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| + | Nyquist's Theorem has many features that are important to know when applying the theorem, such as aliasing. Though aliasing has already been mentioned briefly, it will be further explained here, along with its opposite, oversampling. Aliasing is the under-sampling of a signal. The main purpose of using Nyquist's Theorem is to eliminate any aliasing that may occur. As shown below on a simple sine wave signal in the time domain (time is on the x-axis), sampling with a sample rate less than twice the frequency of even just one of two signals could result in two signals appearing to be the same. Aliasing can be prevented with a variety of anti-aliasing tools, such as low-pass filters that filter out high frequencies. | ||
| + | [[File:AliasedSineWaves.JPG|400px|frameless|center|The red sine wave is under-sampled, resulting in the two distinct waves having the same data points. [1]]] | ||
| + | <small>The red sine wave is under-sampled, resulting in the two distinct waves having the same data points. [1]</small> | ||
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| + | Nyquist's Theorem can also be used to prevent excessive oversampling. Of course, there must be some small amount of oversampling, in order to prevent aliasing. The image below represents a case of excessive oversampling. The sample rate is much higher than the Nyquist Rate, which is not necessarily a horrible thing. However, the frequency of this signal could easily be found with a much lower sampling rate. Though it might be "nice" to have a detailed idea of the looks of the exact waveform, this high sample rate would likely result in large amounts of unnecessary data that would take more time to analyze. [5] | ||
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| + | [[File:Over sampling.JPG|400px|frameless|center|This degree of oversampling would only increase the amount of time necessary to analyze the signal. [5]]] | ||
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| + | <small>This degree of oversampling would only increase the amount of time necessary to analyze the signal. [5]</small> | ||
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| + | Another important consideration is whether the Nyquist Rate itself can be used as the sample rate. In other words, it is essential in certain cases to consider whether the sample rate must be greater than 2''B'' or greater than ''or equal to'' 2''B''. This would likely be more of a theoretical issue, but is still important to think about. For most cases, the sample rate could in fact be equal to the Nyquist Rate, but there are in fact certain situations in which this would create issues. [4] One instance of this is shown in the third [[Worked Examples Using Nyquist’s Theorem|example problem]]. [1] | ||
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| + | [[Worked Examples Using Nyquist’s Theorem|Next Page]] | ||
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[[ Walther MA271 Fall2020 topic21|Back to Walther MA271 Fall2020 topic21]] | [[ Walther MA271 Fall2020 topic21|Back to Walther MA271 Fall2020 topic21]] | ||
Latest revision as of 22:32, 6 December 2020
Features of Nyquist's Theorem
Nyquist's Theorem has many features that are important to know when applying the theorem, such as aliasing. Though aliasing has already been mentioned briefly, it will be further explained here, along with its opposite, oversampling. Aliasing is the under-sampling of a signal. The main purpose of using Nyquist's Theorem is to eliminate any aliasing that may occur. As shown below on a simple sine wave signal in the time domain (time is on the x-axis), sampling with a sample rate less than twice the frequency of even just one of two signals could result in two signals appearing to be the same. Aliasing can be prevented with a variety of anti-aliasing tools, such as low-pass filters that filter out high frequencies.
![The red sine wave is under-sampled, resulting in the two distinct waves having the same data points. [1]](/rhea/index.php/File:AliasedSineWaves.JPG)
The red sine wave is under-sampled, resulting in the two distinct waves having the same data points. [1]
Nyquist's Theorem can also be used to prevent excessive oversampling. Of course, there must be some small amount of oversampling, in order to prevent aliasing. The image below represents a case of excessive oversampling. The sample rate is much higher than the Nyquist Rate, which is not necessarily a horrible thing. However, the frequency of this signal could easily be found with a much lower sampling rate. Though it might be "nice" to have a detailed idea of the looks of the exact waveform, this high sample rate would likely result in large amounts of unnecessary data that would take more time to analyze. [5]
![This degree of oversampling would only increase the amount of time necessary to analyze the signal. [5]](/rhea/index.php/File:Over_sampling.JPG)
This degree of oversampling would only increase the amount of time necessary to analyze the signal. [5]
Another important consideration is whether the Nyquist Rate itself can be used as the sample rate. In other words, it is essential in certain cases to consider whether the sample rate must be greater than 2B or greater than or equal to 2B. This would likely be more of a theoretical issue, but is still important to think about. For most cases, the sample rate could in fact be equal to the Nyquist Rate, but there are in fact certain situations in which this would create issues. [4] One instance of this is shown in the third example problem. [1]
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