What are the differences between discrete and analog signals? Types of signals: analog, digital, discrete. Application of digital signal
In technical branches of knowledge, the term signal is
1) a technical means for transmitting circulation and using information.
2) the physical process of displaying an information message (changing any parameter of the information carrier)
3) the semantic content of a certain physical state or process.
Signal – information/messages/information about any processes/states or physical quantities of objects of the material world, expressed in a form convenient for transmission, processing, storage and use of this information.
From a mathematical point of view, a signal is a function, that is, the dependence of one quantity on another.
Purpose of Signal Processing
The purpose of signal processing is considered to be the study of certain information information, which are displayed as target information and transform this information into a form convenient for further use.
Purpose of Signal Analysis
By “analysis” of signals we mean not only their purely mathematical transformations, but also drawing conclusions about the specific features of the corresponding processes and objects based on these transformations. The goals of signal analysis are usually: - Determination or evaluation of numerical parameters of signals (energy, average power, root mean square value, etc.). - Decomposition of signals into elementary components to compare the properties of different signals. - Comparison of the degree of proximity, “similarity”, “relatedness” of various signals, including with certain quantitative estimates.
Signal registration
The concept of signal is inextricably linked with the term signal registration, the use of which is as broad and ambiguous as the term signal itself. In the most in a general sense this term can be understood as the operation of isolating a signal and converting it into a form convenient for further use, processing and perception. Thus, when receiving information about the physical properties of any objects, signal registration is understood as the process of measuring the physical properties of an object and transferring the measurement results to the material carrier of the signal or the direct energy transformation of any properties of the object into information parameters of the material carrier of the signal (usually electrical ). But the term signal recording is also widely used for the processes of separating already formed signals carrying certain information from the sum of other signals (radio communications, telemetry, etc.), and for the processes of recording signals on media long-term memory, and for many other processes related to signal processing.
Internal and external sources noise
Noises, as a rule, are stochastic (random) in nature. Interference includes distortion of useful signals under the influence of various destabilizing factors (electrical interference, vibration, types of noise and interference are distinguished by their sources, energy spectrum). Depending on the nature of the impact on the signal, sources of noise and interference can be internal or external.
Internal interference is inherent in the physical nature of signal sources and detectors, as well as material media. External sources of interference can be of artificial or natural origin. Artificial noise includes industrial noise and interference from operating equipment.
What gives mathematical model signal
The theory of analysis and processing of physical data is based on mathematical models of the corresponding physical fields and physical processes on the basis of which mathematical models of signals are created; they make it possible to generally abstract from the physical nature to judge the properties of signals, predict changes in signals in various conditions, in addition, it becomes possible to ignore big number secondary signs. Knowledge of mathematical models makes it possible to classify signals according to various criteria (for example, signals are divided into deterministic and stochastic).
Signal classification
Signal classification carried out on the basis of essential features of the corresponding mathematical models of signals . All signals are divided into two large groups: deterministic and random.
Harmonic signals
Harmonic signals (sinusoidal), are described by the following formulas:
s(t) = A×sin (2f o t+f) = A×sin ( o t+f), s(t) = A×cos( o t+), (1.1.1)
Rice. 5. Harmonic signal and spectrum of its amplitudes
where A, f o , o , f are constant values that can act as information parameters of the signal: A is the signal amplitude, f o is the cyclic frequency in hertz, o = 2f o is the angular frequency in radians , and f are the initial phase angles in radians. The period of one oscillation is T = 1/f o = 2/ o. When j = f-p/2, sine and cosine functions describe the same signal. The frequency spectrum of the signal is represented by the amplitude and initial phase value of the frequency f o (at t = 0).Polyharmonic signals
Polyharmonic signals constitute the most widespread group of periodic signals and are described by the sum of harmonic oscillations:
s(t) =A n sin (2f n t+ n) ≡ A n sin (2B n f p t+ n), B n ∈ I, (1.1.2)
or directly by the function s(t) = y(t ± kT p), k = 1,2,3,..., where T p is the period of one complete oscillation of the signal y(t), specified over one period. The value f p =1/T p is called the fundamental oscillation frequency.
Rice. 6. Signal model Fig. 7. Signal spectrum
Polyharmonic signals are the sum of a certain constant component (f o =0) and an arbitrary (in the limit - infinite) number of harmonic components with arbitrary values of amplitudes A n and phases j n , with frequencies that are multiples of the fundamental frequency f p . In other words, on the period of the fundamental frequency f p , which is equal to or a multiple of the minimum harmonic frequency, a multiple number of periods of all harmonics fits, which creates the periodicity of the signal repetition. The frequency spectrum of polyharmonic signals is discrete, and therefore the second common mathematical representation of signals is in the form of spectra (Fourier series).Almostperiodic signal
Almost periodic signals are close in their form to polyharmonic. They also represent the sum of two or more harmonic signals (in the limit - to infinity), but not with multiples, but with arbitrary frequencies, the ratios of which (at least two frequencies minimum) do not relate to rational numbers, as a result of which the fundamental period of the total oscillations is infinite big rice 9.
Rice. 9. Almost periodic signal and spectrum of its amplitudes
Analog signals
Analog signal (analog signal) is a continuous or piecewise continuous function y=x(t) of a continuous argument, i.e. both the function itself and its argument can take any value within a certain interval y 1 £y £ y 2 , t 1 £t £ t 2 . If the intervals of signal values or its independent variables are not limited, then by default they are assumed to be equal to -¥ to +¥. The set of possible signal values forms a continuum - a continuous space in which any signal point can be determined with infinity accuracy.
Sources of analog signals are physical processes and phenomena; examples of analog signals are most often given by changes in the strength of the electric, magnetic and electromagnetic fields over time.
Discrete signals
Discrete signal
Rice. 13. Discrete signal
Discrete signal (discrete signal) – fig. 13 in its values is also a continuous function, but defined only by discrete values of the argument. According to the set of its values, it is finite (countable) and is described by a discrete sequence of samples (samples) y(nt), where y 1 £y £ y 2, t is the interval between samples (interval or sampling step, sample time) , n = 0, 1, 2,...,N. The reciprocal of the sampling step: f = 1/t is called the sampling frequency. If a discrete signal is obtained by sampling an analog signal, then it represents a sequence of samples whose values are exactly equal to the values of the original signal.Digital signal
Digital signal (digital signal) is quantized in its values and discrete in its argument. It is described by a quantized lattice function y n = Q k, where Q k is a quantization function with the number of quantization levels k, and the quantization intervals can be either uniform or uneven, for example, logarithmic. A digital signal is specified, as a rule, in the form of a discrete series of numerical data - a numerical array of successive values of the argument with t = const, but in general case the signal can also be specified in the form of a table for arbitrary argument values.
Rice. 14. Digital signal
Essentially, a digital signal in its values (counts) is a formalized version of a discrete signal when the latter’s counts are rounded to a certain number of digits, as shown in Fig. 14. A digital signal is finite in its many values. The process of converting analog samples with infinite values into a finite number of digital values is called level quantization, and the rounding errors of samples (discarded values) that arise during quantization are called noise or quantization errors.Kotelnikov-Shannon theorem
Physical meaning of the theorem Kotelnikov-Shannon: if the maximum frequency in the signal is f, then it is enough to have at least 2 samples with known values of t 1 and t 2 on one period of this harmonic, and it becomes possible to write a system of two equations (y 1 =a cos 2ft 1 and y 2 =a cos 2ft 2) and solve the system with respect to 2 unknowns - amplitude a and frequency f of this harmonic. Therefore, the sampling frequency should be 2 times higher maximum frequency f in the signal. For lower frequencies this condition will be satisfied automatically.
In practice, this theorem is widely used, for example, in converting audio recordings. The range of frequencies perceived by humans is from 20 Hz to 20 kHz; therefore, for lossless conversion it is necessary to perform sampling at a frequency of more than 40 kHz; therefore, cd dvd mp3 is digitized at a frequency of 44.1 kHz. The quantization operation (analog-to-digital conversion of the ADC ADC) consists of converting a discrete signal into a digital signal encoded in a binary system. dead reckoning
System concept
A system for any purpose always has an input to which input signal or an input effect (generally multidimensional) and an output from which the processed output signal is removed. If the design of the system and internal transformation operations are not of fundamental importance, then the system as a whole can be perceived as a black box in a formalized form.
A formalized system represents a specific system operator(algorithm) for converting the input signal – impact s(t), into the signal at the system output y(t) – response or output reaction systems. Symbolic designation of the transformation operation:
For deterministic input signals, the relationship between input and output signals is uniquely specified by the system operator.
System Operatort
System operator T is a rule (set of rules, algorithm) for converting signal s(t) into signal y(t). For well-known signal conversion operations, extended symbols of transformation operators are also used, where the second symbol and special indices indicate a specific type of operation (for example, TF - Fourier transform, TF -1 - inverse Fourier transform).
Linear and non-linear systems
In the case of implementing a random input signal at the input of the system, there is also a one-to-one correspondence between the processes at the input and output, but in this case the statistical characteristics of the output signal change. Any signal transformations are accompanied by changes in their spectrum and, according to the nature of these changes, they are divided into 2 types: linear and nonlinear
Nonlinear is when new harmonic components appear in the signal spectrum, and when the signals change linearly, the amplitudes of the component spectrum change. Both types of changes can occur with the preservation and distortion of useful information. Linear systems constitute the main class of signal processing systems.
The term linearity means that the signal conversion system must have an arbitrary, but necessarily linear relationship between the input and output signals.
A system is considered linear if, within a specified area of input and output signals, its response to input signals is additive (the principle of superposition of signals is fulfilled) and homogeneous (the principle of proportional similarity is fulfilled).
Additivity principle
Principle additivity requires that the reaction to the sum of two input signals be equal to the sum of the reactions to each signal separately:
T = T+T.
Principle of homogeneity
Principle uniformity or proportional similarity requires maintaining the unambiguity of the transformation scale for any amplitude of the input signal:
T= c T.
Basic system operations
The basic linear operations from which any linear transformation operators can be formed include the operations of scalar multiplication, shift and addition of signals:
y(t) = b x(t), y(t) = x(t-t), y(t) = a(t)+b(t).
Rice. 11.1.1. System Operation Graphics
Addition and multiplication operations are linear only for discrete and analog signals.
For systems with a dimension of 2 or more, there is also one more basic operation, which is called the operation spatial masking, which can be considered as a generalization of scalar multiplication. So, for two-dimensional systems:
z(x,y) = c(x,y)u(x,y),
where u(x,y) is a two-dimensional input signal, c(x,y) is a spatial mask of constant (weighting) coefficients. Spatial masking is the element-wise product of signal values with mask coefficients.
Differential equations as a universal tool for studying signals
Differential equations are a universal tool for specifying a specific relationship between input and output signals, both in one-dimensional and multidimensional systems, and can describe the system both in real time and a posteriori. Thus, in an analog one-dimensional linear system, such a relationship is usually expressed by a linear differential equation
a m = b n . (11.1.1)
When normalized to a o = 1, it follows
y(t) =b n –a m . (11.1.1")
Essentially right side This expression in the most general mathematical form displays the content of the input signal conversion operation, i.e. the operator for transforming the input signal into the output signal is specified. To uniquely solve equations (11.1.1), in addition to the input signal s(t), certain initial conditions must be specified, for example, the values of the solution y(0) and its time derivative y"(0) at the initial time.
A similar connection in a digital system is described by difference equations
a m y((k-m)t) =b n s((k-n)t). (11.1.2)
y(kt) =b n s((k-n)t) –a m y((k-m)t). (11.1.2")
The last equation can be considered as an algorithm for sequentially calculating the values y(kt), k = 0, 1, 2, …, from the values of the input signal s(kt) and the previous calculated values y(kt) with known values of the coefficients a m , b n and taking into account the task initial conditions- values of s(kt) and y(kt) at k< 0. Интервал дискретизации в цифровых последовательностях отсчетов обычно принимается равным 1, т.к. выполняет только роль масштабного множителя.
Recursive systems
In practice, they strive to simplify systems of interdependent models and bring them to the so-called recursive form. To do this, first select an endogenous variable (internal indicator), which depends only on exogenous variables (external factors), and denote it 1. Then an internal indicator is selected, which depends only on external factors and on y 1, etc.; thus, each subsequent indicator depends only on external factors and on the previous internal ones. Such systems are called recursive. The parameters of the first equation of recursive systems are found by the least squares method, they are substituted into the second equation and the least squares method is applied again, etc.
Access and backbone networks
Backbone wide-area networks are used to form peer-to-peer connections between large local networks belonging to large departments of an enterprise. Backbone territorial networks must provide high throughput, since the backbone combines the flows of a large number of subnets. In addition, backbone networks must be constantly available, that is, provide a very high availability factor, since they carry the traffic of many business-critical applications. Due to the special importance of highways, they can be forgiven for their high cost. Since an enterprise usually does not have many large networks, then backbone networks there are no requirements to maintain an extensive access infrastructure.
Access networks are understood as territorial networks necessary for connecting small local networks and individual remote computers with the central local network of an enterprise. If great attention has always been paid to the organization of backbone connections when creating a corporate network, then the organization of remote access for enterprise employees has become a strategically important issue only recently. Fast access access to corporate information from any geographical location determines the quality of decision-making by its employees for many types of enterprise activities. The importance of this factor is growing with the increase in the number of employees working at home (telecommuters) who are often on business trips, and with the increase in the number of small branches of enterprises located in different cities and, perhaps, different countries.
Multiplexing
Multiplexing is the use of one communication channel to transmit data to several subscribers. Communication lines (channel) consist of a physical medium through which information signals of data transmission equipment are transmitted.
Types of communication channels
simplex - when the receiver communicates with the transmitter over one channel, with unidirectional transmission of information (for example, in television and radio broadcasting networks);
half-duplex - when two communication nodes are connected by one channel, through which information is transmitted alternately in one direction, then in the opposite direction (in information-reference and request-response systems);
duplex - allows you to transmit data simultaneously in two directions through the use of a four-wire communication line (two wires for transmitting, the other two for receiving data), or two frequency bands.
Characteristics of communication lines
The main characteristics of the communication channel - throughput and reliability of data transmission
Channel capacity (the amount of information transmitted per unit of time) is estimated by the number of bits of data transmitted over the channel per second BIT/sec
The reliability of data transmission is assessed by the bit error rate (BER), which is determined by the probability of distortion of the transmitted data bit. The value of bit error intensity for communication channels without additional protection from errors is 10 -4 to 10 -6
Main characteristics of cables
Computer networks use cables that comply with international standards ISO 11801. These standards regulate the following basic characteristics of cables:
– attenuation (dB/m);
– resistance of the cable to internal sources of interference (if there is more than one pair of wires in the cable);
Impedance (characteristic impedance) - the effective input resistance of the cable for alternating current;
The level of external EM radiation in the conductor characterizes the cable’s noise immunity.
The degree of attenuation of external interference from various sources. The most widely used types of cables are unshielded twisted pair/ shielded twisted pair / coaxial cable / fiber optic.
Unshielded-
Shielded is better than unshielded
Cable (RG8 and RG11 - thick coaxial cable has a characteristic impedance of 8 Ohms and an outer diameter of 2.5 cm)
RG58 & RG59 cables – thin coaxial cables with characteristic impedance 75 Ohm
Data transmission media (wired and wireless)
Depending on the physical medium of data transmission, communication lines can be divided:
wired communication lines without insulating and shielding braids;
cable, where communication lines such as twisted pair cables, coaxial cables or fiber optic cables are used to transmit signals;
wireless (terrestrial and satellite radio channels) using for signal transmission electromagnetic waves, which spread over the airwaves.
Information signal - physical process that has for a person or technical device informational meaning. It can be continuous (analog) or discrete
The term “signal” is very often identified with the concepts of “data” and “information”. Indeed, these concepts are interrelated and do not exist one without the other, but belong to different categories.
Signal- This information function, carrying a message about the physical properties, condition or behavior of any physical system, object or environment, and the purpose of signal processing can be considered to be the extraction of certain information information that is displayed in these signals (in short - useful or target information) and the transformation of this information into a form convenient for perception and further use.
Information is transmitted in the form of signals. A signal is a physical process that carries information. The signal can be sound, light, in the form postal item and etc
The signal is material carrier information that is transmitted from source to consumer. It can be discrete and continuous (analog)
Analog signal- a data signal in which each of the representing parameters is described by a function of time and a continuous set of possible values.
Analog signals are described by continuous functions of time, which is why an analog signal is sometimes called a continuous signal. Analog signals are contrasted with discrete (quantized, digital).
Examples of continuous spaces and corresponding physical quantities: (line: electrical voltage; circle: rotor position, wheel, gears, arrows analog clock, or phase of the carrier signal; segment: position of a piston, control lever, liquid thermometer or electrical signal limited in amplitude various multidimensional spaces: color, quadrature-modulated signal.)
The properties of analog signals are largely opposite properties of quantized or digital signals.
The absence of clearly distinguishable discrete signal levels leads to the impossibility of applying the concept of information in the form as it is understood in Russian literature to describe it. digital technologies. The “amount of information” contained in one reading will be limited only by the dynamic range of the measuring instrument.
No redundancy. From the continuity of the value space it follows that any noise introduced into the signal is indistinguishable from the signal itself and, therefore, the original amplitude cannot be restored. In reality, filtering is possible, for example, using frequency methods, if some Additional Information about the properties of this signal (in particular, the frequency band).
Application:
Analog signals are often used to represent continuously changing physical quantities. For example, an analog electrical signal taken from a thermocouple carries information about temperature changes, a signal from a microphone - about rapid changes pressure in a sound wave, etc.
Discrete signal is composed of a countable set (i.e. a set whose elements can be counted) of elements (they say - information elements). For example, the “brick” signal is discrete. It consists of the following two elements (this is the syntactic characteristic of this signal): a red circle and a white rectangle inside the circle, located horizontally in the center. It is in the form of a discrete signal that the information that the reader is currently mastering is presented. You can distinguish the following elements: sections (for example, “Information”), subsections (for example, “Properties”), paragraphs, sentences, individual phrases, words and individual characters (letters, numbers, punctuation marks, etc.). This example shows that depending on the pragmatics of the signal, different information elements can be distinguished. In fact, for a person studying computer science from a given text, larger information elements, such as sections, subsections, and individual paragraphs, are important. They allow him to more easily navigate the structure of the material, better assimilate it and prepare for the exam. For the one who prepared this methodological material, in addition to the indicated information elements, smaller ones are also important, for example, individual sentences, with the help of which this or that idea is presented and which implement this or that method of accessibility of the material. The set of the smallest elements of a discrete signal is called an alphabet, and the discrete signal itself is also called message.
Sampling is the conversion of a continuous signal into a discrete (digital) one.
The difference between discrete and continuous representation of information is clearly visible in the example of a clock. IN electronic watch With a digital dial, information is presented discretely - in numbers, each of which is clearly different from each other. IN mechanical watch with a pointer dial, information is presented continuously - by the positions of two hands, and two different positions The hands are not always clearly distinguishable (especially if there are no minute markers on the dial).
Continuous signal- reflected by some physical quantity, changing in a given time interval, for example, timbre or sound intensity. Represented as a continuous signal real information for those students - consumers who attend lectures on computer science and through sound waves(in other words, the lecturer’s voice), which are continuous in nature, perceive the material.
As we will see later, a discrete signal is more amenable to transformation, and therefore has advantages over a continuous one. At the same time, in technical systems and in real processes it prevails continuous signal. This forces us to develop ways to convert a continuous signal into a discrete one.\
To convert a continuous signal into a discrete one, a procedure called quantization.
A digital signal is a data signal in which each of the representing parameters is described by a discrete time function and a finite set of possible values.
A discrete digital signal is more difficult to transmit to long distances than an analog signal, so it is pre-modulated on the transmitter side and demodulated on the information receiver side. Use in digital systems ah verification and recovery algorithms digital information allows you to significantly increase the reliability of information transmission.
Comment. It should be kept in mind that a real digital signal is analog in its physical nature. Due to noise and changes in transmission line parameters, it has fluctuations in amplitude, phase/frequency (jitter), and polarization. But this analog signal (pulse and discrete) is endowed with the properties of a number. As a result, processing it becomes possible use numerical methods (computer processing).
Discrete signals naturally arise in cases where the source of messages provides information at fixed points in time. An example is information about air temperature transmitted by broadcasting stations several times a day. The property of a discrete signal is manifested here extremely clearly: in the pauses between messages there is no information about the temperature. In fact, the air temperature changes smoothly over time, so that the measurement results arise from the sampling of a continuous signal - an operation that records the reference values.
Discrete signals have acquired particular importance in recent decades under the influence of improvements in communication technology and the development of methods for processing information with high-speed computing devices. Great success achieved in the development and use of specialized devices for processing discrete signals, the so-called digital filters.
This chapter is devoted to consideration of the principles of mathematical description of discrete signals, as well as theoretical foundations construction of linear devices for their processing.
15.1. Discrete Signal Models
The distinction between discrete and analog (continuous) signals was emphasized in Chap. 1 upon classification radio signals. Let us recall the main property of a discrete signal: its values are not determined at all times, but only at a countable set of points. If an analog signal has a mathematical model of the form of a continuous or piecewise continuous function, then the corresponding discrete signal is a sequence of sample signal values at points, respectively.
Sampling sequence.
In practice, as a rule, samples of discrete signals are taken in time through equal interval A, called the sampling interval (step):
The sampling operation, i.e. the transition from an analog signal to a discrete signal, can be described by introducing the generalized function
called the sampling sequence.
Obviously, a discrete signal is a functional (see Chapter 1), defined on the set of all possible analog signals and equal to the scalar product of the function
Formula (15.3) indicates the path practical implementation devices for sampling an analog signal. The operation of the sampler is based on the gating operation (see Chapter 12) - multiplication of the processed signal and the “comb” function. Since the duration of the individual pulses that make up the sampling sequence is zero, sample values of the processed analog signal appear at the output of an ideal sampler at equally spaced moments in time .
Rice. 15.1. Structural scheme pulse modulator
Modulated pulse sequences.
Discrete signals began to be used back in the 40s when creating radio systems with pulse modulation. This type of modulation is different in that, as a “ carrier vibration» Instead of a harmonic signal, a periodic sequence of short pulses serves.
A pulse modulator (Fig. 15.1) is a device with two inputs, one of which receives the original analog signal. The other input receives short synchronizing pulses with a repetition interval. The modulator is constructed in such a way that at the moment of applying each synchronizing pulse, the instantaneous value of the signal x(t) is measured. A sequence of pulses appears at the output of the modulator, each of which has an area proportional to the corresponding reference value of the analog signal.
The signal at the output of the pulse modulator will be called a modulated pulse sequence (MPS). Naturally, the discrete signal is a mathematical model of the MIP.
Note that from a fundamental point of view, the nature of the impulses from which the MIP is composed is indifferent. In particular, these pulses can have the same duration, while their amplitude is proportional to the sample values of the signal being sampled. This type of continuous signal conversion is called pulse amplitude modulation (PAM). Another method is possible - pulse width modulation (PWM). Here, the amplitudes of the pulses at the modulator output are constant, and their duration (width) is proportional to the instantaneous values of the analog oscillation.
The choice of one or another pulse modulation method is dictated by a number of technical considerations, the convenience of circuit implementation, as well as characteristic features transmitted signals. For example, it is inappropriate to use AIM if the useful signal varies within very wide limits, i.e., as is often said, has a wide dynamic range. For undistorted transmission of such a signal, a transmitter with a strictly linear amplitude characteristic is required. The creation of such a transmitter is independent, technically complex problem. PWM systems do not impose any requirements on the linearity of the amplitude characteristics of the transmitting device. However, their circuit implementation may be somewhat more complicated compared to AIM systems.
A mathematical model of an ideal MIP can be obtained as follows. Let's consider the formula for the dynamic representation of a signal (see Chapter 1):
Since the MIP is defined only at points, integration in formula (15.4) should be replaced by summation over index k. The role of the differential will be played by the sampling interval (step). Then the mathematical model of a modulated pulse sequence formed by infinitely short pulses will be given by the expression
where are sample values of the analog signal.
Spectral density of a modulated pulse sequence.
Let us examine the spectrum of the signal arising at the output of an ideal pulse modulator and described by expression (15.5).
Note that a signal of the type MIP, up to the proportionality coefficient A, is equal to the product of the function and the sampling sequence
It is known that the spectrum of the product of two signals is proportional to the convolution of their spectral densities (see Chapter 2). Therefore, the laws of correspondence between signals and spectra are known:
then the spectral density of the MIP signal
To find the spectral density of the sampling sequence, we expand the periodic function into a complex Fourier series:
The coefficients of this series
Turning to formula (2.44), we obtain
that is, the spectrum of the sampling sequence consists of an infinite collection of delta pulses in the frequency domain. This spectral density is a periodic function with a period
Finally, substituting formula (15.8) into (15.7) and changing the order of the integration and summation operations, we find
So, the spectrum of the signal obtained as a result of ideal sampling with infinitely short gate pulses is the sum infinite number“copies” of the spectrum of the original analog signal. Copies are located on the frequency axis at equal intervals equal to the value of the angular frequency of the first harmonic of the sampling pulse sequence (Fig. 15.2, a, b).
Rice. 15.2. Spectral density of a modulated pulse sequence at different values of the upper limit frequency: a - the upper limit frequency is high; b - the upper limit frequency is low (the color indicates the spectral density of the original signal subjected to sampling)
Reconstruction of a continuous signal from a modulated pulse sequence.
In what follows, we will assume that the real signal has a low-frequency spectrum, symmetrical with respect to the point and limited by the upper limit frequency. From Fig. 15.2, b it follows that if , then individual copies spectra do not overlap each other.
Therefore, an analog signal with such a spectrum, subjected to pulse sampling, can be completely accurately restored using an ideal low-pass filter, the input of which is a pulse sequence of the form (15.5). In this case, the largest permissible sampling interval is , which is consistent with Kotelnikov’s theorem.
Indeed, let the filter restoring a continuous signal have a frequency transfer coefficient
The impulse response of this filter is described by the expression
Taking into account that the MIP signal of the form (15.5) is a weighted sum of delta pulses, we find the response at the output of the reconstruction filter
This signal, up to a scale factor, repeats the original oscillation with a limited spectrum.
An ideal low-pass filter is physically unrealizable and can only serve as a theoretical model to explain the principle of reconstructing a message from its discrete pulse samples. A real low-pass filter has an frequency response that either covers several lobes of the MIP spectral diagram, or, concentrating near the zero frequency, turns out to be significantly narrower than the central lobe of the spectrum. For example in Fig. Figure 15.3, b-e shows curves characterizing the signal at the output of the RC circuit used as a reconstruction filter (Fig. 15.3, a).
Rice. 15.3. Reconstruction of a continuous signal from its pulse samples using an RC circuit: a - filter circuit; b - discrete input signal; c, d - frequency response of the filter and the signal at its output in the case of ; d, e - the same, for the case
From the above graphs it can be seen that a real reconstruction filter inevitably distorts the input oscillation.
Note that to reconstruct the signal, you can use either the central or any side lobe of the spectral diagram.
Determination of the spectrum of an analog signal from a set of samples.
Having the MIP representation, you can not only restore the analog signal, but also find its spectral density. To do this, you should first of all directly connect the spectral density of the SMIP with the reference values:
(15.13)
This formula exhaustively solves the problem posed under the above limitation.
Any system digital processing signals, regardless of its complexity, contains a digital computing device - a universal digital computer, microprocessor or specially designed to solve specific task computing device. The signal arriving at the input of a computing device must be converted to a form suitable for processing on a computer. It must be in the form of a sequence of numbers represented in the machine code.
In some cases, the task of representing the input signal in digital form is relatively simple to solve. For example, if you need to transmit verbal text, then each symbol (letter) of this text needs to be associated with a certain number and, thus, represent transmitted signal as a number sequence. The ease of solving the problem in this case is explained by the fact that the verbal text is discrete in nature.
However, most of the signals encountered in radio engineering are continuous. This is due to the fact that the signal is a reflection of some physical process, and almost everything physical processes continuous in nature.
Let's consider the process of sampling a continuous signal into specific example. Let's say that air temperature is being measured on board a certain spacecraft; The measurement results must be transmitted to Earth to a data processing center. Temperature
Rice. 1.1. Types of signals: a - continuous (continuous) signal; 6 - discrete signal; c - AIM oscillation; g - digital signal
air is measured continuously; The temperature sensor readings are also a continuous function of time (Fig. 1.1, a). But the temperature changes slowly; it is enough to transmit its values once a minute. In addition, there is no need to measure it with an accuracy higher than 0.1 degrees. Thus, instead of a continuous function, you can transmit a sequence at intervals of 1 minute numerical values(Fig. 1.1, d), and in the intervals between these values you can transmit information about pressure, air humidity and other scientific information.
The considered example shows that the process of sampling continuous signals consists of two stages: sampling by time and sampling by level (quantization). A signal sampled only in time is called discrete; it is not yet suitable for processing in digital device. A discrete signal is a sequence whose elements are exactly equal to the corresponding values of the original continuous signal (Fig. 1.1, b). An example of a discrete signal can be a sequence of pulses with varying amplitude - an amplitude-pulse-modulated oscillation (Fig. 1.1, c). Analytically, such a discrete signal is described by the expression
where is the original continuous signal; single pulse of AIM oscillation.
If we reduce the pulse duration while keeping its area unchanged, then in the limit the function tends to the - function. Then the expression for the discrete signal can be represented as
To convert an analog signal to a digital signal, time sampling must be followed by level sampling (quantization). The need for quantization is caused by the fact that any computing device can only operate with numbers that have a finite number of digits. Thus, quantization is the rounding of transmitted values with a given accuracy. So in the example considered, temperature values are rounded to three significant figures (Fig. 1.1, d). In other cases, the number of bits of the transmitted signal values may be different. A signal that is sampled in both time and level is called digital.
The correct choice of sampling intervals in terms of time and level is very important when developing digital signal processing systems. The smaller the sampling interval, the more closely the sampled signal corresponds to the original continuous signal. However, as the sampling interval decreases in time, the number of samples increases, and in order to keep the total signal processing time unchanged, it is necessary to increase the processing speed, which is not always possible. As the quantization interval decreases, more bits are required to describe the signal, as a result of which the digital filter becomes more complex and cumbersome.
Analog, discrete and digital signals
INTRODUCTION TO DIGITAL SIGNAL PROCESSING
Digital signal processing (DSP or digital signal processing) is one of the newest and most powerful technologies that is being actively implemented in a wide range of fields of science and technology, such as communications, meteorology, radar and sonar, medical imaging, digital audio and television broadcasting, exploration of oil and gas fields, etc. We can say that there is a widespread and deep penetration of digital signal processing technologies into all spheres of human activity. Today, DSP technology is among the basic knowledge that is necessary for scientists and engineers in all industries without exception.
Signals
What is a signal? In the most general formulation, this is the dependence of one quantity on another. That is, from a mathematical point of view, the signal is a function. Dependencies on time are most often considered. The physical nature of the signal may be different. Very often this is electrical voltage, less often – current.
Signal presentation forms:
1. temporary;
2. spectral (in the frequency domain).
The cost of digital data processing is less than analogue and continues to decline, while the performance of computing operations is continuously increasing. It is also important that DSP systems are highly flexible. They can be supplemented with new programs and reprogrammed to perform various operations without changing the equipment. Therefore, interest in scientific and applied issues of digital signal processing is growing in all branches of science and technology.
PREFACE TO DIGITAL SIGNAL PROCESSING
Discrete signals
The essence of digital processing is that physical signal (voltage, current, etc.) is converted into a sequence numbers, which is then subjected to mathematical transformations into a computer.
Analog, discrete and digital signals
The original physical signal is a continuous function of time. Such signals, determined at all times t, are called analog.
What signal is called digital? Let's consider some analog signal (Fig. 1.1 a). It is specified continuously over the entire time interval under consideration. An analog signal is considered to be absolutely accurate, unless measurement errors are taken into account.
Rice. 1.1 a) Analogue signal
Rice. 1.1 b) Sampled signal
Rice. 1.1 c) Quantized signal
In order to receive digital signal, you need to perform two operations - sampling and quantization. The process of converting an analog signal into a sequence of samples is called sampling, and the result of such a transformation is discrete signal.T. arr., sampling consists in compiling a sample from an analog signal (Fig. 1.1 b), each element of which is called countdown, will be separated in time from neighboring samples over a certain interval T, called sampling interval or (since the sampling interval is often unchanged) – sampling period. The reciprocal of the sampling period is called sampling rate and is defined as:
(1.1)
When processing a signal in computing device its readings are presented in the form binary numbers having limited number discharges. As a result, the samples can only take on a finite set of values and, therefore, when presenting a signal, it inevitably rounds off. The process of converting signal samples into numbers is called quantization. The resulting rounding errors are called errors or quantization noise. Thus, quantization is the reduction of the levels of the sampled signal to a certain grid (Fig. 1.1 c), most often by the usual rounding up. A signal discrete in time and quantized in level will be digital.
Conditions under which it is possible full recovery analog signal from its digital equivalent, preserving all the information originally contained in the signal, are expressed by the theorems of Nyquist, Kotelnikov, Shannon, the essence of which is almost the same. To sample an analog signal with full preservation of information in its digital equivalent, the maximum frequencies in the analog signal must be no less than half the sampling frequency, that is, f max £ (1/2)f d, i.e. There must be at least two samples per period of the maximum frequency. If this condition is violated, the effect of masking (substitution) of actual frequencies of more than low frequencies. In this case, instead of the actual one, an “apparent” frequency is recorded in the digital signal, and, therefore, restoration of the actual frequency in the analog signal becomes impossible. The reconstructed signal will appear as if frequencies above half the sampling frequency have reflected from the frequency (1/2)f d to the lower part of the spectrum and superimposed on the frequencies already present in that part of the spectrum. This effect is called aliasing or aliasing(aliasing). A clear example aliasing can be an illusion that is quite often found in movies - a car wheel begins to rotate against its movement if between successive frames (analogous to the sampling frequency) the wheel makes more than half a revolution.
Converting the signal to digital form performed analog-to-digital converters(ADC). Typically they use binary system notation with a certain number of digits in a uniform scale. Increasing the number of bits improves measurement accuracy and expands the dynamic range of measured signals. Information lost due to a lack of ADC bits is irrecoverable, and there are only estimates of the resulting error in the “rounding” of samples, for example, through the noise power generated by an error in the last ADC bit. For this purpose, the concept of signal-to-noise ratio is used - the ratio of signal power to noise power (in decibels). The most commonly used are 8-, 10-, 12-, 16-, 20- and 24-bit ADCs. Each additional digit improves the signal-to-noise ratio by 6 decibels. However, increasing the number of bits reduces the sampling rate and increases the cost of the equipment. An important aspect is also the dynamic range, determined by the maximum and minimum value signal.
Digital Signal Processing either fulfilled special processors, or on mainframe computers and special programs. Easiest to consider linear systems. Linear are systems for which the principle of superposition takes place (response to the sum of input signals equal to the sum responses to each signal separately) and homogeneity (a change in the amplitude of the input signal causes a proportional change in the output signal).
If the input signal x(t-t 0) generates a unique output signal y(t-t 0) for any shift t 0, then the system is called time invariant. Its properties can be studied at any arbitrary time. For description linear system a special input signal is introduced - single impulse(impulse function).
Single impulse(single count) u 0(n) (Fig. 1.2):
Rice. 1.2. Single impulse
Due to the properties of superposition and homogeneity, any input signal can be represented as a sum of such pulses supplied to different moments time and multiplied by appropriate coefficients. The output signal of the system in this case is the sum of the responses to these pulses. The response to a unit pulse (pulse with unit amplitude) is called impulse response of the systemh(n). Knowing the impulse response allows one to analyze the passage through discrete system any signal. Indeed, an arbitrary signal (x(n)) can be represented as a linear combination of unit samples.
