In this section we will consider the spectrum of a periodic sequence of rectangular pulses, as one of the most important signals used in practical applications.
Spectrum of a periodic sequence of rectangular pulses
Let the input signal be a periodic sequence of rectangular pulses of amplitude , duration of seconds following with a period of seconds, as shown in Figure 1
Figure 1. Periodic sequence of rectangular pulses
The unit of measurement for signal amplitude depends on the physical process that the signal describes. This could be voltage, or current, or any other physical quantity with its own unit of measurement, which changes over time as . In this case, the units of measurement of spectrum amplitudes , , will coincide with the units of measurement of the amplitude of the original signal.
Then the spectrum , , of this signal can be represented as:
.
Properties of the spectrum of a periodic sequence of rectangular pulses
Let us consider some properties of the spectrum envelope of a periodic sequence of rectangular pulses.
The constant component of the envelope can be obtained as a limit:
To reveal uncertainty, we use L'Hopital's rule:
Where is called the duty cycle of the pulses and specifies the ratio of the pulse repetition period to the duration of a single pulse.
Thus, the value of the envelope at zero frequency is equal to the pulse amplitude divided by the duty cycle. As the duty cycle increases (i.e., when the pulse duration decreases at a fixed repetition period), the value of the envelope at zero frequency decreases.
Using the duty cycle of the pulses, expression (1) can be rewritten as:
The zeros of the spectrum envelope of a sequence of rectangular pulses can be obtained from the equation:
The denominator goes to zero only when , however, as we found out above 
, then the solution to the equation will be
Then the envelope vanishes if
Figure 2 shows the spectrum envelope of a periodic sequence of rectangular pulses (dashed line) and the frequency relationships between the envelope and the discrete spectrum.
Figure 2. Spectrum of a periodic sequence of rectangular pulses
Also shown are the amplitude envelope, amplitude spectrum, as well as the phase envelope and phase spectrum.From Figure 2 you can see that the phase spectrum takes on values when the envelope has negative values. Note that and correspond to the same point of the complex plane equal to .
Example of a spectrum of a periodic sequence of rectangular pulses
Let the input signal be a periodic sequence of rectangular pulses of amplitude, following with a period of a second and different duty cycle. Figure 3a shows time oscillograms of these signals, their amplitude spectra (Figure 3b), as well as continuous envelopes of the spectra (dashed line).
Figure 3. Spectrum of a periodic sequence of rectangular pulses at different duty cycle values
a - time oscillograms; b - amplitude spectrum
As can be seen from Figure 3, as the signal duty cycle increases, the pulse duration decreases, the spectrum envelope expands and decreases in amplitude (dashed line). As a result, the number of spectrum harmonics within the main lobe increases.
Spectrum of a time-shifted periodic sequence of rectangular pulses
Above, we studied in detail the spectrum of a periodic sequence of rectangular pulses for the case when the original signal was symmetrical with respect to . As a result, the spectrum of such a signal is real and is given by expression (1). Now we will look at what happens to the spectrum of the signal if we shift the signal in time, as shown in Figure 4.
Figure 4. Time-shifted periodic sequence of rectangular pulses
The offset signal can be thought of as a signal delayed by half the pulse duration 
. The spectrum of the shifted signal can be represented according to the cyclic time shift property as:
Thus, the spectrum of a periodic sequence of rectangular pulses, shifted relative to zero, is not a purely real function, but acquires an additional phase factor 
. The amplitude and phase spectra are shown in Figure 5.
Figure 5. Amplitude and phase spectra of a time-shifted periodic sequence of rectangular pulses
From Figure 5 it follows that the shift of a periodic signal in time does not change the amplitude spectrum of the signal, but adds a linear component to the phase spectrum of the signal.
conclusions
In this section, we obtained an analytical expression for the spectrum of a periodic sequence of rectangular pulses.
We examined the properties of the spectrum envelope of a periodic sequence of rectangular pulses and gave examples of spectra at different duty cycle values.
The spectrum was also considered when a sequence of rectangular pulses was shifted in time and it was shown that the time shift changes the phase spectrum and does not affect the amplitude spectrum of the signal.
Moscow, Soviet radio, 1977, 608 p.Dötsch, G. A guide to the practical application of the Laplace transform. Moscow, Nauka, 1965, 288 p.
The concept of "signal" can be interpreted in different ways. This is a code or sign transmitted into space, an information carrier, a physical process. The nature of the alerts and their relationship to noise influences its design. Signal spectra can be classified in several ways, but one of the most fundamental is their variation over time (constant and variable). The second main classification category is frequencies. If we consider in more detail in the time domain, among them we can distinguish: static, quasi-static, periodic, repeating, transitional, random and chaotic. Each of these signals has certain properties that can influence related design decisions.
Signal types
Static by definition is unchanged over a very long period of time. Quasi-static is determined by the DC level, so it must be handled in low-drift amplifier circuits. This type of signal does not occur at radio frequencies because some such circuits can produce a constant voltage level. For example, a continuous wave alert with constant amplitude.
The term "quasi-static" means "almost unchanged" and therefore refers to a signal that changes unusually slowly over a long period of time. It has characteristics more similar to static alerts (persistent) than dynamic ones.
Periodic signals
These are the ones that are accurately repeated on a regular basis. Examples of periodic waveforms include sine, square, sawtooth, triangle, etc. The nature of a periodic waveform indicates that it is identical at similar points along a time line. In other words, if the time line 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 by the formula: V (t) = V (t + T).
Repeated signals
They are quasi-periodic in nature, so they have some similarities 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 warning period. Unlike periodic alerts, in repeating sounds these points may not be identical, although they will be very similar, as will the overall waveform. The alert in question may contain either temporary or stable symptoms, which vary.
Transient signals and pulse signals
Both types are either a one-time event or a periodic event in which the duration is very short compared to the period of the waveform. This means that t1<<< t2. Если бы эти сигналы были переходными процессами, то в радиочастотных схемах намеренно генерировались бы в виде импульсов или переходного режима шума. Таким образом, из вышеизложенной информации можно сделать вывод, что фазовый спектр сигнала обеспечивает колебания во времени, которые могут быть постоянными или периодическими.
Fourier series
All continuous periodic signals can be represented by a fundamental sine wave of frequency and a set of cosine harmonics that sum linearly. These vibrations contain swell forms. An elementary sine wave is described by the formula: v = Vm sin(_t), where:
- v - instantaneous amplitude.
- Vm - peak amplitude.
- "_" - angular frequency.
- t - time in seconds.
The period is the time between repetitions 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 a given quantity is decomposed into its component frequencies, either by a bank of frequency-selective filters 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) = a o/2+ _ n -1 )
