Name: decimation and sampling rate reduction
Uploaded: 2024-09-26T13:17:07.000Z
Duration: 1 min 20 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

Downsampling

Considérons une séquence échantillonnée avec des valeurs nulles entre les instants d’échantillonnage. Remplacez-le en prenant chaque N-ième valeur de la séquence échantillonnée.
Les séquences d’origine et échantillonnées sont égales à des multiples entiers de N.
La décimation extrait chaque Nième échantillon d’une séquence, ce qui rend la nouvelle séquence plus efficace.
La transformée de Fourier de la séquence décimée est une combinaison de versions mises à l’échelle et décalées du spectre d’origine.
Cette transformation simplifie l’analyse en se concentrant sur les intervalles non nuls.
La relation finale montre que la transformée de Fourier de la séquence décimée est une version mise à l'échelle de la transformée de l'original.
Cette échelle met l’accent sur la nature périodique introduite par la décimation, les spectres ne différant que par l’échelle de fréquence.
Si le spectre d’origine est limité en bande passante sans repliement, la décimation étend le spectre sur une bande de fréquences plus grande.
La décimation d’une séquence à partir d’un signal en temps continu réduit la fréquence d’échantillonnage d’un facteur N, évitant ainsi le repliement si le signal d’origine est suréchantillonné.
Lors de l’interprétation de la séquence originale comme des échantillons d’un signal en temps continu, la décimation est appelée sous-échantillonnage.

sous-échantillonnage

Lorsque l'on considère une séquence échantillonnée avec des valeurs nulles entre les instants d'échantillonnage, on peut la remplacer en prenant chaque N-ième valeur de la séquence. À ces multiples entiers de N, les séquences d'origine et échantillonnées coïncident. Ce processus, appelé décimation, consiste à extraire chaque N-ième échantillon d'une séquence, créant ainsi une séquence plus efficace.

La transformée de Fourier de la séquence décimée révèle une combinaison de versions mises à l'échelle et décalées du spectre d'origine. Cette transformation se concentre sur les intervalles non nuls de la séquence, ce qui simplifie l'analyse. La relation entre les transformées de Fourier des séquences d'origine et décimées montre que cette dernière est une version mise à l'échelle de la première, soulignant la nature périodique introduite par la décimation. Les spectres de la séquence décimée ne diffèrent de l'original qu'en termes de mise à l'échelle des fréquences.

Si le spectre d'origine est limité en bande et exempt de repliement, la décimation étale efficacement le spectre sur une bande de fréquence plus large. Cette propagation se produit parce que la décimation réduit le taux d'échantillonnage d'un facteur N. Pour éviter l'aliasing, il est essentiel que le signal d'origine soit suréchantillonné, ce qui signifie que la fréquence d'échantillonnage soit suffisamment élevée par rapport à la composante de fréquence la plus élevée du signal.

En termes pratiques, la décimation d'une séquence dérivée d'un signal en temps continu est également connue sous le nom de sous-échantillonnage. Ce processus réduit le débit de données, le rendant plus gérable tout en préservant les caractéristiques essentielles du signal d'origine. Lorsque la séquence d'origine est interprétée comme des échantillons d'un signal en temps continu, une attention particulière doit être accordée au théorème d'échantillonnage pour garantir l'absence de perte d'informations due à l'aliasing.

La décimation est une technique précieuse dans le traitement du signal numérique, qui permet une gestion et une analyse des données plus efficaces. En réduisant le nombre d'échantillons et en conservant les informations spectrales critiques, la décimation permet un traitement et une transmission efficaces des signaux dans diverses applications, notamment les télécommunications, le traitement audio et la compression de données. Il est essentiel de s'assurer que le signal d'origine soit correctement suréchantillonné avant la décimation pour éviter l'aliasing et préserver l'intégrité du signal reconstruit.

Considérons une séquence échantillonnée avec des valeurs nulles entre les instants d’échantillonnage. Remplacez-le en prenant chaque N-ième valeur de la séquence échantillonnée.
L’original et ...

Education

Enseignement

JoVE Core

Electrical Engineering

Génie électrique

Unit 1

Unité 1

Sampling

Échantillonnage

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

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 pattern.

Decimation also changes the apparent frequency range. If the original signal is band-limited and has no aliasing, the spectrum spreads over a larger frequency band after the sampling rate is reduced by a factor of N. This is why the original signal must be oversampled, meaning the sampling frequency is high enough compared with the signal’s highest frequency component.

When a sequence comes from a continuous-time signal, decimation is also called downsampling. It lowers the data rate while keeping the important features of the signal. Care is needed to follow the sampling theorem, because aliasing can cause information loss. Decimation is widely used in telecommunications, audio processing, and data compression because it helps make signals easier to store, transmit, and analyze.

Decimation and Sampling Rate Reduction

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 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 decimation keeps every N-th sample, reduces data rate, and changes the spectrum. See why oversampling helps prevent aliasing.

Decimation and Sampling Rate Reduction | 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.

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HS-PS4-4

Decimation and Sampling Rate Reduction

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