Name: z-transform rules for signals
Uploaded: 2024-09-26T13:17:10.000Z
Duration: 1 min 17 s
Channel: JoVE

K12 Standards

Next Generation Science Standards

High School

Physical Science

waves-and-their-applications-in-technologies-for-information-transfer

Waves and Their Applications in Technologies for Information Transfer

HS-PS4-2

Properties of the z-Transform I

Некоторые свойства обеспечивают прочную основу для анализа систем дискретного времени с использованием Z-преобразования.
Рассматривая два сигнала дискретного времени, свойство линейности гласит, что Z-преобразование линейной комбинации сигналов равно линейной комбинации их отдельных Z-преобразований.
Свойство time-shifting означает, что если сигнал смещается во времени, его Z-преобразование умножается на коэффициент, который зависит от величины сдвига.
Это свойство помогает анализировать влияние задержек или прогонов во временной области на сигнал в частотной области.
Умножение сигнала во временной области на экспоненциальный коэффициент соответствует масштабированию в z-области.
Это свойство, известное как частотное масштабирование, помогает проанализировать, как изменяются частотные характеристики сигнала.
При обращении времени реверсирование оси времени сигнала соответствует взятию обратной величины переменной Z-преобразования в z-области.
Модуляция сигнала косинусной или синусоидальной функцией приводит к тому, что Z-преобразование сигнала оценивается в смещенных положениях, показывая, как частотная составляющая влияет на его Z-преобразование.

 Свойства z-преобразования

Z-преобразование является фундаментальным инструментом в цифровой обработке сигналов, позволяющим анализировать дискретные системы с помощью его различных свойств. Это бесценный инструмент для анализа дискретных систем, предлагающий ряд свойств, которые упрощают сложные манипуляции сигналами. Одним из фундаментальных свойств является линейность. Для любых двух дискретных сигналов z-преобразование их линейной комбинации равно той же линейной комбинации их отдельных z-преобразований. Это свойство имеет важное значение для анализа систем, в которых сигналы объединяются или накладываются друг на друга.

Другим важным свойством является сдвиг по времени. Если сигнал подвергается сдвигу по времени, его z-преобразование умножается на коэффициент, зависящий от величины сдвига. Это свойство помогает понять, как задержки или продвижения во временной области влияют на сигнал в частотном домене. Оно особенно полезно для анализа реакции систем на входные сигналы с задержкой во времени.

Масштабирование частоты является еще одним важным свойством. Когда сигнал умножается на экспоненциальный множитель во временном домене, это приводит к операции масштабирования в z-домене. Это свойство помогает изучить, как изменения частотных характеристик сигнала отражаются в его z-преобразовании. Это жизненно важно для приложений, включающих модуляцию и демодуляцию сигналов.

Обращение времени также имеет значение. Обращение оси времени сигнала соответствует взятию обратной величины переменной z-преобразования в z-домене. Это свойство полезно для анализа систем, в которых сигналы обращаются или воспроизводятся в обратном порядке, что дает представление о влиянии инверсии времени на поведение системы.

Кроме того, свойства модуляции важны для понимания того, как частотные компоненты влияют на z-преобразование сигнала. Модуляция сигнала функцией косинуса или синуса приводит к тому, что z-преобразование сигнала оценивается в смещенных положениях. Это иллюстрирует, как различные частотные компоненты влияют на общее z-преобразование, облегчая анализ систем, включающих методы модуляции.

Эти свойства в совокупности обеспечивают надежную основу для анализа и понимания систем с дискретным временем. Они позволяют переводить операции из временного домена в z-домен, упрощая анализ и проектирование систем в системах цифровой обработки сигналов и управления. Используя эти свойства, инженеры могут более эффективно проектировать, анализировать и оптимизировать дискретные системы для различных приложений, обеспечивая точную и эффективную обработку сигналов.

 Свойства Z-преобразования

Некоторые свойства обеспечивают прочную основу для анализа систем дискретного времени с использованием Z-преобразования.
Рассматривая два сигнала дискретного времени, ...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

z-Transform

z-преобразование

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

z-transform rules for signals

The z-transform rules for signals help students analyze discrete-time systems in digital signal processing. These rules turn time-domain changes into simpler expressions in the z-domain, which makes signal work easier to follow.

Linearity is one key rule. It means the z-transform of a linear combination of two discrete-time signals equals the same linear combination of their individual z-transforms. This is useful when signals are added together or superimposed.

Time-shifting is another important rule. If a signal is delayed or advanced in time, its z-transform is multiplied by a factor that depends on the size of the shift. This helps show how a time delay changes the signal in the frequency domain. It is especially helpful when studying the response of systems to delayed inputs.

The transcript also covers frequency scaling. When a signal is multiplied by an exponential factor in the time domain, the result is a scaling operation in the z-domain. This makes it easier to study changes in a signal’s frequency features. It is important in modulation and demodulation.

Time reversal is another useful property. When the time axis is reversed, the z-transform uses the reciprocal of the z-transform variable. This helps explain what happens when a signal is reversed or played backward. It gives insight into how time inversion affects system behavior.

Modulation properties also play a major role. When a signal is modulated by a cosine or sine function, its z-transform is evaluated at shifted positions. This shows how frequency components affect the overall z-transform. Together, these rules provide a strong framework for analyzing discrete-time systems and for designing and optimizing digital signal processing and control systems.

Z-Transform Rules for Signals

definition of z-transform

z-Transform for Discrete-Time Signals

The z-transform is a key math tool for analyzing discrete-time signals and systems. It is the discrete-time counterpart of the Laplace transform used for continuous-time systems. In signal and systems analysis, it works alongside the discrete-time Fourier transform. The z-transform changes a discrete-time signal into a power series with a complex variable, z. Each term in the series represents one sampling instant of the signal. This form makes it easier to study the signal in detail and solve...

region of convergence

Z-Transform Convergence and Stability

The z-transform and its Region of Convergence (ROC) are key ideas in discrete-time signals and systems. The z-transform is a mathematical tool for analyzing signals that change at separate time steps. Its values do not always converge, so the ROC identifies the range of complex numbers z where the transform is valid. The ROC can appear in different shapes in the complex plane. It may lie inside a circle, outside a circle, or within an annulus, which is the space between two circles. For an...

properties of the z-transform i

The z-transform rules for signals help students analyze discrete-time systems in digital signal processing. These rules turn time-domain changes into simpler expressions in the z-domain, which makes signal work easier to follow. Linearity is one key rule. It means the z-transform of a linear combination of two discrete-time signals equals the same linear combination of their individual z-transforms. This is useful when signals are added together or superimposed. Time-shifting is another...

properties of the z-transform ii

Z-Transform Rules for Sum and Convolution

The z-transform rules for sum and convolution help analyze discrete-time signals in signal processing. These rules show how changes in the time domain connect to the z-domain, which is the transform domain used for discrete-time systems. The accumulation property comes from the accumulated sum of a discrete-time signal. By applying the time-shifting property, the z-transform of the summed signal can be found from the z-transform of the original signal. The result is a simple multiplicative...

inverse z-transform by partial fraction expansion

inverse-z-transform-by-partial-fraction-expansion

Inverse z-Transform Using Poles and Residues

The inverse z-transform converts a function from the z-domain back to the time domain. One useful approach is the partial fraction method. It breaks a complex function into simpler fractions with distinct coefficients. The process starts by identifying the poles of the function. A pole is a value that makes the function grow without a finite limit. The function is then written in terms of these poles, and each pole adds one term to the partial fraction form. Next, the coefficients for each...

difference equation solution using z-transform

difference-equation-solution-using-z-transform

Solving Discrete-Time Responses with z-Transform

The z-transform helps solve discrete-time systems that are written as linear difference equations. These equations are common in digital signal processing and control systems. For a higher-order difference equation, you need the input signal and the initial conditions up to one term less than the order of the equation. The method also makes delayed signals easier to handle. A delay in time becomes a shift in the z-domain. An advance in time is shifted in the opposite direction. This link...

relation of dft to z-transform

DFT Through the Unit Circle

The Discrete Fourier Transform (DFT) shows how a discrete-time signal breaks into frequency parts. It takes N samples from the time domain and gives N values in the frequency domain. Each value represents one frequency component. The z-transform helps explain this process. A z-transform represents a discrete sequence in the complex frequency domain by summing sequence terms multiplied by powers of a complex number. For causal sequences, which start at a fixed time and move forward, the sum can...

key terms and concepts

Explore key z-transform rules for discrete-time signals, including linearity, time-shifting, frequency scaling, time reversal, and modulation.

Z-Transform Rules for Signals | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

HS-PS4-5

Travelling Waves

A wave is a disturbance that propagates from its source, repeating itself periodically, and is typically associated with simple harmonic motion. ...

Wave Parameters

The simplest mechanical waves are associated with simple harmonic motion and repeat themselves for several cycles. These simple harmonic waves can be ...

Equations of Wave Motion

Mathematically, the motion of a wave can be studied using a wavefunction. Consider a string oscillating up and down in simple harmonic motion, having a ...

Graphing the Wave Function

Consider the wave equation for a sinusoidal wave moving in the positive x-direction. The wave equation is a function of both position and time. From the ...

/files/media/science-education/science-education-thumbs/14721.jpg

Velocity and Acceleration of a Wave

A wave propagates through a medium with a constant speed, known as a wave velocity. It is different from the speed of the particles of the medium, which ...

Speed of a Transverse Wave

The speed of a wave depends on the characteristics of the medium. For example, in the case of a guitar, the strings vibrate to produce the sound. The ...

/files/media/science-education/science-education-thumbs/14722.jpg

Kinetic and Potential Energy of a Wave

All forms of waves carry energy; this is directly visualized in nature. For instance, the waves of earthquakes are so intense that they can shake huge ...

Energy and Power of a Wave

The total energy associated with a wavelength is the sum of the potential energy and the kinetic energy. The average rate of energy transfer associated ...

Propagation of Waves

When a wave propagates from one medium to another, part of it may get reflected in the first medium, and part of it may get transmitted to the second ...

Sound Waves

Sound waves can be thought of as fluctuations in the pressure of a medium through which they propagate. Since the pressure also makes the medium&#39;s ...

Sound as Pressure Waves

Sound waves, which are longitudinal waves, can be modeled as the displacement amplitude varying as a function of the spatial and temporal coordinates. As ...

/files/media/science-education/science-education-thumbs/14724.jpg

Speed of Sound in Solids and Liquids

Most solids and liquids are incompressible&#8212;their densities remain constant throughout. In the presence of an external force, the molecules tend to ...

Speed of Sound in Gases

The speed of sound in a gaseous medium depends on various factors. Since gases constitute molecules that are free to move, they are highly compressible. ...

Doppler Effect - I

The Doppler effect and Doppler shift were named after the Austrian physicist and mathematician Christian Johann Doppler in 1842, who conducted experiments ...

Doppler Effect - II

The Doppler effect has several practical, real-world applications. For instance, meteorologists use Doppler radars to interpret weather events based on ...

Shock Waves

While deriving the Doppler formula for the observed frequency of a sound wave, it is assumed that the speed of sound in the medium is greater than the ...

Echo

The human ear cannot distinguish between two sources of sound if they happen to reach within a specific time interval, typically 0.1 seconds apart. More ...

/files/media/science-education/science-education-thumbs/14729.jpg

Electromagnetic Waves

James Clerk Maxwell formulated a single theory combining all the electric and magnetic effects scientists knew during that time, calling the phenomena his ...

Generating Electromagnetic Radiations

The German physicist Heinrich Hertz (1857&#8211;1894) was the first to generate and detect certain types of electromagnetic waves in the laboratory. ...

Electromagnetic Wave Equation

Maxwell&#39;s equations for electromagnetic fields are related to source charges, either static or moving. These fields act on a test charge, whose ...

propagation-speed-of-electromagnetic-waves

Propagation Speed of Electromagnetic Waves

Electromagnetic waves are consistent with Ampere&#39;s law. Assuming there is no conduction current Ampere&#39;s law is given as:


Consider a plane ...

Electromagnetic Waves in Matter

Electromagnetic waves can travel in the vacuum as well as in matter. For example light, which is an electromagnetic wave, can travel through air, water, ...

Energy Carried By Electromagnetic Waves

Anyone who has used a microwave oven knows there is energy in electromagnetic waves. Sometimes, this energy is obvious, such as in the summer sun&#39;s ...

The Wave Nature of Light

The nature of light has been a subject of inquiry since antiquity. In the seventeenth century, Isaac Newton performed experiments with lenses and prisms ...

Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling in various media. [Clarification Statement: Examples of data could include electromagnetic radiation traveling in a vacuum and glass, sound waves traveling through air and water, and seismic waves traveling through the earth.] [Assessment Boundary: Assessment is limited to algebraic relationships and describing those relationships qualitatively.]

HS-PS4-1

The Electromagnetic Spectrum

Electromagnetic waves are categorized according to their wavelengths and frequencies, giving the electromagnetic spectrum. These waves are classified as ...

Classification of Signals

Signals are categorized as Continuous-time or discrete-time, Periodic or aperiodic, Analog or digital, and Causal or noncausal.
A continuous-time signal ...

Discrete Fourier Transform

Consider a vibration sensor that continuously captures data in the form of a continuous time-dependent signal.
However, in reality, the sensor can only ...

Discrete-Time Fourier Series

The Discrete-Time Fourier Series is a counterpart to the Fourier-series expansion of continuous-time periodic signals.
Calculating the expansion ...

Sampling Theorem

Consider the continuous-time signal x(t), and a train of impulses where Ts,&#xA0; is the sampling interval, and fs, is the sampling frequency.
Multiplying ...

Properties of Fourier Transform I

In radio broadcasting, Fourier Transform properties are applied in simultaneous multi-channel transmission, adjusting audio clip speeds, live broadcast ...

Signal and System

Signals are sets of data or information.
One-dimensional signals include single-variable functions, such as the variations in air pressure over time in ...

Methods of Documentation VII: EMR

Electronic Medical Records (EMRs) primarily center around electronically documenting patients&#39; health information within a single healthcare ...

Fast Fourier Transform

The Fast Fourier Transform, FFT, is a computational algorithm for calculating the Discrete Fourier Transform by breaking the calculations into smaller, ...

reconstruction-of-signal-using-interpolation

Reconstruction of Signal using Interpolation

Consider a continuous signal and its sampled counterpart with a sampling period T.
Applying a zero-order hold to the sampled signal results in a piecewise ...

Downsampling

Consider a sampled sequence with zero values between sampling instants. Replace it by taking every N-th value of the sampled sequence.
The original and ...

Design Example

An engineer designs a circuit for a speaker system with a 2 kHz crossover frequency requirement, considering the speaker&#39;s resistance of 8 ohms.
The ...

Upsampling

Consider a decimated signal with a reduced frequency range due to its lower sampling rate.
Insert zeros between each sample to upsample, that introduces ...

Properties of DTFT I

Consider two discrete-time signals, each with their respective Discrete-Time Fourier Transforms DTFTs.
The signals are first multiplied by constants a and ...

Relation of DFT to z-Transform

The Discrete Fourier Transform (DFT) analyzes the frequency content of discrete-time signals.
It maps the N-sampled discrete time-domain sequence to its ...

Feedback control systems

Feedback control systems are categorized as linear or nonlinear, time-varying or time-invariant, and classified by signal types as continuous or ...

Certain properties provide a solid foundation for analyzing discrete-time systems using the Z-transform.
Considering two discrete-time signals, the ...

Magnetic Resonance Imaging

Magnetic resonance imaging (MRI) is a noninvasive medical imaging technique based on a phenomenon of nuclear physics discovered in the 1930s, in which ...

Evaluate questions about the advantages of using digital transmission and storage of information. [Clarification Statement: Examples of advantages could include that digital information is stable because it can be stored reliably in computer memory, transferred easily, and copied and shared rapidly. Disadvantages could include issues of easy deletion, security, and theft.]

Interference and Superposition of Waves

When two waves of the same nature occur in the same region simultaneously, they result in interference. Interference of waves implies that the net effect ...

Sound Waves: Interference

Sound waves can be modeled either as longitudinal waves, wherein the molecules of the medium oscillate around an equilibrium position, or as pressure ...

Interference: Path Lengths

Consider two sources of sound, that may or may not be in phase, emitting waves at a single frequency, and consider the frequencies to be the same.
Two ...

/files/media/science-education/science-education-thumbs/14727.jpg

concept-of-resonance-and-its-characteristics

Concept of Resonance and its Characteristics

If a driven oscillator needs to resonate at a specific frequency, then very light damping is required. An example of light damping includes playing piano ...

/files/media/science-education/science-education-thumbs/12772.jpg

Sound Waves: Resonance

Resonance is produced depending on the boundary conditions imposed on a wave. Resonance can be produced in a string under tension with symmetrical ...

Reflection of Waves

When a wave travels from one medium to another, it gets reflected at the boundary of the second medium. A common example of this is when a person yells at ...

Standing Waves

Sometimes waves do not seem to move; rather, they just vibrate in place. Unmoving waves can be seen on the surface of a glass of milk kept in a ...

/files/media/science-education/science-education-thumbs/12783.jpg

Modes of Standing Waves - I

A close look at earthquakes provides evidence for the conditions appropriate for resonance, standing waves, and constructive and destructive interference. ...

/files/media/science-education/science-education-thumbs/12784.jpg

Modes of Standing Waves: II

The starting point for expressing the modes of standing waves is understanding the boundary conditions that the waves must follow. The boundary conditions ...

/files/media/science-education/science-education-thumbs/14723.jpg

Beats

The study of music provides many examples of the superposition of waves and the constructive and destructive interference that occurs. Very few examples ...

/files/media/science-education/science-education-thumbs/14728.jpg

Interference and Diffraction

Source: Yong P. Chen, PhD, Department of Physics &#38; Astronomy, College of Science, Purdue University, West Lafayette, IN
Interference and diffraction ...

Reflection and Refraction

Source: Derek Wilson,&#160;Asantha Cooray, PhD, Department of Physics &#38; Astronomy, School of Physical Sciences, University of California, Irvine, CA
...

Electromagnetic Fields

Electric fields generated by static charges, often referred to as electrostatic fields, are characteristically different from electric fields created by ...

Photoelectric Effect

Source: Yong P. Chen, PhD, Department of Physics &#38;&#38; Astronomy, College of Science, Purdue University, West Lafayette, IN
Photoelectric effect ...

Plane Electromagnetic Waves I

The existence of combined electric and magnetic fields that propagate through space as electromagnetic (EM) waves is the most significant prediction of ...

Plane Electromagnetic Waves II

Consider a plane wavefront traveling in position x-direction with a constant speed. This wavefront can be utilized to obtain the relationship between ...

Evaluate the claims, evidence, and reasoning behind the idea that electromagnetic radiation can be described either by a wave model or a particle model, and that for some situations one model is more useful than the other. [Clarification Statement: Emphasis is on how the experimental evidence supports the claim and how a theory is generally modified in light of new evidence. Examples of a phenomenon could include resonance, interference, diffraction, and photoelectric effect.] [Assessment Boundary: Assessment does not include using quantum theory.]

HS-PS4-3

The electromagnetic spectrum consists of all the types of electromagnetic radiation arranged according to their frequency and wavelength. Each of the ...

Biological Effects of Radiation

All radioactive nuclides emit high-energy particles or electromagnetic waves. When this radiation encounters living cells, it can cause heating, break ...

Intensity Of Electromagnetic Waves

The energy transport per unit area per unit time, or the Poynting vector, gives the energy flux of an electromagnetic wave at any specific time. For a ...

When light of a particular wavelength strikes a metal surface, electrons are emitted. This is called the photoelectric effect. The minimum frequency of ...

Mutations

Mutations are changes in the sequence of DNA. These changes can occur spontaneously or they can be induced by exposure to environmental factors. Mutations ...

Radiation: Applications

The average temperature of Earth is the subject of much current discussion. Earth is in radiative contact with both the Sun and dark space; it receives ...

determination-of-nox-in-automobile-exhaust-using-uv-vis-spectroscopy

Determination Of NOx in Automobile Exhaust Using UV-VIS Spectroscopy

Source: Laboratories of Margaret Workman and Kimberly Frye - Depaul University
In the troposphere, ozone is naturally formed when sunlight splits nitrogen ...

Introduction to Fluorescence Microscopy

Fluorescence microscopy is a very powerful analytical tool that combines the magnifying properties of light microscopy with visualization of fluorescence. ...

Evaluate the validity and reliability of claims in published materials of the effects that different frequencies of electromagnetic radiation have when absorbed by matter. [Clarification Statement: Emphasis is on the idea that photons associated with different frequencies of light have different energies, and the damage to living tissue from electromagnetic radiation depends on the energy of the radiation. Examples of published materials could include trade books, magazines, web resources, videos, and other passages that may reflect bias.] [Assessment Boundary: Assessment is limited to qualitative descriptions.]

HS-PS4-4

Z-Transform Rules for Signals

Description

Transcript

More videos by Standards