IN radio circuits load resistances are usually large and do not affect the four-terminal network, or the load resistance is standard and already taken into account in the four-terminal circuit.
Then the quadripole can be characterized by one parameter that establishes a relationship between the output and input voltages when the load current is neglected. With a sinusoidal signal, such a characteristic is the transfer function of the circuit (transfer coefficient), equal to the ratio of the complex amplitude of the output signal to the complex amplitude of the signal at the input: , where is the phase-frequency characteristic, is the amplitude-frequency characteristic of the circuit.
Transmission function linear circuit due to the validity of the superposition principle, it allows you to analyze the passage of a complex signal through a circuit, decomposing it into sinusoidal components. Another possibility of using the superposition principle is to decompose the signal into a sum of time-shifted d-functions d(t). The response of the circuit to the action of a signal in the form of d-functions is impulse response g(t), i.e., this is the output signal if the input signal is a d-function. at . In this case, g(t) = 0 for t< 0 – выходной сигнал не может возникнуть ранее момента появления input signal.
Experimentally, the impulse response can be determined by applying a short pulse of unit area to the input and reducing the pulse duration while maintaining the area until the output signal stops changing. This will be the impulse response of the circuit.
Since there can be only one independent parameter connecting the voltages at the output and input of the circuit, there is a connection between the impulse response and the transfer function.
Let the input be a signal in the form of a d-function w spectral density. At the output of the circuit there will be an impulse response , while all the spectral components of the input signal are multiplied by the transfer function of the corresponding frequency: . Thus, the impulse response of the circuit and the transfer function are related by the Fourier transform:
Sometimes the so-called transient response of the circuit h(t) is introduced, which is a response to a signal called a unit jump:
I(t) = 1 for t ³ 0
I(t) = 0 at t< 0
in this case , h(t) = 0 for t< 0.
Due to the relationship between the transfer function and the impulse response, the following restrictions are imposed on the transfer function:
· The condition that g(t) must be real leads to the requirement that , i.e., the modulus of the transfer function (AFC) is even, and the phase angle (PFC) is an odd function of frequency.
The condition that at t< 0, g(t) = 0 приводит к критерию Пэли-Винера: .
For example, consider an ideal low-pass filter with a transfer function.
Here, the integral in the Paley-Wiener criterion diverges, as for any vanishing on a finite segment of the frequency axis.
The impulse response of such a filter is
g(t) is not equal to zero at t< 0, тем сильнее, чем меньше время задержки , которое определяет ее угол наклона . Это указывает на нереализуемость идеального ФНЧ, имеющего близкое приближение при достаточно больших .
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
.
Since the right side of this equation is equal to zero everywhere except at 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
is 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 the 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.
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 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 )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 feasibility 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 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. Define frequency characteristics circuit, the transfer function of which
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 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
b. decomposition theorem,
V. 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.
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