Name: signal reconstruction with low-pass filtering
Uploaded: 2024-09-26T13:17:07.000Z
Duration: 1 min 10 s
Channel: JoVE

K12 Standards

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New Jersey Student Learning Standards for Science

High School

Physical Science

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Waves and Their Applications in Technologies for Information Transfer

HS-PS4-2

reconstruction-of-signal-using-interpolation

Reconstruction of Signal using Interpolation

Betrachten Sie ein kontinuierliches Signal und sein abgetastetes Gegenstück mit einer Abtastperiode T.
Das Anwenden eines Holds nullter Ordnung auf das abgetastete Signal führt zu einem stückweise konstanten Signal.
Die Kaskadierung des Nullhaltepunkts mit einem Rekonstruktionsfilter ergibt das rekonstruierte Signal.
Ein idealer Tiefpassfilter entfernt alle Spektralrepliken im Frequenzbereich für eine optimale Rekonstruktion.
Die Impulsantwort dieses idealen Filters im Zeitbereich ist eine Sinc-Funktion.
Die Faltung des abgetasteten Signals mit der sinc-Funktion führt zum Zeitbereichssignal.
Dieser Prozess, der als bandbegrenzte Interpolation bezeichnet wird, gewährleistet eine genaue Signalrekonstruktion.
Die Faltung des abgetasteten Signals mit der dreieckigen Impulsantwort führt zu einem glatteren, spitzenfreien Zeitbereichssignal.
Dies wird als Hold erster Ordnung bezeichnet, der allgemein als lineare Interpolation bezeichnet wird.
Im Frequenzbereich wird die Kurve in der Mitte geglättet und an den Seiten gestaucht, wodurch seitliche Nachbildungen effektiver reduziert werden als bei einem Halten nullter Ordnung.

 rekonstruktion eines signals durch interpolation

Signalverarbeitungstechniken sind unerlässlich, um kontinuierliche Signale präzise in digitale Formate und umgekehrt umzuwandeln. Wenn ein kontinuierliches Signal mit einer Periode T abgetastet wird, weist das resultierende abgetastete Signal Nachbildungen des ursprünglichen Spektrums im Frequenzbereich auf, die in Abständen angeordnet sind, die der Abtastfrequenz entsprechen. Zur Verarbeitung dieses abgetasteten Signals kann ein Abtast-Halte-Glied angewendet werden, das ein stückweise konstantes Signal erzeugt, indem der Wert jeder Abtastung bis zur nächsten Abtastperiode beibehalten wird.

Im Frequenzbereich modifiziert das Abtast-Halte-Glied  das Signal, indem es aufgrund ihrer Fourier-Transformation eine Sinc-Funktion einführt. Diese Sinc-Funktion moduliert die Amplitude der spektralen Replikate und dämpft sie. Trotz dieser Modulation muss das abgetastete Signal noch weiter verarbeitet werden, um die resultierende Wellenform zu glätten.

Die Faltung des abgetasteten Signals mit einer dreieckigen Impulsantwort verfeinert das Signal weiter. Diese Faltungsoperation führt zu einem Zeitbereichssignal, das glatter und frei von abrupten Spitzen ist. Im Frequenzbereich glättet dieser Prozess den zentralen Teil der Kurve und komprimiert die seitlichen Nachbildungen effektiver als das Nullordnungshalten allein, wodurch unerwünschte Spektralkomponenten reduziert werden.

Für eine optimale Signalrekonstruktion wird ein idealer Tiefpassfilter verwendet. Dieser Filter entfernt alle Spektralnachbildungen und lässt nur das ursprüngliche Spektrum durch. Die Impulsantwort im Zeitbereich des idealen Filters ist durch eine Sinc-Funktion gekennzeichnet, die, wenn sie mit dem abgetasteten Signal gefaltet wird, ein glattes und kontinuierliches Zeitbereichssignal erzeugt.

Diese Methode, bekannt als bandbegrenzte Interpolation, stellt sicher, dass das rekonstruierte Signal dem ursprünglichen kontinuierlichen Signal sehr nahe kommt. Durch sorgfältiges Filtern des abgetasteten Signals und Ausnutzen der Eigenschaften der Sinc-Funktion minimiert die bandbegrenzte Interpolation Verzerrungen und Artefakte und erreicht so eine genaue Signalrekonstruktion. Dieser Prozess ist in Anwendungen wie digitalen Audio- und Kommunikationssystemen von entscheidender Bedeutung, bei denen die Aufrechterhaltung der Signalintegrität während der Digital-Analog-Umwandlung für die Bewahrung der Qualität und Wiedergabetreue des Originalsignals von entscheidender Bedeutung ist.

 Rekonstruktion eines Signals durch Interpolation

Betrachten Sie ein kontinuierliches Signal und sein abgetastetes Gegenstück mit einer Abtastperiode T.
Das Anwenden eines Holds nullter Ordnung auf das abgetastete Signal führt zu einer stückweisen ...

Education

Lehre

JoVE Core

Electrical Engineering

Elektrotechnik

Unit 1

Einheit 1

Sampling

Stichprobenerhebung

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

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 domain, the zero-order hold adds a sinc function because of its Fourier transform. That sinc shape changes the amplitudes of the spectral replicas and reduces them.

Even after zero-order hold, the waveform still needs more smoothing. Convolution with a triangular impulse response improves the signal further. In the time domain, this produces a smoother signal without sharp peaks. In the frequency domain, it smooths the central part of the curve and compresses the side replicas more effectively than the zero-order hold alone.

For the best reconstruction, an ideal low-pass filter is used. This filter removes the spectral replicas and lets only the original spectrum pass. Its time-domain impulse response is a sinc function. When this filter is convolved with the sampled signal, it creates a smooth and continuous result. This process is called band-limited interpolation, and it helps the reconstructed signal closely match the original continuous signal. It is important in digital audio and communication systems, where signal quality must be preserved during digital-to-analog conversion.

Signal Reconstruction with Low-Pass Filtering

signal-reconstruction-with-low-pass-filtering

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

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

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 sampled signals are smoothed with zero-order hold, triangular convolution, and ideal low-pass filtering for accurate signal reconstruction.

Signal Reconstruction with Low-Pass Filtering | 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

Next Generation Science Standards

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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problem-solving-tuning-of-a-guitar-string

Problem-Solving: Tuning of a Guitar String

In the case of stringed instruments like the guitar, the elastic property that determines the speed of the sound produced is its linear mass density or ...

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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 ...

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 ...

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 ...

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 ...

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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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

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. ...

Sinusoidal Sources

Direct current refers to unidirectional current, whereas alternating current refers to current that reverses its direction periodically.
Alternating ...

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 ...

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 ...

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 ...

Evaluate questions about the advantages of using a 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.

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Dual Nature of Electromagnetic (EM) Radiation

Electromagnetic (EM) radiation consists of electric and magnetic field components oscillating in planes perpendicular to each other and mutually ...

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, ...

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. ...

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

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 ...

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

Control Systems

Control systems influence many applications, from space shuttles to automated manufacturing. They exist naturally, such as our body&#39;s blood sugar ...

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

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 ...

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 ...

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 ...

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 ...

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 ...

Standing Electromagnetic Waves

Electromagnetic waves can be reflected; the surface of a conductor or a dielectric can act as a reflector. As electric and magnetic fields obey the ...

Standing Waves in a Cavity

A household microwave and lasers are examples of standing electromagnetic waves in a cavity. When two conducting metal plates are placed parallel at the ...

Communicate technical information about how some technological devices use the principles of wave behavior and wave interactions with matter to transmit and capture information and energy. 

Clarification statement: Examples could include solar cells capturing light and converting it to electricity; medical imaging; and communications technology. 

Assessment Boundary: Assessments are limited to qualitative information. Assessments do not include band theory.

Signal Reconstruction with Low-Pass Filtering

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