Graphic representation of a series of Fourier spectrum. Practical application Fourier transformation for signals analysis. Introduction for beginners. Spectral diagram of a periodic signal
Currently, the following methods of organizing radio channels (radio technologies) are known: FDMA, TDMA, CDMA, FH-CDMA. Possible their combinations (for example, FDMA / TDMA). The time deadlines for the use of these technologies are largely coincided with the stages of the development of mobile systems. In the equipment of the mobile radiotelephone coupling of the first generation, multiple-dimensioning channels with frequency separation of channels were used (FDMA). The radio technology of FDMA has so far been successfully used in the advanced equipment of the first generation cellular communication, as well as in simpler systems of mobile radiotelephone communications with non-cellular structure. As for the mobile communication standards of the first stage, for the first radial systems, the concept of standards was not used, and the equipment differed by the names of the systems (Altai, Volvetot, Actionet, etc.). Cellular communication systems began to differ according to standards. On FDMA technology, such standards of the first generation cellular systems are based, as NMT-450, NMT-900, AMPS, TACS. In second-generation cellular communication systems, a transition to digital processing of transmitted voice messages was made, for which the radio technology of multiple access to the time separation of channels began to be used (TDMA). As a result of the transition to TDMA: the noise immunity of the radio pain increases, it became better to be better protected from listening, etc. TDMA applies in systems such as standards as GSM, D-amps (The last in the American version is often referred to as TDMA). Radio technology of multiple access with code division of CDMA channels, or in the English version of CDMA, has actively become embedded on public radio telephone networks Only the last five years. This radio technology has its advantages, because In CDMA equipment: - the efficiency of using the radio frequency spectrum 20 times higher than the radio equipment of the AMPS standard (FDMA technology) and 3 times - with GSM (TDMA technology); - significantly better than in other 2nd generation systems TDMA, quality, reliability and confidentiality of communication; - It is possible to use small-sized low-power terminals with for a long time work; - with the same distance from the base station, the radiation power of CDMA subscriber terminals is lower than 5 times with respect to the same indicator in the networks of standards based on other radio technologies; - It is possible to optimize the topology of the networks when calculating the coverage areas. CDMA technology was first implemented in the IS-95 cellular cellular equipment. According to its service capabilities, existing CDMA systems refer to second-generation cellular systems. According to statistical data of the National Telecommunications Institute (ETRI), the number of CDMA network subscribers increases for 2,000 people. In terms of the growth rate of the number of subscribers, these networks are superior to networks of other existing cellular standards, ahead of the development of cellular networks of even such a popular standard as GSM. Currently, CDMA networks have at least 30 million subscribers. The world telecommunication community is inclined to the fact that in future system of wireless access of subscriber lines (third-generation personal communication systems) CDMA will occupy a leading position. Such a conclusion was made due to the fact that CDMA technology is mostly able to ensure the fulfillment of the requirements for the equipment of the third generation IMT-2000, in particular, to ensure the exchange of information with high transmission rates. However, in future wireless access systems, it is planned to use the so-called CDMA broadband systems, where the frequency band on the channel will be at least 5 MHz (in modern CDMA systems of the second generation, the channel bar is 1.23 MHz). In the past few years, wireless communication means began to appear, which are based on the extended frequency spectrum technology with frequency jumps (FH-CDMA). This technology combines the specifics of TDMA, where there is a division of each frequency into several time intervals, and CDMA, where each transmitter uses a certain sequence of noise-like signals. This technology has found its application in systems intended for the organization of fixed communications.
Where to look for their characteristics I dick knows him
44. Presentation of periodic signals in the form of Fourier series
http://scask.ru/book_brts.php?id\u003d8.
Periodic signals and Fourier Rows
The mathematical model of the process recurring in time is the periodic signal with the following property:
Here T is a signal period.
The task is to find the spectral decomposition of such a signal.
Fourier row.
Let us set the time discussed in ch. I Ortonormated basis formed by harmonic functions with multiple frequencies;
Any function from this basis satisfies the condition of frequency (2.1). Therefore, - by performing orthogonal decomposition of the signal in this basis, i.e. computing coefficients
we get spectral decomposition
fair at all infinity of the time axis.
A series of species (2.4) is called near the Fourier of the DanRGO signal. We introduce the main frequency of the sequence forming a periodic signal. Calculating the decomposition coefficients (2.3), write a Fourier series for a periodic signal
with coefficients
(2.6)
So, in the general case, the periodic signal contains the constant constant component and an infinite set of harmonic oscillations, the so-called harmonic with frequencies to multiple the main frequency of the sequence.
Every harmonica can be described by its amplitude and initial phase for this, the coefficients of the Fourier series should be written as
Substitting these expressions in (2.5), we get another, - the equivalent form of the Fourier series:
which is sometimes more convenient.
Spectral diagram of a periodic signal.
So it is customary to call a graphic image of a Fourier series coefficients for a specific signal. The amplitude and phase spectral diagrams distinguish (Fig. 2.1).
Here, along the horizontal axis, the frequencies of the harmonics are postponed on some scale, and their amplitudes and initial phases are presented along the vertical axis.
Fig. 2.1. Spectral diagrams of some periodic signal: A - amplitude; B - Phase
Particularly interested in an amplitude diagram, which allows you to judge the percentage content of certain harmonics in the spectrum of the periodic signal.
We study several specific examples.
Example 2.1. Row Fourier periodic sequence of rectangular video pulses with known parameters even relative to the point T \u003d 0.
In radio engineering, the ratio is called the wellness of the sequence. According to formulas (2.6) we find
The final formula of the Fourier series is conveniently written in the form
In fig. 2.2 The amplitude charts of the sequence in two extreme cases are presented.
It is important to note that the sequence of short impulses, the following rarely, has a rich spectral composition.
Fig. 2.2. The amplitude spectrum of the periodic sequence of rhryaturgical video pulses: A - with high duty; B - with low duty
Example 2.2. A series of Fourier periodic sequence of pulses formed by a harmonic signal of the species limited at level (it is assumed that).
We introduce a special parameter - the cut-off angle determined from the ratio from where
In the correspondence with this, the value is equal to the duration of one impulse, expressed in angle measure:
Analytical recording of a pulse generating the sequence under consideration has the form
The constant component of the sequence
The amplitude coefficient of the first harmonic
Similarly calculate amplitudes - harmonic components when
The results are usually recorded as follows:
where the so-called Berg functions:
Graphs of some functions of Berg are shown in Fig. 2.3.
Fig. 2.3. Graphs of several first functions of Berg
Spectral density of signals. Direct and reverse Fourier transform.
Periodic signal of any shape with a period of t can be presented as a sum
harmonic oscillations with different amplitudes and initial phases, whose frequencies are multiple of the main frequency. The harmonic of this frequency is called the main or first, the rest - the highest harmonics.
Trigonometric form of a row of Fourier:
,
where 
- constant component;
- amplitudes of cosine formal components;
- amplitudes of sinusoidal components.
An even signal ( 
) It has only cosineidal, and odd ( 
- Only sinusoidal terms.
More convenient is the equivalent trigonometric shape of the Fourier series:
,
where 
- constant component;
- The amplitude of the N-th harmonic of the signal. The combination of amplitudes of harmonic components is called the range of amplitudes;
- The initial phase of the N-th harmonic of the signal. The combination of phases of harmonic components is called the phase spectrum.
The spectrum of the periodic sequence of rectangular pulses. The dependence of the spectrum on the period of the impulses and their duration. Spectrum width. Decomposition in Fourier PPPI
Calculate the amplitude and phase spectra of the PPPI with amplitude 
, Duration 
, The period of follow 
and located symmetrically relative to the start of the coordinates (signal - even function).
Figure 5.1 - Temporary diagram of PPPI.
The signal on the interval of one period can be written:
Calculations:
,
Fourier series for PPPI has the form:.
Figure 5.2 - amplitude spectral diagram of PPPI.
Figure 5.3 - Phase spectral diagram of PPPI.
The spectrum of the PPPI (discrete) (seems to be a set of individual spectral lines), harmonic (spectral lines are at the same distance from each other Ω 1), decrease (the amplitudes of harmonics decrease with the growth of their number), has a petal structure (the width of each petal is 2π / τ), unlimited (frequency interval, in which spectral lines are located, is infinite);
With integer beds, the frequency components with frequencies, multiple duty in the spectrum (their frequencies coincide with zeros of the envelope spectrum of amplitudes);
With an increase in the strength of the amplitude of all harmonic components decrease. In this case, if it is associated with an increase in the repetition period T, the spectrum becomes denser (ω 1 decreases), with a decrease in the duration of the pulse τ - the width of each petal becomes greater;
The width of the Spectrum of the PPPI adopted the frequency interval containing 95% of the signal energy, (equal to the width of the two first envelope petals):
or 
;
All harmonics that are in one envelope petal have the same phases equal to either 0 or π.
Using the Fourier transformation to analyze the spectrum of non-periodic signals. Spectrum of a single rectangular pulse. Integral Fourier transformations
Communication signals are always limited in time and therefore are not periodic. Among the non-separation signals are the greatest interest representing single impulses (OI). OI can be considered as an extreme case of a periodic sequence of pulses (PPI) duration 
With an infinitely large period of their repetition 
.
Figure 6.1 - PPP and OI.
The non-periodic signal can be represented by the sum of the infinitely large number of infinitely close to the frequency of oscillations with fadingly small amplitudes. The AI \u200b\u200bspectrum is continuous and entered by Fourier integrals:
-
(1) - Direct Fourier transformation. Allows you to analytically find the spectral function according to a given signal form;
-
(2) - Fourier reverse transformation. Allows you to analytically find the form at a given spectral signal function.
Complex form of integral transformation Fourier (2) gives a two-way spectral representation (having negative frequencies) of the non-periodic signal 
in the form of harmonic oscillations 
with infinitely small complex amplitudes 
whose frequencies are continuously filled with the entire frequency axis.
Complex spectral signal density - comprehensive function Frequencies at the same time carrying information both about amplitude and the phase of elementary harmonics.
The modulus of spectral density is called the spectral density of amplitudes. It can be considered as a leak of the solid spectrum of the non-periodic signal.
Argument of spectral density 
It is called the spectral density of the phases. It can be considered as the FCH solid spectrum of the non-periodic signal.
We transform formula (2):
Trigonometric Fourier integral conversion form Gives a unilateral spectral representation (no negative frequency) of the non-periodic signal:
.
Digital filters (lecture)
By type of impulse characteristics, digital filters are divided into two large classes:
· Filters with finite pulse characteristic (Filters, transversal filters, non-ejective filters). An denominator of the transfer function of such filters is a certain constant.
Filters are characterized by expression:
· Filters with an infinite pulse characteristic (BIX - filters, recursive filters) use one or more of their outputs as an input, that is, form feedback. The main property of such filters is that their impulse transition characteristic has an infinite length in the time domain, and the transfer function has a fractional rational view.
Bih - filters are characterized by an expression:
The difference between the filters from Bih - filters is that the kih filters, the output reaction depends on the input signals, and in Bih filters, the output reaction depends on the current value.
Pulse characteristic - This is a diagram reaction to a single signal.
E.dinich signal
Thus, the unit signal is only at one point equal to one - at the point of origin of the coordinates.
Detained E.dinich signal Determined as follows:
Thus, the delayed single signal delays on k periods of discretization.
Signals and spectra
Duality (duality) of signals.
All signals can be represented in a temporary or frequency plane.
Moreover, frequency planes are several.
Temporary plane.
Conversion.
Frequency plane.
To view the signal in the time plane there is a device:
Imagine that there is a sufficiently long sinusoidal signal (1 sec. 1000 times the sinusoid repeatedly):
Take a signal with a frequency, twice as many:
Moving these signals. We will not get a sinusoid, but a distorted signal:
Transformations from the time plane to the frequency plane are made using Fourier transformations.
To view the signal in the frequency plane there is a device:
Cyclic or circular frequency ( f.).
The frequency plane will show serif:
The magnitude of the scene is proportional to the amplitude of the sinusoid, and the frequency:
For the second signal, the frequency region will show another point:
In the time domain of the total signal, 2 serfs will appear:
Both indications of the signal are equivalent and use either the first or other representation, depending on what is more convenient.
Conversion from the time plane to the frequency plane can be made in various ways. For example: using Laplace transformations or using Fourier transformations.
Three forms of records of Fourier series.
There are three forms of records of Fourier series:
· Sinus is a cosine form.
· Real shape.
· Comprehensive form.
1.) In sinus - cosine form Fourier series has the form:
Included in the formula for multiple frequency kΩ.1 called harmonies; Harmonics are numbered in accordance with the index k.; frequency ωk \u003d.kΩ.1Name k.- Harmonic signal.
This expression indicates the following: that any periodic function can be represented as a sum of harmonics, where:
T. - a period of repetitions of this function;
ω - Circular frequency.
where
t.- current time;
T. - period.
When expanding the Fourier, the most important thing is periodicity. Due to her discretization in frequency, some number of harmonics begins.
In order to establish the possibility of trigonometric decomposition for a given periodic function, you need to proceed from a specific set of coefficients. Reception for their definition came up in the second half of the 18th century Euler and regardless of him at the beginning of the XIX century - Fourier.
Three Euler formulas to determine the coefficients:
; 
; 
The Euler formulas do not need any evidence. These formulas are accurate with infinite number of harmonics. Fourier series - truncated row, since there is no infinite number of harmonics. The coefficient of truncated row is calculated according to the same formulas as for the full range. In this case, the average quadratic error is minimal.
The harmonic power drops with an increase in their number. If add / discard some harmonic components, then recalculation of other members (other harmonics) is not required.
Almost all functions are even or odd:
Sight function
Odd function
Characterized by the equation:
For example, a function Cos.:
which: T \u003d -t
An even function is symmetrical relative to
the axes of the ordinate.
If the function is even, then all sinus coefficients bk. cosine Signed.
Characterized by the equation:
For example, a function Sin.:
An odd feature is symmetric about the center.
If the functions are odd, then all cosine coefficients aK will be zero and in the formula of the Fourier series will be present only sine Signed.
2.) Real form Records of the Fourier series.
Some inconvenience of the sine-cosine shape of a Fourier series is that for each values \u200b\u200bof the summation index k. (i.e. for each harmonic with frequency kΩ.1) The formula appears two terms - sinus and cosine. Taking advantage of the formulas of trigonometric transformations, the sum of these two terms can be transformed into the cosine of the same frequency with a different amplitude and some initial phase:
where
;
If a S.(t.) is an even function, phases φ can only take values \u200b\u200b0 and π , what if S.(t.) - the functions are odd, then possible values \u200b\u200bfor the phase φ equal + π /2.
If a bk. \u003d 0, then TG φ \u003d 0 and angle φ = 0
If a aK \u003d 0, then TG φ - endless and angle φ =
In this formula, there may be a minus (depending on which direction is taken).
3.) Comprehensive form Records of the Fourier series.
This form of submission of a number of Fourier is perhaps the most used in radio engineering. It is obtained from a real form of a cosine presentation in the form of a semi-displacement exponent (such a representation follows from the Euler formula eJθ. = Cosθ. + jsinθ.):
Applying this transformation to the real form of a series of Fourier, we obtain the amount of integrated exponentials with positive and negative indicators:
And now we will interpret the exhibitors with the "minus" sign in the indicator as members of a number with negative numbers. Within the framework of the same general approach Permanent terms a.0/2 will be a member of a number with a zero number. As a result, a comprehensive form of recording of the Fourier series will be obtained:
Formula for calculating coefficients Ck. Fourier series:
If a S.(t.) is an even function, row coefficients Ck.will be clean real, what if S.(t.) - Function odd, the coefficients of the series will be pure mnimami.
Aggregate Amplitude Harmonic Row Fourier is often called amplitude spectrum, and the aggregate of their phases - phase spectrum.
The spectrum of amplitudes is the actual part of the coefficients Ck. Fourier series:
Re ( Ck.) - spectrum of amplitudes.
Spectrum of rectangular signals.
Consider a signal in the form of a sequence of rectangular pulses with an amplitude A., duration τ and repetition period T.. The beginning of the countdown of time is visible located in the middle of the pulse.
This signal is an even function, so it is more convenient to use the sinus-cosine form of the Fourier series - only cosine terms will be present in it. aKequal:
From the formula, it can be seen that the duration of the pulses and the period of their follower are not separated into it, but solely as a relationship. This parameter is the ratio of the period to the pulse duration - call studacity Pulse sequences and denote letter: G: G \u003d T./ τ. We introduce this parameter to the resulting formula for the coefficients of the Fourier series, and then we give the formula to the form Sin (x) / x:
Note: In foreign literature, instead of a duty, a reverse value called the filling coefficient (Duty Cycle) and equal to τ / T..
With such a form of recording, it becomes clearly visible, which is equal to the value of the constant terms of the series: since x. → 0 sin ( x.)/x. → 1, then
Now you can record the presentation of the sequence of rectangular pulses in the form of a series of Fourier:
The amplitudes of the harmonic terms of the series depend on the harmonic number under the SIN law ( x.)/x..
SIN function graph ( x.)/x.it has a petal character. Speaking about the width of these petals, it should be emphasized that for graphs of discrete spectra of periodic signals there are two options for grading the horizontal axis - in the rooms of harmonics and in frequencies.
In the figure, the gradation of the axis corresponds to the number of harmonics, and the frequency parameters of the spectrum are applied to the chart using dimensional lines.
So, the width of the petals, measured in the amount of harmonics, is equal to the wellness of the sequence (when k. = nG. have Sin. (π k /g.) \u003d 0 if n. ≠ 0). From here it follows the important property of the spectrum of the sequence of rectangular pulses - there are no (have zero amplitudes) of harmonics with numbers, multiple duty diseases).
The distance in the frequency between adjacent harmonics is equal to the frequency of the pulses - 2 π /T.. The width of the spectrum petals measured in units of frequency is equal to 2 π /τ , i.e. inversely proportional to the pulse duration. This is a manifestation of a general law - the shorter the signal, the wider its spectrum.
Output : For any signal, its decomposition is known in the Fourier series. Knowing τ and T. We can calculate how many harmonics need to pass power.
Methods for analyzing linear systems with constant coefficients.
Task in setting:
There is a linear system (independent of the amplitude of the signal):
Coeffs: DS B0, B1, B3
…………………
Port_vvod equ y: ffc0; We define the input ports.
Port_vivod equ y: ffc1; Determine the output ports.
ORG P: 0; Organization of P-memory.
RESET: JMP Start; Unconditional transition to Start label.
P: 100; The program will start with a cell of the cell.
START: MOVE BUF_X, R0; The initial address x is introduced into R0.
Move # ORDFIL─1, M0; RESET. to mod. Arif. (Zap. number 1man. than order. This is a buff.)
Move # Coeffs, R4; Cycle organization. Buffer for coefficients. In Y-memory.
Move # M0, M4; t. K.Tlin must coincide, then Perez. From M0 to M4.
CLRA; Remove the battery.
REP # ORDFIL; Repeat the chain operation.
MOVE A, X: (R4) +; Preol. Auto-Crepe and all cells Buff. zero.
Loop: Movep y: port_vvod, x─ (r0); beat. shipment of readings (later. UMN. On b0.).
REP # ORDFIL─1; Rep. Charpecific operation (39 sorts UMN. without round)
Mac X0, Y0, A X: (R0) +, X0 Y: (R4) +, Y0; UMN. X0Na0, cut. in AK; Podg. sl. opera.
Movep a, y: port_vivod; Pull forwarding content. battery.
Jmp Loop; Unconditional transition to Loop label.
Procedure for designing digital filters.
The order of designing digital filters is primarily associated with the type of filter along the frequency characteristics line. One of the often arising in the practice of tasks is to create filters that transmit signals in a specific frequency band and delaying the remaining frequencies. There are four types:
1.) Lower Frequency Filters (FNH; English term - Low-Pass Filter) transmitting frequencies less than some cutoff frequency ω 0.
2.) Upper frequency filters (FVCH; English term - High-Pass Filter) transmitting frequencies, large slice slice ω 0.
3.) Strip filters (PF; English term - Band-Pass Filter) transmitting frequencies in some range ω 1…. ω 2 (they can also be characterized by an average frequency ω 0 = (ω 1 + ω ω = ω 2 – ω 1).
4.) Recorder Filters (Other Possible Names - Barring Filter, Filter Cork, Low-Detention Filter; English Term - Band-Stop Filter) everything frequency besides lying in some range ω 1…. ω 2 (they can also be characterized by an average frequency ω 0 = (ω 1 + ω 2) / 2 and bandwidth width Δ ω = ω 2 – ω 1).
The ideal form of frequency response filters of these four types:
However, such an ideal (rectangular) Form Ahh cannot be physically implemented. Therefore, a number of methods have been developed in the theory of analog filters. approximationrectangular frequency response.
In addition, having calculated the FGC, you can change its frequency of the cut with simple transformations, turn it into a PVCH, strip or a steam filter with specified parameters. Therefore, the calculation of the analog filter begins with the calculation of the so-called filter prototype, representing a VFC with a cutoff frequency of 1 rad / s.
1.) Batterworth filter:
The batterworth filter prototype transmission function (Butterworth Filter) does not have zeros, and its poles are uniformly located on s.-Lelosity in the left half of the circumference of a single radius.
For a batterworth filter, the cutoff frequency is determined by level 1 /. Batterworth filter provides maximum flat Top in the bandwidth.
2.) Chebyshev filter first kind:
Chebyshev filter transmission function (Chebyshev Type i Filter) also does not have zeros, and its poles are located in the left half of the ellipse on s.-Lelos. For the Chebyshev filter of the first kind, the slice frequency is determined by the level of ripples in the bandwidth.
Compared with the butterworth filter of the same order, the Chebyshev filter provides a sharper response of the ACH in the transition area from the bandwidth to the stroke of the delay.
3.) Filter Chebyshev second kind:
The second-kind Chebyshev filter transfer function (Chebyshev Type II Filter), unlike previous cases, has zeros, and poles. Chebyshev filters of the second kind are called Chebyshev inverse filters (Inverse Chebyshev Filter). The frequency of the curing filter Chebyshev second is in the end of the bandwidth, but start of stall of delay. The coefficient of the filter transmission on zero frequency is 1, on the cutting frequency - the specified level of ripples in the stroke of the delay. For ω → ∞ The transmission coefficient is zero with an odd order of the filter and the level of ripples - with even. For ω \u003d 0 Achm Filter Chebyshev The second kind is the maximum flat.
4.) Elliptical filters:
Elliptic filters (Kauer filters; English terms - Elliptic Filter, Cauer Filter) In a sense, the properties of the first and second kind filters are combined in some sense, since the ACH elliptic filter has pulsations of a given value, both in the bandwidth and in the stall of the delay. Due to this, it is possible to ensure the maximum possible (with a fixed manner of the filter) the steepness of the SCH scope, i.e. the transition zone between bandwidth and detention bands.
The transmission function of the elliptical filter has both poles and zeros. Zeros, as in the case of the Chebyshev filter of the second kind, are purely imaginary and form comprehensive-conjugate pairs. The number of zeros of the transmission function is equal to the maximum even number that does not exceed the order of the filter.
MATLAB functions for calculating batterworth filters, first and second chebyshev, as well as elliptic filters, allow you to calculate both analog and discrete filters. Filter calculation functions require tasks as input parameters of the filter order and its cut-off frequency.
The order of the filter depends:
- From permissible non-uniformity in the bandwidth from the value of the uncertainty zone. (The smaller the uncertainty zone, the steeper the decline in the frequency response).
For Qih filters, the order is several dozen or hundreds, and for BIH filters, the order does not exceed several units.
Pictograms make it possible to see all the coefficients. Filter design is made on one window.
The signal is called periodicIf its form is cyclically repeated over time. The periodic signal in general is written as follows:
Here is the signal period. Periodic signals can be both simple and complex.
For the mathematical representation of periodic signals with a period, it is often used by this next, in which harmonic (sinusoidal and cosine and cosine) oscillations are selected as basic functions:
where. - The main angular frequency of the function of functions. With harmonic basic functions, a series of Fourier receives from this series, which in the simplest case can be written in the following form:
where coefficients are
From a number of Fourier, it can be seen that in the general case a periodic signal contains a constant component and a set of harmonic oscillations of the main frequency and its harmonics with frequencies. Each harmonious oscillation of the Fourier series is characterized by amplitude and initial phase.
Spectral Chart and Periodic Signal Spectrum.
If any signal is presented in the form of the sum of harmonic oscillations with different frequencies, this means that it was carried out spectral decomposition Signal.
Spectral diagram The signal is called a graphic image of the Fourier series coefficients of this signal. There are amplitude and phase diagrams. To build these diagrams, at some scale along the horizontal axis, the values \u200b\u200bof the frequency of the harmonic are laid, and their amplitudes and phases are added along the vertical axis. Moreover, the amplitudes of harmonics can only take positive values, phases - both positive and negative values \u200b\u200bin the interval.
Spectral Periodic Signal Charts:
a) - amplitude; b) - Phase.
Signal spectrum - This is a combination of harmonic components with specific frequency values, amplitudes and initial phases forming a signal in the amount. In practice, spectral diagrams are called more briefly - amplitude spectrum, phase spectrum. The greatest interest is shown to the amplitude spectral diagram. It can be estimated by the percentage of harmonics in the spectrum.
Spectral characteristics in telecommunication techniques play a big role. Knowing the spectrum of the signal can be correctly calculated and install the bandwidth, filters, cables and other communication channel nodes. Signal spectra knowledge is necessary to build multichannel systems with frequency separation of channels. Without the knowledge of the spectrum of interference, it is difficult to take measures to suppress it.
From this we can conclude that the spectrum needs to be known to carry out a non-current signal transmission over the communication channel to ensure the separation of signals and weakening interference.
To monitor the spectra of signals there are devices that are called spectrum Analyzers. They allow you to observe and measure the parameters of the individual components of the periodic signal spectrum, as well as measure the spectral density of the continuous signal.
Often a mathematical description even uncomplicated by the structure and form of deterministic signals is a difficult task. Therefore, an original reception is used, in which real complex signals are replaced (represented by approximated) with a set (weighted amount, i.e. nearby) of mathematical models described by elementary functions. This gives an important tool for analyzing the passage of electrical signals through electronic circuits. In addition, the presentation of the signal can also be used as initial when it descriptions and analyze. At the same time, it is possible to significantly simplify the inverse task - synthesis complex signals from the set of elementary functions.
Spectral representation of periodic signals ranks Fourier
General Fourier series.
The fundamental idea of \u200b\u200bthe spectral representation of signals (functions) rises to times more than 200 years ago and belongs to physics and mathematics J. B. Fourier.
Consider the system of elementary orthogonal functions, each of which is obtained from one source - the prototype function. This feature prototype acts as a "construction block", and the desired approximation is suitable for the combination of the same blocks. Fourier showed that any complex function can be represented (approximated) in the form of a finite or infinite sum of a number of multiple harmonic oscillations with certain amplitudes, frequencies and initial phases. This function can be, in particular, current or voltage in the chain. The sunbeam, unfolded by priscious on the color spectrum, is a physical analogue of Fourier mathematical transformations (Fig. 2.7).
The light coming out of the prism is divided into space on separate clean colors, or frequencies. The spectrum has an average amplitude at each frequency. Thus, the intensity function from time was transformed into the amplitude function depending on the frequency. A simple example of Fourier reasoning illustrations is shown in Fig. 2.8. Periodic, quite complicated curved curve (Fig. 2.8, but) - This is the sum of two harmonics of different, but multiple frequencies: single (Fig. 2.8, b) and doubled (Fig. 2.8, in).
Fig. 2.7.
Fig. 2.8.
but - complex oscillation; b, 1st and 2nd approximating signals
With the help of the Fourier spectral analysis complex function It seems the sum of harmonics, each of which has its frequency, amplitude and start-up phase. Fourier transformation determines the functions representing the amplitude and phase of harmonic components corresponding to a specific frequency, and the phase is the initial point of the sinusoid.
The transformation can be obtained by two different mathematical methods, one of which is used when the initial function is continuous, and the other - when it is specified by the set of separate discrete values.
If the function under study is obtained from values \u200b\u200bwith certain discrete intervals, it can be divided into a sequential row of sinusoidal functions with discrete frequencies - from the lowest, main or main frequency, and then with frequencies twice, tripled, etc. Above the main one. Such a sum of the components and is called near Fourier.
Orthogonal signals. A convenient way to spectral descriptions of the Fourier signal is its analytical representation using the system of orthogonal elementary time functions. Let there be a Hilbert Signal Space u 0 (T) Y g /, (?), ..., u N (T) with finite energy defined on a finite or infinite time interval (T V 1 2). On this segment, we set the infinite system (subset) of interrelated elementary functions of time and call it basic. "
where r \u003d. 1, 2, 3,....
Functions u (T) and v (T) Orthogonal on the interval (?,? 2), if their scalar product, provided that none of these functions are identical to zero.
In mathematics, so define in the Hilbert Signal Space orthogonal coordinate basis. The system of orthogonal basic functions.
The property of the orthogonality of functions (signals) is associated with the interval of their definition (Fig. 2.9). For example, two harmonic signals m, (?) \u003d \u003d Sin (2NR / 7 '0) and u., (t) \u003d SIN (4 nT / T Q) (i.e., with frequencies / 0 \u003d 1/7 '0 and 2/0, respectively) orthogonal at any time interval, the duration of which is equal to an integer number of half-periods T 0. (Fig. 2.9, but). Therefore, in the first period, the signals and ((1) and u 2 (t) Orthogonal on the interval (0, 7 "0/2); but on the interval (o, ZG 0/4) they are unorthogonal. PA Fig. 2.9, b. Signals are orthogonal due to the abundance of their appearance.
Fig. 2.9.
but - on the interval; b - Due to the time of the appearance of the signal representation u (T) Elementary models are greatly simplified if the system of basic functions is selected. vFF) possessing a property orthonormality. From mathematics it is known if a condition is satisfied for any pair of functions from the orthogonal system (2.7)
then the functions system (2.7) ortonormated.
In mathematics, such a system of basic functions of the form (2.7) is called or-thin-mounted basis.
Let on the specified time interval | r, t 2. | There is an arbitrary signal u (T) And for its presentation uses an orthonormal system of functions (2.7). Designing an arbitrary signal u (T) on the axis of the coordinate basis is called decomposition into a generalized Fourier series. This decomposition has a view.
where C, - some permanent coefficients.
To determine the coefficients with K. Select one of the basic functions (2.7) v k (t) with arbitrary number to. Multiply both parts of the decomposition (2.9) on this function and integrate the result in time:
Due to the orthonormality of the basis of the selected functions in the right-hand side of this equality, all members of the amount when i. ^ to Turn into zero. Not only the only member of the number with the number will remain non-zero i. = to, so
Production of species c K V K (T), included in the generalized Fourier series (2.9), is spectral component Signal u (T), And the combination of coefficients (projections of the signal vectors on the axis of the coordinate) (from 0, s, ..., with k,..., C ") Fully determines the analyzed signal iI (T) and called it spectrum (from lat. spectrum - image).
Essence spectral representation (analysis) The signal is to determine the coefficients with I in accordance with formula (2.19).
The choice of the rational orthogonal system of the coordinate basis of functions depends on the purpose of the research and is determined by the desire of the maximum simplification of the mathematical apparatus of analysis, transformations and data processing. As basic functions, Chebyshev, Hermita, Lagerre, Lejander, and others are currently used. The greatest distribution received signals in the bases of harmonic functions: complex exponential eXP (J 2LFT) and real trigonometric sinus-cosine functions related to the formula Euler e\u003e H. \u003d COSX + Y "SINX. This is due to the fact that the harmonic oscillation theoretically preserves its form when passing through linear chains with constant parameters, and only its amplitude and the initial phase are changed. The symbolic method well developed in the chain theory is also widely used. The operation of the representation of deterministic signals in the form of a set of constant component ( cONSTANT COMPONENT) and the sum of harmonic oscillations with multiple frequencies is called called spectral decomposition. Fully common use in the theory of signals of a generalized Fourier series is also associated with its very important property: with the chosen orthonormal function system v k (t) and the fixed number of the categories of the series (2.9) it provides the best representation of the specified signal u (T). This property of Fourier series is widely known.
When the spectral view of the signals, orthonormal bases of trigonometric functions were obtained the greatest application. This is due to the following: Harmonic oscillations are most simply generated; Harmonic signals are invariant with respect to transformations carried out by stationary linear electrical circuits.
We estimate the temporary and spectral representation of the analog signal (Fig. 2.10). In fig. 2.10, but A temporary diagram of complex in the form of a continuous signal is shown, and in fig. 2.10, b - His spectral decomposition.
Consider the spectral representation of periodic signals in the form of the sum or harmonic functions, or complex exponentials with frequencies forming arithmetic progression.
Periodic call the signal and "(?). Repeating at regular intervals of time (Fig. 2.11):
where r - a period of repetition or following impulses; n \u003d 0,1, 2,....
Fig. 2.11. Periodic signal
If a T. It is a signal period u (T), then the periods will also be multiple values: 2g, 3 T. etc. Periodic sequence of pulses (they are called video pulses) Described by expression
Fig. 2.10.
but - temporary diagram; b. - amplitude spectrum
Here u Q (T) - a single impulse form characterized by amplitude (height) h \u003d e, duration t ", a period of following T \u003d. 1 / F (F - frequency), the position of pulses in time relative to the clock points, for example t \u003d. 0.
When spectally analyzes periodic signals, an orthogonal system (2.7) is convenient as harmonic functions with multiple frequencies:
where co, \u003d 2p / t- Frequency of pulses.
Computing the integrals, by formula (2.8) it is easy to ensure the orthogonality of these functions on the interval [-G / 2, g / 2 |. Any function satisfies the periodicity condition (2.11), since their frequencies are multiple. If the system (2.12) write as
we will get an orthonormal basis of harmonic functions.
Imagine a periodic signal most common in theory of signals trigonometric (sinus-cosine) form Fourier series:
From the course of mathematics it is known that decomposition (2.11) exists, i.e. A number converges if the function (in this case, the signal) u (T) At the interval [-7/2, 7/2] satisfies dirichlet conditions (Unlike the Dirichlet Theorem, they are often interpreted simplified):
- There should be no breakdowns of the 2nd kind (with branches leaving in infinity);
- The function is limited and has a finite number of gaps of the 1st genus (jumps);
- The function has a finite number of extremes (i.e. maxima and minima).
The following components of the analyzed signal are available in formula (2.13):
Permanent component
The amplitudes of cosine formal components
Amplitude sinusoidal components
The spectral component with the CO frequency in the theory of communication is called first (main) harmonica, and components with ISO frequencies, (p\u003e 1) - higher harmonies Periodic signal. Step at the frequency of ASO between two adjacent sinusoids from Fourier decomposition is called frequency resolution spectrum.
If the signal is an even time function u (T) \u003d U (-t), then in the trigonometric record of the Fourier series (2.13) there are no sinusoidal coefficients B n, since in accordance with formula (2.16) they turn to zero. For signal u (T), Described by an odd function of time, on the contrary, according to formula (2.15), zero is equal to cosine coefficients a P. (constant component a 0. also absent), and the range contains components B n.
The limits of integration (from -7/2 to 7/2) should not necessarily be such as in Formulas (2.14) - (2.16). Integration can be performed at any time interval width 7 - the result will not change from this. Specific limits are chosen due to considerations of the convenience of calculations; For example, it may be easier to continue to integrate from about 7 or from -7 to 0, etc.
Mathematics section setting the ratio between the time function u (T.) and spectral coefficients and p, b n, Call harmonic analysis due to communication function u (T) With sinusoidal and cosineidal members of this amount. Further, the spectral analysis is mainly limited by the framework of harmonic analysis, which is exceptional application.
Often, the use of the sinus-cosine shape of the Fourier series is not entirely convenient, because for each value of the summation index p (i.e., for each harmonic with the Moj frequency) in Formula (2.13), two terms are included - cosine and sinus. From a mathematical point of view, it is more convenient to present this formula equivalent to Fourier equivalent real form /.
where A 0. = a 0 /2; And n \u003d yja 2 n + B - amplitude; P-th harmonic signal. Sometimes in the ratio (2.17) before the Wedl l sign "Plus", then the initial phase of the harmonic is recorded as CP and \u003d -ARCTG ( b N Fa. n).
In the theory of signals, the complex form of the Fourier series is widely used. It is obtained from the real form of a row of a cosine representation in the form of a semi-emissible exponential exhibitor by the Euler formula:
By applying this transformation to the real form of a Fourier series (2.17), we obtain the amount of complex exponentials with positive and negative indicators:
And now we will interpret the exhibitors at the Formula (2.19) at the frequency of CO, with the "minus" sign in the indicator as members of a number with negative numbers. As part of the same approach, the coefficient A 0. It will become a member of a number with a zero number. After uncomplicated transformations come to complex form Row Fourier
Comprehensive amplitude p-d harmonics.
Values With P. on positive and negative numbers p are complex-conjugate.
Note that Fourier series (2.20) is an ensemble of complex exponential eXP (JN (O (T) With frequencies forming arithmetic progression.
We define the relationship between the coefficients of the trigonometric and complex forms of the Fourier series. It's obvious that
You can also show that the coefficients a P. \u003d 2C W COSCP "; b n \u003d 2C / I SINCP, F.
If a u (T) is an even function, the coefficients of the C, will real what if u (T) - The function is odd, the coefficients of the row will become imaginary.
The spectral representation of the periodic signal with a complex form of a series of Fourier (2.20) contains both positive and negative frequencies. But negative frequencies in nature do not exist, and this is a mathematical abstraction (the physical meaning of the negative frequency is the rotation in the direction opposite to the one that is taken for positive). They appear as a result of the formal representation of harmonic oscillations with a comprehensive form. When switching from a comprehensive form of recording (2.20) to real (2.17), the negative frequency disappears.
Clearly about the spectrum of the signal is judged but his graphic Image - spectral diagram (Fig. 2.12). Distinguish amplitude-frequencyand phase-frequency spectra. Aggregate amplitude harmonic A P. (Fig. 2.12, but) Call amplitude spectrum, their phases (Fig. 2.12, b) CP I - phase spectrum. Total With P. = |With P. is an complex amplitude spectrum (Fig. 2.12, in). On spectral diagrams, the abscissa axis is laying down the current frequency, and but the ordinate axes are either a real or complex amplitude or phase of the corresponding harmonic components of the analyzed signal.
Fig. 2.12.
but - amplitude; b - phase; in - Amusement range of complex Fourier
The spectrum of the periodic signal is called linely or discreteSince it consists of separate lines with a height equal to amplitude A P. Harmonic. Of all the types of spectra, the most informative is amplitude, since it allows you to estimate the quantitative content of certain harmonics in the frequency composition of the signal. In the theory of signals, it is proved that the amplitude spectrum is even Frequency Function, and phase - odd.
Note equidistance (Equifference from the origin) of the complex spectrum of periodic signals: symmetrical (positive and negative) frequencies on which the spectral coefficients of the trigonometric series of Fourier are located, form an equidistant sequence (... - V. ..., -2So P -So P 0, V. 2 o, ..., nCO V. ...) containing the frequency of CO \u003d 0 and having a step CO T \u003d 2L / 7 '. Coefficients can take any values.
Example 2.1
Calculate the amplitude and phase spectra of a periodic sequence of rectangular pulses with an amplitude?, Duration T and and a period of repetition T. The signal is even function (Fig. 2.13).
Fig. 2.13.
Decision
It is known that the perfect rectangular video pulse is described by the following equation:
those. It is formed as a difference between two single functions A (?) (Inclusion functions) shifted over time on t n.
The sequence of rectangular pulses is a certain amount of single pulses:
Since the specified signal is an even time function and for one period acts only on the interval [T and / 2, T and / 2], according to formula (2.14)
where q. = T / T " 
Analyzing the resulting formula, it can be noted that the period of the following and the duration of pulses are included in it in the form of a relationship. This parameter q - The ratio of the period to the duration of pulses is called studacity The periodic sequence of pulses (in foreign literature instead of duty, use the reverse value - filling coefficient, from English, duty Cycleequal to T and / 7); for q \u003d 2 The sequence of rectangular pulses, when the duration of pulses and gaps between them become equal, called meander (from Greek. PAIAV5POQ is a pattern, geometric ornament).
Due to the parity of the function describing the analyzed signal, in a number of Fourier, along with the constant component, only cosine components will be present (2.15):
In the right-hand side of formula (2.22), the second factor has the form of an elementary function (sinx) / x. In mathematics, this function is indicated as SINC (X), and only when h. \u003d 0 it is equal to one (lim (sinx / x) \u003d 1) passes
through zero at points x \u003d ± l, ± 2l, ... and fades with the growth of the argument X (Fig. 2.14). Finally trigonometric Fourier series (2.13), which approximates the specified signal, recorded in the form
Fig. 2.14. Schedule function sINX / X.
SINE function has a petal character. Speaking about the width of the petals, it should be emphasized that for graphs of discrete spectra of periodic signals there are two options for grading the horizontal axis - in the rooms of harmonic and frequencies. For example, in Fig. 2.14 Graduation of the axis of the ordinate corresponds to frequencies. The width of the petals, measured among the harmonic, is equal to the wellness of the sequence. From here it follows the important property of the spectrum of the sequence of rectangular pulses - there are no (have zero amplitudes) of harmonics with numbers, multiple duty diseases). When the pulses of the pulses, equal to three, disappears each third harmonic. If the diet would be equal to two, then only odd harmonics of the main frequency would remain in the spectrum.
From formula (2.22) and Fig. 2.14 It follows that the coefficients of a number of higher signal harmonics have a negative sign. This is due to the fact that the initial phase of these harmonics is equal p. Therefore, formula (2.22) is taken to submit in a modified form:
With such a record of the Fourier series, the amplitudes of all higher harmonic components on the graph of the spectral diagram are positive (Fig. 2.15, but).
The amplitude spectrum of the signal largely depends on the relationship of the repetition period T. and the pulse duration T and, i.e. from duty q.The distance in the frequency between adjacent harmonics is equal to the frequency of the pulses from 1 \u003d 2l / t. The width of the spectrum petals, measured in frequency units, is equal to 2nd / t n, i.e. Inversely proportional to the pulse duration. Note that with the same duration of the pulse T and with increasing
Fig. 2.15.
but - amplitude;b. - Phase
roda of their repetition T. The main frequency of CO, decreases and the spectrum becomes denser.
The same picture is observed if the pulse duration is shortened and with a constant period. T. The amplitudes of all harmonics are reduced. This is a manifestation of a general law (the principle of uncertainty V. Heisenberg - Uncertainty principle) ', The shorter the duration of the signal, the wider its spectrum.
Phases of components Determine from the formula CP P \u003d ArCTG (b n / a n). As here coefficients B " \u003d 0, then
where m \u003d. 0, 1, 2,....
The ratio (2.24) shows that when calculating the phases of spectral components are dealing with mathematical uncertainty. For its disclosures, we turn to formula (2.22), according to which the harmonic amplitudes periodically change the sign in accordance with the change of the sign of the SIN function (NCO 1 x 1i / 2). Changing the sign in formula (2.22) is equivalent to the phase shift of this function on p.Therefore, when this function is positive, the harmonic phase (p and \u003d 2 tPand when negative - \u003d (2T. + 1 )to (Fig. 2.15, b). Note that although the amplitudes of the components in the spectrum of rectangular pulses and decrease with increasing frequency (see Fig. 2.15, but), This decline is quite slow (amplitudes decrease in inversely proportionally frequency). To transfer such pulses without distortion, an infinite band of the communication channel is required. For relatively low distortions, the boundary value of the frequency band should be many times greater than the value, the inverse duration of the pulse. However, all the real channels have a finite bandwidth, which leads to distortions of the shape of the transmitted pulses.
Fourier series of arbitrary periodic signals may contain infinitely a large number of members. When calculating the spectra of such signals, the calculation of the infinite sum of the Fourier series causes certain difficulties and is not always required, therefore, limited by the summation of the final number of the terms (the series "trunks").
The accuracy of the signal approximation depends on the number of summable components. Consider this on the example of approximation by the sum of the eight first harmonics of the sequence of rectangular pulses (Fig. 2.16). The signal has a view of a unipolar meander with a period of repetition. T U. amplitude E. \u003d 1 and pulse duration T and \u003d T./ 2 (the specified signal is even - Fig. 2.16, but; Squater q. \u003d 2). Approximation is shown in Fig. 2.16, B, and the number of summable harmonics is shown on the charts. In the approximation of a given periodic signal under the approximation (see Fig. 2.13) trigonometric near (2.13), the summation of the first and higher harmonics will be carried out only by odd coefficients PU Since with even their values \u200b\u200band pulse duration T and \u003d T./ 2 \u003d \u003d TT / CO, the value of sin (Mo, T H / 2) \u003d sin (WT / 2) is vanished.
The trigonometric shape of the Fourier series (2.23) for a given signal has the view
Fig. 2.16.
but - specified signal; 6 - Intermediate stages of summation
For convenience, the Fourier series (2.25) can be written simplified:
From formula (2.26), it is obvious that harmonics, approximating meander, are odd, have alternating signs, and their amplitudes are inversely proportional to the numbers. It should be noted that the sequence of rectangular pulses is poorly suitable for the presentation near Fourier - approximation contains ripples and jumps, and the sum of any number of harmonic components with any amplitudes will always be a continuous function. Therefore, the behavior of a Fourier series in the vicinity of gaps is of particular interest. From graphs. 2.16, it is not difficult to notice, as with an increase in the number of summable harmonics, the resulting function is increasingly approaching the form of the source signal u (T) Everywhere, except for its gaps. In the surroundings of the rupture points, the summation of the Fourier series gives an inclined section, and the tilt the resultant function increases with increasing the number of summable harmonics. At the very point of the rupture (we denote it as t. = t 0) Fourier row u (T 0) It converges to the half of the right and left limits:
On the adjacent areas of the approximated curve, the sum of the row gives noticeable ripples, and in fig. 2.16 It can be seen that the amplitude of the main emission of these ripples does not decrease with the increasing number of summable harmonics - it is only compressed horizontally, approaching the break point.
For p -? At the discontinuity points, the emission amplitude remains constant,
and its width will be infinitely narrow. The relative amplitude of ripples (with respect to the amplitude of the jump), and relative damping are not changed; Only the frequency of pulsations is changed, which is determined by the frequency of the latest summable harmonics. This is due to the convergence of the Fourier series. Let us turn to the classic example: will you ever achieve the walls if you take half the remaining distance with each step? The first step will lead to half the path, the second - to the mark on three of its quarters, and after the fifth step, almost 97% of the path will pass. You almost reached the target, but no matter how many steps come forward, never reach it in a strict mathematical sense. It is only possible to prove mathematically that in the end you can get closer to any given arbitrarily distance. This proof will be equivalent to the demonstration of the amount of numbers 1/2, 1,1 / 8.1 / 16, etc. She strives for one. This phenomenon inherent in all Fourier rows for signals with the ruptures of the 1st genus (for example, jumps, as on the fronts of rectangular pulses), are called the effect of Gibbs* In this case, the value of the first (largest) emission of amplitude in the approximated curve is about 9% of the jump level (see Fig. 2.16, p = 4).
The Gibbs effect leads to a fatal error of approximation of periodic pulse signals with ruptures of the 1st genus. The effect takes place with sharp impaired monotony of functions. At jumps, the effect of maximum, in all other cases, the amplitude of pulsations depends on the nature of the monotony violation. For a number of practical applications, the Gibbs effect causes certain problems. For example, in sound-reproducing systems, this phenomenon is called "ringing" or "rat." In this case, each sharp consonant or other sudden sound may be accompanied by a short sound unpleasant to hearing.
Fourier row can be applied not only for periodic signals, but also for the end duration signals. However, it is stipulated by time
noise interval for which a series of Fourier is being built, and during the remaining moments of time the signal is considered to be zero. To calculate the coefficients of a number, this approach means periodic continuation Signal outside the interval under consideration.
Note that nature (for example, human hearing) uses the principle of harmonic analysis of signals. Virtual Fourier transformation The person makes it possible to hear the sound: the ear automatically performs it, representing the sound in the form of a spectrum of consistent volume values \u200b\u200bfor tones of different heights. The human brain turns this information into a perceived sound.
Harmonic synthesis. In the theory of signals, along with harmonic analysis, the signals are widely used harmonic synthesis - obtaining given oscillations of complex form by summing up a number of harmonic components of their spectrum. Essentially above the synthesis of a periodic sequence of rectangular pulses amount from a number of harmonics was carried out. In practice, these operations are performed on a computer, as shown in Fig. 2.16, b.
- Jean Batist Joseph Fourier (J. V. J. Fourier; 1768-1830) - French mathematician and physicist.
- Josayia Gibbs (J. Gibbs, 1839-1903) is an American physicist and mathematician, one of the founders of chemical thermodynamics and statistical physics.