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

Considere uma sequência amostrada com valores zero entre os instantes de amostragem. Substitua-o tomando cada N-ésimo valor da sequência amostrada.
As sequências original e amostrada são iguais em múltiplos inteiros de N.
A dizimação extrai todas as N-ésimas amostras de uma sequência, tornando a nova sequência mais eficiente.
A transformada de Fourier da sequência dizimada é uma combinação de versões dimensionadas e deslocadas do espectro original.
Essa transformação simplifica a análise, concentrando-se em intervalos diferentes de zero.
A relação final mostra que a transformada de Fourier da sequência dizimada é uma versão em escala da transformada do original.
Essa escala enfatiza a natureza periódica introduzida pela dizimação, com espectros diferindo apenas na escala de frequência.
Se o espectro original for limitado por banda sem aliasing, a dizimação espalha o espectro por uma banda de frequência maior.
A dizimação de uma sequência de um sinal de tempo contínuo reduz a taxa de amostragem por um fator de N, evitando aliasing se o sinal original for sobreamostrado.
Ao interpretar a sequência original como amostras de um sinal de tempo contínuo, a dizimação é chamada de downsampling.

downsampling

Ao considerar uma sequência amostrada com valores zero entre instantes de amostragem, pode-se substituí-la tomando cada N-ésimo valor da sequência. Nesses múltiplos inteiros de N, as sequências original e amostrada coincidem. Esse processo, conhecido como dizimação, envolve extrair cada N-ésima amostra de uma sequência, criando assim uma sequência mais eficiente. 


A transformada de Fourier da sequência dizimada revela uma combinação de versões escalonadas e deslocadas do espectro original. Essa transformação se concentra nos intervalos diferentes de zero da sequência, simplificando a análise. A relação entre as transformadas de Fourier das sequências original e dizimada mostra que a última é uma versão escalonada da primeira, enfatizando a natureza periódica introduzida pela dizimação. Os espectros da sequência dizimada diferem do original apenas em termos de escala de frequência. 


Se o espectro original for limitado por banda e livre de aliasing, a dizimação efetivamente espalha o espectro por uma banda de frequência maior. Essa dispersão ocorre porque a dizimação reduz a taxa de amostragem por um fator de N. Para evitar aliasing, é crucial que o sinal original seja superamostrado, o que significa que a frequência de amostragem é suficientemente alta em relação ao componente de frequência mais alto do sinal. 

Em termos práticos, dizimar uma sequência derivada de um sinal de tempo contínuo também é conhecido como downsampling. Esse processo reduz a taxa de dados, tornando-a mais gerenciável, preservando as características essenciais do sinal original. Quando a sequência original é interpretada como amostras de um sinal de tempo contínuo, deve-se considerar cuidadosamente o teorema de amostragem para garantir que não haja perda de informações devido ao aliasing. 

A dizimação é uma técnica valiosa no processamento de sinais digitais, permitindo um tratamento e análise de dados mais eficientes. Ao reduzir o número de amostras e manter informações espectrais críticas, a dizimação permite o processamento e a transmissão eficazes de sinais em várias aplicações, incluindo telecomunicações, processamento de áudio e compressão de dados. Garantir que o sinal original seja adequadamente superamostrado antes da dizimação é essencial para evitar aliasing e preservar a integridade do sinal reconstruído.

Considere uma sequência amostrada com valores zero entre os instantes de amostragem. Substitua-o tomando cada N-ésimo valor da sequência amostrada.
O original e ...

Education

JoVE Core

Electrical Engineering

Engenharia Elétrica

Unit 1

Unidade 1

Sampling

Amostragem

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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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