[Introduction] See a question on Zhihu, what is the connection between the Fourier transform, the Laplace transform, and the z-transform? Why are these transformations being done? I think it's a very good question, and I can't seem to answer it at once, so I will organize my learning and share it.
To understand these transformations, you first need to understand what mathematical transformations are! If you don't understand the concept of mathematical transformation, then I don't think other concepts are understood either.
A mathematical transformation is a reversible transformation function of a mathematical function from the original vector space in its own function space, or mapped to another function space, or for a set x to itself (such as a linear transformation) or from x to another set y. For example (image source Wikipedia):
Rotation
Reflection
Translation
There are many other mathematical transformations in mathematics, the essence of which can be regarded as a mathematical mapping of the function f(x) using the transformation factor, the result of which is that the independent variables of the function may be the original geometric vector space, or may become other geometric vector spaces, such as the Fourier transform from the time domain to the frequency domain.
While both the Fourier and Laplace transforms are essentially an integral transformation of continuous or finite first-class discontinuous functions, so what is an integral transformation?
The integral transformation maps a function from its original function space to another function space by integrating the original function to the mapping function space argument in a specific interval. As a result, some properties of the original function in the mapping function space may be easier to characterize or analyze than the original function space. It is common to use an inverse transformation to map the transformed function back to the original function space, which is called a reversible transformation.
Suppose that for functions f(t) where the function is an independent variable t, the integral transformation usually has a normal form similar to the following:
The function f(t) is the input to the transformation, and (tf)(u) is the output of the transformation, so the integral transformation is also generally referred to as a specific mathematical operator. The function k(t,u) is called the kernel function.
There is a concept of a symmetric kernel function here, what does this mean? That is, the two independent variables of function k are swapped at the same position as equal:
Some transformations are reversible, what is this concept? It is through the inverse transformation after the transformation, and it can also be restored!
Looking at the positive and inverse transformations, you will find:
The kernel function swaps exactly two independent variables
A positive transformation is an integral of the original function f(t) in the temporal dimension
The inverse transformation is the integration of the transformed function in the u dimension
Before talking about the Fourier transform, it is easier to understand the Fourier transformation by talking about the Fourier series. In mathematics , fourier series are the way to represent wave-like functions as simple sine waves. More formally, it can decompose any periodic function or periodic signal into a set of simple oscillatory functions (which may consist of infinite frequency components), i.e., sine and cosine functions (or, equivalently, the complex exponential), from a mathematical definition:
K in the formula represents the kth harmonic, what is this concept? It is not easy to understand, and it is easier to understand the first 4th harmonic synthesis GIF of a square wave. The concept of synthesis here refers to the concept of superposition on the time domain, picture from Wikipedia
From the above figure, it can be intuitively seen that the periodic square wave can be seen as a linear superposition of multiple harmonics, and its amplitude spectrum is a discrete spectral line, and the amplitude value is getting lower and lower, from this point of view, it can be seen that the component of the high harmonic is getting smaller and smaller. The position of the spectral line is:
Application: Here can be associated with the clock signal in our electronic system, do hardware friends or have experience, in the emc radiation test, found that the product circuit board exceeded the standard at some frequency points, experienced students will soon locate the radiation source. In fact, the probability here is caused by the periodic clock signal, from the frequency point of view can be seen as the linear superposition of multiple harmonics of its fundamental frequency, and a certain harmonic component in the circuit line size meets the radiation conditions, it escapes from the circuit board, into electromagnetic wave energy propagation to space. Therefore, the fundamental frequency of the periodic clock signal that may correspond to this frequency can be quickly located to the radiation source, thus solving the problem.
Speaking of Fourier series, periodic signals can be expanded by Fourier series, so is it possible that any periodic signal can be expanded by Fourier series? The answer is no, the famous dirichlet condition must be met:
If there are breaks in a cycle, the number of breaks needs to be a finite number
The number of maximal and minima values is a finite number in a period
Within a period, a signal or function is absolutely integral. See formula above.
The Fourier series was mentioned earlier, and then the Fourier transformation is looked at. The Fourier transform is called the Fourier transform because in 1822, the French mathematician J.Fourier first proved the theory of expanding the periodic function into Fourier series when studying heat conduction theory, and then evolved into a powerful scientific analysis tool.
Assuming that the periodic signal period t gradually becomes larger, the interval between the spectral lines will gradually become smaller, and if the extrapolated period t is infinitely amplified and becomes infinite, then the signal or function becomes an aperiodic signal or function, and the spectral line becomes continuous, rather than a discrete spectral line! Then the Fourier transform is precisely this general mathematical definition:
The two independent variables of its kernel function are t, which is generally called angular velocity (which can be figuratively understood as the speed of rotational motion) and characterizes the frequency space.
What do these two formulas mean? In the measurement space can be understood that its energy in the measurement space is limited, that is, its independent variable integral (equivalent to the area) is a definite value, then such a function or signal can be expanded by the Fourier transform, and the function obtained by the expansion becomes a function of the frequency domain, if the frequency of the function value is drawn out of the curve is what we call the spectrum map, and its inverse transformation is easier to understand, if we know a signal or function spectral density function, you can correspond to the function of reducing its time domain, It is also possible to plot waveforms for the time domain.
The Fourier transform formula, from the point of view of understanding, can be seen as the sum of infinitesimally many infinitesimal energies, and the Fourier series is also the sum of the harmonic components, except that the former is continuous with respect to the frequency variable, while the latter is discrete with respect to frequency!
Of course, this article limits the discussion of time domain signals because the most common application in our electronic systems is a time domain signal. By extension, other multi-dimensional signals can also be generalized using the above definitions, and signals in multi-dimensional space are also very valuable, such as 2-dimensional image processing, 3-dimensional image reconstruction and so on.
Fourier series corresponds to a periodic signal, while a Fourier transform corresponds to a time-continuous integrable signal (not necessarily a periodic signal)
Fourier series require that the signal has limited energy over a period, which in turn requires limited energy over the entire interval
The correspondence of Fourier series is discrete, while the Fourier transform corresponds to continuous.
Therefore, the physical meanings of the two are different, and their dimensions are also different, representing the magnitude of the kth harmonic amplitude of the periodic signal, but the concept of spectral density. So the answer is that the two are not essentially a concept, and fourier series is another time-domain representation of periodic signals, namely orthogonal series, which is a time-domain superposition of waveforms of different frequencies. The Fourier transform is a complete frequency domain analysis, Fourier series is suitable for mathematical analysis of periodic phenomena, Fourier transforms can be seen as the limiting form of Fourier series, can also be seen as a mathematical analysis of periodic phenomena, but also suitable for the analysis of non-periodic phenomena.
Therefore, the connection between the Fourier transform and the Laplace transform is relatively easy to relate.
The Laplace transform maps the original function from a time dimension (not necessarily a time dimension, but is convenient to understand in this article described in terms of common time dimension signals) to a complex plane
The Fourier transform is a special case of the Laplace transform, that is, when the kernel function is transformed, the Laplace transform becomes the Fourier transform. It is equivalent to taking only the virtual part, and the real part is 0.
The Fourier transform is transformed from the original dimension to the frequency dimension, which is equivalent to converting the time domain signal into the frequency domain for analysis for signal processing, providing a powerful mathematical theoretical basis and tools for signal processing.
Laplace transform, the original dimension into a complex frequency domain, in the analysis of electronic circuits and control theory, for the establishment of a mathematical description of the system provides a strong mathematical theoretical basis, learned control theory all day long with the transfer function, its essence is Laplace transform on the system of a mathematical modeling description. It provides mathematical tools for analyzing the stability and controllability of the system.
The z-transform is essentially a discrete form of the Laplace transform. Also known as fisher-z transform. Sampling transformations on continuous signals yield a discrete sequence of the original function:
So what is the significance of the z-transform? In digital signal processing as well as digital control systems, the z-transform provides the mathematical basis. Using the z-transform can quickly describe a transfer function as a difference equation, which provides a mathematical basis for programming implementation, such as a digital filter knows its z-transform form, writing code is a matter of minutes, and also knows the z-transformation form of a control algorithm, and the same code is also a matter of course.
Here to talk about the discrete form of the z-transform, then here is also mentioned, Fourier transform number landing, that is, the discrete form is discrete Fourier transform (discrete fourier transform), and the well-known fast Fourier transform (fast fourier transform) is the efficient implementation of dft.
To understand the difference between the connections between the three transformations, we must first understand what a mathematical transformation is and what an integral transformation is. The Fourier transform and the Laplace transform are essentially integral transformations of continuous or finite first-class discontinuous functions, while the Fourier transform is a special form of the Laplace transform, and the z-transform is the discrete form of the Laplace transform. Each type of transformation has its application value, fourier transform provides a powerful mathematical tool in the frequency domain analysis of signal processing, while the Laplace transform provides a mathematical analysis tool for modeling and analysis in the fields of electronics, control engineering, aerospace and other fields; the z-transform provides a mathematical theoretical basis for digital implementation. dft is a discretized form of fft, and fft is the algorithm optimization implementation of dft.
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