What is the connection between Fourier transform, Laplace transform and Z transform? Why do we need to perform transforms?

Latest update time：2022-10-10 15:22

Reads：

What is a mathematical transformation?

To understand these transformations, you first need to understand what mathematical transformations are! If you don’t understand the concept of mathematical transformations, then I don’t think you understand other concepts either.

Mathematical transformation refers to the transformation of mathematical functions from the original vector space to its own function space, or mapping to another function space, or a reversible transformation function from set X to itself (such as linear transformation) or from X to another set Y. For example (image source wikipedia):

Rotation

Reflection

Translation Transformation( Translation )

There are many other mathematical transformations in mathematics, and their essence can be regarded as a mathematical mapping of the function f(x) using transformation factors. The transformation result is that the independent variable of the function may still be the original geometric vector space, or it may become other geometric vector spaces. For example, the Fourier transform transforms from time domain to frequency domain.

The essence of Fourier transform and Laplace transform is an integral transform of continuous or finite first-kind discontinuous point functions. So what is integral transform?

What is integral transformation?

The integral transform maps a function from its original function space to another function space by integrating the original function with respect to the independent variable of the mapping function space in a specific interval. In this way, some properties of the original function may be easier to represent or analyze in the mapping function space than in the original function space. Usually, an inverse transform can be used to map the transformed function back to the original function space. Such a transform is called a reversible transform.

Assuming that for a function f(t) with independent variable t, the integral transformation usually has a similar paradigm as follows:

The function f(t) is the input of the transformation, and (Tf)(u) is the output of the transformation, so the integral transformation is also generally called a specific mathematical operator. The function K(t,u) is called the integral kernel function.

Here is the concept of a symmetric kernel function. What does this mean? It means that the two independent variables of the function K are still equal when they are swapped:

Some transformations are reversible. What does this mean? After the transformation, it can be restored through the inverse transformation!

Observe the forward and inverse transformations and you will find:

The kernel function just swaps the positions of the two independent variables
Forward transformation is to integrate the original function f(t) in the time dimension.
The inverse transformation is to integrate the transformed function in the u dimension.

What are Fourier series?

Before talking about Fourier transform, it will be easier to understand Fourier transform if we talk about Fourier series. In mathematics, Fourier series is a way to represent wave-like functions as simple sine waves. More formally speaking, it can decompose any periodic function or periodic signal into a set of simple oscillating functions (possibly composed of infinite frequency components), namely sine functions and cosine functions (or, equivalently, using complex exponentials). From the mathematical definition:

Assume f(t) is a periodic signal, and assume its period is T. If the energy of f(t) in one period is finite, it is:

Then, f(t) can be expanded into a Fourier series. How to expand it? The calculation is as follows:

The coefficients of the Fourier series are calculated as follows:

For f(t), we can also use Euler's formula to write it as the sum of sine function and cosine function, which we will not write here. Euler's formula is as follows:

The k in the formula represents the kth harmonic. What is this concept? It is not easy to understand. It will be easier to understand if you look at the animation of the synthesis of the first 4 harmonics of a square wave . The concept of synthesis here refers to the concept of superposition in the time domain. The picture comes from wikipedia

It can be intuitively seen from the above figure that a periodic square wave can be regarded as a linear superposition of multiple harmonics. Its amplitude spectrum is a discrete spectrum line, and the amplitude value is getting lower and lower. From this perspective, it can It can be seen that the component of higher harmonics accounts for an increasingly smaller proportion. The position of its spectral line is:

The first root is:
The second root is:
The nth root is:

The interval between its spectral lines is

Application: Here we can think of the clock signal in our electronic system. Friends who work on hardware or have experience will find that the product circuit board exceeds the standard at certain frequency points when doing EMC radiation testing. Experienced students will quickly locate the radiation source. In fact, it is most likely caused by the periodic clock signal. From the perspective of frequency, it can be regarded as the linear superposition of multiple harmonics of its fundamental frequency. When the circuit line size meets the radiation conditions, a certain harmonic component escapes from the circuit board and becomes electromagnetic wave energy to propagate into space. Therefore, by reversely checking the fundamental frequency of the periodic clock signal that may correspond to this frequency, the radiation source can be quickly located, thereby solving the problem.

It is said that Fourier series is a periodic signal that can be expanded by Fourier series. So, can any periodic signal be expanded by Fourier series? The answer is no, the famous Dirichlet condition must be met :

If there are discontinuities in a cycle, the number of discontinuities must be finite.
In a cycle, the number of maximum and minimum values is finite.
Within one period, the signal or function is absolutely integrable. See the previous formula.

What is Fourier transform?

We talked about the Fourier series earlier, and next we’ll look at the Fourier transform. The Fourier transform is called the Fourier transform because in 1822, the French mathematician J. Fourier first proved the theory of expanding periodic functions into Fourier series when studying the theory of heat conduction, and then continued to develop into a powerful Scientific research analysis tools.

Assuming that the period T of a periodic signal gradually becomes larger, the interval between spectral lines will gradually become smaller. If the extrapolation period T is infinitely enlarged and becomes infinity, the signal or function will become a non-periodic signal or function. At this time, the spectral lines It becomes continuous instead of discrete spectral lines! Then the Fourier transform is this general mathematical definition:

For a continuous time signal f(t), if f(t) is integral in the time dimension, (actually it does not have to be the time t dimension, it can be any dimension, as long as it is integral in the corresponding dimensional space), Right now:

Then, the Fourier transform of x(t) exists and is calculated as:

Its inverse transformation is:

As mentioned above, Fourier transform is essentially an integral transform of a continuous function. From the above formula, we can see that the kernel function of Fourier transform is:

The two independent variables of its kernel function are t, , which is generally called angular velocity (can be understood vividly as the speed of rotation), which characterizes the frequency space.

What do the above two formulas mean? Being integrable in the metric space can be understood as having limited energy in the metric space, that is, the integral of its independent variable (equivalent to finding the area) is a certain value. Then such a function or signal can be expanded by Fourier transform, and the expanded result is It becomes a function in the frequency domain. If the function value is plotted against the frequency, it is what we call a spectrogram, and its inverse transformation is easier to understand. If we know a signal or function spectral density function , we can correspond The time domain function can be restored and the time domain waveform graph can also be drawn.

From the perspective of understanding, the Fourier transform formula can be seen as the sum of infinitely many infinitesimal energies, and the Fourier series is also the sum of all harmonic components. The difference is that the former is continuous with respect to the frequency variable, while the latter is discrete with respect to the frequency!

Of course, this article limits the discussion to 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 promoted using the above definition. They are also very valuable in multi-dimensional space signals, such as 2-dimensional image processing, 3-dimensional image reconstruction, etc.

What is the difference between Fourier series and transform?

The Fourier series corresponds to a periodic signal, while the Fourier transform corresponds to a time-continuous integrable signal (not necessarily a periodic signal)
The Fourier series requires that the signal has finite energy in one cycle, while the latter requires that the energy in the entire interval is finite.
The Fourier series corresponds to 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 k-th harmonic amplitude of the periodic signal, and is the concept of spectral density. So the answer is that the two are not essentially the same concept. The Fourier series is another time domain expression of periodic signals, that is, the orthogonal series, which is the time domain superposition of waveforms of different frequencies. The Fourier transform is a complete frequency domain analysis. The Fourier series is suitable for mathematical analysis of periodic phenomena. The Fourier transform can be regarded as the limit form of the Fourier series, and can also be regarded as the Periodic phenomena are mathematically analyzed, and it is also suitable for the analysis of non-periodic phenomena.

What is Laplace transform?

In 1814, French mathematician Pierre-Simon Laplace gave a reliable mathematical basis for Laplace while studying probability theory, thus developing the Laplace transform theory. For function f(t) we know that its Fourier transform is:

Then if the Fourier transform of the function is:

The above formula is sorted out:

Let , then the above transformation

We know from the previous article that Laplace is essentially an integral transformation, so the above formula will be regarded as the kernel function of the integral transformation, and its transformation kernel function is:

The factors introduced above will make it easier for the function to converge, and the absolutely integrable restriction of the Fourier transform will be easier to satisfy. The conditions for the existence of Laplace transform are:

Fourier Laplace transform connection difference

Therefore, the connection between Fourier transform and Laplace transform is relatively easy.

Laplace transform maps the original function from the time dimension (not necessarily the time dimension, but just to facilitate understanding, this article describes it with common time dimension signals) to the 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 imaginary part and the real part is 0.
Fourier transform transforms from the original dimension to the frequency dimension. For signal processing, it is equivalent to transforming the time domain signal into the frequency domain for analysis. It provides a powerful mathematical theoretical basis and tool for signal processing.
The Laplace transform transforms the original dimension into the complex frequency domain. In electronic circuit analysis and control theory, it provides a strong mathematical theoretical basis for establishing a mathematical description of the system. After studying control theory, it is related to the transfer function all day long. Its essence is a mathematical modeling description of the system by Laplace transform. It provides mathematical tools for analyzing the stability and controllability of the system.

What is Z-Transform?

The Z transform is essentially the discrete form of the Laplace transform. Also called Fisher-Z transform. For the continuous signal, the discrete sequence of the original function is obtained by sampling transformation:

Where T is the sampling period, and the signal and system are called impulse sampling. In fact, in layman's terms, it is to convert the continuous signal into a discrete sequence of samples at equal intervals. Look at the figure below to understand that AD converters are often used in electronic systems to achieve this.

Perform Laplace transform on the above formula:

This formula takes advantage of the sampling characteristics of the impulse function and can be simplified to:

Introduce , introduce a new independent variable Z, then the above formula becomes like this:

This is the Z transform. From the above process description, we know the relationship between the Z transform and the Laplace transform. Therefore, the connection between the two is that the Z transform is the discrete form of the Laplace transform.

So what is the significance of Z transform? In digital signal processing and digital control systems, Z transform provides a mathematical basis. Using Z transform, a transfer function can be quickly described in the form of a differential equation, which provides a mathematical basis for programming implementation. For example, if you know the Z transform form of a digital filter, writing code is a matter of minutes. Similarly, knowing the Z transform form of a control algorithm, writing code is also a matter of course.

Here we are talking about the discrete form of Z transform, so let's also mention here that the digital implementation of Fourier transform, that is, the discrete form is discrete Fourier transform DFT (Discrete Fourier Transform), and the well-known fast Fourier transform FFT (Fast Fourier Transform) is a highly efficient implementation of DFT.

in conclusion

To understand the connections and differences among the three transformations, we must first understand what mathematical transformations are and what integral transformations are. The Fourier transform and the Laplace transform are essentially integral transformations of continuous or finite first-kind discontinuous point functions. The Fourier transform is a special form of the Laplace transform, and the Z transform is a discrete form of the Laplace transform. form. Each transformation has its application value. Fourier transform provides a powerful mathematical tool in frequency domain analysis of signal processing, while Laplace transform provides modeling and analysis tools in electronics, control engineering, aerospace and other fields. Mathematical analysis tool; Z transformation implements these transformations to provide mathematical theoretical basis for digital implementation. DFT is the discretized form of FFT, and FFT is the algorithm optimization implementation of DFT.