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

Stellen Sie sich eine Stichprobensequenz mit Nullwerten zwischen den Sampling-Zeitpunkten vor. Ersetzen Sie ihn, indem Sie jeden N-ten Wert der abgetasteten Sequenz verwenden.
Die ursprüngliche und die abgetastete Sequenz sind bei ganzzahligen Vielfachen von N gleich.
Bei der Dezimierung wird jede N-te Probe aus einer Sequenz extrahiert, wodurch die neue Sequenz effizienter wird.
Die Fourier-Transformation der dezimierten Sequenz ist eine Kombination aus skalierten und verschobenen Versionen des ursprünglichen Spektrums.
Diese Transformation vereinfacht die Analyse, indem sie sich auf Intervalle ungleich Null konzentriert.
Die letzte Beziehung zeigt, dass die Fourier-Transformation der dezimierten Sequenz eine skalierte Version der Transformation des Originals ist.
Diese Skalierung betont die periodische Natur, die durch die Dezimierung eingeführt wird, wobei sich die Spektren nur in der Frequenzskalierung unterscheiden.
Wenn das ursprüngliche Spektrum ohne Aliasing bandbegrenzt ist, wird das Spektrum durch Dezimierung auf ein größeres Frequenzband verteilt.
Durch das Dezimieren einer Sequenz aus einem zeitkontinuierlichen Signal wird die Abtastrate um den Faktor N reduziert, wodurch Aliasing vermieden wird, wenn das Originalsignal überabgetastet wird.
Bei der Interpretation der ursprünglichen Sequenz als Samples aus einem zeitkontinuierlichen Signal wird die Dezimierung als Downsampling bezeichnet.

downsampling

Wenn man eine abgetastete Sequenz mit Nullwerten zwischen den Abtastzeitpunkten betrachtet, kann man sie ersetzen, indem man jeden N-ten Wert der Sequenz nimmt. Bei diesen ganzzahligen Vielfachen von N stimmen die ursprüngliche und die abgetastete Sequenz überein. Bei diesem als Dezimierung bezeichneten Prozess wird jede N-te Probe aus einer Sequenz extrahiert, wodurch eine effizientere Sequenz erstellt wird.

Die Fourier-Transformation der dezimierten Sequenz zeigt eine Kombination aus skalierten und verschobenen Versionen des ursprünglichen Spektrums. Diese Transformation konzentriert sich auf die von Null verschiedenen Intervalle der Sequenz und vereinfacht so die Analyse. Die Beziehung zwischen den Fourier-Transformationen der ursprünglichen und der dezimierten Sequenz zeigt, dass Letztere eine skalierte Version der Ersteren ist, was die durch die Dezimierung eingeführte periodische Natur betont. Die Spektren der dezimierten Sequenz unterscheiden sich vom Original nur in Bezug auf die Frequenzskalierung.

Wenn das ursprüngliche Spektrum bandbegrenzt und frei von Aliasing ist, verteilt die Dezimierung das Spektrum effektiv über ein größeres Frequenzband. Diese Spreizung tritt auf, weil die Dezimierung die Abtastrate um den Faktor N reduziert. Um Aliasing zu vermeiden, ist es wichtig, dass das Originalsignal überabgetastet wird, d. h. die Abtastfrequenz im Verhältnis zur höchsten Frequenzkomponente des Signals ausreichend hoch ist.

In der Praxis wird die Dezimierung einer aus einem kontinuierlichen Signal abgeleiteten Sequenz auch als Downsampling bezeichnet. Dieser Prozess reduziert die Datenrate und macht sie überschaubarer, während wesentliche Eigenschaften des Originalsignals erhalten bleiben. Wenn die Originalsequenz als Samples aus einem kontinuierlichen Signal interpretiert wird, muss das Abtasttheorem sorgfältig beachtet werden, um sicherzustellen, dass kein Informationsverlust durch Aliasing auftritt.

Dezimierung ist eine wertvolle Technik in der digitalen Signalverarbeitung, die eine effizientere Datenhandhabung und -analyse ermöglicht. Durch die Reduzierung der Anzahl der Samples und die Beibehaltung kritischer Spektralinformationen ermöglicht die Dezimierung eine effektive Verarbeitung und Übertragung von Signalen in verschiedenen Anwendungen, einschließlich Telekommunikation, Audioverarbeitung und Datenkomprimierung. Um Aliasing zu verhindern und die Integrität des rekonstruierten Signals zu bewahren, muss sichergestellt werden, dass das Originalsignal vor der Dezimierung ausreichend überabgetastet wird.

Stellen Sie sich eine Stichprobensequenz mit Nullwerten zwischen den Sampling-Zeitpunkten vor. Ersetzen Sie ihn, indem Sie jeden N-ten Wert der abgetasteten Sequenz verwenden.
Das Original und die ...

Education

Lehre

JoVE Core

Electrical Engineering

Elektrotechnik

Unit 1

Einheit 1

Sampling

Stichprobenerhebung

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

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...

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-PS4-5

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HS-PS4-4

Decimation and Sampling Rate Reduction

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