Name: zero-inserted signal reconstruction
Uploaded: 2024-09-26T13:17:07.000Z
Duration: 1 min 22 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

Upsampling

Рассмотрим ослабленный сигнал с уменьшенным частотным диапазоном из-за более низкой частоты дискретизации.
Вставляйте нули между каждым сэмплом для повышения дискретизации, что вводит повторяющиеся спектральные реплики с интервалами, определяемыми новой частотой Найквиста.
Пропустите вставленную в ноль последовательность через фильтр нижних частот с частотой среза на новом пределе Найквиста. Этот фильтр ослабляет высокочастотные реплики, сохраняя только исходные частотные компоненты.
Отфильтрованный выходной сигнал дает сигнал с более высокой частотой дискретизации, эффективно поддерживая исходный сигнал и обращая вспять процесс понижения дискретизации.
Возьмем последовательность с преобразованием Фурье, показывающую ненулевые значения от -4π/9 до +4π/9.
Уменьшите его на два, в результате чего спектр будет от -8π/9 до + 8π/9.
Увеличьте его на восемь, сжимая преобразование Фурье до диапазона от -π/9 до + π/9.
Уменьшите его на девять, масштабируя преобразование Фурье, чтобы увеличить его от -π до +π.
Сочетание повышающей дискретизации на четыре и даунсемплинга на девять дает максимальную даунсемплинг без наложения.

 Повышение частоты дискретизации

Управление частотой дискретизации сигнала имеет важное значение в цифровой обработке сигналов для поддержания целостности сигнала. Прореженный сигнал, характеризующийся уменьшенным диапазоном частот из-за его более низкой частоты дискретизации, может быть повышен путем вставки нулей между каждым образцом. Этот процесс повышения частоты дискретизации расширяет исходный спектр и вводит повторяющиеся спектральные копии с интервалами, предписываемыми новой частотой Найквиста. Чтобы очистить эту последовательность с вставленными нулями, ее пропускают через фильтр нижних частот с частотой среза, установленной на новом пределе Найквиста. Этот фильтр ослабляет более высокочастотные копии, сохраняя только исходные частотные компоненты.

Результатом этого процесса фильтрации является сигнал с более высокой частотой дискретизации, который эффективно обращает процедуру понижения частоты дискретизации. Например, рассмотрим последовательность с преобразованием Фурье, демонстрирующую ненулевые значения от −2π/9 до 2π/9. Если для этой последовательность частота дискретизации понижается в четыре раза, ее спектр охватывает диапазон от −8π/9 до 8π/9. Впоследствии повышение частоты дискретизации последовательности в два раза сжимает преобразование Фурье, теперь в диапазоне от −π/9 до π/9.

Дальнейшее понижение частоты дискретизации в 9 раз для этой последовательности с повышенной частотой масштабирует преобразование Фурье, расширяя его от −2π/9 до 2π/9. Эта комбинация повышения частоты дискретизации в два раза и понижения частоты дискретизации в девять раз эквивалентна понижению частоты дискретизации в 9/2 раза, что позволяет достичь максимального понижения частоты дискретизации без введения наложения спектров.

Процесс повышения частоты дискретизации путем вставки нулей и последующей фильтрации нижних частот с последующими точными комбинациями повышения частоты дискретизации и понижения частоты дискретизации позволяет эффективно управлять частотой дискретизации сигнала. Этот метод гарантирует сохранение целостности исходного сигнала, предотвращая наложение спектров и искажение при адаптации к различным требованиям к выборке.

Такие методы имеют решающее значение в приложениях цифровой обработки сигналов, где баланс между эффективностью выборки и точностью сигнала имеет первостепенное значение. Тщательно регулируя частоты выборки с помощью этих процессов, можно сохранить основные характеристики исходного сигнала, что облегчает точную обработку и реконструкцию сигнала в различных технологических областях, включая связь, аудиотехнику и сжатие данных.

Рассмотрим ослабленный сигнал с уменьшенным частотным диапазоном из-за более низкой частоты дискретизации.
Вставьте нули между каждым сэмплом для повышения сэмплинга, что вводит ...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

Sampling

Выборка

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

zero-inserted signal reconstruction

Zero-inserted signal reconstruction is a digital signal processing method used to raise a signal’s sampling rate while keeping its important frequency content. A decimated signal has fewer samples and a narrower frequency range. To upsample it, zeros are inserted between the original samples.

This zero-stuffed sequence changes the spectrum. It creates repeated spectral replicas at intervals set by the new Nyquist frequency, which is the highest frequency allowed by the new sampling rate. A lowpass filter is then applied with a cutoff at that new Nyquist limit. The filter reduces the higher-frequency replicas and preserves the original signal components.

The filtered result is a higher-rate signal that effectively reverses the earlier downsampling step. The transcript also shows how sampling-rate changes can be combined. A signal with non-zero Fourier values from −2π/9 to 2π/9 becomes wider after downsampling by four. Upsampling that signal by two compresses the Fourier transform to −π/9 to π/9, and a later downsampling by nine extends it back to −2π/9 to 2π/9.

That sequence of upsampling by two and downsampling by nine is equivalent to downsampling by 9/2. It gives the maximum downsampling possible without aliasing, which is frequency overlap that can distort a signal. These steps are important in communications, audio engineering, and data compression, where signal fidelity must be protected while sampling rates are adjusted.

Zero-Inserted Signal Reconstruction

sampling theorem

Sampling Signals for Digital Use

Sampling signals for digital use turns a continuous-time signal, x(t), into a discrete-time signal. This step is essential when a real-world signal must be stored or processed by a digital system. A train of impulses sets the sampling interval, and the sampling frequency describes how often the signal is measured. The sampling process works by multiplying the continuous-time signal by the impulse train. This creates a series of discrete impulses called the sampled signal. Engineers then study...

sampling continuous time signal

Impulse-Train Sampling and Zero-Order Hold

Impulse-train sampling and zero-order hold are two key steps in digital signal processing. Together, they show how a continuous-time signal can be turned into samples and then built back into a continuous-looking waveform. Impulse-train sampling uses a periodic impulse train. This train is a series of delta functions placed at regular intervals set by the sampling period. When the continuous-time signal is multiplied by the impulse train, the result is a set of impulses whose amplitudes match...

reconstruction of signal using interpolation

reconstruction-of-signal-using-interpolation

Signal Reconstruction with Low-Pass Filtering

Signal reconstruction with low-pass filtering helps turn a sampled signal back into a smooth continuous signal. When a continuous signal is sampled with a period T, the sampled version creates replicas of the original spectrum in the frequency domain. These replicas are spaced by the sampling frequency. A zero-order hold can be used to process the sampled signal. This method keeps each sample value until the next sampling period, so the output becomes piecewise constant. In the frequency...

aliasing

Sampling Signals Without Aliasing

Sampling a signal at the right rate is essential for accurate signal reconstruction. A time-domain signal can be analyzed with its Fourier transform, which reveals its spectrum. When the signal is sampled, the spectrum becomes multiple scaled replicas in the frequency domain. The spacing between these replicas depends on the sampling frequency. If the sampling frequency is below the Nyquist rate, the replicas overlap. The Nyquist rate is twice the highest frequency in the original signal. Once...

downsampling

Decimation and Sampling Rate Reduction

Decimation reduces a sampled sequence by keeping every N-th sample. At those integer multiples of N, the original and reduced sequences match. This makes the sequence shorter and more efficient for digital signal processing. The Fourier transform of a decimated sequence is not just a simple copy of the original. It becomes a scaled and shifted combination of the original spectrum. In effect, the reduced sequence reflects the same frequency content, but in a form shaped by the new sampling...

upsampling

Zero-inserted signal reconstruction is a digital signal processing method used to raise a signal’s sampling rate while keeping its important frequency content. A decimated signal has fewer samples and a narrower frequency range. To upsample it, zeros are inserted between the original samples. This zero-stuffed sequence changes the spectrum. It creates repeated spectral replicas at intervals set by the new Nyquist frequency, which is the highest frequency allowed by the new sampling rate. A...

bandpass sampling

Sampling Narrow Frequency Bands

Bandpass sampling is a signal processing method for signals whose energy is concentrated in a narrow frequency band. Such a signal is called a bandpass signal. The method works when the sampling rate is greater than twice the signal bandwidth, which helps prevent aliasing. A bandpass signal has a spectrum with a lower limit, ω1, and an upper limit, ω2. Its spectrum is nonzero only between those two frequencies. To sample it, the time-domain signal is multiplied by an impulse train. This...

key terms and concepts

Learn how zero insertion and lowpass filtering rebuild a signal at a higher sampling rate while reducing spectral replicas and avoiding aliasing.

Zero-Inserted Signal Reconstruction | 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 ...

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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 ...

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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 ...

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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 ...

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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

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 ...

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 ...

Properties of the z-Transform I

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 ...

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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 ...

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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 ...

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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. ...

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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 ...

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Beats

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

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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. ...

Introduction to the Spectrophotometer

The spectrophotometer is a routinely used instrument in scientific research.  Spectrophotometry is the quantitative measurement of how much a chemical ...

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

Zero-Inserted Signal Reconstruction

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