Relationship between impulse and transfer characteristics. General properties of the transfer function. Lecture Time Allocation
The time and frequency characteristics of the circuit are interconnected by the Fourier transform formulas. According to the transient response found in clause 2.1, the impulse response of the circuit is calculated (Figure 1)
The calculation result coincides with the formula H(jш) obtained in Section 2.2
Input signal sampling and impulse response
Let be taken as the upper limit of the input signal spectrum. Then, according to the Kotelnikov theorem, the sampling frequency is kHz. Where is the sampling period T=0.2ms
According to the graph shown in Fig. 2, we determine the values of discrete samples of the input signal U 1 (n) for t sampling times.
Discrete values of the impulse response are calculated by the formula
where T=0.0002 s; n=0, 1, 2,…., 20.
Table 3. Discrete values of the input signal function and impulse response
The discrete values of the signal at the output of the circuit are calculated for the first 8 samples using the discrete convolution formula.
Table 4. Discrete signal at the output of the circuit.
Comparison of the calculation results with the data in Table 1 shows that the difference in the values of U 2 (t), calculated using the Duhamel integral and by sampling the signal and impulse response, differ by several tenths, which is an acceptable deviation for the given initial parameters.
Figure 9. The value of the discrete signal at the input of the circuit.
Figure 10. The value of the discrete signal at the output of the circuit.
Figure 11. The value of discrete readings of the impulse response of the circuit H(n).
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
The transfer function of a general circuit 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. 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,
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.
Student of the group 220352 Chernyshev D. A. Reference - report on patent and scientific and technical research Theme of graduation qualification 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. Selection of the duration and number of elementary signals used to generate the output signal In real communication channels, a signal of the form is used to transmit signals over a frequency-limited channel, but it is infinite in time, so it is smoothed according to the cosine law. , Where - ...
In radio circuits, the load resistances are usually large and do not affect the quadripole, or the load resistance is standard and already taken into account in the quadripole circuit.
Then the four-terminal network can be characterized by one parameter that establishes the relationship between the output and input voltages while neglecting the load current. 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.
The transfer function of a linear circuit, due to the validity of the superposition principle, makes it possible to analyze the passage of a complex signal through the 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 the 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 входного сигнала.
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 with a 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 .
Let an arbitrary impulse system be given by a block diagram, which is a set of standard connections from the simplest impulse systems (connections of the feedback type, serial and parallel). Then, in order to obtain the transfer function of this system, it is enough to be able to find the transfer function of standard connections from the transfer functions of the connected impulse systems, since the latter are known (either exactly or approximately) (see § 3.1).
Connections of purely impulse systems.
Formulas for calculating -transfer functions of standard connections of purely impulsive systems by z-transfer functions of connected purely impulsive elements coincide with similar formulas from the theory of continuous systems. This coincidence occurs because the structure of formula (3.9) coincides with the structure of a similar formula from the theory of continuous systems; formula (3.9) describes the operation of a purely impulsive system exactly.
Example. Find the z-transfer function of a purely impulse system given by the block diagram (Fig. 3.2).
Taking into account (3.9) from the block diagram shown in fig. 3.2, we get:
Substitute the last expression into the first:
(compare with the well-known formula from the theory of continuous systems).
Connections of impulse systems.
Example 3.2. Let the impulse system be represented by a block diagram (see Fig. .3.3, without taking into account the dotted line and dash-dotted line). Then
If you need to determine the discrete values of the output (see the fictitious synchronous key at the output - dotted line in Fig. 3.3), then in a way similar to that used in the derivation of (3.7), we get the connection:
Let's consider another system (Fig. 3.4, excluding the dotted line), which differs from the previous one only in the location of the key. For her
With a fictitious key (see dotted line in Fig. 3.4)
From the relations obtained in this example, conclusions can be drawn.
Conclusion 1. The type of analytical connection of the input as with continuous [see. (3.10), (3.12)], and with discrete ones [see (3.11), (3.13)] values of the output of an arbitrary impulsive system essentially depends on the location of the switch.
Conclusion 2. For an arbitrary impulse system, as well as for the simplest one, which is described in 3.1, it is not possible to obtain a characteristic similar to the transfer function that connects the input and output at all times. It is not possible to obtain a similar characteristic that connects the input and output and at discrete times multiple of , which was done for the simplest impulsive system (see § 3.1). This is evident from relations (3.10), (3.12) and (3.11), (3.13), respectively.
Conclusion 3. For some particular cases of connections of impulse systems, for example, for an impulse system, the block diagram of which is shown in fig. 3.5 (without the dotted line), it is possible to find the transfer function connecting the input and output at discrete times that are multiples of . Indeed, from (3.10) follows But then [see derivation of formula (3.7)]
The connection structure of the z-transfer function of open and closed systems in this case is the same as in the theory of continuous systems.
It should be noted that although this is a special case, it is of great practical importance, since many systems from the class of pulse servo systems are reduced to it.
Conclusion 4. To obtain a convenient expression similar to the z-transfer function in the case of an arbitrary impulsive system (see, for example, Fig. 3.3), it is required to introduce synchronous fictitious keys not only at the output of the system (see the dashed line in Fig. 3.3), but also at its other points (see, for example, the dash-dotted section instead of the solid one in Fig. 3.3). Then
and formulas (3.10), (3.11) will take the following form, respectively:
and hence
The consequences of introducing the keys shown in fig. 3.3 dash-dotted line and dotted line are significantly different, since the latter does not change the nature of the operation of the entire system, it simply provides information about it at discrete times.
The first one, converting the continuous signal that enters the feedback link into a pulse, turns the original system into a completely different one. This new system will be able to represent the operation of the original system quite well, if accepted (see § 5.4) and if
1) the conditions of the Kotelnikov theorem (2.20) are satisfied;
2) the bandwidth of the feedback link is less:
where is the cutoff frequency of the feedback link;
3) the amplitude frequency response (AFC) of the link in the region of the cutoff frequency decreases quite steeply (see Fig. 3.6).
Then only that part of the pulse signal spectrum that corresponds to a continuous signal passes through the feedback link.
Thus, formula (3.16) in the general case only approximately represents the operation of the original system even at discrete times. Moreover, it does this the more accurately, the more reliably the conditions (2.20), (3.17) and the conditions for a steep drop in the amplitude-frequency characteristic for the link, the normal operation of which is violated by a fictitious key, are met.
So, using the z-transform, you can accurately investigate the operation of a purely impulsive system; using the Laplace transform - to accurately investigate the operation of a continuous system.
Impulse system with the help of one (any) of these transformations can be studied only approximately, and even then under certain conditions. The reason for this is the presence in the pulse system of both continuous and pulse signals (therefore, such pulse systems are continuous-pulse and are sometimes called continuous-discrete). In this regard, the Laplace transform, which is convenient when operating with continuous signals, becomes inconvenient when it comes to discrete signals. The z-transform, which is convenient for discrete signals, is inconvenient for continuous ones.
So in this case, the one noted in the aporias appears)