Signals, Systems, and Processes: Understanding Their Interactions
A) Signal
A signal can be defined as characteristic information about any physical event that has occurred or is occurring. These include electric, visual, mechanical, electromagnetic, or even analog and discrete signals, among other classifications assigned by the field of study. The importance of a signal is that it can provide valuable information about an uncertain event, thus enabling prediction and even control.
B) System
A system is a set of interrelated elements that interact among themselves to play a role.
C) Process
A process is a set of operations or ordered sequential events that aim to shape, adapt, change, or transform something. Its interpretation depends on the area to which it applies, such as the evolutionary process, the thermodynamic process, a chemical process, or a mechanical process.
1.2 Interactions Between Signal, System, and Process
The words defined above are strongly linked. All cohere as a system is the physical environment that runs a process, which delivers and receives signals from the environment to deliver a particular task or achieve a desired final product.
2. Signals
A) Main Characteristics of a Signal
- Size: Defined as the maximum value achieved by a wave or signal in a given period.
- Frequency: Indicated by the number of occurrences of some phenomenon occurring at a time. One way to calculate the frequency of a signal is to measure the time in which an event repeats, which is equal to the reciprocal of the frequency, where T is the period of the signal.
- Period: The time it takes for a signal to go through the same point, the same event that occurs at a given time. It is defined as the inverse of the frequency.
- Phase: The angle with which a sinusoidal source begins.
- Waveform: Represented in a graphical visual signal as time passes in a given period.
- Wavelength: Defined as the distance between two consecutive crests, it describes the wave’s length and is inversely proportional to the wave frequency.
The Greek letter “E” is used for mathematical representation.
B)
- A: Amplitude
- T: Period
- λ: Wavelength
- f: Frequency, as defined above, is the inverse of the period.
C) Relationship Between Characteristics and Fourier Transform
When we transform a signal, we get a different signal from the original. This can have multiple purposes, such as obtaining information that can only be found in the processed signal or facilitating the transformed signal display for future interpretation.
For example, the Fourier transform allows us to obtain information about the distribution of the frequency components of the signal. It informs us about its amplitude and phase information, but time information is lost.
The cosine transformation gives us information on the magnitude of the frequency components.
3. Sampling Process
A) Explanation of the Sampling Process
To define the sampling process, we know that sampling involves taking samples at instants of time. This misses the information contained between two points (sampled), but these points are now at a finite distance given by the sampling time, which can be constant or variable, but it is more common to use a constant.
The sampling process would determine the relationship between the analog signal coming from the channel and the digital signal sampled. The fundamental parameter in this process is the sampling frequency, which defines the minimum frequency we can display a signal to keep information about it.
The sampling theorem states that a band-limited signal with a bandwidth of W needs at least a sampling frequency fm = 2W. This shows the importance of the proper choice of frequency.
B) Details to Consider When Sampling a Signal
When sampling a signal, there is a risk that the frequency at which it is sampled is less than the frequency of the signal to be sampled. This leads to the appearance of alias components that can cause disturbances on the device using the signals. Signals must be sampled as the following example illustrates:
Sampling maintaining minimal loss of information of the original signal (continuous).
Sampling that has lost the information of the original signal form (continuous).
Or as in this case, where the frequency of the signal you want to sample is much higher than the sampling frequency fs, and as a result, information about the form, frequency, and phase of the original signal is lost (continuous).
The main problems related to sampling refer to the loss of signal information, either phase, frequency, and/or amplitude. Therefore, it is essential to study the sampling to avoid these problems.
D) Characteristics of Continuous and Discrete Signals
First, let’s describe the differences between them. A continuous signal is a “soft” signal that is defined for all points of a given interval in the set of real numbers. For example, the sine function is a continuous example, as are the exponential function and the constant function.
A portion of the sine function in the time range of 0 to 6 seconds is also continuous. If we want examples from nature, we have current, voltage, sound, light, etc.
A discrete signal is characterized by being discontinuous, i.e., in an interval A and B, there exists a finite number of values. Therefore, we can say that for some range, the function is undetermined, and this is because this type of function does not obey time.
Graphically, it is represented like this:
4. Purpose of Transforms
In mathematical analysis, transforms are used to change the domain with which the function works, like passing a function in the time domain to a function in the frequency domain or a complex frequency domain. This obtains several advantages, whether for discrete or continuous transforms, such as facilitating the calculation of linear operations and avoiding working directly with differential equations, which can have a high degree of complexity or be indefinite in some cases. In addition, as in the case of the Fourier transform, when applied to a function of time, it can obtain important information about its frequency response, which is desirable in electronics. In digital signal processing, large profits are obtained in the area of science.
5. Fourier Transform: Description and Use
The Fourier transform is a special case of the Laplace transform. Both can transform a linear integro-differential equation of order n into a polynomial equation of degree n.
Its immediate application is in the resolution of such equations. It is useful in troubleshooting for electrical engineering, electronic engineering, control theory, and in general, any problem that can be modeled as a linear system.
It allows you to pass from the time domain to the frequency domain. This is because the equations are transformed with an independent variable of the inverse dimension of the independent variable of the original equation.
Often applied to transform equations where the independent variable is time, the Fourier transform is described as a change in the problem domain from time to frequency.
In the case of the Fourier transform, it is a real frequency.
7. Frequency Spectrum: Definition and Information
A)
Any periodic signal can be decomposed as a sum of sinusoidal waves whose amplitudes and frequencies are obtained from Fourier analysis. Thus, any signal can be represented as an infinite sum of its harmonics or frequency components. The frequency spectrum of a signal is this decomposition, where the frequency of the signal is expressed as a Fourier series. The spectrum is represented graphically using the classical coordinate axes, assigning the value of the frequencies to the zero axis and plotting the amplitude on the same coordinate axis.
B) How to Obtain the Frequency Spectrum
Mathematically, spectrum analysis is related to a tool called the Fourier transform or Fourier analysis. Such analysis can be carried out for small time intervals, or less frequently for long intervals, or it may be a spectral analysis of a deterministic function. The Fourier transform of a function not only allows a spectral decomposition of a wave or oscillatory signal, but with the spectrum generated by the Fourier analysis, it is even possible to reconstruct (synthesize) the original function by the inverse transform. To do that, the transform not only contains information about the intensity of a certain frequency but also its phase. This information can be represented as a two-dimensional vector or as a complex number.