The concept of "signal" can be interpreted in different ways. This is a code or sign transmitted into space, a storage medium, a physical process. The nature of the alerts and their relationship to noise affect its design. Spectra of signals can be classified in several ways, but one of the most fundamental is their change in time (constants and variables). The second main classification category is frequencies. If we consider the types of signals in the time domain in more detail, we can distinguish among them: static, quasistatic, periodic, repeating, transient, random and chaotic. Each of these signals has certain properties that can affect the corresponding design decisions.
Signal types
Static by definition is unchanged for a very long period of time. The quasistatic is determined by the level of direct current, so it must be processed in low-drift amplifier circuits. This type of signal does not occur at radio frequencies, because some similar circuits can create a level of unchanging voltage. For example, continuous wave notification with a constant amplitude.
The term “quasistatic” means “almost unchanged,” therefore, refers to a signal that changes unusually slowly over time. It has characteristics more similar to static alerts (permanent) than dynamic alerts.
Periodic signals
These are those that are accurately repeated on a regular basis. Examples of periodic signals include sinusoidal, square, sawtooth, triangular waves, etc. The nature of the periodic shape indicates that it is identical at the same points along the time line. In other words, if there is an advance along the time line for exactly one period (T), then the voltage, polarity and direction of the waveform change will be repeated. For the voltage form, this can be expressed by the formula: V (t) = V (t + T).
Repeating signals
They are quasiperiodic in nature, therefore, they have some similarities with the periodic waveform. The main difference between them is detected by comparing the signal at f (t) and f (t + T), where T is the notification period. Unlike periodic alerts, in repeating sounds these points may not be identical, although they will be very similar, just like the general waveform. The alert in question may contain either temporary or stable symptoms that vary.
Transitional signals and pulse signals
Both types are either a one-time event or a periodic one in which the duration is very short compared to the period of the waveform. This means that t1 <<< t2. If these signals were transients, then in radio frequency circuits they would intentionally be generated in the form of pulses or transient noise. Thus, from the above information, we can conclude that the phase spectrum of the signal provides oscillations in time, which can be constant or periodic.
Fourier Series
All continuous periodic signals can be represented by a fundamental sinusoidal frequency wave and a set of cosine harmonics, which are summed linearly. These oscillations contain a Fourier series of swell shapes. The elementary sine wave is described by the formula: v = Vm sin (_t), where:
- v is the instantaneous amplitude.
- Vm is the peak amplitude.
- "_" Is the angular frequency.
- t is the time in seconds.
The period is the time between the repetition of identical events or T = 2 _ / _ = 1 / F, where F is the frequency in cycles.
The Fourier series that makes up the waveform can be found if the given value is decomposed into its frequency components either by a bank of frequency-selective filters or by an algorithm for digital signal processing called fast conversion. A construction method from scratch can also be used. The Fourier series for any waveform can be expressed by the formula: f (t) = a o / 2 + _ n –1 [a n cos (n_t) + b n sin (n_t). Where:
- an and bn are component deviations.
- n is an integer (n = 1 is fundamental).
Amplitude and phase spectrum of a signal
Deviating coefficients (an and bn) are expressed by writing: f (t) cos (n_t) dt. Moreover, an = 2 / T, b n = 2 / T, f (t) sin (n_t) dt. Since only certain frequencies are present, fundamental plus harmonics determined by an integer n, the spectrum of a periodic signal is called discrete.
The term ao / 2 in the expression of the Fourier series is the average value of f (t) for one complete cycle (one period) of the waveform. In practice, it is a DC component. When the form in question has half-wave symmetry, that is, the maximum amplitude spectrum of the signal is above zero, it is equal to the deviation of the peak below the specified value at each point in t or (+ Vm = _ – Vm_), then there is no DC component, therefore ao = 0.
Waveform symmetry
Some postulates about the spectrum of Fourier signals can be deduced by examining its criteria, indicators, and variables. From the above equations it can be concluded that harmonics propagate to infinity on all waveforms. Clearly, in practical systems, there are far fewer infinite bandwidths. Therefore, some of these harmonics will be removed by the usual action of electronic circuits. In addition, it is sometimes found that higher ones may not be very significant, so they can be ignored. As n increases, the amplitude coefficients an and bn tend to decrease. At some point, the components are so small that their contribution to the waveform is either negligible for practical purposes or impossible. The value of n at which this occurs partially depends on the rise time of the quantity in question. The period of increase is defined as the interval necessary for the wave to increase from 10% to 90% of its final amplitude.
A square wave is a special case because it has an extremely fast rise time. Theoretically, it contains an infinite number of harmonics, but not all of them are definable. For example, in the case of a square wave, only odd 3, 5, 7 were found. According to some standards, accurate reproduction of a square swell requires 100 harmonics. Other researchers claim that 1000 are needed.
Fourier Series Components
Another factor determining the profile of the system in question for a particular waveform is a function that must be identified as odd or even. The second is the one in which f (t) = f (–t), and for the first –f (t) = f (–t). In the even function, only cosine harmonics are present. Therefore, the sine amplitude coefficients bn are equal to zero. Similarly, in an odd function, only sinusoidal harmonics are present. Therefore, the cosine amplitude coefficients are equal to zero.
Both symmetry and opposite values can appear in several ways in the form of a wave. All these factors are capable of influencing the nature of the Fourier series of the swell type. Or, in terms of the equation, the term ao is nonzero. The DC component is a case of asymmetry in the signal spectrum. This bias can seriously affect the measuring electronic circuits, which are connected by unchanging voltage.
Steadiness in deviations
Symmetry with a zero axis occurs when, based on a wave point and amplitude is above the zero base. The lines are equal to the deviation below the base, or (_ + Vm_ = _ –Vm_). When the swell has symmetry with a zero axis, it usually does not contain even harmonics, and only odd ones are present. Such a situation occurs, for example, in square waves. However, symmetry with a zero axis does not occur only in sinusoidal and rectangular swells, as the sawtooth value under consideration shows.
There is an exception to the general rule. In a symmetrical form, the zero axis will be present. If the even harmonics are in phase with the main sine wave. This condition will not create a DC component and will not violate the symmetry of the zero axis. Half-wave immutability also implies the absence of even harmonics. With this type of invariance, the waveform is above the zero baseline and is a mirror image of the swell.
The essence of the other correspondences
Quarterly symmetry exists when the left and right halves of the sides of the waveforms are mirror images of each other on the same side of the zero axis. Above the zero axis, the waveform is similar to a square wave, and indeed, the sides are identical. In this case, there is a complete set of even harmonics, and any odd ones that are present are in phase with the main sinusoidal wave.
Many pulsed signal spectra meet the period criterion. From the point of view of mathematics, they are actually periodic. Temporary alerts are not properly represented by the Fourier series, but can be set forth by sine waves in the signal spectrum. The difference is that the transient alert is continuous rather than discrete. The general formula is expressed as: sin x / x. It is also used for repeated impulse alerts and for transient form.
Sampled signals
A digital computer is not capable of receiving analog input sounds, but requires a digital representation of this signal. An analog-to-digital converter changes the input voltage (or current) into a representative binary word. If the device works clockwise or can start asynchronously, then it will receive a continuous sequence of signal samples, depending on the time. When combined, they represent the original analog signal in binary form.
The waveform in this case is a continuous function of the voltage of time, V (t). The signal is sampled by another signal p (t) with a frequency of Fs and a sampling period of T = 1 / Fs, and then reconstructed later. Although this may be representative enough for the waveform, it will be reconstructed with greater accuracy if the sampling frequency (Fs) increases.
It happens that the sine wave V (t) is sampled by a pulsed sampling notification p (t), which consists of a sequence of equally spaced narrow values spaced in time T. Then the frequency of the signal spectrum Fs is 1 / T. The result obtained is another impulse response, where the amplitudes are a sample version of the original sinusoidal alert.
According to the Nyquist theorem, the sampling frequency Fs should be twice the maximum frequency (Fm) in the Fourier spectrum of the applied analog signal V (t). To restore the original signal after sampling, it is necessary to pass a selective waveform through a low-pass filter, which limits the passband to Fs. In practical RF systems, many engineers determine that the minimum Nyquist speed is not sufficient for good sample reproductions, so increased speed should be indicated. In addition, some oversampling techniques are used to dramatically reduce noise.
Signal spectrum analyzer
The sampling process is similar to the form of amplitude modulation, in which V (t) is a constructed alert with a spectrum from DC to Fm, and p (t) is the carrier frequency. The result obtained is reminiscent of a double sideband with a carrier value of AM. Spectra of modulation signals appear around the frequency Fo. The actual value is a little more complicated. Like the unfiltered AM radio transmitter, it appears not only around the fundamental frequency (Fs) of the carrier, but also at harmonics located at intervals Fs up and down.
Provided that the sampling frequency corresponds to the equation Fs ≥ 2Fm, the original response is restored from the sample version, passing it through a low-pass filter with a changing cutoff Fc. In this case, it is possible to transmit only the spectrum of analog sound.
In the case of the inequality Fs <2Fm, a problem arises. This means that the spectrum of the frequency signal is similar to the previous one. But the sections around each harmonic overlap so that “–Fm” for one system is less than “+ Fm” for the next lower oscillation region. This overlap results in a discretized signal whose spectrum width is restored by low pass filtering. It will not generate the initial frequency of the sine wave Fo, but a lower one equal to (Fs - Fo), and the information transferred in the form of a wave is lost or distorted.