The concept of "signal" can be interpreted in different ways. This is a code or a sign transferred into space, a carrier of information, a physical process. The nature of alerts and their relationship to noise influence its design. Signal spectra can be classified in several ways, but one of the most fundamental is their change over time (constant and variable). The second main classification category is frequencies. If we consider the types of signals in the time domain in more detail, among them we can distinguish: static, quasi-static, periodic, repetitive, transient, random and chaotic. Each of these signals has specific properties that can influence the respective design decisions.
Signal types
Static, by definition, is unchanged for a very long period of time. Quasi-static is determined by the DC level, so it needs to be handled in low-drift amplifier circuits. This type of signal does not occur at radio frequencies because some of these circuits can produce a steady voltage level. For example, continuousconstant amplitude wave alert.
The term "quasi-static" means "almost unchanged" and therefore refers to a signal that changes unusually slowly over a long time. It has characteristics that are more like static alerts (permanent) than dynamic alerts.
Periodic signals
These are the ones that repeat exactly on a regular basis. Examples of periodic waveforms include sine, square, sawtooth, triangular waves, etc. The nature of the periodic waveform indicates that it is identical at the same points along the timeline. In other words, if the timeline advances exactly one period (T), then the voltage, polarity, and direction of the waveform change will repeat. For the voltage waveform, this can be expressed as: V (t)=V (t + T).
Repeating signals
They are quasi-periodic in nature, so they bear some resemblance to a periodic waveform. The main difference between them is found by comparing the signal at f(t) and f(t + T), where T is the alert period. Unlike periodic alerts, in repeated sounds these dots may not be identical, although they will be very similar, as will the overall waveform. The alert in question may contain either temporary or permanent indications, which vary.
Transient signals and impulse signals
Both types are either one-time events orperiodic, in which the duration is very short compared to the period of the waveform. This means that t1 <<< t2. If these signals were transients, they would be intentionally generated in RF circuits as pulses or transient noise. Thus, from the above information, we can conclude that the phase spectrum of the signal provides fluctuations in time, which can be constant or periodic.
Fourier series
All continuous periodic signals can be represented by a fundamental frequency sine wave and a set of cosine harmonics that add up linearly. These oscillations contain the Fourier series of the swell shape. An elementary sine wave is described by the formula: v=Vm sin(_t), where:
- v – instantaneous amplitude.
- Vm is the peak amplitude.
- "_" – angular frequency.
- t – time in seconds.
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 a waveform can be found if a given value is decomposed into its component frequencies either by a frequency selective filter bank or by a digital signal processing algorithm called fast transform. The method of building from scratch can also be used. The Fourier series for any waveform can be expressed by the formula: f(t)=ao/2+_ –1[a cos(n_t) + b sin(n_t). Where:
- an and bn –component deviations.
- n is an integer (n=1 is fundamental).
Amplitude and phase spectrum of the signal
Deviating coefficients (an and bn) are expressed by writing: f(t)cos(n_t) dt. Here an=2/T, bn =2/T, f(t)sin(n_t) dt. Since only certain frequencies are present, fundamental positive harmonics, defined by an integer n, the spectrum of a periodic signal is called discrete.
The term ao / 2 in the Fourier series expression is the average of f(t) over one complete cycle (one period) of the waveform. In practice, this is a DC component. When the waveform under consideration is half-wave symmetric, i.e., the maximum amplitude spectrum of the signal is above zero, it is equal to the peak deviation below the specified value at each point in t or (+ Vm=_–Vm_), then there is no DC component, so ao=0.
Waveform symmetry
It is possible to derive some postulates about the spectrum of Fourier signals by examining its criteria, indicators and variables. From the equations above, we can conclude that harmonics propagate to infinity on all waveforms. It is clear that there are far fewer infinite bandwidths in practical systems. Therefore, some of these harmonics will be removed by the normal operation 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 forpractical purpose, or impossible. The value of n at which this occurs depends in part on the rise time of the quantity in question. The rise period is defined as the amount of time needed for a wave to rise from 10% to 90% of its final amplitude.
The 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 the possible ones are definable. For example, in the case of a square wave, only the odd 3, 5, 7 are found. According to some standards, the exact reproduction of a square swell requires 100 harmonics. Other researchers claim that they need 1000.
Components for the Fourier series
Another factor that determines the profile of the considered system of a particular waveform is the function to 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 an even function, there are only cosine harmonics. Therefore, the sine amplitude coefficients bn are equal to zero. Likewise, only sinusoidal harmonics are present in an odd function. Therefore, the cosine amplitude coefficients are zero.
Both symmetry and opposites can manifest in several ways in a waveform. All these factors can influence the nature of the Fourier series of the swell type. Or, in terms of the equation, the term ao is non-zero. The DC component is a case of signal spectrum asymmetry. This offset can seriously affect measurement electronics that are coupled to a non-varying voltage.
Stability in deviations
Zero-axis symmetry occurs when the base point of the wave and the amplitude are above the zero base. The lines are equal to the deviation below the baseline, or (_ + Vm_=_ –Vm_). When a swell is zero-axis symmetrical, it usually contains no even harmonics, only odd harmonics. This situation occurs, for example, in square waves. However, zero-axis symmetry does not occur only in sinusoidal and rectangular swells, as shown by the sawtooth value in question.
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 fundamental sine wave. This condition will not create a DC component and will not break the symmetry of the zero axis. Half-wave invariance 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 pattern.
Essence of other correspondences
Quarter symmetry exists when the left and right halves of the waveform sides are mirror images of each other on the same side of the zero axis. Above the zero axis, the waveform looks like a square wave, and indeed the sides are identical. In this case, there is a full set of even harmonics, and any odd ones that are present are in-phase with the fundamental sinusoidal.wave.
Many impulse spectra of signals meet the period criterion. Mathematically speaking, they are in fact periodic. Temporal alerts are not properly represented by Fourier series, but can be represented 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 repetitive pulse alerts and for transitional form.
Sampled signals
A digital computer is not capable of receiving analog input sounds, but requires a digitized representation of that signal. An analog-to-digital converter changes the input voltage (or current) into a representative binary word. If the device is running clockwise or can be started asynchronously, then it will take 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 time voltage, V(t). The signal is sampled by another signal p(t) with frequency Fs and sampling period T=1/Fs and then later reconstructed. While this may be fairly representative of the waveform, it will be reconstructed with greater accuracy if the sample rate (Fs) is increased.
It happens that a sine wave V (t) is sampled by the sampling pulse alert p (t), which consists of a sequence of equallyspaced narrow values separated in time T. Then the signal spectrum frequency Fs is 1 / T. The result is another impulse response, where the amplitudes are a sampled version of the original sinusoidal alert.
The sampling frequency Fs according to the Nyquist theorem should be twice the maximum frequency (Fm) in the Fourier spectrum of the applied analog signal V (t). To recover the original signal after sampling, the sampled waveform must be passed through a low pass filter that limits the bandwidth to Fs. In practical RF systems, many engineers find that the minimum Nyquist speed is not sufficient for good sampling shape reproductions, so increased speed must be specified. In addition, some oversampling techniques are used to drastically reduce the noise level.
Signal spectrum analyzer
The sampling process is similar to a form of amplitude modulation in which V(t) is the built alert with a spectrum from DC to Fm and p(t) is the carrier frequency. The result obtained resembles a double sideband with the carrier quantity AM. The spectra of the modulation signals appear around the frequency Fo. The actual value is a little more complicated. Like an unfiltered AM radio transmitter, it appears not only around the fundamental frequency (Fs) of the carrier, but also on harmonics spaced Fs up and down.
Assuming that the sampling frequency corresponds to the equation Fs ≧ 2Fm, the original response is reconstructed from the sampled version,passing it through a low oscillation filter with a variable cutoff Fc. In this case, only the analog audio spectrum can be transmitted.
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 region of oscillation. This overlap results in a sampled signal whose spectral width is restored by low-pass filtering. It will not generate the original frequency of the sine wave Fo, but lower, equal to (Fs - Fo), and the information carried in the waveform is lost or distorted.
