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Nyquist's Theorem states that if a signal contains no frequencies higher than a certain value ''B'', then the all of the necessary information in the signal can be captured with a sampling frequency of 2''B'' or higher. [5] This means that to obtain an accurate understanding of a signal, the sampling period must be at most half the length of the period of oscillation of the signal. But why is this the case?<br /> | Nyquist's Theorem states that if a signal contains no frequencies higher than a certain value ''B'', then the all of the necessary information in the signal can be captured with a sampling frequency of 2''B'' or higher. [5] This means that to obtain an accurate understanding of a signal, the sampling period must be at most half the length of the period of oscillation of the signal. But why is this the case?<br /> | ||
| − | The proof for Nyquist's Theorem requires use of a concept known as the Fourier Transform, which converts a signal or other sinusoid from the time domain (that is, the graph is a graph of amplitude of the wave versus time) to the frequency domain (where the graph is of amplitude versus frequency). More information about how the Fourier Transform works can be found [https://en.wikipedia.org/wiki/Fourier_transform here]. As a result of the Fourier Transform, the graph of a waveform can be converted into one similar to the image below, where the frequency is graphed on the x-axis. In order for Nyquist's Theorem to be valid, one particular condition must be satisfied. The function must be of finite bandwidth, meaning that there should be no positive or negative frequencies beyond a certain constant value. Waveforms that satisfy this criterion are referred to as bandlimited signals. In practical cases, bandwidth is limited by filters, such as the one in a car radio, that filters out frequencies that are not very close to say, 104.5 Hertz.<br /> | + | The proof for Nyquist's Theorem requires use of a concept known as the Fourier Transform, which converts a signal or other sinusoid from the time domain (that is, the graph is a graph of amplitude of the wave versus time) to the frequency domain (where the graph is of amplitude versus frequency). More information about how the Fourier Transform works can be found [https://en.wikipedia.org/wiki/Fourier_transform here]. As a result of the Fourier Transform, the graph of a waveform can be converted into one similar to the image below, where the frequency is graphed on the x-axis. In order for Nyquist's Theorem to be valid, one particular condition must be satisfied. The function must be of finite bandwidth, meaning that there should be no positive or negative frequencies beyond a certain constant value. Waveforms that satisfy this criterion are referred to as bandlimited signals. In practical cases, bandwidth is limited by filters, such as the one in a car radio, that filters out frequencies that are not very close to say, 104.5 Hertz. In the image below, there is no frequency beyond +''B'' or -''B'', meaning that the sampling frequency, according to the theorem, should be<br /> |
Revision as of 14:51, 6 December 2020
Definition of Nyquist’s Theorem
In order to fully understand Nyquist's Theorem, one needs to understand the basics of waves and signals. The frequency of a wave is the rate at which waves are observed; that is, it is the number of complete oscillations of the wave per second. The period of a wave is the inverse of the frequency: it is the number of seconds required for one complete oscillation. The frequency of a wave is measured in Hertz, which is defined as 1/seconds, or cycles per second. The period is thus measured in seconds, sometimes phrased as seconds per cycle.
Nyquist's Theorem states that if a signal contains no frequencies higher than a certain value B, then the all of the necessary information in the signal can be captured with a sampling frequency of 2B or higher. [5] This means that to obtain an accurate understanding of a signal, the sampling period must be at most half the length of the period of oscillation of the signal. But why is this the case?
The proof for Nyquist's Theorem requires use of a concept known as the Fourier Transform, which converts a signal or other sinusoid from the time domain (that is, the graph is a graph of amplitude of the wave versus time) to the frequency domain (where the graph is of amplitude versus frequency). More information about how the Fourier Transform works can be found here. As a result of the Fourier Transform, the graph of a waveform can be converted into one similar to the image below, where the frequency is graphed on the x-axis. In order for Nyquist's Theorem to be valid, one particular condition must be satisfied. The function must be of finite bandwidth, meaning that there should be no positive or negative frequencies beyond a certain constant value. Waveforms that satisfy this criterion are referred to as bandlimited signals. In practical cases, bandwidth is limited by filters, such as the one in a car radio, that filters out frequencies that are not very close to say, 104.5 Hertz. In the image below, there is no frequency beyond +B or -B, meaning that the sampling frequency, according to the theorem, should be
Once a signal is bandlimited, it is fairly simple to understand why Nyquist's Theorem is true.
