Physical Interpretation and difference between Fourier transform and Laplace Transform

Fourier transform is widely used in physics, number theory, composite mathematics, signal processing, probability theory, statistics, cryptography, acoustics, optics, oceanic and structural dynamics., fourier transformation is typically used to break down signals into amplitude and frequency components ).
Fourier transformation can represent a function that meets certain conditions as a linear combination of trigonometric functions (sine and/or Cosine Functions) or their points. In different research fields, Fourier transformation has different variants, such as continuous Fourier transformation and discrete Fourier transformation.

Fourier transform is a method to solve the problem, a tool, and a view of the problem. The key to understanding is that a continuous signal can be considered as a small signal superposition, and the original signal can be formed from the time domain superposition and from the frequency domain superposition. After the signal is decomposed, it will be helpful for processing.

We used to understand a signal from the perspective of time. Without knowing it, we actually split the signal by time. Each part is just a time point corresponding to a signal value, A signal is the superposition of such components. After Fourier transformation, it is actually a superposition problem, but it is only from the frequency perspective, but each small signal is a signal covering the entire interval in a time domain, but he does have a fixed cycle, or, given a cycle, we can draw a signal for the entire interval. Then, given a set of cycle values (or frequency values ), we can plot the corresponding curve, just like the signal value at each point in the time domain. However, if the signal is a period, the frequency domain is simpler, you only need a few or even one, and the time domain needs to map a function value at each point on the entire timeline.

Fourier transformation maps the time domain representation of a signal to a frequency domain representation. Inverse Fourier transformation is the opposite. This is a different representation of a signal. Its formula can be used. Of course, the proof is better understood.

Perform Fourier transformation on a signal to obtain its frequency-domain characteristics, including amplitude and phase. The amplitude represents the size of the frequency component. What physical significance does the phase have? Is the phase in the frequency domain related to the phase in the time domain? Whether the phase changes in the first part of the signal (frequency domain) are proportional to the signal frequency.

Fourier transform is to divide a signal into countless sine waves (or cosine waves. That is to say, using countless sine waves can synthesize any signal you need.

Think about this question: how can you synthesize the signals you need for many sine signals? 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 now we should understand that the phase in the frequency domain is the phase between each sine wave.

Fourier transform is used to analyze the frequency domains of signals. Generally, we describe electrical signals as mathematical models of time domains, while digital signal processing is more interested in the frequency features of signals, however, the frequency domain characteristics of signals are easily obtained through Fourier transformation.

The simple and common understanding of Fourier transform is to combine seemingly chaotic signals into basic sine (Cosine) signals of a certain amplitude, phase, and frequency, the purpose of Fourier transformation is to find out the frequency corresponding to the large amplitude (High Energy) signals in these basic sine (Cosine) signals, so as to find out the main vibration frequency characteristics of chaotic signals. For example, if a gear machine fails, Fourier transform is used for spectrum analysis. The gear damage level can be quickly determined based on the comparison of gear speed, number of teeth, and amplitude in the noise spectrum.

Laplace transformation is an integral transformation commonly used in Engineering Mathematics. It is a function transformation between a real-variable function and a complex variable function established to simplify computing. Perform Laplace transformation on a real Variable Function, perform various operations in the complex number field, and then perform Laplace inverse transformation on the calculation result to obtain the corresponding results in the real number field, it is usually easier to calculate the same result than simply finding the same result in the real number field. This operation step of Laplace transformation is particularly effective for solving linear differential equations. It can convert the differential equation into an easily solved algebraic equation for processing, thus simplifying the calculation. In the classical control theory, the analysis and synthesis of the control system are based on Laplace transformation.

One of the main advantages of Laplace transformation is that the system features can be described by using a transfer function instead of a differential equation. This is to use an intuitive and simple graphical method to determine the overall characteristics of the Control System (see the signal flowchart and dynamic structure diagram) analysis of the Movement Process of the Control System (see the nequest stability criterion and root trajectory method), and the calibration device of the integrated control system (see the control system correction method.

Application of Laplace transformation in Engineering: using Laplace transformation to solve homogeneous differential equations with constant variables, the differential equations can be converted into algebraic equations to solve the problem. In engineering, the great significance of Laplace transformation is to convert a signal from a time domain to a complex frequency domain (s domain) for representation. In linear systems, control Automation is widely used.

In digital signal processing, ztransform is a very important analysis tool. However, in common applications, we usually only need to analyze the signal or the system's frequency response, that is, we usually only need to perform Fourier transformation. So why should we introduce ztransformation?

What is the relationship between the ztransform and Fourier transform? The physical meaning of Fourier transform is very clear: the signals commonly expressed in the time domain are decomposed into multiple sine signals. Each sine signal can be completely characterized by amplitude, frequency, and phase. The signal after Fourier transformation is usually called the spectrum. The spectrum includes the amplitude spectrum and the phase spectrum, which respectively indicate the distribution of the amplitude with frequency and the distribution of the phase with frequency. In nature, frequency has a clear physical significance. For example, the voice signal and the voice of male compatriots are low and arrogant. This is mainly because there are more low frequencies in male voices and more high-definition female siblings, this is mainly because female voice has more high-frequency components. For a signal, in terms of the information contained, the time domain signal and the corresponding Fourier transform signal are exactly the same. What is the function of Fourier transformation? Some signals present their characteristics mainly in the time domain, such as the capacitor charge and discharge process, while some signals present their characteristics in the frequency domain, such as mechanical vibration and human speech. If the signal features are mainly expressed in the frequency domain, the corresponding time domain signals may seem messy, but the interpretation in the frequency domain is very convenient. In reality, when we collect a signal without any prior information, intuition is to try to discover some features in the time domain. If there is no such thing in the time domain, naturally, the signal is converted to the frequency domain to see what features are there. The Time Domain description and frequency domain description of a signal are like the two sides of a coin. Although they look different, they are actually the same thing. Because of this, we are very concerned about Fourier transformation in the Process of signal and system analysis.

Since people only care about Signal Representation in the frequency domain, what is the ztransform? When it comes to the ztransform, it may be traced back to the Laplace transform first. Laplace transformation is a transformation method named by the French mathematician Laplace, mainly for the analysis of continuous signals. Laplace and Fourier are contemporary people. They live in France in the Napoleon age, and their national strength is flourishing. In science, he also replaced Britain as the center of the world at that time. Among the numerous scientific masters at that time, Laplace, Lagan, and Fourier were the most brilliant three stars among them. Fourier's thesis on signal can be decomposed into sine signal superposition. The reviewers include Laplace and Laplace.

Back to the question, although Fourier transform is easy to use and has a clear physical significance, the biggest problem is that it has harsh conditions, for example, Fourier transformation may exist for absolute Product signals in the time domain. Laplace transformation can be said to promote this concept. In nature, the exponential signal exp (-x) is one of the fastest attenuation signals. After the signal is multiplied by the exponential signal, it is easy to meet the conditions of absolute product. Therefore, after the original signal is multiplied by the exponent signal, it can generally meet the conditions of Fourier transformation. This transformation is the Laplace transformation. This transformation can transform a differential equation into an algebraic equation, which is of great significance when computers were not invented in the 18th century. From the above analysis, we can see that Fourier transformation can be seen as a special form of Laplace, that is, the exponent signal multiplied is exp (0 ). That is to say, Laplace transformation is a general expression of Fourier transformation. In the process of signal and system analysis, we can first obtain the more general result of Laplace transformation, and then obtain the special result of Fourier transformation. This general-to-special solution has proved great convenience in the analysis of continuous Signals and Systems.

Ztransformation can be said to be the Laplace transformation of Discrete Signals and Systems. Therefore, we can easily understand the importance of ztransformation and the relationship between ztransformation and Fourier transformation. The zplane in the ztransform has a ing relationship with the S plane in the Laplace, Z = exp (TS ). In the ztransform, the result on the unit circle corresponds to the result of the discrete time Fourier transformation.

Physical Interpretation and difference between Fourier transform and Laplace Transform