourier transform is a special orthogonal transform in linear space! He is special because the exponential function is the characteristic function of the differential operator!