Calculation and construction of the temporal characteristics of the analog filter. Impulse response and transfer function Impulse transfer response

By definition Transmission function(PF) is an operator equal to the ratio of the images of the output and input coordinates under zero initial conditions:

W(p) = R(p) / Q(p)

Service assignment. The control object (CO) is described by a linear differential equation order n. For an oscillatory link of the nth order, the following are determined:

1. Transmission function;
2. frequency characteristics (amplitude (AFC), phase (PFC), logarithmic (LFC));
3. transient and impulse transient (weight) functions;
4. graphs of transient and frequency responses.

To find the transfer function online, you must select the type of link and enter the degree of the link.

Example. The control object (OC) is described by a linear differential equation of the third order:
(2)
1) The transfer function of the op-amp in the general case can be represented as a ratio
W(iω) = A(ω)e iφ(ω) = U(ω) + iV(ω),
where R(p) and Q(p) are the Laplace images of the output and input variables of the CO corresponding to the left and right parts of Equation 1. Hence, the transfer function will look like:
(3)
or
. (4)

2) Let's determine the frequency characteristics of the op-amp. It is known that the frequency transfer function W(ω) can be represented as:
, (5)
where A(ω) is the amplitude frequency response (AFC);
φ(ω) – phase frequency response (PFC);
U(ω) – real frequency response (VCH);
V(ω) is the imaginary frequency response;
Let us substitute iω into expression (3) instead of p . We get:
(6)
Based on expressions (5) and (6), we single out the amplitude and phase frequency characteristics separately and substitute the numerical values ​​of the coefficients. Based on the fact that:
A(ω) = |W(iω)|
φ(ω) = arg(W(iω))
(see complex numbers). Finally we get: (7)

3) Define the logarithmic amplitude frequency response (LAFC).
It is known that LACH is determined from the ratio:
L(ω) = 20lg(A(ω)) (8)
This characteristic has the dimension of dB (decibels) and shows the change in the ratio of the output power to the input value. For convenience, the LACH is plotted on a logarithmic scale.
The phase frequency response plotted on a logarithmic scale will be referred to as the logarithmic phase frequency response (LPFC).
Examples of constructing LAFC and LPFC for our initial data are shown in Figure 1.
Let us define the impulse transition (weight) function. The weight function w(t) is the response of the system to the unit impulse function applied to its input. The weight function is related to the transfer function by the Laplace transform.
. (9)
Therefore, the weight function can be found by applying the inverse Laplace transform to the transfer function.
w(t) = L -1 (10)

Pulse transient function (weight function, impulse response) is the output signal of the dynamic system as a response to the input signal in the form of the Dirac delta function. In digital systems, the input signal is a simple pulse of minimum width (equal to the sampling period for discrete systems) and maximum amplitude. When applied to signal filtering, it is also called filter kernel. It finds wide application in control theory, signal and image processing, communication theory and other areas of engineering.

Definition [ | ]

impulse response system is called its response to a single impulse at zero initial conditions.

Properties [ | ]

Application [ | ]

Systems analysis [ | ]

Frequency response restoration[ | ]

An important property of the impulse response is the fact that on its basis a complex frequency response can be obtained, defined as the ratio of the complex spectrum of the signal at the output of the system to the complex spectrum of the input signal.

The complex frequency response (CFC) is an analytical expression of a complex function. The CFC is built on the complex plane and is a curve of the trajectory of the end of the vector in the operating frequency range, called hodograph of KChKh. To construct the CFC, usually 5-8 points are required in the operating frequency range: from the minimum realized frequency to the cutoff frequency (frequency of the end of the experiment). KCHH, as well as the time characteristic will give full information on the properties of linear dynamical systems.

The frequency response of the filter is defined as the Fourier transform (discrete Fourier transform in the case digital signal) from the impulse response.

H (j ω) = ∫ − ∞ + ∞ h (τ) e − j ω τ d τ (\displaystyle H(j\omega)=\int \limits _(-\infty )^(+\infty )h( \tau)e^(-j\omega \tau )\,d\tau )

To determine the impulse response g(t,τ), where τ is the exposure time, t- the time of occurrence and action of the response, directly according to the given parameters of the circuit, it is necessary to use the differential equation of the circuit.

To analyze the method of finding g(t,τ), consider a simple chain described by a first-order equation:

where f(t) - impact, y(t) - response.

By definition, the impulse response is the response of the circuit to a single delta pulse δ( t-τ) supplied to the input at the moment t=τ. It follows from this definition that if we put on the right side of the equation f(t)=δ( t-τ), then on the left side we can take y(t)=g(t,).

Thus, we arrive at the equation

.

Because right part of this equation is equal to zero everywhere, except for the point t=τ, function g(t) can be sought in the form of a solution of a homogeneous differential equation:

under the initial conditions following from the previous equation, as well as from the condition that by the moment of application of the impulse δ( t-τ) there are no currents and voltages in the circuit.

In the last equation, the variables are separated:

where
- values ​​of the impulse response at the moment of impact.

D To determine the initial value
Let's go back to the original equation. It follows from this that at the point
function g(t) must make a jump by 1/ a 1 (τ), because only under this condition the first term in the original equation a 1 (t)[dg/dt] can form a delta function δ( t-τ).

Since at

, then at the moment

.

Replacing the indefinite integral with a definite one with a variable upper limit of integration, we obtain relations for determining the impulse response:

Knowing the impulse response, it is not difficult to determine the transfer function of a linear parametric circuit, since both axes are connected by a pair of Fourier transforms:

where a=t-τ - signal delay. Function g 1 (t,a) is obtained from the function
replacement τ= t-a.

Along with the last expression, one more definition of the transfer function can be obtained, in which the impulse response g 1 (t,a) does not appear. To do this, we use the inverse Fourier transform for the response S EXIT ( t):

.

For the case when the input signal is a harmonic oscillation, S(t)=cosω 0 t. Corresponding S(t) there is an analytical signal
.

The spectral plane of this signal

Substituting
instead of
into the last formula, we get

From here we find:

Here Z EXIT ( t) - analytical signal corresponding to the output signal S EXIT ( t).

Thus, the output signal under harmonic action

defined in the same way as for any other linear circuits.

If the transfer function K(jω 0 , t) changes in time according to a periodic law with a fundamental frequency Ω, then it can be represented as a Fourier series:

where
- time-independent coefficients, generally complex, which can be interpreted as transfer functions of some quadripoles with constant parameters.

Work

can be considered as the transfer function of a cascade (serial) connection of two quadripoles: one with the transfer function
, independent of time, and the second with the transfer function
, which varies in time, but does not depend on the frequency ω 0 of the input signal.

Based on the last expression, any parametric circuit with periodically changing parameters can be represented as the following equivalent circuit:

Where is the process of formation of new frequencies in the spectrum of the output signal.

The analytical signal at the output will be equal to

where φ 0 , φ 1 , φ 2 ... are the phase characteristics of the quadripoles.

Passing to a real signal at the output, we get

This result indicates the following property of a circuit with variable parameters: when changing the transfer function according to any complex, but periodic law with a fundamental frequency

Ω, harmonic input signal with frequency ω 0 forms at the output of the circuit a spectrum containing frequencies ω 0 , ω 0 ±Ω, ω 0 ±2Ω, etc.

If a complex signal is applied to the input of the circuit, then all of the above applies to each of the frequencies ω and to the input spectrum. Of course, in a linear parametric circuit, there is no interaction between the individual components of the input spectrum (superposition principle) and no frequencies of the form n ω 1 ± mω 2 where ω 1 and ω 2 - different frequencies of the input signal.

2.3 General properties of the transfer function.

The stability criterion of a discrete circuit coincides with the stability criterion of an analog circuit: the poles of the transfer function must be located in the left half-plane of the complex variable , which corresponds to the position of the poles within the unit circle of the plane

Transfer function of the circuit general view is written, according to (2.3), as follows:

where the signs of the terms are taken into account in the coefficients a i , b j , while b 0 =1.

It is convenient to formulate the properties of the transfer function of a general circuit in the form of requirements for the physical realizability of a rational function of Z: any rational function of Z can be implemented as a transfer function of a stable discrete circuit up to a factor of H 0 PH Q if this function satisfies the requirements:

1. coefficients a i , b j - real numbers,

2. roots of the equation V(Z)=0, i.e. the poles H(Z) are located within the unit circle of the Z plane.

The multiplier H 0 × Z Q takes into account the constant amplification of the signal H 0 and the constant signal shift along the time axis by QT.

2.4 Frequency characteristics.

Discrete circuit transfer function complex

determines the frequency characteristics of the circuit

AFC, - PFC.

Based on (2.6), the general transfer function complex can be written as

Hence the formulas for the frequency response and phase response

The frequency characteristics of a discrete circuit are periodic functions. The repetition period is equal to the sampling frequency w d.

Frequency characteristics are usually normalized along the frequency axis to the sampling frequency

where W is the normalized frequency.

In calculations with the use of a computer, frequency normalization becomes a necessity.

Example. Determine the frequency characteristics of the circuit, the transfer function of which is

H(Z) \u003d a 0 + a 1 × Z -1.

Transfer function complex: H(jw) = a 0 + a 1 e -j w T .

taking into account frequency normalization: wT = 2p × W.

H(jw) = a 0 + a 1 e -j2 p W = a 0 + a 1 cos 2pW - ja 1 sin 2pW .

Formulas for frequency response and phase response

H(W) =, j(W) = - arctan .

graphs of the frequency response and phase response for positive values ​​a 0 and a 1 under the condition a 0 > a 1 are shown in Fig. (2.5, a, b.)

Logarithmic scale of the frequency response - attenuation A:

; . (2.10)

The zeros of the transfer function can be located at any point of the Z plane. If the zeros are located within the unit circle, then the characteristics of the frequency response and phase response of such a circuit are connected by the Hilbert transform and can be uniquely determined one through the other. Such a circuit is called a minimum phase circuit. If at least one zero appears outside the unit circle, then the circuit belongs to a nonlinear phase type circuit for which the Hilbert transform is not applicable.

2.5 impulse response. Convolution.

The transfer function characterizes the circuit in the frequency domain. In the time domain, the circuit has an impulse response h(nT). The impulse response of a discrete circuit is the response of the circuit to a discrete d-function. The impulse response and the transfer function are system characteristics and are interconnected by Z-transformation formulas. Therefore, the impulse response can be considered as a certain signal, and the transfer function H(Z) - Z is the image of this signal.

The transfer function is the main characteristic in the design, if the norms are set relative to the frequency characteristics of the system. Accordingly, the main characteristic is the impulse response if the norms are given in the time domain.

The impulse response can be determined directly from the circuit as the circuit's response to the d-function, or by solving the circuit's difference equation, assuming x(nT) = d(t).

Example. Determine the impulse response of the circuit, the scheme of which is shown in Fig. 2.6, b.

Difference circuit equation y(nT)=0.4 x(nT-T) - 0.08 y(nT-T).

The solution of the difference equation in numerical form, provided that x(nT)=d(t)

n=0; y(0T) = 0.4 x(-T) - 0.08 y(-T) = 0;

n=1; y(1T) = 0.4 x(0T) - 0.08 y(0T) = 0.4;

n=2; y(2T) = 0.4 x(1T) - 0.08 y(1T) = -0.032;

n=3; y(3T) = 0.4 x(2T) - 0.08 y(2T) = 0.00256; etc. ...

Hence h(nT) = (0 ; 0.4 ; -0.032 ; 0.00256 ; ...)

For a stable circuit, the counts of the impulse response tend to zero over time.

The impulse response can be determined from a known transfer function by applying

a. inverse Z-transform,

b. decomposition theorem,

in. the delay theorem to the results of dividing the numerator polynomial by the denominator polynomial.

The last of the listed methods refers to numerical methods for solving the problem.

Example. Determine the impulse response of the circuit in Fig. (2.6, b) from the transfer function.

Here H(Z) = .

Divide the numerator by the denominator

Applying the delay theorem to the result of division, we obtain

h(nT) = (0 ; 0.4 ; -0.032 ; 0.00256 ; ...)

Comparing the result with the calculations using the difference equation in the previous example, one can verify the reliability of the calculation procedures.

It is proposed to independently determine the impulse response of the circuit in Fig. (2.6, a), applying successively both considered methods.

In accordance with the definition of the transfer function, the Z - image of the signal at the output of the circuit can be defined as the product of the Z - image of the signal at the input of the circuit and the transfer function of the circuit:

Y(Z) = X(Z) x H(Z). (2.11)

Hence, by the convolution theorem, the convolution of the input signal with the impulse response gives the signal at the output of the circuit

y(nT) =x(kT)Chh(nT - kT) =h(kT)Chx(nT - kT). (2.12)

The definition of the output signal by the convolution formula is used not only in calculation procedures, but also as an algorithm for the functioning of technical systems.

Determine the signal at the output of the circuit, the circuit of which is shown in Fig. (2.6, b), if x (nT) = (1.0; 0.5).

Here h(nT) = (0 ; 0.4 ; -0.032 ; 0.00256 ; ...)

Calculation according to (2.12)

n=0: y(0T) = h(0T)x(0T) = 0;

n=1: y(1T) = h(0T)x(1T) + h(1T) x(0T) = 0.4;

n=2: y(2T)= h(0T)x(2T) + h(1T) x(1T) + h(2T) x(0T) = 0.168;

Thus y(nT) = ( 0; 0.4; 0.168; ... ).

In technical systems, instead of linear convolution (2.12), circular or cyclic convolution is more often used.

Student of the group 220352 Chernyshev D. A. Reference - report on the patent and scientific and technical research Theme of the final qualifying work: television receiver with digital signal processing. Start of search 2. 02. 99. End of search 25.03.99 Search subject Country, Index (MKI, NKI) No. ...

Carrier and amplitude-phase modulation with a single sideband (AFM-SBP). 3. The choice of the duration and number of elementary signals used to form the output signal In real communication channels for transmitting signals by frequency limited channel a signal of the form is used, but it is infinite in time, so it is smoothed according to the cosine law. , where - ...

The temporal characteristics of the circuit are called responses to the typical components of the original signal.

The transient response of the circuit is the response of the circuit with zero initial conditions to the action identity function(Heaviside functions). The transition response is determined from the operator transfer function by dividing it by the operator , and finding the original from the resulting image using the inverse Laplace transform through residues.

The impulse response of a circuit is the response of the circuit to the action of the delta function. - an infinitely short in duration and infinitely large in amplitude pulse of unit area. The impulse response is determined by finding the residues from the transfer function of the circuit.

We will also search for the temporal characteristics of the circuit by the operator method. To do this, you need to find the operator image of the input signal, multiply it by the transfer coefficient in operator form and find the original from the resulting expression, that is, knowing the transfer coefficient of the circuit, we can find a response to any impact.

Finding the impulse response comes down to finding the response of the circuit to the delta function. It is known that for the delta function, the image is 1. Applying the inverse Laplace transform, we find the impulse response.

.

We select the integer part for the transfer function of the circuit, since the degrees of the highest coefficients in the numerator and in the denominator are equal:

Find the singular points of the transfer function by equating the denominator to zero.

We have only one singular point, now we take the residue at this singular point.

The expression for the impulse response is written as follows:

Similarly, we find the transient response of the circuit, knowing that for the Heaviside function, the image is the function .

; , ;

Transient and impulse responses are interconnected, as well as input actions:

Let us check the fulfillment of the limiting relations between the frequency and time characteristics of the circuit, i.e. fulfillment of the following conditions:

We substitute specific expressions for the characteristics of circuits into the system.

.

As you can see, the conditions are met, which indicates the correctness of the found formulas.

Let us write down the final formulas for the time characteristics, taking into account the normalization

According to the above formulas, we construct graphs of these functions.

Fourier signal analog linear

Figure 2.5 - Impulse response analog filter prototype

Figure 2.6 - Step response of analog prototype filter

Temporal characteristics exist only at , since the responses cannot lead the impact.

Our chain is differentiating, so transient response behaves like this. The differentiating circuit sharpens the transient and skips the leading edge. Those who passed are responsible for the “throw” high frequencies, and for the blockage - low frequencies that have not passed.

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