The spectrum of a direct signal with a negative amplitude. Amplitude spectra of the phases of periodic signals. Zastosuvannya analysis Four'є up to two-world hour series
The concept of "signal" can be interpreted differently. This is a code or a sign that is transmitted to space, information, physical process. The nature of the spirit of that їх zv'yazok іz vplyvayut noise on yoga design. Spectra of signals can be classified in decalcom ways, but also by one of the most fundamental changes at the hour (constant and change). Another main classification category is frequencies. If you take a look at the hourly area of the report, among them you can name: static, quasi-static, periodic, repetition, transitional, fluctuating and chaotic. Skin from these signals may be the power of authority, as they can contribute to the relevant design decisions.
Types of signals
Static for the appointment of an unchanging stretch through the last hour. Quasi-static signifies equal fast strumu Therefore, it is necessary to work it out in the schemes of a pilot with a low drift. This type of signal cannot be blamed on radio frequencies, so that similar circuits can create a voltage equalization that does not change. For example, without interruption, hvilyove alerts from a constant amplitude.
The term "quasi-static" means "mayzhe imminent", that comes before the signal, which is supra-lingually correctly changed by the protracted hour. Vіn maє characteristics, more similar to static alerts (postyni), lower dynamic ones.
Periodic signals
These are exactly what are repeated on a regular basis. Apply periodic signals to include sinusoidal, square, sawtooth, tricot fluctuations, etc. The nature of the periodic form is indicated by those that are identical at the same points of the temporal line. In other words, if you go through the clock line exactly for one period (T), then the voltage, polarity, and directly change the shape of the flutter. For the shape of the tension, you can use the formula: V (t) \u003d V (t + T).
Signals that repeat
Quasi-periodic for nature, that may deak similarity with the periodic form of sickness. The main difference between them is shown by the path of the signal at f(t) and f(t+T), where T is the notification period. At the sight of a periodic alert, in the sounds that are repeated, the q points may not be identical, although the stinks will be more similar, just like the wild form of wheezing. Look at the details, you can take revenge either timchasov, or stable signs that vary.
Transitional signals and pulse signals
Offended, you see either a one-time podієyu, or a periodic one, in which the trivality is even short against the period of the form of a wheeze. Tse means that t1<<< t2. Если бы эти сигналы были переходными процессами, то в радиочастотных схемах намеренно генерировались бы в виде импульсов или переходного режима шума. Таким образом, из вышеизложенной информации можно сделать вывод, что фазовый спектр сигнала обеспечивает колебания во времени, которые могут быть постоянными или периодическими.
Ryadi Four'e
All continuous periodic signals can be represented by a fundamental sinusoidal wave frequency and a set of cosine harmonics, which are sumuyutsya linearly. Tsі kolivannya mіstat form zibi. An elementary sinusoidal wave is described by the formula: v = Vm sin (_t), de:
- v - mitteva amplitude.
- Vm is the peak amplitude.
- "_" – cutoff frequency.
- t - hour in seconds.
Period - tse hour between repetitions of identical pods or T = 2_/_=1/F, de F - frequency in cycles.
A number of Fur'є, which is to become a form of fluctuations, can be determined, as the value is set to її warehouse frequencies either by a bank of frequency-selective filters, or by an algorithm for digital signal processing, which is called a swedish transformation. Also, you can find a way to get started from scratch. A number of Fur'є be-yakoy forms hvili can be expressed by the formula: f(t) = a o/2+ _ n -1 .
9. Powerful transformation of Fur'є. The power of the line, change the hour, іnshі. Theorems about the spectrum of the future. Theorem about the spectrum of the integral.
10. Discrete transformation of Fur'є. Change the radio reception. Classification of the transition.
Discrete transformation of Fur'є can be taken directly from the integral transformation of discretization arguments (t k = kt, f n = nf):
S(f) =s(t) exp(-j2ft) dt, S(f n) = ts(t k) exp(-j2f n kt), (6.1.1)
s(t) =S(f) exp(j2ft) df, s(t k) = fS(f n) exp(j2nft k). (6.1.2)
Let's guess that the discretization of the function by the hour leads to the periodization of the spectrum, and the discretization of the spectrum by frequency - the periodization of the function. It should also not be forgotten that the value (6.1.1) of the numerical series S(f n) is discretization of the interruptless function S"(f) of the spectrum of the discrete function s(t k), so the value (6.1.2) of the numerical series s(t k ) є discretization of the interruptless function s "(t), and with the introduction of interruptless functions S" (f) and s "(t) for їx discrete variables, the viability S" (f) = S (f) and s "(t) = s (t)
For discrete transformations s(kt) S(nf), і function, і її the spectrum is discrete and periodic, and the numerical arrays of їх occurrences correspond to the task on the head periods T = Nt (vіd 0 to T or vіd - T / 2 to T / 2), ta 2f N = Nf (vіd -f N to f N), de N - number of vіdlіkіv, with tsomu:
f = 1/T = 1/(Nt), t = 1/2f N = 1/(Nf), tf = 1/N, N = 2Tf N . (6.1.3)
Spіvvіdnoshennia (6.1.3) є minds іnformаtsіynoї іnformаtsіyої іnоnоnіnії іnіnаіnіїї аnd frequency forms оf discrete signals. In other words: the number of variables of the function and її of the spectrum may be the same. Ale skin expressions of the complex spectrum are represented by two speech numbers i, obviously, the number of expressions of the complex spectrum is 2 times more than the number of expressions of the complex spectrum? Tse so. However uyavlennya spectrum kompleksnіy formі - not bіlsh nіzh zruchne ically mathematical uyavlennya spektralnoї funktsії, realnі vіdlіki yakoї utvoryuyutsya dodavannyam dvoh pov'yazanih Complex vіdlіkіv and Povny іnformatsіya about the range funktsії in kompleksnіy formі way tіlki in odnіy Yogo polovinі - vіdlіkah dіysnoї that uyavnoї Chastain Complex numbers in the frequency interval from 0 to f N since information of the other half of the range from 0 to -f N є received from the first half and no additional additional information is carried.
When a discrete signal is given, the argument t k is put down by the numbers of the variables k (for locking t = 1, k = 0.1,…N-1), and the transformations Fur'є are counted by the argument n (the number of the clock by the frequency) on the main periods. For N values divisible by 2:
S(f n) S n = k exp(-j2kn/N), n = -N/2,…,0,…,N/2. (6.1.4)
s(t k) s k = (1/N)S n exp(j2kn/N), k = 0,1,…,N-1. (6.1.5)
Head period of the spectrum (6.1.4) for cyclic frequencies vіd -0.5 to 0.5, for peak frequencies vіd - to . With an unpaired value of N between the leading period behind the frequency (value f N) is half a croc behind the frequency behind the variables (N / 2) i, obviously, the upper limit of the summation (6.1.5) is equal to N / 2.
In enumeration operations on the EOM, for the inclusion of negative frequency arguments (negative values of the numbers n) and the selection of identical algorithms in the direct and reverse transformation of Four's, the leading period of the spectrum is taken in the interval of 0 to 2f N (0 n n N), and the subsummation in (6.1 .5) oscillates normally from 0 to N-1. With the help of the following, which are complexly related to the S n * interval (-N,0) of the two-sided spectrum in the interval 0-2f N, it is necessary to match the S N + 1- n words (this is related to the intervals 0-2f N є examples S n and S N+1- n).
Butt: At intervals T=,N=100, tasks discrete signals s(k) =(k-i) - a direct pulse with single values at points k in 3 to 8. The signal shape is the modulus of the th spectrum in the main frequency range, calculated using the formula S( n) = s(k)exp(-j2kn/100) with numeration from -50 to +50 with a crochet for frequency, obviously, =2/100, pointing at fig. 6.1.1.
Rice. 6.1.1. Discrete signal is the module of the 1st spectrum.
On fig. 6.1.2 the initial value of the other form of the input to the head band of the spectrum is set. Regardless of the form of the representation, the spectrum is periodic, for which it is not important to reconsider, so calculate the spectrum values for a larger interval of the argument n with savings of the same croc for frequency, as shown in Fig. 6.1.3 for the original value of the spectrum.
Rice. 6.1.2. Spectrum module. Rice. 6.1.3. Spectrum module.
On fig. 6.1.4. shows the reversal of Fur'є for a discrete spectrum, not following the formula s "(k) \u003d (1/100) S (n) exp (j2 kn / 100), as it shows the periodization of the output function s (k), but the main period =( 0.99) function function will again work with the output signal (k).
Rice. 6.1.4. Zvorotne transformation Fur'є.
The reincarnations (6.1.4-6.1.5) are called discrete Four's reincarnations (DFT). For the DFT, in principle, all the power of the Four's integral transformations is fair, while maintaining the periodicity of discrete functions and spectra. Variation of the spectra of two discrete functions (when there are no longer any operations when processing signals from a frequency data, like, for example, filtering signals without a middle in a frequency form), it is necessary to remember a bunch of periodized functions from a time data (and navpacki). Such a cluster is called cyclic (div. section 6.4) and the results on the final plots of information intervals can be considered as a cluster of finite discrete functions (linear cluster).
It is possible to say that for counting the skin harmonics, N operations of complex multiplication and addition and, of course, N 2 operations for removing the DFT are necessary. With great obsyagi arrays of data, we can produce up to a hundred hours. The quickest calculation is available to the victories of the Swedish transformation of Fur'є.
Transmissions are called third-party electrical devices that are superimposed on the signal that is transmitted and make it difficult to receive. Due to the great intensity, the transition becomes practically impossible.
Classification of the shift code:
a) transfers to the mainstream radio transmitters (stations);
b) shifting from industrial installations;
c) atmospheric obstacles (thunderstorms, fall);
d) crossing, zooming in the passage of electromagnetic waves through the spheres of the atmosphere: the troposphere, the ionosphere;
e) thermal and shot noise in the elements of radio lanterns, coated with thermal electrons.
Mathematically, the signal at the input of the receiver can be detected, or at the sight of the sum of the signal transmitted to the signal and the transition, and then the transition is called additive, but just noise, otherwise, when looking at the creation of the transmitted signal, that transition is called, and even such a transition is called multiplicative. To cause a transition to significant changes in the intensity of the signal at the input of the receiver and explain such phenomena, like zavmirannya.
The presence of a crossover makes it easier to receive signals due to the high intensity of the crossover, recognizing a signal can be practically impossible. The building of the system resists the change, which is important, to bear the name arrogance.
Sounds of natural active disturbances are noises that are blamed on the radio industry of the earth's surface and space objects, robots and other radio electronic devices. The complex of entrances, investing in the change of the mutual REM transitions, is called the electromagnetic sum. This complex includes both technical and complete radio equipment, choosing a signal and the method of processing, and organizing it: frequency regulation, REM separation in space, normalization of the level of inter-smoking and side effects and other.
11. Discretization of continuous signals. Kotelnikov's theorem (for example). Understanding the Nyquist frequency. The concept of discretization interval.
Discretization of analog signals. Kotelnikov series
Be-yaké without interruption s(t), which takes the end of the hour T h, may be transmitted with sufficient accuracy by the last number N vіdlіkіv (vibration) s(nT), then. a sequence of short impulses separated by a pause.
The discretization of the update by the hour is a procedure that replaces the unresolved multiplier of the mitt’s value of the signal with its own (discrete) multiplier, in order to avenge information about the value of the uninterrupted signal at the time of the hour.
With a discrete method of transmitting an uninterrupted alert, you can spend an hour by stretching a channel to connect a session by transmitting that alert, T h to, de - trivality of the impulse, zastosovuvannogo transmission vibirki; it is possible to set up an hourly transmission by the channel of the communication of a decal call (time-hour amplification of signals).
The most simple way of discretization, which is based on the theorem of V.A. Kotelnikov, formulated for signals with an interleaved spectrum (theorem of conclusions):
the most important frequency of the spectrum function s(t) is less, lower F m , then the function s(t) is again assigned to the sequence of its values at the moment that one or the other is not more than one, lower for seconds and the order can be presented:
|
|
.Here the value means the interval between the waveforms on the axis of the hour, and
vibrating hour, 
- The value of the signal at the moment of observation.
Row (1) is called the order of Kotelnikov, and the vibrators (followers) of the signal ( s(nT)) It is sometimes called the time spectrum of the signal.
may yet have power:
a) at the point t=nT the function is older than 1, because at this point, the argument of the function is 0, and the value of її is 1;
b) at the points t=kT, Function, because the argument of the sine at these points is equal, and the sine itself is equal to zero;
c) spectral width of the function u n (nT) equal in smooth frequencies and more expensive. Tsej vysnovok zrobleno on the basis of the theorem of reciprocity of frequency and the hour of the wager the transformation of Fur'є. PFC of the spectral width is linear and long 
(similar to the theorem about the sound signal). in such a manner,
.
Time and frequency representation of the function u n (t) given in Fig.3.
The graphic interpretation of Kotelnikov's low is presented in Fig.4.
The Kotelnikov series (1) may have all the power of the narrowed Four's series with basic functions u n (nT), and therefore assign the function s(t) not only at the points of view, but at any hour.
Function orthogonality interval u n dorіvnyuє neskіchennostі. Normie Square
The coefficients of the series, which are assigned to the common formula for the Fur'є series, are equal (from parseval's victories):
later 
When the spectrum is interleaved by the signal with the highest frequency, row (1) goes down to the function s(t) for any meaning t.
How to take an interval T between smaller, lower, then the width of the spectrum of the basic function will be greater than the width of the spectrum of the signal, so the accuracy of the signal will be higher, especially in cases where the spectrum of the signal is not obscured by the frequency and I will find the frequency F m to be brought to vibirati z energy chi іnformatsiynyh mirkuvani, zalushayuchi nepravannym "tails" of the spectrum of the signal.
With an increase in the distance between vibrators (), the spectrum of the basic function becomes narrower than the spectrum of the signal, the coefficient C n will be the choice of another function s 1 (t), The spectrum of such oblezheniya frequency .
Like a trivality signal T c Kіntseva, then the swarm of yoga frequencies is strictly inconsistency, tk. Wash away the final frivolities and the unsettled smugi. However, in practice, you can choose the highest frequency so that the "tails" avenge either a small part of the energy, or weakly poured into the form of an analog signal. With such an omission N at the hour T h be alone T h /T, then. N=2F m T c. Row (1) at different times between 0 , N.
Number N sometimes called the number of steps of freedom of the signal, or base signal. To increase the base accuracy of analog signal replacement from a discrete one increases.
12. Timing and frequency characteristics of linear radio engineering lances. The concept of impulse response. The concept of transition characteristics. Understanding the input and transmission frequency characteristics.
When looking at radio technical signals, it was found that the signal can be presented both in the clock (dynamic performance) and in the frequency (spectral manifestation) areas. Obviously, when analyzing the processes of converting the signals of the lancer, it is also necessary to describe the timing and frequency characteristics.
Let's look at the timing characteristics of the linear lancets from the constant parameters. Yakshcho linear lancer zdijsnyuє reworked to the operator and a signal is sent to the input of the lancer 
If the delta function is visible (in practice, the pulse is short), then the output signal (lanceg reaction)
called impulse response lanceugs. The impulse response becomes the basis of one of the methods for analyzing signal transformation, which will be discussed below.
If you want to enter a linear lance, you need to get a signal, tobto. a signal of the form "single drop", then the output signal of the lancer
called transition characteristic.
Between an impulse and a transitional characteristic, there is an unambiguous link. Oscilki delta function (div. update 1.3):
,
then substituting this virase (5.5), we take:
At my heart, a transitional characteristic
.
(5.8)
Let's move on to looking at the frequency indicators of linear lances. Zastosuєmo to the input and output signals directly transforming Fur'є
The extension of the complex spectrum of the output signal to the complex spectrum of the input signal is called complex transfer coefficient
(5.9)
Why go out
in such a manner, operator transformation of the signal by a linear lance in the frequency domain to serve as a complex transmission coefficient.
Imagine the complex transmission coefficient of the view
de іvіdpovіdno module and argument of a complex function. The module of the complex transfer coefficient as a function of the frequency is called amplitude-frequency characteristic (frequency response), and the argument - phase-frequency characteristic (PFC). Amplitude-frequency characteristic є steam room, and the phase-frequency characteristic - unpaired frequency function.
Hours and frequency characteristics of linear lances related to themselves by the transformations of Fur'є
which is fully understood, shards of stink describe the same object - a linear lancet.
13. Analysis of the injection of deterministic signals on linear lances with constant parameters. Timchasovy, frequency, operator methods.
