The Fourier transform has a very wide range of applications in physics, number theory, combinatorics, signal processing, probability theory, statistics, cryptography, acoustics, optics, oceanography, structural dynamics, and other fields (for example, in signal processing, the typical use of the Fourier transform is to decompose a signal into amplitude components and frequency components).
The Fourier transform can express a function satisfying certain conditions as a trigonometric function (sine and/or cosine functions) or a linear combination of their integrals. In different fields of research, the Fourier transform takes many different variant forms, such as the continuous Fourier transform and the discrete Fourier transform.
The Fourier transform is a way of solving problems, a tool, an angle from which to look at a problem. The key to understanding it is this: a continuous signal can be viewed as the superposition of many small signals, and superposing them in the time domain or in the frequency domain both reconstructs the original signal. Decomposing the signal this way makes it easier to process.
We originally understood a signal from the perspective of time; without realizing it, we were in fact dividing the signal up by time, where each part is just one time point corresponding to one signal value, and a signal is the superposition of a set of such components.
After the Fourier transform, it is still a superposition problem — just one from the perspective of frequency. The only difference is that each small signal is a signal that spans the entire interval in the time domain, but it does have a fixed period; put another way, given a period, we can draw a sub-signal over the whole interval. Then, given a set of period values (or frequency values), we can draw the corresponding curve, just as giving the signal value at every point in the time domain does. If the signal is periodic, though, the frequency domain is simpler — only a few values, or even just one, are needed, whereas the time domain requires mapping a function value to every point along the entire time axis.
The Fourier transform maps a signal’s time-domain representation to a frequency-domain representation; the inverse Fourier transform does exactly the opposite. These are all different representations of one signal. You only need to know how to use its formula — though it’s better if you understand the proof.
Taking the Fourier transform of a signal gives you its frequency-domain characteristics, including both amplitude and phase. Amplitude expresses the size of that frequency component; then what about phase — what physical meaning does it have? Is the phase in the frequency domain related to the phase in the time domain? Is the change between the phase (frequency domain) of an earlier part of the signal and the phase of a later part proportional to the signal’s frequency? The Fourier transform decomposes a signal into countless sine waves (or cosine waves). That is, with countless sine waves you can synthesize any signal you need.
Think about this: given many sine signals, how can you synthesize the signal you need? The answer is that two conditions are required — one is the amplitude of each sine wave, and the other is the phase difference between each sine wave. So now it should be clear: the phase in the frequency domain is the phase between each sine wave. The Fourier transform is used for frequency-domain analysis of signals. We generally describe electrical signals as time-domain mathematical models, but digital signal processing is more interested in the frequency characteristics of signals, and the Fourier transform makes it easy to obtain a signal’s frequency-domain characteristics.
A simple, plain-language way to understand the Fourier transform is to regard a seemingly chaotic signal as being composed of basic sine (cosine) signals with a certain amplitude, phase, and frequency. The purpose of the Fourier transform is to find the frequency corresponding to the basic sine (cosine) signals with larger amplitude (higher energy), and thereby find the main vibration-frequency characteristics within a chaotic signal. For example, when a reducer fails, doing spectrum analysis via the Fourier transform lets you quickly determine which stage of gears is damaged, by comparing the rotational speed and tooth count of each gear stage against the large-amplitude parts of the noise spectrum.
The Laplace transform is an integral transform commonly used in engineering mathematics. It is a transform between functions of a real variable and functions of a complex variable, created to simplify computation. Applying the Laplace transform to a function of a real variable, performing various operations in the complex domain, and then applying the inverse Laplace transform to the result to obtain the corresponding result in the real domain is often computationally far easier than obtaining the same result directly in the real domain. This procedure of the Laplace transform is especially effective for solving linear differential equations: it can turn a differential equation into an algebraic equation that is easy to solve, thereby simplifying the computation. In classical control theory, the analysis and synthesis of control systems are built on the foundation of the Laplace transform.
One major advantage of introducing the Laplace transform is that a transfer function can be used in place of a differential equation to describe the characteristics of a system. This makes it possible to determine the overall characteristics of a control system by intuitive and convenient graphical methods (see signal flow graphs, dynamic block diagrams), to analyze the motion process of a control system (see the Nyquist stability criterion, the root locus method), and to synthesize the compensation devices of a control system (see control system compensation methods).
The application of the Laplace transform in engineering: using the Laplace transform to solve homogeneous differential equations in constant variables turns the differential equation into an algebraic equation, allowing the problem to be solved.
In engineering, the great significance of the Laplace transform lies in: converting a signal from the time domain into a representation in the complex frequency domain (the s-domain); it is widely used in linear systems and automation control. In digital signal processing, the Z-transform is a very important analytical tool. But in ordinary applications, we often only need to analyze the frequency response of a signal or system — in other words, we usually only need to perform a Fourier transform. So why introduce the Z-transform at all? And what relationship exists between the Z-transform and the 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 sine signals. Each sine signal can be completely characterized by amplitude, frequency, and phase.
The signal after the Fourier transform is usually called a spectrum. The spectrum includes the amplitude spectrum and the phase spectrum, which represent the distribution of amplitude with frequency and the distribution of phase with frequency, respectively. In the natural world, frequency has a clear physical meaning. Take sound signals, for instance: men’s voices are deep and resonant, mainly because men’s voices have more low-frequency components; women’s voices are mostly high-pitched and crisp, mainly because women’s voices have more high-frequency components.
For a signal, in terms of the amount of information it contains, the time-domain signal and its corresponding signal after the Fourier transform are completely identical. So what use is the Fourier transform? Because some signals mainly exhibit their characteristics in the time domain, such as the process of a capacitor charging and discharging; while other signals mainly exhibit their characteristics in the frequency domain, such as mechanical vibration and human speech. If a signal’s characteristics are mainly represented in the frequency domain, the corresponding time-domain signal may look chaotic, yet it is very convenient to interpret in the frequency domain.
In practice, after we have acquired a stretch of signal and have no prior information whatsoever, our intuition is to try to find some characteristics in the time domain; if we find nothing there, it is natural to convert the signal to the frequency domain and see what characteristics show up. The time-domain description and the frequency-domain description of a signal are like two sides of a coin: they look different, but they are in fact the same thing. Precisely because of this, in the usual analysis of signals and systems, we care a great deal about the Fourier transform.
Since people only care about the frequency-domain representation of a signal, then what is the Z-transform about? To talk about the Z-transform, we probably have to trace back to the Laplace transform first. The Laplace transform is a transform method named after the French mathematician Laplace, mainly aimed at the analysis of continuous signals. Laplace and Fourier were contemporaries; the era they lived in was the Napoleonic era in France, when the nation was at its zenith. In science, France had replaced Britain as the center of the world at the time, and among the many great scientific minds of that era, Laplace, Lagrange, and Fourier were the three brightest stars.
Fourier’s paper on decomposing signals into a superposition of sine signals had reviewers that included Laplace and Lagrange. Back to the point: although the Fourier transform is convenient to use and has clear physical meaning, one of its biggest problems is that the conditions for its existence are rather strict — for example, only signals that are absolutely integrable in the time domain can have a 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. Once a signal is multiplied by an exponential signal, it easily satisfies the condition of absolute integrability. So multiplying the original signal by an exponential signal generally satisfies the conditions for the Fourier transform — and this transform is the Laplace transform. This transform can convert differential equations into algebraic equations, which was of enormous significance in the 18th century, when computers were far from being invented. From the analysis above it can be seen that the Fourier transform can be regarded as a special form of the Laplace transform, namely the one where the exponential signal multiplied in is exp(0).
In other words, the Laplace transform is a generalization of the Fourier transform, a more universal form of expression. In the process of analyzing signals and systems, one can first obtain the more general result, the Laplace transform, and then obtain the special result, the Fourier transform. This approach of going from the general to the specific has proven to bring great convenience in the analysis of continuous signals and systems.
The Z-transform can be said to be the Laplace transform for discrete signals and systems, from which we can easily understand the importance of the Z-transform and easily understand the relationship between the Z-transform and the Fourier transform.
The Z-plane in the Z-transform and the S-plane in the Laplace transform have a mapping relationship: z=exp(Ts). In the Z-transform, the result on the unit circle corresponds to the result of the discrete-time Fourier transform. This article comes from kevinhg’s blog on CSDN.
