Fourier images - complex coefficients of the Fourier series F(jw k) periodic signal (1) and spectral density F(jw) non-periodic signal (2) - have a number of common properties.
1. Linearity . Integrals (1) and (2) perform a linear transformation of the function f(t). Therefore, the Fourier image of a linear combination of functions is equal to a similar linear combination of their images. If a f(t) = a 1 f 1 (t) + a 2 f 2 (t), then F(jw) \u003d a 1 F 1 (jw) + a 2 F 2 (jw), where F 1 (jw) and F 2 (jw) - Fourier images of signals f 1 (t) and f 2 (t), respectively.
2. Delay (change the time reference for periodic functions) . Consider the signal f 2 (t), delayed for a time t 0 relative to signal f 1 (t), which has the same form: f 2 (t) = f 1 (t – t 0). If the signal f 1 has an image F 1 (jw), then the Fourier image of the signal f 2 equals F 2 (jw) \u003d \u003d . After multiplying and dividing by, we group the members as follows:
Since the last integral is F 1 (jw), then F 2 (jw) \u003d e -j w t 0 F 1 (jw) . Thus, when the signal is delayed for a time t 0 (change in the time origin), the modulus of its spectral density does not change, and the argument decreases by the value w t 0 proportional to the delay time. Therefore, the amplitudes of the signal spectrum do not depend on the origin, and the initial phases with a delay of t 0 decrease by w t 0 .
3. Symmetry . For valid f(t) image F(jw) has conjugate symmetry: F(– jw) \u003d . If a f(t) is an even function, then Im F(jw) \u003d 0; for an odd function Re F(jw) \u003d 0. Module | F(jw) | and the real part Re F(jw) - even frequency functions, argument arg F(jw) and Im F(jw) are odd.
4. Differentiation . From the direct transformation formula, integrating by parts, we obtain the connection between the image of the signal derivative f(t) with the image of the signal itself
For an absolutely integrable function f(t) outside the integral term is equal to zero, and, therefore, at, and the last integral represents the Fourier image of the original signal F(jw) . Therefore, the Fourier image of the derivative df/dt associated with the image of the signal itself by the ratio jw F(jw) - when differentiating the signal, its Fourier image is multiplied by jw. The same relation is valid for the coefficients F(jw k), which are determined by integration within finite limits from - T/ 2 to + T/2. Indeed, the product within the appropriate limits
Since due to the periodicity of the function f(T/2) = f(– T/ 2), a \u003d \u003d \u003d (- 1) k, then in this case the term outside the integral vanishes, and the formula
where the arrow symbolically denotes the operation of the direct Fourier transform. This relationship is generalized to multiple differentiation: for n-th derivative we have: d n f/dt n (jw) n F(jw).
The formulas obtained make it possible to find the Fourier image of the derivatives of a function from its known spectrum. It is also convenient to apply these formulas in cases when, as a result of differentiation, we arrive at a function, the Fourier image of which is calculated more simply. So if f(t) is a piecewise linear function, then its derivative df/dtis piecewise constant, and for it the integral of the direct transformation is found elementary. To obtain the spectral characteristics of the integral of the function f(t) its image should be divided into jw.
5. The duality of time and frequency . Comparison of the integrals of the direct and inverse Fourier transforms leads to the conclusion about their peculiar symmetry, which becomes more obvious if the inverse transformation formula is rewritten by transferring the factor 2p to the left side of the equality:
For signal f(t), which is an even function of time f(– t) = f(t) when the spectral density F(jw) is a real value F(jw) \u003d F(w), both integrals can be rewritten in trigonometric form by the Fourier cosine transform:
When interchangeable t and w, the integrals of the direct and inverse transformations transform into each other. Hence it follows that if F(w) represents the spectral density of an even function of time f(t), then the function 2p f(w) is the spectral density of the signal F(t). For odd functions f(t) [f(t) = – f(t)] spectral density F(jw) purely imaginary [ F(jw) \u003d jF(w)]. In this case, the Fourier integrals are reduced to the form of sine transforms, from which it follows that if the spectral density jF(w) corresponds to an odd function f(t), then the quantity j2p f(w) represents the spectral density of the signal F(t). Thus, the graphs of the time dependence of the signals of the indicated classes and its spectral density are dual to each other.
Integral (1)
Integral (2)
Spectral and temporal representation of signals is widely used in radio engineering. Although signals are inherently random processes, however, individual implementations of a random process and some special (for example, measurement) signals can be considered deterministic (that is, known) functions. The latter are usually divided into periodic and non-periodic, although strictly periodic signals do not exist. A signal is called periodic if it satisfies the condition
on a time interval, where T is a constant value called a period, and k is any integer.
The simplest example of a periodic signal is a harmonic oscillation (or harmonic for short).
where is the amplitude, \u003d is the frequency, is the angular frequency, is the initial phase of the harmonic.
The importance of the concept of harmonics for the theory and practice of radio engineering is explained by a number of reasons:
- harmonic signals retain their shape and frequency when passing through stationary linear electrical circuits (for example, filters), changing only the amplitude and phase;
- harmonic signals can be easily generated (eg with LC auto-generators).
A non-periodic signal is a signal that is nonzero at a finite time interval. A non-periodic signal can be considered as a periodic one, but with an infinitely large period. One of the main characteristics of a non-periodic signal is its spectrum. The signal spectrum is a function that shows the dependence of the intensity of various harmonics in the signal, on the frequency of these harmonics. The spectrum of a periodic signal is the dependence of the coefficients of the Fourier series on the frequency of the harmonics to which these coefficients correspond. For a non-periodic signal, the spectrum is the direct Fourier transform of the signal. So, the spectrum of a periodic signal is a discrete spectrum (discrete function of frequency), while a non-periodic signal is characterized by a continuous spectrum (continuous) spectrum.
Note that the discrete and continuous spectra have different dimensions. The discrete spectrum has the same dimension as the signal, while the dimension of the continuous spectrum is equal to the ratio of the signal dimension to the frequency dimension. If, for example, a signal is represented by an electrical voltage, then the discrete spectrum will be measured in volts [V], and the continuous spectrum in volts per hertz [V / Hz]. Therefore, the term "spectral density" is also used for the continuous spectrum.
Let us first consider the spectral representation of periodic signals. It is known from the course of mathematics that any periodic function satisfying the Dirichlet conditions (one of the necessary conditions is that the energy is finite) can be represented by a Fourier series in trigonometric form:
where determines the average value of the signal over the period and is called the constant component. The frequency is called the fundamental frequency of the signal (the frequency of the first harmonic), and its multiples are called the higher harmonics. Expression (3) can be represented as:
The inverse dependences for the coefficients a and b have the form
Figure 1 shows a typical graph of the spectrum of the amplitudes of a periodic signal for the trigonometric form of the series (6):
Using an expression (Euler's formula).
instead of (6), we can write the complex form of the Fourier series:
where the coefficient is called the complex amplitudes of the harmonics, the values \u200b\u200bof which, as follows from (4) and the Euler formula, are determined by the expression:
Comparing (6) and (9), we note that when using the complex form of writing the Fourier series, negative values \u200b\u200bof k make it possible to speak about components with "negative frequencies". However, the appearance of negative frequencies is of a formal nature and is associated with the use of a complex notation to represent a valid signal.
Then instead of (9) we get:
has the dimension [amplitude / hertz] and shows the signal amplitude per 1 Hertz bandwidth. Therefore, this continuous function of the frequency S (jw) is called the spectral density of the complex amplitudes or simply the spectral density. Let us note one important circumstance. Comparing expressions (10) and (11), we note that for w \u003d kwo they differ only by a constant factor, and
those. complex amplitudes of a periodic function with a period T can be determined from the spectral characteristic of a non-periodic function of the same shape, specified in the interval. The above is also true for the spectral density modulus:
From this relation it follows that the envelope of the continuous amplitude spectrum of the non-periodic signal and the envelope of the amplitudes of the line spectrum of the periodic signal coincide in shape and differ only in scale. Let us now calculate the energy of the non-periodic signal. Multiplying both sides of inequality (14) by s (t) and integrating in infinite limits, we obtain:
where S (jw) and S (-jw) are complex conjugate quantities. Because
This expression is called Parseval's equality for a non-periodic signal. It determines the total energy of the signal. Hence it follows that there is nothing more than the signal energy per 1 Hz of the frequency band around the frequency w. Therefore, the function is sometimes called the spectral energy density of the signal s (t). We now present, without proof, several theorems on spectra that express the basic properties of the Fourier transform.
General remarks
Among the various systems of orthogonal functions that can be used as bases for the presentation of radio engineering signals, harmonic (sinusoidal and cosine) functions occupy an exceptional place. The importance of harmonic signals for radio engineering is due to a number of reasons.
In radio engineering, one has to deal with electrical signals that are associated with the transmitted messages by the adopted coding method.
We can say that an electrical signal is a physical (electrical) process that carries information. The amount of information that can be transmitted using a certain signal depends on its main parameters: duration, frequency band, power and some other characteristics. The level of interference in the communication channel is also important: the lower this level, the more information can be transmitted using a signal with a given power. Before talking about the informational capabilities of the signal, you need to familiarize yourself with its main characteristics. It is advisable to consider separately deterministic and random signals.
Any signal is called deterministic, the instantaneous value of which at any moment of time can be predicted with a probability of one.
Examples of deterministic signals are pulses or bursts of pulses, the shape, magnitude and position in time of which are known, as well as a continuous signal with specified amplitude and phase relationships within its spectrum. Deterministic signals can be divided into periodic and non-periodic.
Any signal for which the condition is satisfied is called periodic
where the period T is a finite segment, and k is any integer.
The simplest periodic deterministic signal is a harmonic oscillation. Strictly harmonic vibration is called monochromatic. This term borrowed from optics emphasizes that the spectrum of a harmonic vibration consists of one spectral line. Real signals that have a beginning and an end inevitably blur the spectrum. Therefore, there is no strictly monochromatic vibration in nature. In what follows, a harmonic and monochromatic signal will conventionally mean oscillation. Any complex periodic signal, as you know, can be represented as a sum of harmonic oscillations with frequencies that are multiples of the fundamental frequency w \u003d 2 * Pi / T. The main characteristic of a complex periodic signal is its spectral function, which contains information about the amplitudes and phases of individual harmonics.
Any deterministic signal for which the condition s (t) s (t + kT) is satisfied is called a non-periodic deterministic signal.
Typically, a non-periodic signal is time limited. Examples of such signals are the already mentioned pulses, bursts of pulses, “scraps” of harmonic oscillations, etc. Non-periodic signals are of main interest, since they are mainly used in practice.
The main characteristic of a non-periodic, like a periodic signal, is its spectral function;
Random signals include signals whose values \u200b\u200bare not known in advance and can be predicted only with a certain probability less than one. Such functions are, for example, the electric voltage corresponding to speech, music, the sequence of characters of the telegraph code when transmitting a non-repeating text. Random signals also include a sequence of radio pulses at the input of a radar receiver, when the amplitudes of the pulses and the phases of their high-frequency filling fluctuate due to changes in propagation conditions, target position, and some other reasons. There are many other examples of random signals. Essentially, any signal that carries information should be considered random. The listed deterministic signals, "fully known", no longer contain information. In the following, such signals will often be referred to as "wobble".
A statistical approach is used to characterize and analyze random signals. The main characteristics of random signals are:
a) the law of probability distribution.
b) spectral distribution of signal power.
On the basis of the first characteristic, it is possible to find the relative residence time of the signal value in a certain interval of levels, the ratio of the maximum values \u200b\u200bto the rms value and a number of other important signal parameters. The second characteristic gives only the frequency distribution of the average signal power. The spectral characteristic of the random process does not provide more detailed information on the individual components of the spectrum - about their amplitudes and phases.
Along with useful random signals, in theory and practice, one has to deal with random interference - noise. As mentioned above, the noise level is the main factor limiting the information transfer rate for a given signal.
Periodic signals, naturally, do not exist, since any real signal has a beginning and an end. However, when analyzing signals in a steady state, one can proceed from the assumption that they exist for an infinitely long time and take a periodic function of time as a mathematical model of such signals. Further, the presentation of such functions is considered, both as a sum of exponential components, and with their transformation into harmonic ones.
Let the function u (t), given in the time interval and satisfying the Dirichlet conditions, repeat with the period T \u003d 2 / \u003d t 2 -t 1 during the time from - to +.
Dirichlet conditions: on any finite interval, the function must be continuous or have a finite number of discontinuity points of the first kind, as well as a finite number of extreme points. At the points of discontinuity, the function u (t) should be considered equal.
If exponential functions are chosen as basic functions, then expression (1.5) can be written in the form
Relation (1.15) is a Fourier series in complex form, containing exponential functions with both positive and negative parameters? (two-way frequency representation). Components with negative frequencies are a consequence of the complex form of writing a real function.
The function A (jk? 1) is usually called the complex spectrum of the periodic signal u (t). This spectrum is discrete, since the function A (jk? 1) is defined on the numerical axis only for integer values \u200b\u200bk. The value of the function A (jk? 1) for a specific k is called the complex amplitude.
The envelope of the complex spectrum A (j?) Has the form
We write the complex spectrum in the form
The modulus of the complex spectrum A (k? 1) is called the spectrum of amplitudes, and the function? (K? 1) is called the spectrum of phases.
If the spectrum of the amplitudes and the spectrum of the phases of the signal are known, then in accordance with (1.15) it can be uniquely reconstructed. In practical applications, the amplitude spectrum is more significant, and information on the phases of the components is often insignificant.
Since A (k? 1) and? (K? 1) differ from zero only for integer k, the spectra of the amplitudes and phases of the periodic signal are discrete.
Using Euler's formula e - jk? t \u003d cosk? t - j sink? t, we express the complex spectrum A (jk? 1) in the form of real and imaginary parts:
The amplitude spectrum is an even function of k, i.e.
Since the parity of A k and B k is opposite, the spectrum of the phases is an odd function, i.e.
For k \u003d 0, we obtain the constant component
It is easy to move from a two-sided spectral representation to a one-sided one (having no negative frequencies), combining complex conjugate components [see. (1.14)]. In this case, we get the Fourier series in trigonometric form. Indeed, separating in (1.15) the constant component A 0/2 and summing up the components of symmetric frequencies? and -?, we have
Taking into account relations (1.15) and (1.16), we write
Using Euler's formula (1.14) and denoting? (K? 1) by? k, we finally get
Another trigonometric form of the Fourier series is also widespread, which has the form
However, it is less convenient for practical use. Individual components in representations (1.23) and (1.24) are called harmonics. It is convenient to visualize both the amplitude spectrum and the spectrum of the phases of a periodic signal by spectral diagrams. On the amplitude spectrum diagram, each harmonic is assigned a vertical segment, the length of which is proportional to the amplitude, and the location on the abscissa axis corresponds to the frequency of this component. Similarly, in the phase spectrum diagram, the values \u200b\u200bof the harmonic phases are indicated. Since the resulting spectra are displayed as collections of lines, they are often called ruled.
Note that the discrete (line) spectrum does not have to belong to a periodic signal. The spectrum of a periodic signal characterizes a set of harmonics that are multiples of the fundamental frequency ??. Line spectra, including harmonics of non-multiple frequencies, belong to the so-called almost periodic signals. The spectrum diagram of the amplitudes of the periodic signal is shown in Fig. 1.4. The envelope A (t) of this amplitude spectrum can be obtained by replacing k? 1 VA (k? 1) on ?, where? \u003d k? 1 for the kth harmonic.
Example 1.1. Determine the spectra of amplitudes and phases of a periodic sequence of rectangular pulses with duration? and amplitude u 0, following with frequency? 1 \u003d 2? /? (fig. 1.5).
The function u (t) describing such a sequence of pulses on a period can be specified in the form:
In accordance with (1.16), we have either
The amplitudes of the harmonics, including the constant component equal to A 0/2, can be determined from the expression at k \u003d O, 1, 2, ....
The choice of the time reference does not affect their value. The amplitude spectrum envelope is determined by the type of function
When? \u003d 0 we get
The character of the amplitude change is dictated by the sin x / x function and does not depend on the pulse repetition rate. At frequencies that are multiples of 2? /?, The envelope of the spectrum is zero.
In fig. 1.6 shows a diagram of the amplitude spectrum for the case
? /? \u003d 3 [? 1 \u003d 2? / (3?)]. The number of components in the spectrum is infinitely large. The steepness of the pulse fronts is due to the presence in the spectrum of components with frequencies significantly higher than the fundamental frequency? 1 .
Based on formula (1.29) and taking into account that the signs of the function sin (k? 1/2) on the sequence of frequency intervals ?? \u003d 2? /? alternate, we write the expression for the spectrum of phases as follows:
where n is the number of the frequency interval? \u003d 2? /?, Counted from? \u003d 0.
The spectrum of the phases depends on the choice of the origin. If the leading edge of the rectangular pulse of the sequence falls at the origin of the time reference, then at each interval ?? \u003d 2? /? the phases of the components increase linearly. The diagram of the spectrum of the phases of a sequence of rectangular pulses for this case (? /? \u003d 3, t1 \u003d 0) is shown in Fig. 1.7.
Example 1.2. Calculate the first few terms of the Fourier series for a periodic sequence of rectangular pulses and trace how their gum converges to the specified signal.
Let us use the results of the previous example for the case of a widely used in practice periodic sequence of pulses, for which the duration? is equal to half of the period T. We also take t 1 \u003d 0.
By the formula (1.32) we will determine the constant component, and by the formulas (1.30) and (1.33) - the amplitudes and phases of the first five harmonics. The calculation data are summarized in table. 1.1. Even harmonics in tab. 1.1 are not specified as they are equal to zero.
Table 1.1
Summing up the indicated components, we obtain a sequence of pulses (Fig. 1.8), which differ from rectangular ones, mainly by insufficiently high steepness of the fronts.
Note that the steepness of the pulse fronts is due to the presence in their spectrum of components with frequencies that are many times higher than the fundamental frequency.
The shape of the amplitude-frequency characteristic is nothing more than a spectral image of a decaying sinusoidal signal. In addition, as is known, the amplitude-frequency transmission characteristic of a single electric oscillatory circuit has a similar shape.
The relationship between the shape of the amplitude-frequency characteristics of certain devices and the properties of the signal is studied in the foundations of theoretical electrical engineering and theoretical radio engineering. In short, what should interest us now from this is the following.
The amplitude-frequency characteristic of the oscillatory circuit in its outlines coincides with the image of the frequency spectrum of the signal that occurs when the shock excitation of this oscillatory circuit. To illustrate this point, Fig. 1-3 is shown, which shows a damped sinusoid that occurs when an impact is applied to an oscillatory circuit. This signal is given in time aboutm ( and) and spectral ( b) image.
Figure: 1-3
According to the branch of mathematics called spectral-temporal transformations, the spectral and temporal images of the same time-varying process are, as it were, synonyms, they are equivalent and identical to each other. This can be compared to the translation of the same concept from one language to another. Anyone familiar with this section of mathematics will tell you that Figures 1-3 and and 1-3 b are equivalent to each other. In addition, the spectral image of this signal obtained upon shock excitation of the oscillatory system (oscillatory circuit) is simultaneously geometrically similar to the amplitude-frequency characteristic of this circuit itself.
It is easy to see that the graph ( b) in Figure 1-3 is geometrically similar to the graph 3 See Figure 1-1. That is, seeing that as a result of measurements, a graph was obtained 3 , I immediately treated it not only as an amplitude-frequency characteristic of sound attenuation in the rocks of the roof, but also as evidence of the presence of an oscillatory system in the rock mass.
On the one hand, the presence of oscillatory systems in rocks lying in the roof of an underground mine did not raise any questions for me, because it is impossible to obtain a sinusoidal (or, in other words, harmonic) signal in other ways. On the other hand, I have never heard of the presence of oscillatory systems in the earth's thickness before.
To begin with, let us recall the definition of an oscillatory system. An oscillatory system is an object that reacts to a shock (impulse) effect with a damped harmonic signal. Or, in other words, it is an object with a mechanism for converting an impulse (shock) into a sinusoid.
The parameters of a damped sinusoidal signal are the frequency f 0 and quality factor Q , the value of which is inversely proportional to the damping coefficient. As can be seen from Fig. 1-3, both of these parameters can be determined from both the temporal and spectral images of this signal.
Spectral-temporal transformations are an independent branch of mathematics, and one of the conclusions that we must draw from knowledge of this section, as well as from the form of the amplitude-frequency characteristic of the sound conductivity of the rock mass, shown in Fig. 1-1 (curve 3), is that that according to the acoustic properties the rock mass under study showed the property of an oscillatory system.
This conclusion is quite obvious to anyone who is familiar with spectral-temporal transformations, but is categorically unacceptable for those who are professionally engaged in solid media acoustics, seismic exploration, or generally geophysics. It so happened that this material is not given in the course of training students of these specialties.
As you know, in seismic prospecting it is generally accepted that the only mechanism that determines the shape of the seismic signal is the propagation of the elastic vibration field according to the laws of geometric optics, its reflection from the boundaries lying in the earth's thickness and the interference between individual signal components. It is believed that the shape of the seismic signals is due to the nature of the interference between many small echoes, that is, reflections from many small boundaries lying in the mountain range. In addition, it is believed that any waveform can be obtained using interference.
Yes, this is all true, but the fact of the matter is that a harmonic (including harmonic decaying) signal is an exception. It is impossible to get it by interference.
A sinusoid is an elementary information brick that cannot be decomposed into simpler components, because a signal in nature does not exist simpler than a sinusoid. That is why, by the way, the Fourier series is a collection of precisely sinusoidal terms. Being an elementary, indivisible information element, a sinusoid cannot be obtained by adding (interference) any other, even simpler components.
You can get a harmonic signal in one and only way - namely, by acting on the oscillatory system. With a shock (impulse) impact on the oscillatory system, a damped sinusoid arises, and with a periodic or noise impact, an undamped sinusoid appears. And consequently, having seen that the amplitude-frequency characteristic of a certain object is geometrically similar to the spectral image of a harmonic damped signal, it is no longer possible to treat this object otherwise than to an oscillatory system.
Before taking my first measurements in the mine, I, like all other people working in the field of acoustics of solid media and seismic exploration, was convinced that there are no oscillatory systems in the rock mass and cannot be. However, having discovered such an amplitude-frequency characteristic of attenuation, I simply had no right to remain with this opinion.
Carrying out measurements similar to those described above is very laborious, and processing the results of these measurements takes a lot of time. Therefore, having seen that the rock mass is an oscillatory system by the nature of sound conductivity, I realized that a different measurement scheme should be used, which is used in the study of oscillatory systems, and which we still use to this day. According to this scheme, the source of the sounding signal is an impulse (shock) impact on the rock mass, and the receiver is a seismic receiver, specially designed for spectral seismic measurements. The scheme of indication and processing of the seismic signal makes it possible to observe it both in time and in spectral form.
Applying this measurement scheme at the same point of the underground workings as in our first measurement, we made sure that when the rock mass of the roof is hit by a shock, the signal arising in this case really has the form of a damped sinusoid, similar to that shown in Fig. 1. -3 a, and its spectral image is similar to the graph shown in Fig. 1-3 b.
Most often, the seismic signal contains not one, but several harmonic components. However, no matter how many harmonic components there are, they all arise solely due to the presence of an appropriate number of oscillatory systems.
Multiple studies of seismic signals received in a variety of conditions - both in underground workings, and on the earth's surface, and in the conditions of a sedimentary cover, and in the study of rocks of the crystalline basement - have shown that in all possible cases, signals received not as a result of the presence of oscillatory systems, and as a result of interference processes, does not exist.
1.2 Spectral characteristics of signals
The signals used in radio engineering have a rather complex structure. Describing such signals mathematically is difficult. Therefore, to simplify the procedure for analyzing signals and passing them through radio engineering circuits, a technique is used that provides for the decomposition of complex signals into a set of idealized mathematical models described by elementary functions.
Harmonic spectral analysis of periodic signals involves an expansion in a Fourier series in trigonometric functions - sines and cosines. These functions describe harmonic oscillations, which retain their shape during conversion by linear devices (only the amplitude and phase change), which allows the theory of oscillatory systems to be used to analyze the properties of radio engineering circuits.
The Fourier series can be represented as
Practical application has another form of writing the Fourier series
where is the amplitude spectrum;
- phase spectrum.
Complex Form of Fourier Series
The above formulas are used to obtain the spectral response of a periodic signal. Fourier transforms are used to obtain the spectrum of a non-periodic signal.
Direct Fourier Transform
Inverse Fourier Transform
Expressions (1.5), (1.6) are the main relations for obtaining spectral characteristics.
1.3 Properties of the Fourier transform
The formulas of the direct and inverse Fourier transforms make it possible to determine its spectral density S (jω) from the signal s (t) and, if necessary, to determine the signal s (t) from the known spectral density S (jω). The symbol s (t) ↔ S (jω) is used to indicate this correspondence between the signal and its spectrum.
Using the properties of Fourier transforms, you can determine the spectrum of the modified signal by transforming the spectrum of the original signal.
Basic properties:
1. Linearity
s 1 (t) ↔ S 1 (jω)
s n (t) ↔ S n (jω)
_____________________
We use the direct Fourier transform
Final Result
Conclusion: the direct Fourier transform is a linear operation, has the properties of homogeneity and additivity. Therefore, the spectrum of the sum of signals is equal to the sum of the spectra.
2. Spectrum of the time-shifted signal
s (t ± t 0) ↔ S c (jω)
Final Result
Conclusion: a shift of the signal in time by an amount of ± t 0 leads to a change in the phase characteristic of the spectrum by an amount of ± ωt 0. The amplitude spectrum does not change.
3. Changing the scale in time
s (αt) ↔ S m (jω)
Final Result
Conclusion: when the signal is compressed (expanded) in time by a certain number, its spectrum expands (contracts) by the same amount along the frequency axis with a proportional decrease (increase) in the amplitudes of its components.
4. Derivative spectrum
ds (t) / dt↔ S п (jω).
To determine the spectrum of the signal derivative, we take the time derivative of the right and left sides of the inverse Fourier transform:
Final Result
Conclusion: the spectrum of the signal derivative is equal to the spectrum of the original signal multiplied by jω. In this case, the amplitude spectrum changes in proportion to the change in frequency, and a constant component is added to the phase characteristic of the original signal, equal to π / 2 at ω\u003e 0 and equal to -π / 2 at ω
5. Spectrum of the integral
Take the integral of the right and left sides of the inverse Fourier transform
Comparing the result with the inverse Fourier transform, we obtain
Final Result
Conclusion: the spectrum of a signal equal to the integral of the original signal is equal to the spectrum of the original signal divided by jω. In this case, the amplitude spectrum changes in inverse proportion to the change in frequency, and a constant component is added to the phase characteristic of the original signal, equal to π / 2 at ω 0.
6. Spectrum of the product of two signals
s 1 (t) ↔ S 1 (jω)
s 2 (t) ↔ S 2 (jω)
s 1 (t) s 2 (t) ↔ S pr (jω).
Find the spectrum of the product of two signals using the inverse Fourier transform
Final Result
Conclusion: The spectrum of the product of two signals is equal to the convolution of their spectra, multiplied by a factor of 1 / (2π).
In the course of calculating the signal spectra, the properties of linearity and integral of the signal will be used.
1 .4 Classification and properties of radio circuits
In the theoretical foundations of radio engineering, an important place is occupied by methods of analysis and synthesis of various radio engineering circuits. In this case, a radio engineering circuit is understood as a set of passive and active elements connected in a certain way, ensuring the passage and functional transformation of signals. Passive elements are resistors, capacitors, inductors and means for connecting them. Active elements are transistors, vacuum tubes, power supplies and other elements capable of generating energy and increasing signal power. If there is a need to emphasize the functionality of a circuit, the term device is used instead of the term circuit. Radio circuits used for signal conversion are very diverse in their composition, structure and characteristics. In the process of their development and analytical research, various mathematical models are used that meet the requirements of adequacy and simplicity. In the general case, any radio engineering circuit can be described by a formalized relation defining the transformation of the input signal x (t) into the output y (t), which can be symbolically represented as
where T is an operator indicating the rule according to which the input signal is converted.
Thus, the set of operator T and two sets X \u003d (), Y \u003d () of signals at the input and output of the circuit can serve as a mathematical model of a radio engineering circuit so that
By the type of conversion of input signals into outputs, i.e. by the type of the operator T, the radio circuits are classified.
1. A radio engineering circuit is linear if the operator T is such that the circuit satisfies the conditions of additivity and homogeneity.
It is characteristic that the linear transformation of a signal of any shape is not accompanied by the appearance of harmonic components with new frequencies in the spectrum of the output signal, i.e. linear transformation does not lead to an enrichment of the signal spectrum.
2. A radio circuit is nonlinear if the operator T does not ensure the fulfillment of the additivity and homogeneity conditions. The functioning of such circuits is described by nonlinear differential equations, i.e. equations, at least one coefficient of which is a function of the input signal or its derivatives. Non-linear circuits do not satisfy the superposition principle. When analyzing the passage of signals through a non-linear circuit, the result is defined as the response to the signal itself. It cannot be decomposed into simpler signals. At the same time, nonlinear circuits have a very important property - to enrich the signal spectrum. This means that with nonlinear transformations in the spectrum of the output signal, harmonic components appear with frequencies that were not in the spectrum of the input signal. The appearance of components with frequencies equal to the combination of frequencies of harmonic components of the spectrum of the input signal is also possible. This property of nonlinear circuits has led to their use for solving a wide class of problems related to the generation and conversion of signals. Structurally, linear circuits contain only linear elements, which include nonlinear elements operating in a linear mode (on the linear sections of their characteristics). Linear circuits are linear amplifiers, filters, long lines, delay lines, etc. Non-linear circuits contain one or more non-linear elements. Nonlinear circuits include generators, detectors, modulators, multipliers and frequency converters, limiters, etc.
