# Device for reproduction of orthogonal functions

FIELD: engineering of specialized computer means, possible use for engineering generator equipment, and also solving boundary problems of mathematical physics.

SUBSTANCE: device contains three integrators, multiplication blocks, adder, inverter, analog-digital converter, five analog-digital converters of first group, three analog-digital converters of second group and two memory blocks.

EFFECT: expanded functional capabilities due to reproduction of all classes of orthogonal functions, being solutions of second-order differential equations.

1 dwg, 1 tbl

The invention relates to specialized tools for computational engineering and can be used to create the generator equipment, as well as when solving boundary value problems of mathematical physics.

Known function generator containing matrix solvers, the first group of blocks dividing, the synchronization unit, the shift register R-1 group of blocks of elements And P-2 groups of blocks dividing the first and second memory blocks, the matrix operating units, R-2 delay elements and the circular shift register (see USSR author's certificate No. 1753463, CL G 06 F 1/02, 1990).

However, the known generator is designed to generate a discrete basis functions, described their own vectors a positive definite symmetric matrix, and does not generate an analog orthogonal functions.

It is also known a device for playing orthogonal oscillations containing integrators, multipliers, first and second integrators, amplifiers (see USSR author's certificate No. 1368899, CL G 06 G 7/26, 1986).

However, the known device generates only one class of functions, which is the solution obtained by the authors of the differential equation, and is not able to generate orthogonal functions of different classes.

The closest in technical essence of the present invention is the device that contains the ASEE five integrators, two block multiplication, the adder unit and the inverter, and the input of the first integrator is an input device and an output connected to the first input of the first unit of multiplication and with the input of the second integrator, the output of which is connected to the first input of the second block multiplication, the output of which is connected to the first input of the adder, the output of which is connected with the second input of the second block multiplication and with the input of the third integrator, the output of which is connected to the input of the fourth integrator and a second input of the first block multiplication, the output of which is connected to the second input of the adder, the output of the first integrator is connected to the fifth input of the integrator, the output which is connected to the first input unit, the output of which is connected to the input of the integrator, the output of which is connected to the fourth input of the adder, the output of the fourth integrator is connected with the second input unit, the output of the fourth integrator is connected to the third input of the adder and an output device (see USSR author's certificate No. 822216, CL G 06 G 7/40, 1979).

However, this device is intended for play only one class of orthogonal functions described adjoint Legendre polynomials, and is not able to form orthogonal functions of other classes.

The aim of the invention is the expansion of fu is clonally opportunities through play all classes of orthogonal functions, which is the solution of the differential equations of the second order.

This objective is achieved in that in the known device, containing three integrator, two block multiplication, the adder and inverter, and the input of the first integrator is the input device, the output of the first block multiplication connected to the first input of the adder, the output of the second block multiplication is connected to the second input of the adder, entered three block multiplication, analog-to-digital Converter, five digital to analog converters of the first group, three digital-to-analogue Converter of the second group and two of the memory block, and the first group of address inputs of the first and second memory blocks is a group of inputs setup class generated functions, the second group of address inputs of the first and second memory blocks is a group of inputs installation sequence numbers generated function, the output of the first integrator connected to the input of an analog-digital Converter whose outputs are connected to the third group of address inputs of the first memory block, the inputs of digital to analog converters of the first group connected to the corresponding data outputs of the first memory block, the output of the first d / a Converter of the first group connected to the first input of the first block multiplication, the second input of which the connection is to the output of the adder, the output of the second d / a Converter of the first group connected to the first input of the second block multiplication, the second input of the second block multiplication is connected to the output of the inverter, the output of the third digital-to-analogue Converter of the first group connected to the first input of the third block multiplication, the second inputs of the third and fourth multiplier units connected to the output of the third integrator, the first input of the fourth block multiplication is connected to the output of the fifth block multiplication, the first input of which is connected to the output of the fourth d / a Converter of the first group and the second input is connected to the output of the fifth d / a Converter of the first group, the output of the third block multiplication connected with the third input of the adder, the output of the fourth block multiplication is connected to the fourth input of the adder, the output of the adder connected to the input of the second integrator, the output of which is connected to the input of the third integrator and the input of the inverter, the input of digital-analog converters of the second group are connected to the corresponding data outputs of the second memory block, the output of the first d / a Converter of the second group connected to the input set initial state of the first integrator, the output of the second d / a Converter of the second group connected to the input of set the initial state of the second integrator, the output of the third d / a Converter of the second group connected to the input set initial state of the third integrator, the output of the third integrator is the output device.

A linear differential operator of second order General form can be written as:

(see Gafken. Mathematical methods in physics. - M.: Atomizdat, 1970, s equation (9.1)).

Differential equation of second order can be written in the form:

(see Gafken. Mathematical methods in physics. - M.: Atomizdat, 1970, s equation (9.6)).

Here λ - a constant, and ω(x) is a known function of x, called the density or weight function.

Demand that ω(x)>0, except, may be isolated points where ω(x)=0. U_{λ}(x), for some λsatisfying the equation (1) with given boundary conditions, is called a private function that corresponds to the eigenvalue λ.

In theory of differential equations it is convenient to define a conjugate operator:

Comparison of equations (1) and (3) gives a necessary and sufficient condition that:

If the condition (4):

the operator L is called self-adjoint. Here p_{0}(x) replaced by p(x), and p_{2}(x) g(x) (see Gafken. Mathematical methods in physics. - M.: Atomizdat, 1970, s equation (9.2), s equation (9.3), (9.4)).

theory of linear self-adjoint differential equations of the second order is of a General nature, as it is always possible to bring nonselfadjoint operator to self-adjoint form (see Gafken. Mathematical methods in physics. - M.: Atomizdat, 1970, s, second paragraph).

Self-adjoint operator is characterized by the following properties:

his own real values;

is its own orthogonal functions

(see Gafken. Mathematical methods in physics. - M.: Atomizdat, 1970, s, first paragraph).

When casting self-adjoint equations to mind the significance of their coefficients and parameters can be easily determined (see Gafken. Mathematical methods in physics. - M.: Atomizdat, 1970, s, second paragraph from the bottom). While the function u_{λ}{x) must satisfy certain boundary conditions.

The relation (2) subject to (5) can be written in the form:

or otherwise

Differential equation of second order can be written in the form:

wherefunction of time, λ - its own value.

Private function U(t), the corresponding eigenvalues λunder certain values of p(t), p'(t), g(t), ω(t)defined by the class of oscillations according to the above, are orthogonal among themselves and can be used in communication systems as vectors.

The values of the functions p, p', g, ω known for all classes of classical fluctuations and can be easily calculated for any class of oscillations described by functions which are solutions of the differential equations of the second order.

The values of the coefficients and parameters for the classical oscillations presented, for example, in source: Gafken. Mathematical methods in physics. - M.: Atomizdat, 1970, s, table 9.1.

These values are presented in table 1. In addition to the coefficients and parameters for the classical fluctuations, in the penultimate row of the table presents values for the vibrations generated by the analog and which are the solutions obtained by the authors of the differential equation of the second order.

Figure 1 shows the structural diagram of the device for playback of orthogonal functions.

The device contains integrators 1, 3, 4, the adder 2, the inverter 5, the multipliers 6, an analog-to-digital Converter 7, partyblack 8 memory digital to analogue Converter 9 of the first group, the second block 10 memory digital to analogue Converter 11 of the second group.

The device operates as follows. Before the formation of orthogonal functions on the group of inputs setup class generated functions served code corresponding to the desired class. For example, for formation of Legendre functions for this group of inputs is fed code 0...001 to form a Chebyshev functions of the first kind - 0...010 to form a Chebyshev functions of the second kind - 0...011 etc. This code is served by the bus on the first group of address inputs of the first block 8 memory and the second memory block 10.

The memory unit 8 is a conventional permanent memory (ROM), in which the memory is divided into areas corresponding to each class of functions generated by the proposed device. Each zone, in turn, is divided into areas corresponding to the number of the generated function. Submitting code to a second group of address inputs of the memory unit 8 can make a choice of function rooms, which will form the device. Carrying the signal current time, converted into digital form, to the third group of address inputs of the memory unit 8, the outputs of digital to analog converters 9, the inputs of which are connected to data outputs of the memory unit 8, can SFD shall inform the values of the coefficients and parameters p(t),
p'(t), g(t), λ_{i}that ω(t).

The second block 10 as well as a memory is a ROM memory which is divided into zones corresponding to the class of functions, each of which includes a region corresponding to the number of the generated function. When submitting code class functions and function rooms on the first and second group of address inputs of the memory block 10 outputs d / a converters 11, the inputs of which are connected to data outputs of the memory block 10, briefly formed the initial conditions of the integrators 1, 3 and 4 are fed to the inputs of the initial setup of the integrators.

With that said, filing code class functions on the first group of address inputs and code number of the required functions for the second group of address inputs of the memory unit 8 and block 10 memory at the outputs of digital to analog converters 11.1, 11.2 and 11.3 short-term will be recorded respectively in the first integrator 1, the second integrator 3 and the third integrator 4 initial conditions. For example, in the case that the functions similar to the generated prototype (see USSR author's certificate No. 822216, CL G 06 G 7/40, 1979) at the output of digital to analogue Converter 11.1 will be a unit, on the output of the digital to analogue Converter 11.2 will receive a zero on the output of the digital to analogue Converter 11.3 appears one of the values

depending on the number function.

When applying a DC voltage U at the input of the integrator 1 at its output, a signal is generated for the current time t at the input of analog-to-digital Converter 7. Code current time with its output fed to the third group of address inputs of the first memory unit 8. As a result, in accordance with the last row of table 1, the output of the Converter 9.1 a signal is generated 1-t^{2}on the inverter output 9.2 formed signal - 2t, the output of the Converter 9.3 a signal is generatedthe output of the Converter 9.4 formed a constant level corresponding to the eigenvalue λ (where λ=n(n+1)), the output of the Converter 9.5 formed unit. As a result, the output of the multiplier 6.5, the first input of which is connected to the inverter output 9.4, and the second to the output of the Converter 9.5 a signal is generated n(n+1).

At the output of the multiplier 6.1 a signal

at the output of the multiplier 6.2 a signal

at the output of the multiplier 6.3 a signal

at the output of the multiplier 6.4 a signal

Each of these signals is fed to a corresponding input of the amounts of the Torah. As a result, the output of the adder 2 is formed oscillation, described by the functionand the output of the second integrator 3, a signal is generated, described by the functionThus, the output of the third integrator 4 is removed oscillation described desired function of the formthat is attached Legendre polynomial of the corresponding order.

Similar is the operation of the device and the formation of other types of vibrations.

For example, the formation of the oscillations generated identical counterpart (see USSR author's certificate No. 1368899, CL G 06 G 7/26, 1986), which is the solution obtained by the authors of the differential equation of the second order.

The code of this class of functions and the number of the desired function is served on the first and second group of address inputs of the block 8 and block 10 memory respectively. The outputs of the digital-to-analog converters 11.1, 11.2 and 11.3 short-term will be recorded respectively in the first integrator 1, the second integrator 3 and the third integrator 4 initial conditions:

where k is the number of the generated function.

When applying a DC voltage + U to the input of the integrator 1 at its output, a signal is generated (π-t)at the input and the scarlet-digital Converter 7.
Code corresponding to the values (π-t), its output is fed to the third group of address inputs of the first memory unit 8. As a result, in accordance with the penultimate row of table 1, the output digital to analogue Converter 9.1 formed unit, the output of the Converter 9.2 formed zero, the output of the Converter 9.3 a signal is generated λ_{n}(π-t-λ_{n}on the inverter output 9.4 formed is λ_{n}at the output of the Converter 9.5 formed unit. As a result, the output of the multiplier 6.5, the first input of which is connected to the inverter output 9.4, and the second to the output of the Converter 9.5, a signal is generated λ_{n}where λ_{t}- the solution of the equation

where n is the number of generated functions (see USSR author's certificate No. 1368899, CL G 06 G 7/26, 1986).

Because the second input of the multiplication 6.1 is connected to the output of the adder 2, on which a signal is generatedand second input units of the multiplication 6.3 and 6.4 is connected to the output of the third integrator 4, which is formed by the signal U(t), the signal U(t) is the solution of the differential equation (11).

The use of the invention allows to extend the functionality of the known devices, because it can play vs the classes of orthogonal functions, are solutions of the differential equations of the second order, which gives the opportunity to generate vibrations, which provide increased robustness of the sealed channels in communication systems, increase throughput, and to solve boundary value problems of mathematical physics.

Table 1 | ||||

The class of functions | p(x) | g(x) | λ | ω(x) |

Legendre | 1-x^{2} | 0 | l(l+1) | 1 |

Chebyshev 1 | (1-x^{2})^{1/2} | 0 | n^{2} | (1-x^{2})^{1/2} |

Chebyshev 2 | (1-x^{2})^{3/2} | 0 | n(n+2) | (1-x^{2})^{1/2} |

Bessel | x | -n^{2}/x | and^{2} | x |

Laguerre | xe^{-x} | 0 | α | e^{-x} |

Laguerre, adjoint equations | x^{k+1}e^{-x} | 0 | α-k | x^{k}e^{-x} |

Hermite | e^{-x2} | 0 | 2α | e^{
-x2} |

Harmonic oscillator | 1 | 0 | ω^{2} | 1 |

Generated by analog | 1 | λ_{n}(π-x)-λ_{n} | λ_{n} | 1 |

Generated by the prototype, i.e. the Legendre functions of the adjoint equations | 1-x^{2} | -m^{2}/(1-x^{2}) | l(l+1) | 1 |

Device for playback of orthogonal functions, containing three integrator, two block multiplication, the adder and inverter, and the input of the first integrator is the input device, the output of the first block multiplication connected to the first input of the adder, the output of the second block multiplication connected with the second input of the adder, characterized in that it introduced three blocks multiplying analog-to-digital Converter, five digital to analog converters of the first group, three digital-to-analogue Converter of the second group and two of the memory block, and the first group of address inputs of the first and second memory blocks is a group of inputs setup class generated functions, the second group of address inputs the first and second memory blocks is a group of inputs installation sequence numbers generated function, the output is, I can pay tithing integrator connected to the input of analog-to-digital Converter,
the outputs of which are connected to the third group of address inputs of the first memory block, the inputs of digital to analog converters of the first group connected to the corresponding data outputs of the first memory block that stores the values of the coefficients and parameters p(t), p'(t), g(t), λ_{i}w(t), where p(t), p'(t), g(t), w(t) - values replicable functions, and λ_{i}is a constant for a given class of functions, all classes of classical oscillations described by functions which are solutions of the differential equations of the second order, the output of the first d / a Converter of the first group connected to the first input of the first block multiplication, the second input is connected to the output of the adder, the output of the second d / a Converter of the first group connected to the first input of the second block multiplication, the second input of the second block multiplication is connected to the output of the inverter, the output of the third digital-to-analogue Converter of the first group connected to the first input of the third block multiplication, the second inputs of the third and fourth multiplier units connected to the output of the third integrator, the first input of the fourth multiplication unit connected to the output of the fifth block multiplication, the first input of which is connected to the output of the fourth d / a Converter of the first group and the second input is connected to the exit of the fifth digital-to-analogue Converter of the first group,
the output of the third block multiplication connected with the third input of the adder, the output of the fourth block multiplication is connected to the fourth input of the adder, the output of the adder connected to the input of the second integrator, the output of which is connected to the input of the third integrator and the input of the inverter, the input of the digital-to-analog converters of the second group are connected to the corresponding data outputs of the second memory block that stores the initial conditions for the integrators of all classes of classical oscillations described by functions which are solutions of the differential equations of the second order, the output of the first d / a Converter of the second group connected to the input set initial state of the first integrator, the output of the second d / a Converter of the second group connected to the input set initial state of the second integrator, the output of the third d / a Converter of the second group connected to the input set initial state of the third integrator, the output of the third integrator is the output device.

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