**Signal Sampling**

To start the topic about signal sampling, it is necessary to understand what are the two types of signal, the analog signal and the digital signal, and what are continuous values and discrete values. After this introduction, we must understand where the Sampling Theorem comes from, who developed it and its application in everyday life.

**What is analog signal?**

** **The analog signal is a continuous signal whose value varies as a function of time. Normally the analog signal is illustrated by a sine wave, a series of parabolas that have their peak alternating about a center.

The analog signal is a continuous signal whose value varies as a function of time. Normally the analog signal is illustrated by a sine wave, a series of parabolas that have their peak alternating about a center.

Assuming that the interval from the peak of the wave to its base is from 0 to 10 Volts, even being represented by fixed values, this signal has infinite values, its wave passes through all possible intervals, such as 0.001, 0.478, 5.721, 9.999 ...That's why we say that the analog signal has continuous values. Faced with its infinite variation in value and its oscillation, analog transmission was subject to interference and noise, making it unreliable.

Analog signals are “seen” all over the place, like the waves of our voice, which could be represented by a drawing similar to the one above.

**What is digital signal?**

The digital signal is a signal that transits between two states, on and off, 0 and 1, but that does not mean that its value varies from 0 Volts to 1 Volts. The digital signal has discrete values, unlike the analog signal that has infinite variation between each value, the digital will have point values, such as 1,2,3,4,5...

Because the digital signal varies abruptly and not gradually to reach its value, for example from 0V (Volts) to 5V (Volts), your signal generation becomes square.

**Harry Nyquist – Sampling Theorem**

**A little about Harry Nyquist**

Harry Nyquist was born in 1889 in Sweden, a few years later he entered the University of North Dakota where he did a Bachelor's and Master's degree in Electrical Engineering. . In 1917, Nyquist began his professional career at AT&T, more specifically in the company's research department, left after 17 years with the company and joined Bell Laboratories and remained there until his retirement in 1954. During his years working at Bell Laboratories, Nyquist has done impressive work on thermal noise in electronic components and amplifier stability and feedback. Along with his friend Herbet E. Ives, Nyquist helped develop the first AT&T fax machine where he published his previously stated research, stability and feedback amplifiers, Nyquist Stability Criteria, on paper.

Physicist and electronics engineer, Harry Nyquist made a very important contribution to the communication we know today.

While studying signal processing, Nyquist established a representation, defined that for the recreation of an original continuous signal to be represented functionally identical, the sampling rate would need to be at least twice as high as the maximum frequency of the original wave.

**Sampling Theorem – Nyquist Theorem**

Instant sampling is the process of transforming an analog signal or wave into a set of discrete numbers, these discrete values are the samples, which are the value of a point on the wave at a given instant. For example, radio signals, which in turn are continuous waves, where the value of its signal can vary infinitely within its wave, when selecting a certain point in time for this wave, its value is now represented by a discrete signal.

A good sample should be one that can accurately recreate the signal, that is, the waveform originally represented in continuous values is now represented by discrete sample values that respecting the time interval does not lose definition.

For a good recreation of the original signal, it is necessary that the sampling rate is at least twice the frequency of the original signal, that is, 2ωm, with ωm being the highest frequency of the original signal. If the signal has a value exactly in ωm and the samples respect the spacing of 1/(2ωm), that is, a sampling rate equal to the frequency of the original signal, a good recovery of the signal will not be achieved.

The perfect copy of the signal is mathematically possible, but in practice its creation only comes close, in any case, it is a very faithful approximation to the original.

**Formulas and concepts applied in mathematics**

Putting the concepts in order, if x(t) is the representation of a continuous signal in time and let X(jω) be the Fourier transform:

X(t) is band bound, being band ω if X(jω) = 0 for any |ω| > ωm.

The condition for perfect construction from samples at a uniform sampling rate ωs, where ωs samples per time: ωs > 2ωm or ωm < (ωs/2).

2ωm is called the Nyquist Rate and ωs/2 is called the Nyquist Frequency. And the Sampling Interval is represented by T = (1/ωs).

**Aliasing**

As we have seen, Nyquist defined what is needed for the ideal construction of an original continuous signal. But if the sample rate of the function is less than twice the frequency of the original continuous signal, it will cause loss of information. Then, at the time of recreating the continuous signal through the obtained discrete values, the Aliasing will be generated,

Observing the figure above, the larger red dots are the samples obtained from the continuous signal that, because they do not obey the Sampling Theorem and have their frequency twice that of the original signal, when processing this information from discrete values into a continuous signal again it is noticeable that its shape changes, generating a totally different data and with several losses of information.

Briefly, Aliasing is caused by the low sampling frequency, thinking of a quick and generic solution, the best thing to do would be to increase this sampling frequency by exceeding twice the original one. Mathematically speaking, having a sampling frequency X times the maximum frequency of the continuous signal, with X being at least twice as high as ωm, would be correct and effective, but this is not the case in the application of signal processing.

*Sampling, Oversampling e Downsampling *

Sampling is when the original signal is sampled twice as high as its maximum frequency, according to the Nyquist Theorem.

Represented by this image, there is a continuous signal (gray line) where samples were collected (red dots), at the frequency of twice the maximum frequency of the signal, making it possible to recreate its original wave after processing the discrete values (line blue).

**Oversampling**, is when the original signal is sampled more times than the minimum, according to the Nyquist Theorem.

As can be analyzed by this image, there is a continuous signal (gray line) where samples were collected (red dots), at the frequency of quadruple the maximum frequency of the signal, making it possible to recreate its original wave after processing the values. discrete (blue line).

The acquisition of more samples does not interfere with the construction of a continuous signal identical to the original one, on the contrary, it favors it to be as faithful as possible. However, processing the continuous signal into discrete values and storing the obtained samples end up taking up a lot of memory, so you should be careful with how much oversampling you need to do, since in some situations its use is feasible.

**Decimation** or **Undersampling**, is when you take samples of the original signal less times than the minimum, according to the Nyquist Theorem.

Analyzing the image above, we have a continuous signal (gray line) where samples were collected (red dots), at a frequency lower than twice the maximum frequency of the signal, in this case a frequency of half the maximum frequency, making the recreation of its original waveform distorted and with loss of information after processing the discrete values (blue line). Generating the Aliasing, seen earlier.

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**References:**

**[01] R. Jagan Mohan Rao, “What are analog and digital signals? Differences, examples”. Inst Tools, ***ano***. Diponível em: <https://instrumentationtools.com/what-are-analog-and-digital-signals-differences-examples/>.**

**[02] Harry Nyquist – Biography. ETHW, ***ano.*** Disponível em: <https://ethw.org/Harry_Nyquist>.**

**[03] J. J. Abdul, “The Shannon Sampling Theorem – Its Various Extensions and Application: A Tutorial Review”, in IEEE Access, vol. 65, NO. 11, Novembro 1977.**

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