Name: sampling signals without aliasing
Uploaded: 2024-09-26T13:17:07.000Z
Duration: 1 min 18 s
Channel: JoVE

考虑一个时域信号及其傅里叶变换，以揭示其频谱。
以特定频率对信号进行采样会生成原始频谱的多个缩放副本。
如果采样频率低于奈奎斯特频率，这些复制频谱将发生重叠，从而无法通过低通滤波器恢复原始信号。
这种重叠效应被称为混叠，会使重建的信号失真。
考虑一个正弦信号及其频谱，分析采样信号的频谱需要在固定采样频率下考察基频的不同取值。
当基频小于采样频率的一半时，提高基频会导致输出频率升高。&#160;
相反，当基频处于采样频率的一半与采样频率之间时，提高基频会降低输出频率。
由于混叠效应，重建后的信号无法恢复到原始形态。
只有当采样频率超过奈奎斯特速率时，才能准确重建原始信号，从而避免混叠现象。

混叠

在各种信号处理的应用中，准确的信号采样和重构都是至关重要的。时域信号的频谱可以通过傅里叶变换来进行表示。当以一个特定的频率对该信号进行采样时，便会在频域中产生原始频谱的多个缩放副本。这些副本的间距是由采样频率来进行决定的。

如果采样频率低于奈奎斯特速率，那么这些副本将会出现重叠，从而导致无法使用低通滤波器来准确的恢复其原始信号。这种重叠效应被称为混叠，同时会扭曲重构信号，并且无法恢复其中的原始信号。

如果要分析采样信号的频谱，则必须考虑基频及其与固定采样频率之间的相互作用。当信号的基频介于采样频率的一半和采样频率本身之间时，基频的任何增加都会适得其反地导致感知输出频率出现降低。这种违反直觉的效果是由于混叠所造成的，即进行采样后的高频与低频将会变得难以区分。因此，重建后的信号将会严重失真，从而使其无法恢复到原来的状态。

相反，如果基频小于采样频率的一半，那么增加基频则会导致输出频率增加。这种行为是符合预期的，并且可以更加清晰地对原始信号进行重建。因此，如果要准确的实现重建，那么其中的采样频率就必须超过奈奎斯特速率，即原始信号中最高频率的两倍。当达到或超过这一频率时，便能够避免混叠，并且频域中的副本不会发生重叠。

遵守奈奎斯特准则可以其具有足够高的采样频率，以此来捕获原始信号中的必要信息，并以此来实现精确的信号重建。这一原则在音频处理、电信和数据采集等各种应用中都是至关重要的，因为在这些应用中，保持信号的完整性是至关重要的。通过使用适当的采样频率来避免混叠，从而使其能够可靠地恢复其原始时域信号，并保持其中的质量和保真度。

考虑一个时域信号及其傅里叶变换以揭示其频谱。  
以特定频率对该信号进行采样会产生多个按比例缩放的……

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

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 overlap occurs, a low-pass filter cannot recover the original signal correctly. This overlap is called aliasing, and it distorts the reconstructed signal.

The effect also changes how frequency is perceived after sampling. If the fundamental frequency is between half the sampling frequency and the sampling frequency itself, raising the fundamental frequency can make the output frequency appear lower. This counterintuitive result happens because higher frequencies become indistinguishable from lower ones after sampling.

If the fundamental frequency is less than half the sampling frequency, the output frequency rises as expected when the fundamental frequency rises. In this range, the reconstructed signal is clearer and stays closer to the original. For accurate recovery, the sampling frequency must meet or exceed the Nyquist criterion so the replicas do not overlap. This idea matters in audio processing, telecommunications, and data acquisition, where preserving signal quality is important.

Sampling Signals Without Aliasing

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 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 sampling frequency, the Nyquist rate, and overlapping spectra affect signal reconstruction and why aliasing distorts sampled signals.

Sampling Signals Without Aliasing | 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

Sampling Signals Without Aliasing

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