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

Si consideri una sequenza campionata con valori pari a zero tra gli istanti di campionamento. Sostituirlo prendendo ogni valore N-esimo della sequenza campionata.
Le sequenze originali e campionate sono uguali a multipli interi di N.
La decimazione estrae ogni N-esimo campione da una sequenza, rendendo la nuova sequenza più efficiente.
La trasformata di Fourier della sequenza decimata è una combinazione di versioni scalate e spostate dello spettro originale.
Questa trasformazione semplifica l'analisi concentrandosi su intervalli diversi da zero.
La relazione finale mostra che la trasformata di Fourier della sequenza decimata è una versione in scala della trasformata originale.
Questa scala enfatizza la natura periodica introdotta dalla decimazione, con spettri che differiscono solo nella scala di frequenza.
Se lo spettro originale è a banda limitata senza aliasing, la decimazione distribuisce lo spettro su una banda di frequenza più ampia.
La decimazione di una sequenza da un segnale a tempo continuo riduce la frequenza di campionamento di un fattore N, evitando l'aliasing se il segnale originale viene sovracampionato.
Quando si interpreta la sequenza originale come campioni da un segnale a tempo continuo, la decimazione è chiamata downsampling.

downsampling

Quando si considera una sequenza campionata con valori zero tra gli istanti di campionamento, è possibile sostituirla prendendo ogni N-esimo valore della sequenza. A questi multipli interi di N, sia la sequenza originale che campionata coincidono. Questo processo, noto come decimazione, comporta l'estrazione di ogni N-esimo campione da una sequenza, creando così una sequenza più efficiente.


La trasformata di Fourier della sequenza decimata rivela una combinazione di versioni scalate e spostate dello spettro originale. Questa trasformazione si concentra sugli intervalli diversi da zero della sequenza, semplificando l'analisi. La relazione tra le trasformate di Fourier della sequenza originale e decimata mostra che quest'ultima è una versione scalata della prima, sottolineando la natura periodica introdotta dalla decimazione. Gli spettri della sequenza decimata differiscono dall'originale solo in termini di scala di frequenza.


Se lo spettro originale è limitato in banda e privo di aliasing, la decimazione distribuisce efficacemente lo spettro su una banda di frequenza più ampia. Questa diffusione avviene perché la decimazione riduce la frequenza di campionamento di un fattore N. Per evitare l'aliasing, è fondamentale che il segnale originale sia sovracampionato, ovvero che la frequenza di campionamento sia sufficientemente alta rispetto alla componente di frequenza più alta del segnale.


In termini pratici, la decimazione di una sequenza derivata da un segnale a tempo continuo è anche nota come sottocampionamento. Questo processo riduce la velocità dei dati, rendendola più gestibile e preservando le caratteristiche essenziali del segnale originale. Quando la sequenza originale viene interpretata come un campione da un segnale a tempo continuo, è necessario prestare molta attenzione al teorema del campionamento per garantire che non vi siano perdite di informazioni dovute all'aliasing.

La decimazione è una tecnica preziosa nell'elaborazione del segnale digitale, che consente una gestione e un'analisi dei dati più efficienti. Riducendo il numero di campioni e mantenendo informazioni spettrali critiche, la decimazione consente un'elaborazione e una trasmissione efficace dei segnali nelle varie applicazioni, tra cui le telecomunicazioni, l’elaborazione audio e la compressione dei dati. Garantire che il segnale originale sia adeguatamente sovracampionato prima della decimazione è fondamentale per prevenire l'aliasing e preservare l'integrità del segnale ricostruito.

Si consideri una sequenza campionata con valori pari a zero tra gli istanti di campionamento. Sostituirlo prendendo ogni valore N-esimo della sequenza campionata.
L'originale e ...

Education

Didattica

JoVE Core

Electrical Engineering

Ingegneria elettrica

Unit 1

Unità 1

Sampling

Campionamento

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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

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sampling continuous time signal

Impulse-Train Sampling and Zero-Order Hold

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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...

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

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HS-PS4-4

Decimation and Sampling Rate Reduction

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