What are the expressions for the Fourier transform and the Z transform?

In general, if the term "Fourier transform" is not preceded by any qualifier, it refers to "continuous Fourier transform". The Continuous Fourier Transform represents the squared quadratic function f(t) as an integral or series form of a complex exponential function.

f(t) = \mathcal^[f(\omega)] = \frac} \int\limits_^\infty f(\omega) e^\,d\omega.

The above equation is actually the inverse transformation of the continuous Fourier transform, i.e., the function f(t) of the time domain is expressed as the integral of the function f( ) of the frequency domain. In turn, its positive transformation happens to be an integral form of the function f( ) of the frequency domain expressed as the function f(t) of the time domain. Generally speaking, the function f(t) can be called the original function, and the function f( ) can be called the image function of the Fourier transform, and the original function and the image function constitute a Fourier transform pair.

A generalization of the continuous Fourier transform is called the fractional fourier transform.

When f(t) is an odd function (or even function), the remaining string (or sinusoidal) components will die out, and the transformation can be called a cosine transform or a sine transform

Another noteworthy property is that when f(t) is a pure real function, f( 6 1 ) = f( ) holds.

Find the solution of the difference equation with the z-transform: y(z) = yzs(z) + yzi(z).

The system with the full response y(n), the zero state response yzs(n) and the zero input response yzi(n) are obtained by inverse transformation of 1, 2, and 3, respectively

System functions: h (z) = y(z) x(z) = n(z) d(z) n(z)=0 is the zero of h(z) d(z)=0 is the root of h(z).

In addition to the proportionality constant, the entire system can be uniquely determined by all of its zeros and poles.

The system h(z) and the unit impulse response h(n) are a z-transform. The complete form of the unit impulse response is determined by the uniqueness of all the zeros and poles of the system function.

The causality and stability of the system were analyzed with the z-transform.

1Fourier transformis 2 δ (t).

Fourier transform.

Indicates that a function that satisfies certain conditions can be expressed as a trigonometric lead-friendly type.

Sine sum or coincident limb chord function.

or a linear combination of their integrals, in different fields of study, the Fourier transform has many different variant forms, such as the continuous Fourier transform and the discrete Fourier transform, and the Fourier analysis was originally proposed as a tool for the analytical analysis of thermal processes.

In the field of mathematics, although Fourier analysis was originally used as a tool for the analytical analysis of thermal processes, its method of thinking still has the characteristic characteristics of typical reductionism and analysis.

Optional"The Huaichai function can be expressed as a sinusoidal function through a certain decomposition.

The form of linear combinations, while the sine function is physically well-studied and relatively simple class of functions

1. The Fourier transform is a linear operator, if given an appropriate norm.

It is also a unitary operator.

2. The inverse transformation of the Fourier transform is easy to find, and the form is very similar to the positive transformation.

3. The sinusoidal basis function is the eigenfunction of differential operations, so that the linear differential square.

Fourier transformLaplace transform, z-transform:

First, let's imagine oneComplex planes

Then the Laplace transform is on top of the complex plane, so if we take the imaginary axis of s then the transformation is now the Fourier transform, that is, the Fourier transform is a special case of the Laplace transform, which is a special form. Relatively speaking, the Laplace transform is a generalization of the Fourier transform. If we take the imaginary axis on the complex plane and fold it into a circle in the imagination, its radian system.

For 2, in fact, it is in polar coordinates.

, the axes on the complex plane are bent and rotated. z-transform.

Its polar diameter = 1, that is, the transformation on the circumference of the unit, is essentially a Fourier transformation, and the relationship between z and Laplace is naturally z=e st.

It is summarized as follows:

The fourier transform is the transformation of a continuous time-domain signal into the frequency domain; It can be said to be a special case of the laplace transformation.

The Laplace transform is a generalization of the Fourier transform, which has weaker conditions than the Fourier transform, which transforms a continuous time-domain signal into a complex frequency grinding domain (the entire complex plane, while the Fourier transform can be seen only in the J-axis).

The Z transform is the Laplace transformation of the discrete signal after the ideal sampling of the continuous signal, and then the transformation result when Z=E ST (t is the sampling period), the corresponding domain is the digital complex frequency domain, and the digital frequency = t.

Mathematically, the three major transformations:

Fourier transform

f(t) is a periodic function of t.

If t satisfies the Dirichlet condition: f(x) is continuous or has only a finite number of first-class discontinuities in a period of 2t, with f(x) monotonic or can be divided into finite monotonic intervals, then f(x) is a Fourier series with a period of 2t.

convergence, and the function s(x) is also a periodic function with periods of 2t, and at these discontinuities, the function is finite; There are a finite number of extreme points in a cycle.

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Laplace transform

The Laplace transform is an integral transformation commonly used in engineering mathematics, also known as the Laplace transform.

The Rass transform is a linear transformation.

You can convert a function with a real number t(t 0) with a parameter to a complex number s with a parameter. The Laplace transform has a wide range of applications in many engineering and scientific research fields, especially in the system science of mechanical systems, electrical systems, automatic control systems, reliability systems, and random service systems.

z-transform.

Z-transformation is a mathematical transformation of a discrete sequence, often used to find solutions to linear time-invariant difference equations. Its position in discrete systems is what the Laplace transform is in a continuous system. The z-transform has become an important tool for analyzing linear time-invariant discrete system problems and in digital signal processing.

It has a wide range of applications in the fields of computer control systems.

The z-transform is a generalization of the Fourier transform. Because when the Fourier transform does not exist, the dense functions defined by the z-transform may converge. Then the Fourier transform is a z-transformation that is performed on a unit circle.

This is equivalent to conceptually a linear frequency axis wound around a unit circle to sell filial piety.

Therefore, the period of the matter in the frequency of the Fourier transform can be obtained naturally. We can get the Fourier transform of the discrete sequence according to the formula of the z-transform. According to the formula of the z-transformation, we take the integral as a line on the unit circle, and we can get that the week on the z-plane unit circle corresponds exactly to one period of the transformation.

To sum up, the z-transform is a generalization of the Fourier transform.

The formula table for the Fourier transform is as follows:

An introduction to the Fourier variation is as follows:

The Fourier transform represents a function that satisfies a certain let condition as a trigonometric function (sine sum or cosine function) or a linear combination of their integrals.

In different fields of study, the Fourier transform has many different variant forms, such as the continuous Fourier transform and the discrete Fourier transform. Originally, Fourier analysis was proposed as a tool for analytical analysis of thermal processes.

The Fourier transform is a basic operation in digital signal processing, which is widely used in the field of expressing and analyzing discrete time-domain signals. However, due to the fact that the amount of computation is proportional to the square of the transformation point n, it is impractical to directly apply the DFT algorithm for spectral transformation when n is large. However, the advent of the fast Fourier transform technique has fundamentally changed the situation.

This article describes the design method of using FPGA to implement 2K 4K 8K point FFT.

Fourier transform or transformée de fourier has multiple Chinese translations, and the common ones are "Fourier transform", "Fu Liye transform", "Fourier transform", "Fu transform", "Fu transform", and so on.

The Fourier transform is a method of analyzing a signal by analyzing the components of the signal and synthesizing a signal from those components. Many waveforms can be used as components of a signal, such as sine waves, square waves, sawtooth waves, etc., and the Fourier transform uses a sine wave as a component of the signal.