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

Considérons un signal décimé avec une gamme de fréquences réduite en raison de son taux d’échantillonnage plus faible.
Insérez des zéros entre chaque échantillon pour suréchantillonner, ce qui introduit des répliques spectrales répétées à des intervalles déterminés par la nouvelle fréquence de Nyquist.
Passez la séquence insérée à zéro à travers un filtre passe-bas avec une fréquence de coupure à la nouvelle limite de Nyquist. Ce filtre atténue les répliques à haute fréquence, en ne conservant que les composants de fréquence d’origine.
La sortie filtrée produit un signal à taux d'échantillonnage plus élevé, ce qui permet de maintenir efficacement le signal d'origine et d'inverser le processus de sous-échantillonnage.
Prenons une séquence avec une transformée de Fourier montrant des valeurs non nulles de -4π/9 à +4π/9.
Sous-échantillonnez par deux, ce qui donne un spectre allant de -8π/9 à + 8π/9.
Suréchantillonnez-le de huit, en compressant la transformée de Fourier pour qu’elle s’étende de -π/9 à + π/9.
Sous-échantillonnez-le par neuf, en mettant à l’échelle la transformée de Fourier pour qu’elle s’étende de -π à +π.
La combinaison du suréchantillonnage par quatre et du sous-échantillonnage par neuf donne le sous-échantillonnage maximal sans aliasing.

suréchantillonnage

La gestion des taux d'échantillonnage du signal est essentielle dans le traitement du signal numérique pour maintenir l'intégrité du signal. Un signal décimé, caractérisé par une plage de fréquences réduite en raison de son taux d'échantillonnage inférieur, peut être suréchantillonné en insérant des zéros entre chaque échantillon. Ce processus de suréchantillonnage élargit le spectre d'origine et introduit des répliques spectrales répétées à des intervalles dictés par la nouvelle fréquence de Nyquist. Pour affiner cette séquence insérée de zéros, on la fait passer par un filtre passe-bas dont la fréquence de coupure est ajustée à la nouvelle limite de Nyquist. Ce filtre atténue les répliques à haute fréquence, en ne conservant uniquement que les composantes de fréquence initiales.

Le résultat de ce processus de filtrage est un signal avec un taux d'échantillonnage plus élevé qui inverse efficacement la procédure de sous-échantillonnage. Par exemple, considérons une séquence avec une transformée de Fourier présentant des valeurs non nulles de −2π/9 à 2π/9. Si cette séquence est sous-échantillonnée d'un facteur quatre, son spectre s'étend de −8π/9 à 8π/9. Par la suite, le suréchantillonnage de la séquence par un facteur deux comprime la transformée de Fourier, dont le spectre s'étend désormais de −π/9 à π/9.

Un sous-échantillonnage supplémentaire de cette séquence suréchantillonnée de neuf étend la transformée de Fourier de −2π/9 à 2π/9. Cette combinaison d’un suréchantillonnage de deux et d’un sous-échantillonnage de neuf équivaut à un sous-échantillonnage d’un facteur 9/2, ce qui permet d'obtenir le sous-échantillonnage maximal sans introduire d’aliasing.

Le processus de suréchantillonnage par insertion de zéros suivi d’un filtrage passe-bas, puis de combinaisons précises de suréchantillonnage et de sous-échantillonnage, permet de gérer efficacement le taux d'échantillonnage du signal. Cette méthode garantit le maintien de l'intégrité du signal d'origine, en évitant le repliement et la distorsion tout en s'adaptant aux différentes exigences d'échantillonnage.

De telles techniques sont cruciales dans les applications de traitement du signal numérique, où l'équilibre entre l'efficacité de l'échantillonnage et la fidélité du signal est primordial. En ajustant soigneusement les taux d’échantillonnage par le biais de ces processus, il est possible de conserver les caractéristiques essentielles du signal d’origine, ce qui facilite le traitement et la reconstruction précise du signal dans divers domaines technologiques, notamment les communications, l’ingénierie audio et la compression de données.

Suréchantillonnage

Considérons un signal décimé avec une gamme de fréquences réduite en raison de son taux d’échantillonnage plus faible.
Insérez des zéros entre chaque échantillon pour suréchantillonner, cela introduit ...

Education

Enseignement

JoVE Core

Electrical Engineering

Génie électrique

Unit 1

Unité 1

Sampling

Échantillonnage

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

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

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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

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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

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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 ...

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Echo

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Electromagnetic Wave Equation

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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

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The Wave Nature of Light

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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.
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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.
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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 ...

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

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Interference: Path Lengths

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Two ...

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Concept of Resonance and its Characteristics

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Modes of Standing Waves: II

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Beats

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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
...

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Photoelectric effect ...

Plane Electromagnetic Waves I

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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

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Intensity Of Electromagnetic Waves

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Radiation: Applications

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Determination Of NOx in Automobile Exhaust Using UV-VIS Spectroscopy

Source: Laboratories of Margaret Workman and Kimberly Frye - Depaul University
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Introduction to Fluorescence Microscopy

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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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