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Fourier transform has a wide range of applications in physics, number theory, combinatorics, signal processing, probability theory, statistics, cryptography, acoustics, optics, oceanography, structural dynamics, etc. A typical use of the Lie transform is to decompose a signal into amplitude and frequency components).
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The Fourier transform can express a function satisfying certain conditions as a trigonometric function (sine and/or cosine function) or a linear combination of their integrals. In different research fields, Fourier transform has many different variants, such as continuous Fourier transform and discrete Fourier transform.
The Fourier transform is a problem-solving method, a tool, and a way of looking at a problem. The key to understanding is: a continuous signal can be regarded as the superposition of small signals. The original signal can be composed from the time domain superposition and the frequency domain superposition. Decomposing the signal in this way is helpful for processing.
We originally understood a signal from the perspective of time. Unconsciously, the signal is actually divided according to time. Each part is only a time point corresponding to a signal value. A signal is a superposition of a group of such components. . After the Fourier transform, it is actually a superposition problem, but it is superimposed from the perspective of frequency, but each small signal is a signal covering the entire interval in the time domain, but it does have a fixed period, or in other words, Given a period, we can draw a sub-signal over the entire interval, then given a set of period values (or frequency values), we can draw its corresponding curve, just like giving each point in the time domain The signal value is the same, but if the signal is periodic, the frequency domain is simpler, only a few or even one is needed, and the time domain needs to map a function value to each point on the entire time axis.
The Fourier transform is the mapping of a time-domain representation of a signal to a frequency-domain representation; the inverse Fourier transform is just the opposite. These are all different representations of a signal. Its formula can be used, of course, it is better to understand the proof.
Fourier transform is performed on a signal, and its frequency domain characteristics can be obtained, including amplitude and phase. Amplitude is the size of the frequency component, so what about the phase, what physical meaning does it have? Is the phase in the frequency domain related to the phase in the time domain? Whether the change of the phase (frequency domain) of the previous segment of the signal and the phase of the latter segment is proportional to the frequency of the signal.
Fourier transform is to decompose a signal into countless sine wave (or cosine wave) signals. That is, with countless sine waves, you can synthesize any signal you need.
Think about this: given you a lot of sinusoidal signals, how can you synthesize the signal you need? The answer is two conditions, one is the amplitude of each sine wave, and the other is the phase difference between each sine wave. So it should be understood now that the phase in the frequency domain is the phase between each sine wave.
Fourier transform is used for frequency domain analysis of signals. Generally, we describe electrical signals as mathematical models in the time domain, while digital signal processing is more interested in the frequency characteristics of signals, and it is easy to obtain the frequency domain characteristics of signals through Fourier transform. .
The simple and popular understanding of Fourier transform is to consider the seemingly disordered signal as a combination of basic sine (cosine) signals of certain amplitude, phase and frequency. The purpose of Fourier transform is to find these basic sine (cosine) signals. The frequency corresponding to the signal with larger amplitude (higher energy) in the middle, so as to find the main vibration frequency characteristics in the cluttered signal. For example, when the reducer is faulty, the Fourier transform is used for spectrum analysis. According to the comparison of the speed of each gear, the number of teeth and the large amplitude in the noise spectrum, it can quickly determine which gear is damaged.
Laplace transform is an integral transform commonly used in engineering mathematics. It is a functional transformation between a real variable function and a complex variable function established to simplify calculations. Perform Laplace transform on a real variable function, perform various operations in the complex number domain, and then perform the inverse Laplace transform of the operation result to obtain the corresponding result in the real number domain, which is often better than directly in the real number domain. Finding the same result is computationally much easier. This operation step of the Laplace transform is particularly effective for solving linear differential equations, which can simplify the calculation by transforming the differential equation into an easy-to-solve algebraic equation. In the classical control theory, the analysis and synthesis of the control system are based on the Laplace transform.
A major advantage of introducing the Laplace transform is that a transfer function can be used instead of a differential equation to characterize the system. This provides an intuitive and simple graphical method to determine the overall characteristics of the control system (see signal flow diagram, dynamic structure diagram), analyze the motion process of the control system (see Nyquist stability criterion, root locus method), and The correction device of the integrated control system (see Control system correction method) offers the possibility.
The application of Laplace transform in engineering: The application of Laplace transform to solve homogeneous differential equations with constant variables can transform the differential equations into algebraic equations, so that the problem can be solved. In engineering, the significance of Laplace transform lies in: converting a signal from the time domain to the complex frequency domain (s domain) to represent it; it has a wide range of applications in linear systems and control automation.
In digital signal processing, Z-transform is a very important analysis tool. But in common applications, we often only need to analyze the frequency response of the signal or system, that is to say, we usually only need to perform Fourier transform. So, why introduce the Z-transform?
What is the relationship between Z transform and Fourier transform? The physical meaning of the Fourier transform is very clear: decompose a signal, usually represented in the time domain, into a superposition of multiple sinusoidal signals. Each sinusoidal signal can be fully characterized by amplitude, frequency, and phase. The signal after the Fourier transform is usually called a spectrum, and the spectrum includes an amplitude spectrum and a phase spectrum, which respectively represent the distribution of amplitude with frequency and the distribution of phase with frequency. In nature, frequencies have clear physical meanings. For example, in sound signals, male voices are low and powerful, mainly because male voices have more low-frequency components; female voices are more high-pitched and crisp, mainly because female voices have more high-frequency components. .
For a signal, in terms of the amount of information contained, the time domain signal and its corresponding Fourier transform signal are exactly the same. What does the Fourier transform do? Because some signals mainly show their characteristics in the time domain, such as the process of capacitor charging and discharging; while some signals mainly show their characteristics in the frequency domain, such as mechanical vibration, human voice and so on. If the characteristics of the signal are mainly represented in the frequency domain, the corresponding time domain signal may look cluttered, but it is very convenient to interpret in the frequency domain.
In practice, when we collect a signal, without any prior information, the intuition is to try to find some features in the time domain. If nothing is found in the time domain, it is natural to convert the signal to frequency. Domain to see what features can be. The time domain description and the frequency domain description of a signal are like two sides of a coin. Although they look different, they are actually the same thing. Because of this, in the usual analysis of signals and systems, we are very concerned with the Fourier transform.
Since people only care about the frequency-domain representation of the signal, what about the Z-transform? When it comes to the Z transform, it may be traced back to the Laplace transform. Laplace transform is a transformation method named after the French mathematician Laplace, mainly for the analysis of continuous signals. Laplace (March 23, 1749 - March 5, 1827) and Fourier (March 21, 1768 - May 16, 1830) were both contemporaries, and their time In France, it was in the Napoleonic era and its national power was at its peak. In science, it also replaced Britain as the center of the world at that time. Among the many masters of science at that time, Laplace, Lagrange (January 25, 1736 - April 10, 1813), Fourier is them The three brightest stars in the middle. Fourier's paper on the decomposition of a signal into a superposition of sinusoids was reviewed by Laplace and Lagrange.
Back to the topic, although Fourier transform is easy to use and has a clear physical meaning, one of the biggest problems is that its existence conditions are relatively harsh. For example, only signals that are absolutely integrable in the time domain may have Fourier transform. The Laplace transform can be said to generalize this concept. In nature, the exponential signal exp(-x) is one of the fastest decaying signals, and it is easy to satisfy the absolute integrability condition after multiplying the exponential signal by the signal. Therefore, after multiplying the original signal by the exponential signal, the conditions of the Fourier transform can generally be satisfied, and this transformation is the Laplace transform. This transformation, which converts differential equations into algebraic equations, was significant in the 18th century, when computers were far from being invented.
It can be seen from the above analysis that the Fourier transform can be regarded as a special form of Laplace, that is, the multiplied exponential signal is exp(0). That is to say, the Laplace transform is a generalization of the Fourier transform and is a more general form of expression. In the process of analyzing signals and systems, the more general result of the Laplace transform can be obtained first, and then the special result of the Fourier transform can be obtained. This general-to-special solution has proven to be very convenient in the analysis of continuous signals and systems.
Z transform can be said to be the Laplace transform for discrete signals and systems, so we can easily understand the importance of Z transform and the relationship between Z transform and Fourier transform. There is a mapping relationship between the Z plane in the Z transformation and the S plane in Laplace, z=exp(Ts). In the Z-transform, the result on the unit circle corresponds to the result of the discrete-time Fourier transform.