The purpose of the story is to show what the concept of “signal” is, what common signals exist and what common characteristics they have.
What is a signal? Even to this question Small child will say that this is “a thing with which you can communicate something.” For example, using a mirror and the sun, you can transmit signals over a line-of-sight distance. On ships, signals were once transmitted using semaphore flags. This was done by specially trained signalmen. Thus, information was transmitted using such flags. Here's how to convey the word "signal":
There are a huge variety of signals in nature. Yes, 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, everything is clear with such signals, so I’ll move on to electrical signals, which in nature are no less numerous than any other. But they can at least be roughly divided into groups: triangular, sinusoidal, rectangular, sawtooth, single pulse, etc. All of these signals are named for what they look like when plotted on a chart.
The signals can be used as a metronome to count clocks (as a timing signal), to keep time, as control pulses, to control motors, or to test equipment and transmit information.
Electrical characteristics signals
In a sense, an electrical signal is a graph showing the change in voltage or current over time. Which in Russian means: if you take a pencil and mark time along the X axis, and voltage or current along 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 ones, the current flows in one direction (or does not flow at all), and in bidirectional ones, the current is alternating and flows either “there” or “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 designated T
- Frequency -- indicates how many times the signal will repeat in 1 second. It is measured in Hertz. For example, 1Hz = 1 repetition per second. Frequency is the reciprocal of the period ( ƒ = 1/T )
- Amplitude -- measured in volts or amperes (depending on whether the signal is current or voltage). Amplitude refers to the "strength" of the signal. How far does the signal graph deviate from the X-axis?
Types of signals
Sine wave
I think that representing 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 1 second by period T, then you will find out how many periods fit into 1 second 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 repeat per sec
Frequency and period are the inverse of each other. The longer the period, the lower the frequency and vice versa. The relationship between frequency and period is expressed by simple relationships:
Signals that resemble rectangles in shape are called “rectangular signals.” They can be divided into simple 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 meander period.
A regular rectangular signal differs from a meander in that it has different pulse and pause durations (no pulse). See the picture below - it will say 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). This narration And fill factor. The ratio (S) is equal to the ratio of the period to the pulse duration and vice versa for the coefficient. filling.
Thus, a square wave is a rectangular signal with a duty cycle of 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 rectangular signals. In life they look slightly different, since in no device can a signal change absolutely instantly from 0 to some value and then go back down to zero.
If we climb a mountain and then immediately descend and record the change in the height of our position on the graph, we will get a triangular signal. A harsh comparison, but a true one. In triangular signals, the voltage (current) first increases and then immediately begins to decrease. And for a classic triangular signal, the increasing time is equal to the decreasing time (and equal to half the period).
If such a signal has an increasing time less or greater than the decreasing time, then such signals are already called sawtooth. And about them below.
Ramp signal
As I wrote above, an asymmetrical triangular signal is called a sawtooth signal. All these names are conditional and are needed simply for convenience.
TOPIC 3 Digital signal processing devices
LECTURE 8_
Basic Concepts of Digital Signal Processing
Lecture questions:
Types of signals. Communication between signals various types.
Number systems and codes used in DAC and ADC converters.
Application areas of DAC and ADC
Basic parameters and classification of DAC and ADC
Types of signals. Relationship between signals of different types
The entire variety of signals can be divided into three main types of signals: analog, discrete and digital.
Analog signal is described by a continuous or piecewise continuous function, and both the argument and the function itself can take any values from certain intervals: , 
.
Examples. 
, speech signal in radio and television.
Discrete signal is described by a lattice function, which can take any value, while the independent variable can only take discrete values ( - sampling interval).
Discrete non-quantized signals include signals with pulse amplitude modulation.
A digital signal is described by a quantized lattice function, that is, a lattice function that takes only a number of discrete values - quantization levels, while the independent variable takes.
Each of the quantization levels is encoded by a binary code, so that the transmission and processing of a digital encoded signal sample is reduced to operations on a dimensionless binary code. The number of quantization levels and the number of binary bits are related by the relationship 
.
Digital signals include, for example, signals used in pulse code modulation communication systems.
Sampling operation connects the analog and discrete signals and consists in the fact that a discrete signal is constructed from the analog signal such that 
.
Recovery operation is that based on a given discrete signal, a analog signal.
The reconstruction and sampling operations are mutually inverse if the sampled analog signal satisfies Kotelnikov's theorem.
The relationship between the spectrum of an analog signal and the spectrum of a discrete signal is determined by the formula
.
This expression describes the “multiplication” of the spectrum of an analog signal during sampling.
Quantization and encoding operation(analog-to-digital conversion) is that, based on a given discrete signal, a coded signal is constructed such that 
, .
Digital-to-analog conversion operation consists in the fact that a discrete signal is constructed from a given digital encoded signal, and 
.
The operations of quantization and encoding and digital-to-analog conversion are not exactly mutually inverse, since quantization in the general case is performed with an unavoidable error. However, if a sufficiently large number of binary signals are used to represent each sample, then the quantization error will be small enough that the discrete signal (and therefore the corresponding analog signal) can be replaced by a digital signal.
Sampling, quantization and encoding operations are performed analog-to-digital converters (ADCs), and digital-to-analog conversion and restoration operations are digital-to-analog converters (ADCs).
Digital signal processing (DSP) devices are devices that implement one or another digital processing algorithm.
Basic advantages DSP compared to analogue:
1) the characteristics of DSP devices are absolutely stable and do not change when external conditions change (temperature, humidity, etc.) as long as these devices remain operational;
2) it is possible to implement a number of operations and algorithms that are fundamentally impossible to implement using analog elements, for example, processing of infra-low-frequency signals, since digital storage devices have an almost unlimited duration of information storage.
DSP devices are conveniently implemented in the form of LSI and VLSI.
Among shortcomings UTsOS can be distinguished as follows:
1) Relatively low processing speed;
2) Relatively high power consumption;
3) Relatively high cost;
4) The need to use ADC and DAC at the input and output of the DSP.
It should be noted that the significance of the first two disadvantages is decreasing due to the development of LSI and VLSI manufacturing technologies. The cost of algorithms and programs is gaining more and more weight in the cost of the DSP. Fundamentally, the accuracy of the DSP is limited by the ADCs and DACs used. The accuracy of the calculations in the device itself is determined by the number of binary digits used to represent the codes.
2. Number systems and codes,
used in DAC and ADC converters
Typically, numbers are represented using a decimal positional number system, in which each number is represented as a sum of powers of 10, although only the coefficients of this expansion are written:
The decimal system uses 10 digits to represent expansion coefficients.
However, digital devices convert information represented by just two digits 0 and 1, so to represent numbers it is convenient to use the binary number system, in which the weights of binary coefficients are powers of 2.
The measured physical quantities can be unipolar or bipolar. Therefore, to represent them in digital form, ADCs and DACs use both unipolar and bipolar codes.
Unipolar codes.
Binary code (regular binary code).
The rightmost digit is the least significant digit (LSB), the leftmost digit is the most significant digit (MSB).
In this code, the contribution of each bit (binary digit) depends on its position:
In a bit sequence, the SZR has a weight of , and the maximum number that can be represented by a bit code is equal to .
Fractional encoding
When considering the operation of an ADC, it is important to consider a binary number as a representation of the fractional part of some integer. In this case, the weight of the MZR is equal to , and the weight of the PPP is . There is a comma before the number:
.
The value of the fractional number corresponding to units in all digits is defined as 1-1MZR. In addition, the MZR determines the resolution of the -bit code of the converter
3. Application areas of DAC and ADC
The level and development of microelectronic DACs and ADCs are determined by the requirements for the technical and operational characteristics of the radio systems in which they are used.
These requirements can vary significantly depending on the purpose, operating principle and operating conditions of the systems.
The need to receive, process, and transmit a large amount of information in real time, as well as the problems of studying fast processes in various installations led to the creation high-speed DAC and ADC integrated circuits.
Solving communication problems required the creation multichannel converters.
Precision measurements, seismic exploration, robotics, high-quality audio and video recording equipment are impossible without high resolution converters.
Strict requirements for energy consumption and weight and size characteristics imposed on on-board systems are satisfied through the use of micropower and functionally complete converters.
For military RTS, required converters resistant to various external factors.
Household electrical and radio appliances require a wide range of inexpensive converters that do not have record values of electrical parameters and performance characteristics.
Some ADC applications:
| Average parameter values | ||||
| Areas of use | number of doors ranks | conversion time (μs) | input frequency band signal, Hz | Differential nonlinearity, MZR |
| Radar | 6-8 | 0.05 | 2 10 7 | 0.5 |
| Radar (long range detection) | 14-16 | 2 10 3 | 0.5 | |
| Aerospace data processing | 0.01 | up to 10 8 | 0.5 | |
| Radio navigation | 8-10 | 0.05-0.1 | 10 7 | 0.5 |
| High quality audio and video recording | 2 10 4 | 0.5 | ||
| Instruments for physical research | 16-18 | 1-5 | 0.5 | |
| Specialist. Digital computers | 3-5 | 10 5 | 0.5 |
Some applications of DAC.
3 Basic parameters and classification of DACs and ADCs
DAC classification is carried out by conversion methods.
There are two conversion methods -
* method of summing a single analog value (quanta);
* summation method taking into account the weight of digits.
By implementation scheme DACs are divided into: DACs with voltage summation, DACs with current summation, and multiplying DACs.
DAC parameters.
Parameters of the nominal conversion function.
The nominal conversion function has the form
Or 
with binary coding.
Graphically interpreted by points on a line. Final output value 
.
The parameters of this function are conversion factor , type of input signal code And number of digits .
Conversion factor is the ratio of the analog signal increment to the digital signal increment. It has the dimension of the output quantity and is numerically equal to the nominal unit of the least significant digit.
Entry code may be natural binary code, binary decimal codes.
Static accuracy parameters.
Conversion error- deviation of the real conversion function from the nominal one.
Systematic conversion error- time-averaged value of the conversion error with a constant value of the control code.
Conversion error is random- random component (noise) of the output signal with a constant value of the input code.
Nonlinearity of transformation- the maximum deviation of the values of the real transformation function from the corresponding points on the straight line that approximates this function.
Differential nonlinearity of transformation- deviation of the output signal increment when the input code transitions to an adjacent value from the value of the MZ unit. Expressed in fractions of a unit of minimum wage.
Dynamic parameters.
Current (voltage) settling time) - the time interval from the moment of a given code change at the input of the DAC until the moment at which the output analog signal finally enters the zone of steady state corresponding to ±1/2 LSB or other specified value.
Output surge- a short burst in the output signal when the input code changes.
Influence function- dependence of changes in parameters on influencing factors (temperature, supply voltage, etc.).
Electrical pairing parameters.
Characterize all inputs and outputs of the DAC from the point of view of interfacing with external devices. Divided into analog pairing parameters and digital pairing parameters.
The first includes input and output resistances, nominal values and tolerances of supply voltages, external reference voltages.
To the second - nominal values and voltage tolerances log. "0" and log. "1", input impedances (currents) from the digital inputs.
There are four types of signals s(t): continuous continuous time, continuous discrete time, discrete continuous time and discrete discrete time.
Continuous-time signals are called continuous-time (analog) signals for short. They can change at arbitrary moments, taking on any of a continuous set of possible values (Fig. 1.3). Such signals include the well-known sinusoid.
Rice. 1.3 Continuous signal
Rice. 1.4 Continuous discrete time signal
Continuous discrete-time signals can take arbitrary values, but change only at certain, predetermined (discrete) moments (Fig. 1.4).
Discrete continuous-time signals differ in that they can change at arbitrary moments, but their values take only allowed (discrete) values (Fig. 1.5).
Discrete time signals (abbreviated discrete) (Fig. 1.6) at discrete times can only take on allowed (non-crete) values.
The signals generated at the output of the discrete message-to-signal converter are, as a rule, discrete in terms of the information parameter, i.e., they are described by a discrete time function and a finite set of possible values. In data transmission technology, such signals are called digital data signals (DDS). The data signal parameter, the change of which reflects a change in the message, is called representing (information). In Fig. Figure 1.7 shows a DSD, the representing parameter of which is amplitude, and the set of possible values of the representing parameter is equal to two. Part of a digital data signal that differs from the other parts in the value of one of its representing ones. parameters is called the DAC element.
The fixed value of the state of the representing parameter of the signal is called the significant position. The moment at which the significant position of the signal changes is called significant (SM).
Rice. 1.5 Discrete continuous time signal
Rice. 1.6 Discrete signal
Rice. 1.7 Digital data signal
The time interval between two adjacent significant moments of the signal is called significant (SI)
The minimum time interval, which is equal to the significant time intervals of the signal, is called unit ( intervals a-b, b-c and others in Fig. 1 7). A signal element having a duration equal to a unit time interval is called a unit element (e e)
The term unit element is one of the main ones in data transmission technology. In telegraphy it corresponds to the term elementary parcel
There are isochronous and anisochronous data signals. For an isochronous signal, any significant time interval is equal to a unit interval or an integer. Anisochronous signals are signals whose elements can have any duration, but not less than. Another feature of anisochronous signals is that they can be separated from each other in time at an arbitrary distance
Lecture 1
Basic types of signals and their mathematical description.
Main types of signals: analog, discrete, digital.
Analog is a signal continuous in time and state (Fig. 1a). The signal is described by a continuous (or piecewise continuous) function X(t). In this case, both the argument and the function itself can take any values from certain 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 X(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 timesn * T and maybe only in these moments take any values from a certain interval x" ≤ x ≤ x"". Values of the lattice function, and, accordingly, the signal itself at moments of time n* T, are called samples. (A discrete signal can be either real or 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 X c( n* T), which can take only a finite number of values from some finite interval x" ≤ x ≤ x"". These values are called quantization levels, and the corresponding functions are called quantized.
When analyzing discrete signals convenient to use standardized time 
, otherwise, i.e. the sample number of a discrete signal can be interpreted as normalized time. When transitioning to normalized time, a discrete signal can be considered as a function of an integer variable n. That is further X(n) is equivalent X(n·
T).
Frequency normalization.
According to Kotelnikov’s theorem, the maximum frequency of an analog signal f there shouldn't be more f D 2. Therefore, it is advisable to consider all discrete signals in the range. This introduces the concept normalized frequency
or 
and consider a discrete signal f in area
or 
The use of normalized frequency allows one 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 frequency that are important, but their ratio, i.e. normalized frequency value.
For example, for 2 discrete cosines:
Where
Eventually: 
Their discrete signals are the same, since their normalized frequencies are equal, they will only be different in time.
In the general case, a discrete cosine wave in the normalized frequency range has the form:
Generalized scheme of Digital Signal Processing.
The DSP process includes 3 stages:
Number Sequence Generator X(n* T) from analog signal x(t) ;
Sequence conversion X(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 the sequence y(n* T).
Sampling frequency f d is selected: f d ≥ 2 f V.
Real signals do not satisfy this requirement. Therefore, they install 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), and the spectra are below:
Quantization levels(Fig. 1.c.) are encoded with binary numbers, so at the output of the ADC we have a sequence of binary numbers 
. Digital signal 
different from discrete 
by the amount:
Quantization error.
To reduce it, it is necessary to increase the number of quantization levels. The discrete signal enters the central processing unit, which, according to the algorithm, assigns a unique correspondence to the output signal for each input report 
. In this case, the number of operations (multiplications, additions, inversions, transfers, etc.) to obtain one sample can be calculated as desired. However, the processing period (computation time) cannot be greater than the sampling period 
. And this can only be if the clock frequency f T CPOS >> f D.
Next, the DAC generates a step analog signal 
(t), the steps of which are smoothed by a filter, obtaining an analog y(t).
By types (types) of signals the following stand out:
- analog
- digital
- discrete
Analog signal
Analog signal is natural. It can be detected using various types of sensors. For example, environmental sensors (pressure, humidity) or mechanical sensors (acceleration, speed). Analog signals in mathematics they are described by continuous functions. Electrical voltage is described using a straight line, i.e. is analog.
Digital signal
Digital the signals are artificial, i.e. they can only be obtained by converting an analog electrical signal.
The process of sequentially converting a continuous analog signal is called sampling. There are two types of discretization:
- by time
- by amplitude
Time sampling is usually called a sampling operation. And sampling by signal amplitude is quantization by level.
Mostly digital signals are light or electrical impulses. A digital signal uses the entire given frequency (bandwidth). This signal still remains analog, only after conversion it is endowed with numerical properties. And you can apply numerical methods and properties to it.
Discrete signal
Discrete signal– this is still the same converted analog signal, only it is not necessarily quantized in level.
This is the basic information about types (types) of signals.
