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Fourier and Laplace transforms.

Lecture



Fourier transform

Ratio

  Fourier and Laplace transforms.

called the direct Fourier transform. Corner frequency function   Fourier and Laplace transforms. -   Fourier and Laplace transforms. called the Fourier transform or frequency spectrum function   Fourier and Laplace transforms. . The spectrum characterizes the ratio of the amplitudes and phases of an infinite set of infinitely small sinusoidal components, which together form a non-periodic signal   Fourier and Laplace transforms. . The Fourier transform operation is mathematically written as follows:

  Fourier and Laplace transforms.

Where   Fourier and Laplace transforms. - the symbol of the direct Fourier transform.

The spectra in the theory of automatic control are represented graphically, depicting separately their real and imaginary parts:

  Fourier and Laplace transforms.

In fig. 1 shows a typical image of the spectrum of a non-periodic signal.

  Fourier and Laplace transforms.

Fig. one

We note the following features of the spectrum of a non-periodic function   Fourier and Laplace transforms. :

    1. The spectrum of the non-periodic function of time is continuous;

    2. The range of valid values ​​of the spectrum argument

  Fourier and Laplace transforms.

  1. The real part of the spectrum is an even frequency function, the imaginary part of the spectrum is an odd function, which allows using one half of the spectrum

  Fourier and Laplace transforms.

The Fourier transform is reversible, that is, knowing the Fourier image, you can determine the original function - the original. The ratio of the inverse Fourier transform is as follows:

  Fourier and Laplace transforms.

or in abbreviated notation   Fourier and Laplace transforms. where   Fourier and Laplace transforms. - the symbol of the inverse Fourier transform. Note that a time function has a Fourier transform if and only if:

  • the function is unambiguous, contains a finite number of maxima, minima and discontinuities;

  • the function is absolutely integrable, i.e.

  Fourier and Laplace transforms.

The inverse Fourier transform is only possible if all poles   Fourier and Laplace transforms. - left.

Consider the examples of determining the spectrum of temporal functions.

Example :

Find the frequency spectrum of the delta function.

  Fourier and Laplace transforms. ,

as when   Fourier and Laplace transforms.

  Fourier and Laplace transforms. ,

and at   Fourier and Laplace transforms.   Fourier and Laplace transforms. and

  Fourier and Laplace transforms. .

Eventually,   Fourier and Laplace transforms. has a single, uniform and frequency independent real spectrum, and the imaginary part of the spectrum will be zero (see Fig. 2).

  Fourier and Laplace transforms.

Fig. 2

Example :

Let us find the frequency spectrum of a single step function.

For this function, the requirement of absolute integrability is not fulfilled, since

  Fourier and Laplace transforms.

therefore   Fourier and Laplace transforms. Fourier image has not.

Laplace transform

Ratio

  Fourier and Laplace transforms.

called the direct Laplace transform. Complex variable   Fourier and Laplace transforms. called the Laplace operator, where   Fourier and Laplace transforms. - angular frequency,   Fourier and Laplace transforms. - some positive constant number. Complex variable function   Fourier and Laplace transforms. called signal image   Fourier and Laplace transforms. according to Laplace. The operation of determining the image from the original is abbreviated -   Fourier and Laplace transforms. where   Fourier and Laplace transforms. - the symbol of the direct Laplace transform.

The Laplace transform is reversible, that is, knowing the image from Laplace, you can determine the original using the ratio of the inverse transform

  Fourier and Laplace transforms.

or   Fourier and Laplace transforms. where   Fourier and Laplace transforms. - the symbol of the inverse Laplace transform.

Note that the Laplace transform represents the original function only when   Fourier and Laplace transforms. , and the behavior of the original function when   Fourier and Laplace transforms. no effect on the image. The class of functions transformed by Laplace is much wider than the class of functions transformed by Fourier. Virtually any time functions in TAU have a Laplace transform.

We obtain Laplace images for impulse functions.

  Fourier and Laplace transforms. ,

because   Fourier and Laplace transforms. at   Fourier and Laplace transforms. ,

  Fourier and Laplace transforms. and   Fourier and Laplace transforms. at   Fourier and Laplace transforms. .

  Fourier and Laplace transforms. .

In practice, the conversion tables are used to perform the direct and inverse Laplace transformations, a fragment of which is shown in Table. one.

Table 1.

  Fourier and Laplace transforms.

  Fourier and Laplace transforms.

  Fourier and Laplace transforms.

  Fourier and Laplace transforms.

  Fourier and Laplace transforms.

  Fourier and Laplace transforms.

  Fourier and Laplace transforms.

  Fourier and Laplace transforms.

  Fourier and Laplace transforms.

one

  Fourier and Laplace transforms.

  Fourier and Laplace transforms.

  Fourier and Laplace transforms.

  Fourier and Laplace transforms.

  Fourier and Laplace transforms.

  Fourier and Laplace transforms.

The Laplace transform tables can be used to determine the Fourier transforms of such absolutely integrable functions that are 0 for   Fourier and Laplace transforms. . To obtain Fourier images in this case, it suffices to put Laplace in the image   Fourier and Laplace transforms. . In general, it looks like

  Fourier and Laplace transforms. ,

if a   Fourier and Laplace transforms. at   Fourier and Laplace transforms. and   Fourier and Laplace transforms.

Consider the statements of the main theorems of the Laplace transform, which are widely used in TAU.

    1. The linearity theorem. Any linear relation between the functions of time is also valid for the Laplace images of these functions;

  Fourier and Laplace transforms. ;

    1. The differentiation theorem of the original.

If a   Fourier and Laplace transforms. and   Fourier and Laplace transforms. then   Fourier and Laplace transforms. ,

Where   Fourier and Laplace transforms. - the initial value of the original.

For the second derivative, use the expression

  Fourier and Laplace transforms. .

For derivative   Fourier and Laplace transforms. th order is true the following relationship:

  Fourier and Laplace transforms. ;

For derivative   Fourier and Laplace transforms. th order with zero initial conditions, the following relationship holds:

  Fourier and Laplace transforms. ;

i.e. differentiation   Fourier and Laplace transforms. degree of the original in time with zero initial conditions corresponds to multiplying the image by   Fourier and Laplace transforms. .

  1. The original integration theorem.

  Fourier and Laplace transforms. ;

Comment

In the field of images according to Laplace, the complex operations of differentiation and integration are reduced to the operations of multiplication and division by   Fourier and Laplace transforms. that allows you to move from differential and integral equations to algebraic. This is the main advantage of the Laplace transformation as a mathematical tool of the theory of automatic control.

    1. Delay theorem. For anyone   Fourier and Laplace transforms. fair value

  Fourier and Laplace transforms. ;

    1. The convolution theorem (multiplication of images).

  Fourier and Laplace transforms. ,

Where

  Fourier and Laplace transforms. ;

  1. Limit value theorem. If a   Fourier and Laplace transforms. then

  Fourier and Laplace transforms.

if a   Fourier and Laplace transforms. exists.

To find the original function in its image using the inverse Laplace transform. The image function must be represented in the form of Heavisite, using the necessary formula for the decomposition of a fractionally rational function. The resulting sum of the simplest fractions is subjected to the inverse Laplace transform. To do this, you can use the Laplace transform tables, which define the images of many temporary functions. A fragment of the Laplace transform table is given in Table. 1. In cases where there are complex conjugate poles of the image, it is necessary to convert the corresponding simple fractions into a form suitable for using the Laplace transform table. The use of a personal computer with packages of mathematical programs containing the functions of the direct and inverse Laplace transformations makes it much easier.

Example

Define the original   Fourier and Laplace transforms. in the image as a fractional rational function

  Fourier and Laplace transforms. .

We use the Hevisite decomposition for a fractional rational function with one zero pole. Then

  Fourier and Laplace transforms. .

Decomposition coefficients are

  Fourier and Laplace transforms. .

The image in the form of Heavisite has the form

  Fourier and Laplace transforms. .

We use the linearity theorem and the transformation table for each term, as a result we get

  Fourier and Laplace transforms. .

The graph of the original function has the form shown in Fig. 3

  Fourier and Laplace transforms.

Fig. 3

We briefly explain the algorithm for solving differential equations by an operator method using the example of solving a differential equation of order 2 in general

  Fourier and Laplace transforms. ,

Where   Fourier and Laplace transforms. ,   Fourier and Laplace transforms. ,   Fourier and Laplace transforms. .

We apply the differentiation theorem to find images of derivatives

  Fourier and Laplace transforms. ,   Fourier and Laplace transforms. .

Let be   Fourier and Laplace transforms. then

  Fourier and Laplace transforms. .

We obtain an operator equation using the linearity theorem

  Fourier and Laplace transforms. ,

  Fourier and Laplace transforms. .

Solve the equation for   Fourier and Laplace transforms. ,

  Fourier and Laplace transforms. .

We find   Fourier and Laplace transforms. using the transition to the Heavisite form (decomposition of Heavisite)

  Fourier and Laplace transforms. ,

Where   Fourier and Laplace transforms. ,   Fourier and Laplace transforms. .

Particular attention should be paid to obtaining the image of the derivative of the step unit function.   Fourier and Laplace transforms. which is defined as follows:

  Fourier and Laplace transforms.

If use

  Fourier and Laplace transforms. ,

This is an erroneous solution, so you should use the “left” initial conditions called

  Fourier and Laplace transforms. .

The validity of this can be easily verified by substituting the solution into the original differential equation.

Test questions and tasks

    1. What restrictions are imposed on the direct and inverse Fourier transforms?

    2. How to get the frequency spectrum of a real signal - a non-periodic function of time using the Laplace transform tables?

    3. If the image according to Laplace has the form of a fractionally rational function, in what form is it more convenient to represent it for obtaining the original, in the form of a Bode or in the form of Hevisite?

    4. Determine the original image of the following Laplace

  Fourier and Laplace transforms. .

Answer :

  Fourier and Laplace transforms. .

    1. Determine the original image of the following Laplace

  Fourier and Laplace transforms. .

Answer :

  Fourier and Laplace transforms. .

    1. Find   Fourier and Laplace transforms. solving a differential equation

  Fourier and Laplace transforms. ,

Where   Fourier and Laplace transforms. .

Answer :

  Fourier and Laplace transforms. .

    1. Find   Fourier and Laplace transforms. solving a differential equation

  Fourier and Laplace transforms. ,

Where   Fourier and Laplace transforms. .

Answer :

  Fourier and Laplace transforms. .


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Mathematical foundations of the theory of automatic control

Terms: Mathematical foundations of the theory of automatic control