The purpose of the story is to show what is the essence of the concept of "signal", what common signals exist and what their general characteristics are.
What is a signal? To this question, even a small child will say that this is "such a thing with the help of which you can communicate something." For example, using a mirror and the sun, signals can be transmitted over a line-of-sight distance. On ships, signals were once transmitted using semaphore flags. Specially trained signalmen were engaged in this. Thus, with the help of such flags, information was transmitted. Here's how to convey the word "signal":
There are many signals in nature. In fact, anything can be a signal: a note left on the table, some sound - can serve as a signal to start a certain action.
Okay, with such signals, everything is clear, so I will turn to electrical signals, which in nature are no less than any others. But at least they can be somehow conventionally divided into groups: triangular, sinusoidal, rectangular, sawtooth, single impulse, etc. All of these signals are named for the way they look when plotted on a chart.
The signals can be used as a metronome for counting beats (as a timing signal), for timing, as control pulses, for controlling motors or for testing equipment and transmitting information.
Characteristics of el. signals
In a sense, an electrical signal is a graph that reflects the change in voltage or current over time. What does it mean in Russian: if you take a pencil and mark the time on the X axis, and the voltage or current on the Y axis, and mark the corresponding voltage values \u200b\u200bat specific times with dots, then the final image will show the waveform:
There are a lot of electrical signals, but they can be divided into two large groups:
- Unidirectional
- Bidirectional
Those. in unidirectional current flows in one direction (or does not flow at all), and in bidirectional current is variable and flows either "there", then "here".
All signals, regardless of type, have the following characteristics:
- Period - the period of time after which the signal begins to repeat itself. Most often denoted by T
- Frequency - indicates how many times the signal will be repeated in 1 second. Measured in hertz. For example 1Hz \u003d 1 repetition per second. Frequency is the inverse of the period (ƒ \u003d 1 / T)
- Amplitude - measured in volts or amperes (depending on which signal: current or voltage). Amplitude refers to the "strength" of the signal. How far the signal graph deviates from the X-axis.
Signal types
Sinusoid
I think that presenting a function whose graph in the picture above makes no sense is well known to you sin (x).Its period is 360 o or 2pi radians (2pi radians \u003d 360 o).
And if you divide to divide 1 sec by the period T, then you will find out how many periods are indicated in 1 sec, or, in other words, how often the period repeats. That is, you will determine the frequency of the signal! By the way, it is indicated in hertz. 1 Hz \u003d 1 sec / 1 rep per sec
Frequency and period are opposite to each other. The longer the period, the lower the frequency and vice versa. The relationship between frequency and period is expressed by simple ratios:
Signals that resemble rectangles in shape are called "rectangular signals". They can be conditionally divided into simply rectangular signals and meanders. A square wave is a square wave with the same pulse and pause durations. And if we add up the duration of the pause and the pulse, we get the period of the meander.
A regular square wave signal differs from a square wave in that it has different pulse and pause durations (no pulse). See the picture below - she speaks better than a thousand words.
By the way, there are two more terms for square-wave signals that you should know. They are inverse to each other (like period and frequency). it fabledness and filling factor.The load factor (S) is equal to the ratio of the period to the pulse duration and vice versa for coeff. filling.
Thus, a square wave is a rectangular signal with a duty cycle equal to 2. Since its period is twice the pulse duration.
S - duty cycle, D - duty cycle, T - pulse period, - pulse duration.
By the way, the graphs above show ideal square wave signals. In real life, they look slightly different, since in no device can the signal change absolutely instantly from 0 to some value and back down to zero.
If you go up the mountain, and then immediately go down and write down the change in the height of our position on the graph, we will get a triangular signal. A crude comparison, but true. In triangular signals, the voltage (current) first rises and then immediately begins to decrease. And for a classic triangle signal, the rise time is equal to the decay time (and is equal to half the period).
If such a signal has a rise time less or more than a decay time, then such signals are already called sawtooth. And about them below.
Sawtooth signal
As I wrote above, an unbalanced triangle waveform is called a sawtooth waveform. All these names are conditional and are needed just for convenience.
Let's classify the signals. Signals are divided into:
deterministic;
random.
Deterministic signals are signals that are precisely defined at any time. In contrast, some parameters of random signals cannot be predicted in advance.
Strictly speaking, since the issuance of a particular message by a message source (for example, a sensor) is random, it is impossible to accurately predict the change in the values \u200b\u200bof the signal parameters. Consequently, the signal is fundamentally random. Deterministic signals have a very limited independent meaning only for the purpose of setting up and adjusting information and computing technologyplaying the role of standards.
Depending on the structure of the parameters, signals are divided into:
discrete;
continuous;
discrete continuous.
A signal is considered discrete for a given parameter if the number of values \u200b\u200bthat this parameter can take is finite (countable). Otherwise, the signal is considered continuous for this parameter. A signal that is discrete in one parameter and continuous in another is called discrete-continuous.
In accordance with this, the following types of signals are distinguished (Fig. 1.4.):
a) Continuous in level and time (analog) are signals at the output of microphones, temperature sensors, pressure, etc.
b) Continuous in level, but discrete in time. Such signals are obtained by time sampling of analog signals.
Figure: 1.4. Varieties of signals.
Sampling means the transformation of a continuous time function (in particular a continuous signal) into a discrete time function representing a sequence of quantities called coordinates, samples or samples (sample value).
The most widespread method is discretization, in which the role of coordinates is played by instantaneous values \u200b\u200bof a continuous function (signal) taken at certain points in time S (t i), where i \u003d 1,…, n. The time intervals between these moments are called sample intervals. This type of sampling is often referred to as pulse amplitude modulation (PAM).
c) Discrete in level, continuous in time. Such signals are obtained from continuous ones as a result of level quantization.
By level quantization (or simply quantization) is meant the transformation of a quantity with a continuous scale of values \u200b\u200b(for example, the signal amplitude) into a quantity with a discrete scale of values.
This continuous scale of values \u200b\u200bis divided into 2m + 1 intervals, called quantization steps. Of the set of instantaneous values \u200b\u200bbelonging to the j-th quantization step, only one value S j is allowed, it is called the j-th quantization levels. Quantization is reduced to replacing any instantaneous value of a continuous signal with one of a finite set of quantization levels (usually the closest one):
S j, where j \u003d -m, -m + 1, ..., -1,0,1, ..., m.
The set of S j values \u200b\u200bforms a discrete scale of quantization levels. If this scale is uniform, i.e. the difference ΔS j \u003d S j - S j-1 is constant, quantization is called uniform. Otherwise, it will be uneven. Due to the simplicity of technical implementation, uniform quantization has become the most widespread.
d) Discrete in level and time. Such signals are obtained by sampling and quantizing simultaneously. These signals are easy to represent in digital form (digital sample), i.e. in the form of numbers with a finite number of digits, replacing each pulse with a number denoting the number of the quantization level that the pulse reached at a particular time. For this reason, these signals are often referred to as digital signals.
The impetus for the presentation of continuous signals in discrete (digital) form was the need to classify speech signals during World War II. An even greater incentive to digital transformation of continuous signals was the creation of computers, which are used as a source or receiver of signals in many information transmission systems.
Let's give examples of digital transformation of continuous signals. For example, in digital telephone systems (standard G.711), the replacement of an analog signal with a sequence of samples occurs with a frequency of 2F \u003d 8000 Hz, T d \u003d 125 μs. (Since the frequency range of the telephone signal is 300-3400 Hz, and the sampling frequency according to the Nyquist -Kotelnikova must be at least twice the maximum frequency of the converted signal F). Further, each pulse is replaced in an 8-bit analog-to-digital converter (ADC - ADC-Analog-to-Digital Converter) with a binary code that takes into account the sign and amplitude of the sample (256 quantization levels). This quantization process is called Pulse Code Modulation (PCM or PCM). In this case, a non-linear quantization law called "A \u003d 87.6" is used, which better takes into account the nature of human perception of speech signals. The transmission speed of one telephone message turns out to be 8 × 8000 \u003d 64 Kbps. A 30-channel telephone messaging system (the first level of the hierarchy of the CCITT standard - PDH-E1) with time division of channels already operates at a speed of 2048 Kbit / s.
When digital music is recorded on a CD (Compact Disk), containing a maximum of 74 minutes of stereo sound, a sampling frequency of 2F≈44.1 KHz is used (since the hearing limit of the human ear is 20 kHz plus 10% margin) bit linear quantization of each sample (65536 levels of the audio signal, 7-8 bits are enough for speech).
The use of discrete (digital) signals dramatically reduces the likelihood of obtaining distorted information, because:
in this case, efficient coding techniques are applicable that provide error detection and correction (see Topic 6);
it is possible to avoid the effect of accumulation of distortions inherent in a continuous signal during their transmission and processing, since the quantized signal can be easily restored to its original level whenever the amount of accumulated distortion approaches half the quantization step.
In addition, in this case, the processing and storage of information can be carried out by means of computer technology.
Analog, discrete and digital signals
One of the development trends modern systems communication is the widespread use of discrete-analog and digital signal processing (DAO and DSP).
The analog signal Z '(t), originally used in radio engineering, can be represented as a continuous graph (Fig. 2.10a). Analog signals include AM, FM, FM signals, telemetry sensor signals, etc. Devices in which analog signals are processed are called analog processing devices. Such devices include frequency converters, various amplifiers, LC filters, etc.
Optimal reception of analog signals, as a rule, provides for an optimal linear filtering algorithm, which is relevant especially when using complex noise-like signals. However, it is in this case that the construction of a matched filter is very difficult. When using matched filters based on multi-tap delay lines (magnetostrictive, quartz, etc.), large attenuation, dimensions and delay instability are obtained. Filters based on surface acoustic waves (SAW) are promising, but the short durations of the signals processed in them and the complexity of tuning the filter parameters limit their scope.
In the 1940s, analog RES were replaced by discrete processing devices for analog input processes. These devices provide discrete analog processing (DAO) of signals and have great capabilities. The signal is discrete in time, continuous in states. Such a signal Z '(kT) is a sequence of pulses with amplitudes equal to the values \u200b\u200bof the analog signal Z' (t) at discrete moments of time t \u003d kT, where k \u003d 0,1,2, ... are integers. The transition from a continuous signal Z '(t) to a pulse train Z' (kT) is called time sampling.
Figure 2.10 Analog, discrete and digital signals
Figure 2.11 Sampling the analog signal
The sampling of the analog signal in time can be performed by the "AND" coincidence stage (Fig. 2.11), at the input of which the analog signal Z '(t) acts. The coincidence cascade is controlled by the clock voltage UT (t) - short pulses of duration tp, following at intervals T \u003e\u003e tp.
The sampling interval T is selected in accordance with the Kotelnikov theorem T \u003d 1 / 2Fmax, where Fmax is the maximum frequency in the analog signal spectrum. The frequency fd \u003d 1 / T is called the sampling rate, and the set of signal values \u200b\u200bat 0, T, 2T, ... - a signal with amplitude-pulse modulation (AMM).
Until the late 1950s, PAM signals were used only for converting speech signals. For transmission over the radio relay channel, the AIM signal is converted into a phase-pulse modulation (PPM) signal. In this case, the amplitude of the pulses is constant, and the information about the speech message is contained in the deviation (phase) Dt of the pulse relative to a certain average position. Using short pulses of one signal, and placing pulses of other signals between them, multi-channel communication is obtained (but not more than 60 channels).
Currently, DAO is intensively developing on the basis of the use of "fire chains" (PC) and devices with charging connections (CCD).
In the early 70s, pulse-code modulation (PCM) systems began to appear on communication networks in various countries and the USSR, where signals in digital form were used.
The PCM process is a conversion of an analog signal into numbers, it consists of three operations: time sampling at intervals T (Figure 2.10, b), level quantization (Figure 2.10, c) and encoding (Figure 2.10, e). The time sampling operation is discussed above. The level quantization operation consists in the fact that a sequence of pulses, the amplitudes of which correspond to the values \u200b\u200bof the analog signal 3 at discrete times, is replaced by a sequence of pulses, the amplitudes of which can take only a limited number of fixed values. This operation leads to a quantization error (Figure 2.10, d).
Signal ZKV '(kT) is a discrete signal both in time and in states. Possible values \u200b\u200bu0, u1, ..., uN-1 of the signal Z '(kT) on the receiving side are known, therefore, not the values \u200b\u200bof uk, which the signal received on the interval T, are transmitted, but only its level number k. On the receiving side, according to the received number k, the value uk is restored. In this case, sequences of numbers in the binary number system - code words - are subject to transmission.
The encoding process is to transform the quantized signal Z '(kT) into a sequence of codewords (x (kT)). In fig. 2.10, d depicts code words in the form of a sequence of binary code combinations using three bits.
The considered PCM operations are used in DSP with DSP, while PCM is necessary not only for analog signals, but also for digital ones.
Let us show the need for PCM when receiving digital signals over a radio channel. So, when transmitting in the decameter range, the element xxxxxxxxxxxxxxxxxxxxxxа of the digital signal xi (kT) (i \u003d 0,1), reflecting the n-th code element, the expected signal at the input of the RFP together with the additive noise ξ (t) can be represented as:
z / i (t) \u003d μx (kT) + ξ (t), (2.2)
at (0 ≤ t ≥ TE),
where μ is the channel transmission coefficient, TE is the duration of the signal element. From (2.2) it can be seen that the interference at the input of the radio control system forms a set of signals, which are analog oscillations.
Examples of digital circuits are logic gates, registers, flip-flops, counters, memory devices, etc. According to the number of nodes on ICs and LSIs, RPUs with DSP are divided into two groups:
1. Analog-digital RPU, which have separate units implemented on the IC: frequency synthesizer, filters, demodulator, AGC, etc.
2. Digital radio receivers (TsRPU), in which the signal is processed after an analog-to-digital converter (ADC).
In fig. 2.12 shows the elements of the main (information channel) of the DAC of the decameter range: the analog part of the receiving path (AFCT), the ADC (consisting of a sampler, quantizer and encoder), the digital part of the receiving path (DAC), a digital-to-analog converter (DAC) and a lower filter frequencies (LPF). Double lines indicate the transmission of digital signals (codes), and single lines - analog and PAM signals.
Figure 2.12 Elements of the main (information channel) TsRPU decameter range
AFCT produces preliminary frequency selectivity, significant amplification and transformation of the Z '(T) signal in frequency. The ADC converts the analog signal Z '(T) into a digital signal x (kT) (Fig. 2.10, e).
In CHPT, as a rule, additional frequency conversion, selectivity (in the digital filter - basic selectivity) and digital demodulation of analog and discrete messages (frequency, relative phase and amplitude telegraphy) are performed. At the output of the TsCHPT, we obtain a digital signal y (kT) (Fig. 2.10, e). This signal, processed according to a given algorithm, from the output of the CHPT enters the DAC or the computer memory (when receiving data).
In a series-connected DAC and a low-pass filter, the digital signal y (kT) is converted first into a continuous in time and discrete in states signal y (t), and then in yF (t), which is continuous in time and in states (Fig. 2.10, g , h).
Digital filtering and demodulation are the most important of the many digital signal processing methods in the digital control center. Let's consider the algorithms and structure of a digital filter (DF) and a digital demodulator (CD).
A digital filter is a discrete system (physical device or computer program). It converts the sequence of numeric samples (x (kT)) of the input signal to the sequence (y (kT)) of the output signal.
The main CF algorithms are: linear difference equation, discrete convolution equation, operator transfer function in the z-plane, and frequency response.
The equations that describe the sequences of numbers (pulses) at the input and output of a digital filter (discrete system with a delay) are called linear difference equations.
The linear difference equation of the recursive CF has the form:
, (2.3)
where x [(k-m) T] and y [(k-n) T] are the values \u200b\u200bof the input and output sequences of numeric samples at times (k-m) T and (k-n) T, respectively; m and n - the number of delayed summed up previous input and output numeric samples, respectively;
a0, a1,…, am and b1, b2,…, bn are real weight coefficients.
In (3), the first term is a linear difference equation of a nonrecursive CF. The discrete convolution equation for CF is obtained from a linear difference non-recursive CF by replacing al in it with h (lT):
, (2.4)
where h (lT) is the impulse response of the DF, which is the response to a single impulse.
The operator transfer function is the ratio of the Laplace-transformed functions at the output and input of the CF:
, (2.5)
This function is obtained directly from the difference equations using the discrete Laplace transform and the displacement theorem.
A discrete Laplace transform, for example, a sequence (x (kT)), is understood as obtaining an L-image of the form
, (2.6)
where p \u003d s + jw is the complex Laplace operator.
The theorem of displacement (shift) in relation to discrete functions can be formulated: the displacement of the independent variable of the original in time by ± mT corresponds to the multiplication of the L-image by. For instance,
Taking into account the linearity properties of the discrete Laplace transform and the displacement theorem, the output sequence of numbers of the non-recursive CF will take the form
, (2.8)
Then the operator transfer function of the non-recursive CF:
, (2.9)
Figure 2.13
Similarly, taking into account formula (2.3), we obtain the operator transfer function of the recursive CF:
, (2.10)
The formulas for operator transfer functions are complex. Therefore, great difficulties arise in the study of fields and poles (roots of Fig. 2.13 of the polynomial of the numerator and the roots of the polynomial of the denominator), which in the p-plane have a periodic structure in frequency.
The analysis and synthesis of CFs is simplified by applying the z-transformation, when we pass to a new complex variable z, related to p by the relation z \u003d epT or z-1 \u003d e-pT. Here the complex plane p \u003d s + jw is displayed by another complex plane z \u003d x + jy. This requires that es + jw \u003d x + jy. In fig. 2.13 shows the complex planes p and z.
Making the change of variables e-pT \u003d z-1 in (2.9) and (2.10), we obtain the transfer functions in the z-plane for the non-recursive and recursive CFs, respectively:
, (2.11)
, (2.12)
The transfer function of a non-recursive CF has only zeros, so it is absolutely stable. A recursive CF will be stable if its poles are located inside the unit circle of the z-plane.
The transfer function of the CF in the form of a polynomial in negative powers of the variable z makes it possible to draw up a structural diagram of the CF directly from the form of the function HTS (z). The variable z-1 is called the unit delay operator, and in structural diagrams it is the delay element. Therefore, the highest powers of the numerator and denominator of the transfer function HTS (z) rivers determine the number of delay elements in the non-recursive and recursive parts of the DF, respectively.
The frequency response of the DF is obtained directly from its transfer function in the z-plane by replacing z with ejl (or z-1 with e-jl) and performing the necessary transformations. Therefore, the frequency response can be written as:
, (2.13)
where KC (l) is the amplitude-frequency (AFC), and φ (l) is the phase-frequency characteristics of the DF; l \u003d 2 f '- digital frequency; f '\u003d f / fD - relative frequency; f is the cyclic frequency.
The characteristic KC (jl) of the DF is a periodic function of the digital frequency l with a period of 2 (or one in relative frequencies). Indeed, ejl ± jn2 \u003d ejl ± jn2 \u003d ejl, since by Euler's formula ejn2 \u003d cosn2 + jsinn2 \u003d 1.
Figure 2.14 Block diagram of the oscillatory circuit
In radio engineering, for analog signal processing, the simplest frequency filter is the LC oscillatory circuit. Let us show that in digital processing the simplest frequency filter is a second-order recursive link whose transfer function in the z-plane is
, (2.14)
and the block diagram has the form shown in Fig. 2.14. Here the operator Z-1 is a discrete delay element for one clock cycle of the DF, the lines with arrows denote multiplication by a0, b2, and b1, "block +" denotes an adder.
To simplify the analysis, in expression (2.14) we take a0 \u003d 1, presenting it in positive powers of z, we obtain
, (2.15)
The transfer function of the digital resonator, as well as the oscillatory LC-circuit, depends only on the parameters of the circuit. Role L, C, R perform the coefficients b1 and b2.
It can be seen from (2.15) that the transfer function of the second-order recursive link has in the z-plane a zero of second multiplicity (at the points z \u003d 0) and two poles
and 
The equation for the frequency response of the second-order recursive link is obtained from (2.14), replacing z-1 with e-jl (for a0 \u003d 1):
, (2.16)
The frequency response is equal to the modulus (2.16):
After the elementary transformations... The frequency response of the second-order recursive link will take the form:
Figure 2.15 Graph of a second order recursive link
In fig. 2.15 shows the graphs in accordance with (2.18) for b1 \u003d 0. It can be seen from the graphs that the second-order recursive link is a narrow-band electoral system, i.e. digital resonator. Shown here only the working section of the frequency range of the resonator f '<0,5. Далее характери-стики повторяются с интервалом fД
Research shows that the resonant frequency f0 'will take on the following values:
f0 '\u003d fD / 4 when b1 \u003d 0;
f0 ’
f0 ’\u003e fД / 4 at b1<0.
The values \u200b\u200bb1 and b2 change both the resonant frequency and the Q-factor of the resonator. If b1 is chosen from the condition
, where, then b1 and b2 will only affect the quality factor (f0 '\u003d const). Tuning of the resonator frequency can be provided by changing fD.
Digital demodulator
A digital demodulator is considered in general communication theory as a computing device that processes a mixture of signal and interference.
Let us define the algorithms for the CD when processing analog signals AM and FM with a high signal-to-noise ratio. To do this, we represent the complex envelope Z / (t) of a narrow-band analog mixture of signal and interference Z ’(t) at the output of the AChPT in exponential and algebraic form:
and
, (2.20)
is the envelope and full phase of the mixture, and ZC (t) and ZS (t) are the quadrature components.
From (2.20) it is seen that the envelope of the signal Z (t) contains complete information about the modulation law. Therefore, the digital algorithm for processing the analog AM signal in the CD using the quadrature components XC (kT) and XS (kT) of the digital signal x (kT) has the form:
It is known that the frequency of a signal is the first derivative of its phase, i.e.
, (2.22)
Then from (2.20) and (2.22) it follows:
, (2.23)
Figure 2.16 Block diagram of the CHPT
Using in (2.23) the quadrature components XC (kT) b XS (kT) of the digital signal x (kT) and replacing the derivatives with the first differences, we obtain a digital algorithm for processing the analog FM signal in the CD:
In fig. 2.16 shows a variant of the block diagram of the CChPT when receiving analog signals AM and FM, which consists of a quadrature converter (QC) and a CD.
In the QP, the quadrature components of the complex digital signal are formed by multiplying the signal x (kT) by two sequences (cos (2πf 1 kT)) and (sin (2πf 1 kT)), where f1 is the center frequency of the lowest frequency display of the signal spectrum z '(t ). At the output of the multipliers, digital low-pass filters (LPF) suppress harmonics with a frequency of 2f1 and extract digital samples of the quadrature components. Here, LPFs are used as a digital basic selectivity filter. The block diagram of the CD corresponds to algorithms (2.21) and (2.24).
The considered algorithms for digital signal processing can be implemented using a hardware method (using specialized computers based on digital ICs, devices with a charging connection, or devices based on surface acoustic waves) and in the form of computer programs.
In the software implementation of the signal processing algorithm, the computer performs arithmetic operations on the coefficients al, bl and the variables x (kT), y (kT) stored in it.
Previously, the disadvantages of computational methods were: limited performance, the presence of specific errors, the need for resettlement, great complexity and cost. Currently, these limitations are being successfully overcome.
The advantages of digital signal processing devices over analog ones are perfect algorithms associated with training and signal adaptation, ease of control of characteristics, high temporal and temperature stability of parameters, high accuracy and the possibility of simultaneous and independent processing of several signals.
Simple and complex signals. Signal base
The characteristics (parameters) of communication systems improved as the types of signals and their methods of receiving, processing (separating) were mastered. Each time there was a need for a competent distribution of a limited frequency resource between operating radio stations. Parallel to this, the issue of reducing the emission bandwidth by signals was being addressed. However, there were problems when receiving signals, which were not solved by simple distribution of the frequency resource. Only the use of a statistical method of signal processing - correlation analysis - allowed these problems to be solved.
Simple signals have a signal base
BS \u003d TS * ∆FS≈1, (2.25)
where TS is the signal duration; ∆FS is the spectrum width of a simple signal.
Communication systems operating on simple signals are called narrowband. For complex (composite, noise-like) signals, additional modulation (keying) in frequency or phase occurs during the signal duration TS. Therefore, the following relationship is applied here for the base of a complex signal:
BSS \u003d TS * ∆FSS \u003e\u003e 1, (2.26)
where ∆FSS is the spectrum width of the complex signal.
It is sometimes said that for simple signals ∆FS \u003d 1 / TS is the message spectrum. For complex signals, the signal spectrum expands by a factor of ∆FSS / ∆FS. This results in redundancy in the signal spectrum, which determines the useful properties of complex signals. If, in a communication system with complex signals, the information transmission rate is increased to obtain the duration of the complex signal TS \u003d 1 / ∆FSS, then a simple signal and a narrow-band communication system are formed again. The useful properties of the communication system disappear.
Ways to spread the signal spectrum
The discrete and digital signals discussed above are time division signals.
Let's get acquainted with wideband digital signals and with methods of multiple access with code (in shape) channel division.
Broadband signals were originally used in military and satellite communications because of their useful properties. Here, their high immunity from interference and secrecy were used. The communication system with broadband signals can work when energy interception of the signal is impossible, and eavesdropping without a signal sample and without special equipment is impossible even with a received signal.
Shannon suggested using pieces of white thermal noise as a carrier of information and a method of broadband transmission. He introduced the concept of the bandwidth of a communication channel. He showed the connection between the possibility of error-free transmission of information with a given ratio and the frequency band occupied by the signal.
The first communication system with complex signals from segments of white thermal noise was proposed by Costas. In the Soviet Union, L.E. Varakin suggested using broadband signals when the code division multiple access method is implemented.
To temporarily represent any variant of a complex signal, you can write the ratio:
where UI (t) and (t) are the envelope and initial phases, which are slowly varying
Functions compared to cosω 0 t; - carrier frequency.
With the frequency representation of the signal, its generalized spectral form has the form
, (2.28)
where are coordinate functions; - expansion coefficients.
Coordinate functions must satisfy the orthogonality condition
, (2.29)
and the expansion coefficients
(2.30)
For parallel complex signals, the coordinate functions were initially used trigonometric functions multiple frequencies
, (2.31)
when everyone i-th option complex signal has the form
Z i (t) \u003d 
t .
(2.32)
Then, taking
A ki = and \u003d - arktg (β ki / ki), (2.33)
Ki, βki - coefficients of expansion in the trigonometric Fourier series of the i-th signal;
i \u003d 1,2,3, ..., m; m is the base of the code, we get
Z i (t) \u003d 
t .
(2.34)
Here, the signal components occupy frequencies from ki1 / 2π \u003d ki1 / TS to ki2 / 2π \u003d ki2 / TS; ki1 \u003d min (ki1) and ki2 \u003d max (ki2); ki1 and ki2 - numbers of the smallest and largest harmonic components, which significantly affect the formation of the i-th signal variant; Ni \u003d ki2 - ki1 + 1 is the number of harmonic components of the complex i-th signal.
Signal bandwidth
∆FSS \u003d (ki2 - ki1 + 1) ω 0 / 2π \u003d (ki2 - ki1 + 1) / TS. (2.35)
It contains the main part of the signal energy spectrum.
It follows from relation (35) that the base of this signal
BSS \u003d TS ∙ ∆FSS \u003d (ki2 - ki1 + 1) \u003d Ni, (2.36)
is equal to the number of harmonic components of the signal Ni, which are formed by the i-th version of the signal
Figure 2.17
b) 
Figure 2.18 Signal spreading scheme with periodic sequence graph
Since 1996-1997, for commercial purposes, Qualcomm began to use for the formation of parallel complex signals based on (28) a subset (φ k (t)) of full Walsh functions orthogonalized on an interval. At the same time, the method of multiple access with code division multiplexing is implemented - the CDMA standard (Code Division Multiple Access)
Figure 2.19 Correlation receiver circuit
Useful properties of wideband (composite) signals
Figure 2.20
When communicating with mobile stations (MS), multipath (multipath) signal propagation is manifested. Therefore, signal interference is possible, which leads to the appearance in the spatial distribution electromagnetic field deep dips (signal fading). So in urban conditions at the receiving point there can be only multiple reflected signals from high-rise buildings, hills, etc., if there is no line of sight. Therefore, two signals with a frequency of 937.5 MHz (l \u003d 32cm), arriving with a time shift of 0.5 ns with a path difference of 16cm, are added in antiphase.
The signal level at the input of the receiver also changes from the transport passing by the station.
Narrowband communication systems cannot operate in multipath conditions. So if at the input of such a system there are three signal beams of one message Si (t) –Si1 (t), Si2 (t), Si3 (t), which overlap in time due to the difference in the length of the path, then they are divided at the output of the strip filter (Yi1 (t), Yi2 (t), Yi3 (t)) is impossible.
Communication systems with complex signals resist the multipath nature of radio propagation. So, choosing the ∆FSS band such that the duration of the folded pulse at the output of the correlation detector or matched filter is less than the delay time of adjacent beams, one can receive one beam or, providing appropriate pulse delays (Gi (t)), add up their energy, which will increase the ratio signal / noise. The American communication system Rake, like a rake, collected the received beams of the signal reflected from the Moon and summarized them.
The principle of signal accumulation can significantly improve the noise immunity and other properties of the signal. The concept of signal accumulation is given by a simple repetition of the signal.
The first element for this purpose was a frequency selective system (filter).
Correlation analysis allows to determine the statistical relationship (dependence) between the received signal and the reference signal located on the receiving side. The concept of the correlation function was introduced by Taylor in 1920. The correlation function is a statistical mean of the second order over time, or a spectral mean, or a probabilistic mean.
If time functions (continuous sequences) x (t) and y (t) have arithmetic mean values
With time division of channels;
Code division multiplexing.
The periodic function is:
f (t) \u003d f (t + kT), (2.40)
where T is the period, k is any integer (k \u003d, 2,…). Periodicity exists along the entire time axis (-< t <+ ). При этом на любом отрезке времени равном T будет полное описание сигнала.
Figure 2.10, a, b, c shows a periodic harmonic signal u1 (t) and its spectrum of amplitudes and phases.
Figure 2.11, a, b, c shows graphs of a periodic signal u2 (t) - a sequence of rectangular pulses and its spectrum of amplitudes and phases.
So, any signals can be represented as a Fourier series for a certain period of time. Then the separation of signals will be represented through the parameters of the signals, i.e., through the amplitudes, frequencies, and phase shifts:
a) signals, the rows of which with arbitrary amplitudes, non-overlapping frequencies and arbitrary phases are separated in frequency;
b) signals whose rows with arbitrary amplitudes overlap in frequency, but phase-shifted between the corresponding components of the rows are separated in phase (the phase shift is proportional to the frequency);
The high capacity of communication systems with composite signals will be shown below.
c) signals, the rows of which with arbitrary amplitudes, with components overlapping in frequency (frequencies may coincide) and arbitrary phases are separated in shape.
Shape separation is a code separation when there are complex signals (samples) specially created from simple signals on the transmitting and receiving sides.
Upon reception, a complex signal is first subject to correlation processing, and then
a simple signal is being processed.
Sharing the frequency resource with multiple access
Nowadays, signals can be transmitted in any medium (in the surrounding space, in a wire, in a fiber-optic cable, etc.). To increase the efficiency of the frequency spectrum, and for one and the transmission lines form group channels for transmitting signals over one communication line. On the receiving side, the opposite process takes place - channel separation. Let's consider the methods of channel separation used:
Figure 2.21 Frequency Division Multiple Access FDMA
Figure 2.22 Time Division Multiple Access TDMA.
Figure 2.23 Code Division Multiple Access CDMA
Encryption in wi-fi networks
Data encryption in wireless networks has received so much attention because of the very nature of wireless networks. Data is transmitted wirelessly using radio waves, and in general, omnidirectional antennas are used. Thus, the data is heard by everyone - not only the one to whom it is intended, but also the neighbor who lives behind the wall or "interested" who stops with a laptop under the window. Of course, the distances over which wireless networks operate (no amplifiers or directional antennas) are short - about 100 meters under ideal conditions. Walls, trees, and other obstructions dull the signal a lot, but that still doesn't solve the problem.
Initially, only the SSID (network name) was used for security. But, generally speaking, this method can be called protection with a big stretch - the SSID is transmitted in clear text and no one bothers the attacker to eavesdrop on it, and then substitute the desired one in their settings. Not to mention that (this applies to access points), the broadcast mode for the SSID can be enabled, i.e. it will be forcibly broadcast to all listeners.
Therefore, there was a need for data encryption. The first such standard was WEP - Wired Equivalent Privacy. Encryption is performed using a 40 or 104 bit key (stream encryption using the RC4 algorithm on a static key). And the key itself is a set of ASCII characters with a length of 5 (for a 40-bit) or 13 (for a 104-bit key) characters. The set of these characters is translated into a sequence of hexadecimal digits, which are the key. Drivers from many manufacturers allow hexadecimal values \u200b\u200b(of the same length) to be entered directly instead of ASCII characters. Please note that the algorithms for translating from an ASCII sequence of characters into hexadecimal key values \u200b\u200bmay differ from manufacturer to manufacturer. Therefore, if your network uses dissimilar wireless equipment and you cannot set up WEP encryption using an ASCII key phrase, try entering the key in hexadecimal instead.
But what about the manufacturers' statements about support for 64 and 128-bit encryption, you ask? That's right, marketing plays a role here - 64 is more than 40, and 128 is 104. In reality, data encryption takes place using a key of length 40 or 104. But besides the ASCII phrase (the static component of the key) there is also such a thing as Initialization Vector - IV Is the initialization vector. It serves to randomize the rest of the key. The vector is selected randomly and dynamically changes during operation. In principle, this is a reasonable solution, since it allows you to introduce a random component into the key. The vector is 24 bits long, so the total key length is 64 (40 + 24) or 128 (104 + 24) bits.
All would be good, but the encryption algorithm used (RC4) is currently not particularly strong - if you want to, you can brute force the key in a relatively short time. Still, the main vulnerability of WEP is related to the initialization vector. The IV is only 24 bits long. This gives us roughly 16 million combinations - 16 million different vectors. Although the figure "16 million" sounds pretty impressive, everything in the world is relative. In real work, all possible key variants will be used in the interval from ten minutes to several hours (for a 40-bit key). After that, the vectors will start repeating. An attacker only needs to collect a sufficient number of packets by simply listening to the wireless network traffic and find these repeats. After that, the selection of a static c