Analog and digital signal. Types of signals and how it works

The concept of "information" (from lat. informatio- clarification, presentation) and "message" are currently inextricably linked.

Information Is information that is the object of transmission, distribution, transformation, storage or direct use. A message is a form of information presentation. It is known that a person receives 80 ... 90% of information through the organs of sight and 10 ... 20% through the organs of hearing. Other senses add up to 1 ... 2% of the information.

Information is transmitted in the form messages. Message - form of expression (presentation) of information, convenient for transmission over a distance. Examples of messages are telegram texts, speech, music, television images, data on the computer output, commands in the automatic control system of objects, etc. Messages are transmitted using signals that are carriers of information. The main type of signals are electrical signals. In recent years, more and more widespread are optical signals, n / a, in fiber-optic data transmission lines. Signal- the physical process displaying the transmitted message. The display of the message is provided by changing the quantity of the physical quantity characterizing the process. The signal transmits (unfolds) the message in time, that is, it is always a function of time. Signals are formed by changing certain parameters of the physical medium in accordance with the transmitted message.

This value is informational parameter of the signal.Informational parameter of the message - the parameter, in the change of which the information is "laid down". For sound messages, the informational parameter is the instantaneous value of the sound pressure, for motionless images - reflection coefficient, for mobile - the brightness of the luminescence of the areas of the screen.

In this case, the concepts quality and speed transmission of information.

The quality of information transmission is the higher, the less distortion of information on the receiving side. With an increase in the speed of information transfer, it is required to take special measures to prevent information loss and reduce the quality of information transfer.

Messaging at a distance using a material carrier, n / a, paper or magnetic tape or a physical process, for example, sound or electromagnetic waves, current, etc.

The transmission and storage of information is carried out using various signs (symbols) that allow it to be presented in some form.

Messages can be functions of time, for example, speech when transmitting telephone conversations, temperature or pressure when transmitting telemetry data, a show when broadcasting on television, etc. In other cases, the message is not a function of time (eg telegram text, still image, etc.). Signal transmits a message in time. Therefore, it is always a function of time, even if the message (for example, a still picture) is not. There are 4 types of signals: continuous signal continuous time. (Figure 2.2, a), continuous discrete time. (Figure 2.2, b), discrete continuous time. (Figure 2.2, c) and discrete discrete time (Figure 2.2, d).

Figure 2.2 - Continuous signal of continuous time (a), continuous signal of discrete time (b), discrete signal of continuous time (c), discrete signal of discrete time (d).

Continuous signals continuous time. called abbreviated continuous (analog.) signals. They can change at arbitrary moments, taking any values ​​from a continuous set of possible values ​​(sinusoid).

Continuous signals of discrete time. can take arbitrary values, but change only at certain, predetermined (discrete) moments t 1, t 2, t 3 … .

Discrete continuous time signals differ in that they can change at arbitrary moments, but their values ​​take only permitted (discrete) values.

Discrete signals of discrete time(abbreviated discrete) at discrete moments of time can only take permissive (discrete) values.

By the nature of the change in information parameters are distinguished continuous and discrete messages.

Analog the signal is a continuous or partially continuous function of time X (t). The instantaneous values ​​of the signal are analogous to the physical quantity of the process under consideration.

Discrete the signal is discrete pulses following each other with a time interval Δt, the pulse width is the same, and the level (pulse area) is an analogue of the instantaneous value of some physical quantity, which is a discrete signal.

Digital the signal is a discrete series of digits following each other with a time interval Δt, in the form of binary digits and representing the instantaneous value of some physical quantity.

A continuous or analog signal is a signal that can take any level of values ​​in a certain range of values. A time-continuous signal is a signal specified on the entire time axis.

For example, speech is a message that is continuous both in level and in time, and the temperature sensor, which issues its values ​​every 5 minutes, serves as a source of messages that are continuous in magnitude, but discrete in time.

The concept of the amount of information and the possibility of measuring it is the basis of information theory. Information theory took shape in the 20th century. Claude Shannon (USA), A.N. Kolmogorov (USSR) R. Hartley (USA) et al. According to Claude Shannon, information is a removed uncertainty. Those. informativeness of the message x-Xia contained in it useful information i.e. that part of the message that reduces the uncertainty that exists before it is received.

Lecture 1

The main types of signals and their mathematical description.

The main types of signals: analog, discrete, digital.

Analog is a signal that is continuous in time and state (Fig. 1a). The signal is described by a continuous (or piecewise continuous) function NS(t). In this case, both the argument and the function itself can take any values ​​from some intervals:

t" ≤ t ≤ t"" , x" ≤ x ≤ x"".

Discrete is a signal that is discrete in time and continuous in state (Fig. 1b). Described by a lattice function NS(n* T), where n- reference number (1,2,3, ...). Interval T is called the sampling period, and the reciprocal f d = 1 / T- sampling frequency. The lattice function is defined only at the moments of timen * T and can only in these moments take any values ​​from some interval x" ≤ x ≤ x"". The values ​​of the lattice function, and, accordingly, of the signal itself at the moments of time n* T are called counts. (A discrete signal can be both real and complex).

Digital is a signal that is discrete both in time and in state (Fig. 1c). Signals of this type are also described by lattice functions NS c ( n* T), which can take only a finite number of values ​​from a certain finite interval x" ≤ x ≤ x"". These values ​​are called quantization levels, and the corresponding functions are called quantized.

When analyzing discrete signals, it is convenient to use normalized time
, otherwise, i.e. the sample number of a discrete signal can be interpreted as a normalized time. When passing to the normalized time, the discrete signal can be considered as a function of an integer variable n... That is further NS(n) is equivalent to NS(n· T).

Frequency normalization.

According to the Kotelnikov theorem, the maximum frequency of an analog signal f there should be no more f D 2. Therefore, it is advisable to consider all discrete signals in the range. In this case, the concept is introduced normalized frequency

or

and consider the discrete signal f in the area of

or

The use of a normalized frequency makes it possible to study the frequency characteristics of discrete systems and the spectra of discrete signals in a single frequency band. For DSP, it is not the absolute values ​​of the signal frequency and sampling rate that are important, but their ratio, i.e. value of the normalized frequency.

For example, for 2 discrete cosine waves:

where

Eventually:

Their discrete signals are the same, since their normalized frequencies are equal, they, only, will be differently in time.

In the general case, a discrete cosine wave in the region of normalized frequencies has the form:

Generalized digital signal processing circuit.

The DSP process includes 3 stages:

Number Sequencer NS(n* T) from analog signal x(t) ;

Sequence transformation NS(n* T) according to a given algorithm by a digital signal processor (DSP) into a new, output numerical sequence y (n* T) ;

Formation of the resulting analog signal y(t) from sequence y(n* T).

Sampling frequency f d is selected: f d ≥ 2 f v.

Real signals do not meet this requirement. Therefore, they put a low-pass filter that limits the spectrum. Since the energy of real signals decreases with increasing frequency, the distortions introduced by the low-pass filter are insignificant (Fig. 3 a and b), as well as the spectra below:

Quantization levels(Fig. 1.c.) are encoded with binary numbers, therefore, at the output of the ADC, we have a sequence of binary numbers
... Digital signal
different from discrete
by the value:

Quantization error.

To reduce it, it is necessary to increase the number of quantization levels. The discrete signal enters the DSP, which, according to the algorithm, sets the output signal in a one-to-one correspondence with each input report
... In this case, the number of operations (multiplications, additions, inversions, transfers, etc.) to obtain one sample can be calculated as much as you like. However, the processing period (computation time) cannot be greater than the sampling period. ... And this can only be if the clock frequency f T TsPOS >> f D.

Next, the DAC generates a stepped analog signal (t), the steps of which are smoothed by a filter, obtaining an analog y(t).

Signals - carriers information in automation tools can differ both in physical nature and parameters, and in the form of information presentation. Within the framework of the GSP (State Instrument System), the following types of signals are used in the serial production of automation equipment:

Electrical signal (voltage, strength or frequency of electrical current);

Pneumatic signal (compressed air pressure);

Hydraulic signal (pressure or differential pressure of fluid).

Accordingly, within the framework of the GSP, electrical, pneumatic and hydraulic branches of automation equipment are formed.

In the form of information presentation, the signal can be analog, pulse and code.

Analog signal characterized by current changes in any physical parameter-carrier (for example, instantaneous values ​​of electric voltage or current). Such a signal exists practically at any given moment of time and can take any values ​​within the specified range of parameter changes.

Pulse signal is characterized by the presentation of information only at discrete moments in time, i.e. the presence of time quantization. In this case, information is presented in the form of a sequence of pulses of the same duration, but different amplitudes (amplitude-pulse modulation of the signal) or the same amplitude, but different durations (pulse-width modulation of the signal).

Code signal is a complex sequence of pulses used to transmit digital information. Moreover, each digit can be represented as a complex sequence of impulses, i.e. code, and the transmitted signal is discrete (quantized) both in time and in level.

Optical signal- a light wave carrying certain information. A feature of a light wave in comparison with a radio wave is that, due to its small wavelength, it can practically be transmitted, received and processed signals that are modulated not only in time, but also in spatial coordinates. This makes it possible to significantly increase the amount of information introduced into the optical signal. The optical signal is a function of four variables (x, y, z, t) - 3 coordinates and time. An electromagnetic wave is a change in time and at every point in space of electric and magnetic fields, which are interconnected according to the law of induction. An electromagnetic wave is characterized by mutually perpendicular vectors of the strengths of the electric E and magnetic H fields, which vary in time according to the same harmonic law.

Analog signal is a continuous function of a continuous argument, i.e. defined for any value of the independent variable. Sources of analog signals, as a rule, are physical processes and phenomena that are continuous in their development (dynamics of changes in the values ​​of certain properties) in time, in space or in any other independent variable, while the recorded signal is similar (analogous) to the process that generates it. An example of a mathematical notation for a specific analog signal: y(t) = 4.8exp [- ( t-4) 2 /2.8]. An example of a graphical display of this signal is shown in Fig. 1, while both the numerical values ​​of the function itself and its arguments can take any values ​​within certain intervals y£ 1 y £ y 2,t£ 1 t £ t 2. If the intervals of values ​​of the signal or its independent variables are not limited, then by default they are assumed to be equal from - ¥ to + ¥. The set of possible signal values ​​forms a continuous space in which any point can be determined with infinite precision.

Rice. 2.2.1. Signal graphical display y(t) = 4.8 exp [- ( t-4) 2 /2.8].

Discrete signal by its values ​​it is also a continuous function, but defined only by the discrete values ​​of the argument. By the set of its values, it is finite (countable) and is described by a discrete sequence y(n× D t), where y£ 1 y £ y 2, D t- interval between samples (signal sampling interval), n = 0, 1, 2, ..., N- numbering of discrete sample values. If a discrete signal is obtained by sampling an analog signal, then it is a sequence of samples, the values ​​of which are exactly equal to the values ​​of the original signal in coordinates n D t.

An example of sampling an analog signal shown in Fig. 1 is shown in Fig. 2.2.2. When D t= const (uniform data sampling) a discrete signal can be described by the abbreviated notation y(n).

With non-uniform signal sampling, the designations of discrete sequences (in text descriptions) are usually enclosed in curly braces - ( s(t i)), and the values ​​of the samples are given in the form of tables with the indication of the values ​​of the coordinates t i... For short, irregular number sequences, the following numeric description also applies: s(t i) = {a 1 , a 2 , ..., a N}, t = t 1 , t 2 , ..., t N.

Digital signal quantized in its values ​​and discrete in argument. It is described by the quantized lattice function y n = Q k[y(n D t)], where Q k- quantization function with the number of quantization levels k, while the quantization intervals can be either uniformly distributed or non-uniform, for example, logarithmic. A digital signal is set, as a rule, in the form of a numeric array based on sequential values ​​of the argument for D t = const, but, in general, the signal can be specified in the form of a table for arbitrary values ​​of the argument.

In essence, a digital signal is a formalized version of a discrete signal when the values ​​of the latter are rounded to a certain number of digits, as shown in Fig. 2.2.3. In digital systems and computers, the signal is always represented with an accuracy of up to a certain number of bits and, therefore, is always digital.Taking into account these factors, when describing digital signals, the quantization function is usually omitted (it is assumed uniform by default), and the rules for describing discrete signals are used to describe signals.

Rice. 2.2.2. Discrete signal Fig. 2.2.3. Digital signal

y(n D t) = 4.8 exp [- ( n D t-4) 2 /2.8], D t= 1. y n = Q k, D t=1, k = 5.

In principle, an analog signal recorded by the corresponding digital equipment can also be quantized in terms of its values ​​(Fig. 2.2.4). But it makes no sense to separate these signals into a separate type - they remain analog piecewise continuous signals with a quantization step, which is determined by the permissible measurement error.

Most of the discrete and digital signals that have to be dealt with are sampled analog signals. But there are signals that initially belong to the class of discrete ones, for example, gamma quanta.

Rice. 2.2.4. Quantized signal y(t)= Q k, k = 5.

Spectral representation of signals. In addition to the usual temporal (coordinate) representation of signals and functions, the description of signals by frequency functions is widely used in the analysis and processing of data, i.e. by arguments opposite to the arguments of the temporal (coordinate) representation. The possibility of such a description is determined by the fact that any signal, arbitrarily complex in its form, can be represented as a sum of simpler signals, and, in particular, as a sum of the simplest harmonic oscillations, the combination of which is called the frequency spectrum of the signal. Mathematically, the spectrum of signals is described by functions of the values ​​of amplitudes and initial phases of harmonic oscillations in a continuous or discrete argument - frequency... The amplitude spectrum is usually called frequency response(AFC) signal, spectrum of phase angles - phase-frequency response(PFC). The frequency spectrum description displays the signal as unambiguously as the coordinate description.

In Fig. 2.2.5 shows a segment of the signal function, which is obtained by summing the constant component (the frequency of the constant component is 0) and three harmonic oscillations. The mathematical description of the signal is determined by the formula:

where A n= (5, 3, 6, 8) - amplitude; f n= (0, 40, 80, 120) - frequency (Hz); φ n= (0, -0.4, -0.6, -0.8) - initial phase angle (in radians) of oscillations; n = 0,1,2,3.

Rice. 2.2.5. Temporary representation of the signal.

The frequency representation of this signal (the spectrum of the signal in the form of frequency response and phase response) is shown in Fig. 2.2.6. Note that the frequency representation of the periodic signal s(t), limited in the number of spectrum harmonics, is only eight samples and is very compact compared to the continuous time representation, defined in the range from - ¥ to + ¥.

Rice. 2.2.6. Frequency representation of the signal.

Graphical display analog signals (Fig. 2.2.1) does not require special explanations. When graphically displaying discrete and digital signals, either the method of direct discrete segments of the corresponding scale length over the axis of the argument is used (Fig. 2.2.6), or the method of the envelope (smooth or broken) by the sample values ​​(dashed curve in Fig. 2.2.2). Due to the continuity of the fields and, as a rule, the secondary nature of digital data obtained by sampling and quantizing analog signals, the second method of graphic display will be considered the main one.

The purpose of the story is to show what is the essence of the concept of "signal", what common signals exist and what common characteristics they have.

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 in Russian means: 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 ​​at 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 time interval 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 = 1 repetition per second. Frequency is the inverse of the period (ƒ = 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.

Types of signals

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 = 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 = 1 sec / 1 repetition 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 rectangular signal in which the pulse and pause durations are equal. 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 fill 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 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 we 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 receive a triangular signal. A crude comparison, but true. In triangular signals, the voltage (current) first increases and then immediately begins to decrease. And for a classic triangular signal, the rise time is equal to the decay time (and is equal to half the period).

If the rise time of such a signal is less or more than the 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.