Name: impulse-train sampling and zero-order hold
Uploaded: 2024-09-26T13:17:07.000Z
Duration: 1 min 11 s
Channel: JoVE

使用冲激串采样，随后采用零阶保持法，对连续时间信号进行采样。
脉冲序列采样使用周期性脉冲序列，即一系列以采样周期的倍数间隔分布的δ函数。
当连续时间信号与脉冲串相乘时，会在采样点处产生幅度等于信号值的脉冲。
在频域中，采样会导致原始信号的频谱与脉冲序列的频谱发生卷积。
脉冲序列的频谱包含信号频谱的平移副本，这些副本以采样频率的整数倍为间隔分布。
这种卷积产生一个周期性函数，其采样频谱具有平移的副本，并按采样周期的倒数进行缩放。
零阶保持法将每个采样值保持恒定，直至下一个采样周期，从而生成一个分段常值信号。
该分段常数信号经过一个具有矩形脉冲响应的系统处理，产生一个稳定的输出。

连续时间信号采样

在信号处理的过程中，可以使用脉冲序列采样技术来对连续时间信号进行采样，然后再使用零阶保持法。脉冲序列采样通常会涉及到使用周期性脉冲序列，该序列是由一系列由采样周期确定的规则间隔来进行排列的 delta 函数所组成的。当连续时间信号与该脉冲序列进行相乘时，便会产生幅度与采样点处的信号值相对应的脉冲。

在频域中，可以将采样表示为原始信号频谱与脉冲序列频谱的卷积。脉冲序列的频谱是由原始信号频谱的移位副本所组成的，其中的间隔等于采样频率。该卷积会产生一个周期函数，从而使得采样信号的频谱能够由这些移位副本组成，并按采样周期的倒数来进行缩放。

通常会将零阶保持法用于在采样后重建信号的过程中。它能够将每个采样值保持在一个恒定的状态，直到下一个采样周期，从而使其能够得到一个分段恒定的信号。这种方法通过在下一个采样到达之前保持每个样本的幅度的方式来近似原始的连续时间信号，从而使其能够有效地创建出一个类似于阶梯状的波形。

然后，通过一个以矩形脉冲响应为特征的系统对该分段恒定信号进行处理。该系统平滑了保持值之间的转换过程，从而使其能够得出一个近似于原始信号的稳定输出。零阶保持法在数模转换中发挥着重要作用，它为从离散采样生成连续时间信号的过程提供了一种简单有效的方法。

从本质上来讲，脉冲序列采样和零阶保持方法共同构成了数字信号处理的基本过程，这使得连续时间信号能够转换为离散时间信号，并在随后进行重建。这一过程在数字音频、电信和数据采集系统等各种应用中是至关重要的，因为这可以确保在数字域中能够准确地表示和恢复模拟信号。

使用冲激串采样法对连续时间信号进行采样，随后采用零阶保持法。  
冲激串采样使用一个周期性冲激……

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Sampling

采样

KEY TERMS AND CONCEPTS

关键术语与概念

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 the signal values at the sampling points.

The same idea can be described in the frequency domain. Sampling is represented by convolution of the original signal spectrum with the spectrum of the impulse train. The impulse train spectrum creates shifted copies of the original spectrum, spaced by the sampling frequency. The result is a periodic spectrum made of these replicas, scaled by the inverse of the sampling period.

After sampling, the zero-order hold method is used to reconstruct the signal. It keeps each sampled value constant until the next sample arrives, so the output becomes piecewise constant. This creates a staircase-like waveform that is then passed through a system with a rectangular impulse response. That system smooths the transitions between held values and produces a steady output that approximates the original signal.

This method is especially useful in digital-to-analog conversion. It provides a simple way to turn discrete samples back into a continuous-time signal. Impulse-train sampling and zero-order hold are both important in digital audio, telecommunications, and data acquisition systems, where accurate signal representation and recovery matter.

Impulse-Train Sampling and Zero-Order Hold

impulse-train-sampling-and-zero-order-hold

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 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 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 impulse-train sampling creates discrete samples from a continuous signal and how zero-order hold rebuilds a staircase-like analog output.

Impulse-Train Sampling and Zero-Order Hold | 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

Impulse-Train Sampling and Zero-Order Hold

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