Aliasing, frequency noun, insignalIn the spectrum, it can be called overlapping frequency;stayimageIt can be called superimposition.It mainly comes from the continuous time signalsamplingwithdigitizationThe sampling frequency is less than twiceNyquist frequency。
stayStatistics、signal processing In and related fields, aliasing refers to the phenomenon that the sampled signals overlap and distort each other when they are restored to continuous signals.When aliasing occurs, the original signal cannot be restored from the sampled signal.Aliasing may occur in the time domain, calledTime aliasing, or occurs in the frequency domain, calledSpatial aliasing。
In the visual imageAnalog digital conversionorMusic signalAliasing is a very important topic.Because if the sampling frequency is not properly selected during analog digital conversion, the high-frequency signal and low-frequency signal will be mixed together, so the original signal cannot be reconstructed perfectly.In order to avoid this situation, sampling must be carried out beforewave filteringOperation of.
(1) Look at aliasing from time domain signal reconstruction
As shown in Figure 1, the signal in Figure 1 is
The frequency domain of
There is a spectral line at.Used
When sampling, a DC curve is obtained.Used
When sampling, a triangle wave signal is obtained.Used
When it is sampled, a lower frequency triangular wave signal is also obtained.The sampled signal can not only reconstruct the original signal, but also has aliased frequencies, that is, the sampled signal can not maintain the spectral characteristics of the original signal.
Figure 1 Aliasing from time-domain signal reconstruction
(2) Aliasing in frequency domain
After the continuous signal is sampled discretely, the Fourier spectrum of the discrete signal is the periodic continuation of SF times of the Fourier spectrum of the original signal. If the original signal contains the highest frequency component
, the spectrum of the corresponding period in the discrete signal spectrum will overlap.Conversely, if
That is, if the sampling frequency is more than 2 times of the highest spectral component in the analyzed signal, there will be no frequency aliasing in the discrete signal spectrum after sampling.[1]
Aliasing in frequency domain
Aliasing in frequency domain
Anti aliasing
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An important guiding significance of the sampling theorem is that it provides the minimum conditions for eliminating aliasing. Aliasing itself is an inevitable effect of sampling. However, if the frequency component of the aliasing within the bandwidth of the original signal is zero, the signal will not be damaged and can be "completely reconstructed".There are two ways to eliminate frequency aliasing:
(1) Increase the sampling frequency fs, that is, reduce the sampling interval.However, the actual signal processing system can not achieve a large sampling frequency.In addition, many signals themselves may contain
It is impossible to increase the sampling frequency to
。Therefore, the method of avoiding aliasing by increasing the sampling frequency is limited.
(2) Anti aliasing filter is adopted.Under the premise that the sampling frequency fs is fixed, pass thelow pass filter Filter out the frequency components higher than fs/2, and avoid frequency aliasing by low-pass filtering.
In the case of ideal filtering, the signal components higher than Nyquist frequency can be filtered out without aliasing.However, the actual filters do not have the characteristics of ideal filters, as shown in Figure 2.Therefore, the following relationships should be met in the actual processing:
Figure 2 Anti aliasing
Compare with real frequency
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According to the aliasing mechanism, the formula for calculating the aliasing frequency of the analyzed signal can be obtained, and the frequency of the actual signal is assumed to be
, sampling frequency is
, and
, the post aliasing frequency obtained through sampling analysis is
, the formula is as follows:
Where,
, where
For rounding operation, only the integer part of the decimal point is reserved.
Aliasing Instances
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A common case of aliasing is movies.This is because the changing image is continuously sampled discretely at a rate of 24 frames per second.Nyquist sampling theoremIt tells us that if there is any aliasing at any point in the image plane
(12 frames/second in this case) higher frequency components or light dark transitions will cause aliasing.But in many cases, this transition between light and dark may occur faster than this - for example, the wheel or propeller of a carriage rotates at high speed.
Consider a wheel with eight spokes rotating at 3 revolutions per second (or 180 rpm).In this case, the wheel moves one spoke per frame because:
Therefore, the truck wheels will appear to be stationary.But this is very rare, because the probability of the wheel rotating at this speed is very small.
Consider if the wheel rotates at a rate lower than this value, such as 2.5 rpm.The wheel will move 83% of the spoke spacing per frame.Therefore, comparing two adjacent frames, we can see the following phenomenon:
The human brain will have two explanations when watching these movie frames.One explanation is that the wheel has moved 83% along the clockwise direction of the spoke spacing.Another explanation is that it has moved 17% of the spoke spacing counterclockwise.It turns out that the brain likes the latter explanation, so the result you feel is that the wheel moves backward (counterclockwise) at a slower speed than the actual speed.[2]
Correlation law
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Nyquist criterion
Nyquist criterion
If the sampling frequency is too low, the sampling result will be different from the original sample.If the spectrum of a sample is band limited, that is, at a certain frequency
The sampling frequency is 0 for all other frequencies
Must be greater than twice
, so that the spectrum will not overlap, and therefore produce distortion.
