Formula of radio pulses of rectangular shape. Mathematical spectrum of a periodic sequence of rectangular radio pulses

dt = 0.01; = 0: dt: 4; = sin (10 * 2 * pi * t). * rectpuls (t-0.5,1); (4,1,1), plot (t, y); (" t "), ylabel (" y (t) ") (" RF pulse with a rectangular envelope ")

Xcorr (y, "unbiased"); (4,1,2), plot (b * dt, Rss); ([- 2,2, -0.2,0.2]) ("\ tau"), ylabel ("Rss (\ tau) ") (" auto-correlation ") = fft (y, 8192); = abs (Y); = 5000 * (0: 4096) / 8192; = 2 * pi * f; (4,1, 3), plot (w, AY (1: 4097)) ("\ omega"), ylabel ("yA (\ omega)") ("Amplitude-frequency characteristic") (4,1,4) = phase (Y ); (w, PY (1: 4097)) ("phase-frequency characteristic")

graphical representation of a radio pulse with a rectangular envelope

all = 0.01; = - 4: dt: 4; = sinc (10 * t); (4,1,1), plot (t, y); ([- 1,1, -0.5,1.5]) (" t "), ylabel (" y (t) "), title (" y = sinc (t) ")

Xcorr (y, "unbiased"); (4,1,2), plot (b * dt, Rss); ([- 1,1, -0.02,0.02]) ("\ tau"), ylabel ("Rss (\ tau) ") (" auto-correlation ") = fft (y, 8192); = abs (Y); = 5000 * (0: 4096) / 8192; = 2 * pi * f; (4,1, 3), plot (w, AY (1: 4097)) () ("\ omega"), ylabel ("yA (\ omega)") ("Amplitude-frequency characteristic") (4,1,4) = phase (Y); (w, PY (1: 4097)) () ("phase-frequency characteristic")

graphical representation of sync

Radio pulse with Gaussian envelope

dt = 0.01; = - 4: dt: 4; = sin (5 * 2 * pi * t). * exp (-t. * t); (4,1,1), plot (t, y); ( "t"), ylabel ("y (t)") ("y (t) = Gaussian function")

Xcorr (y, "unbiased"); (4,1,2), plot (b * dt, Rss); ([- 4,4, -0.1,0.1]) ("\ tau"), ylabel ("Rss (\ tau) ") (" auto-correlation ") = fft (y, 8192); = abs (Y); = 5000 * (0: 4096) / 8192; = 2 * pi * f; (4,1, 3), plot (w, AY (1: 4097)) ("\ omega"), ylabel ("yA (\ omega)") ("Amplitude-frequency characteristic") = phase (Y); (4,1, 4)

plot (w, PY (1: 4097))

graphical representation of a radio pulse with a Gaussian envelope

Pulse train of the "meander" type

dt = 0.01; = 0: dt: 4; = square (2 * pi * 1000 * t); (4,1,1), plot (t, y); ("t"), ylabel ("y (t ) ") (" y = y (x) ")

Xcorr (y, "unbiased"); (4,1,2), plot (b * dt, Rss); ("\ tau"), ylabel ("Rss (\ tau)") ("auto-correlation") = fft (y, 8192); = abs (Y); = 5000 * (0: 4096) / 8192; = 2 * pi * f; (4,1,3), plot (w, AY (1: 4097) ) ("\ omega"), ylabel ("yA (\ omega)") ("Amplitude-frequency characteristic") = phase (Y); (4,1,4)

plot (w, PY (1: 4097))

graphical representation of a meander pulse train

Phase-Manipulated Sequence

xt = 0.5 * sign (cos (0.5 * pi * t)) + 0.5;

y = cos (w0 * t + xt * pi);

subplot (4,1,1), plot (t, y);

axis () ("t"), ylabel ("y (t)"), title ("PSK")

plot (w, PY (1: 4097))

a graphical representation of a phase shift keyed sequence

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Call the file AmRect. dat... Sketch the signal and its spectrum. Determine the width of the radio pulse, its height U o , the carrier frequency f о, the amplitude of the spectrum C max and the width of its lobes. Compare them with the parameters of the modulating video pulse, which you can see Fig. 14. call from the file RectVideo.dat.

3.2.7. Sequence of radio pulses

A. Call the file AmRect. dat.

B. Click on and set the window width Wx = 250 μs

V. Key<8>, set the "Periodic" type of signal, and by pressing<Т>or , enter the period T = 100 μs. Sketch the signal.

* If you activate the vertical menu button<7, F7 –T>, then the signal period can be changed using the horizontal arrows on the keyboard.

G. Go to the spectra window and with the key<0>(zero) move the origin to the left. Sketch the spectrum. Write down the interval value df between spectral lines and the number of lines in the spectral lobes. Compare this data with, T and the so-called signal duty cycle Q = T/ .

E. Record the C max value and compare with that for a single signal.

Explain all results.

* 3.2.8. Formation and study of AM signals

The SASWin program allows you to generate signals with various and rather complex types of modulation. You are offered, using the acquired experience of working with the program, to form an AM-signal, the parameters and shape of the envelope of which you set yourself.

A. In the Plot option, use the mouse or cursor to create the desired type of modulation signal. It is recommended not to get carried away with its too complex form. Sketch the spectrum of your signal.

B. Memorize the signal by pressing the vertical menu button<R AM> and assigning a name or number to the signal.

V. Enter the Instal option and specify the signal type<Радио>. In the menu of modulation types that opens, select the Normal option of Amplitude modulation and press the button<Ок>.

G. To the request "The law of amplitude change" specify<1.F(t) из ОЗУ>.

D. The vertical menu of signals in RAM appears.

Select your signal and press the button .

For example: Carrier frequency, kHz = 100,

Carrier phase = 0,

Frequency window boundaries fmin and fmax for spectrum display

Press the button

The generated signal is displayed in the left window, and its spectrum - in the right.

J. Sketch the generated signal and its spectrum. Compare them with the shape and spectrum of the modulation signal.

Z. The signal can be written to RAM memory or to a file and then used as needed.

AND. Repeat the test with other modulation signals if desired.

3.3. Angle modulation

3.3.1. Low index harmonic modulation 

A. Call up the signal (Fig. 15)) from the file FMB0"5. dat... Sketch its spectrum. Compare the spectrum with the theoretical one (see Fig. 10, a). Note how it differs from the AM spectrum.

B. Determine the carrier frequency from the spectrum f o, modulation frequency F, initial phases  O and  ... Measure the amplitudes of the components of the spectrum, from them find the index

Rice. 15.modulation  . Determine the width of the spectrum.

3.3.2. Harmonic FM with index  >1

A. Call the file FMB"5. dat, where the signal with the index = 5 is written (Fig. 16). Sketch the signal and its spectrum.

B. Determine the modulation frequency F, the number of side components of the spectrum and its width. Find the frequency deviation  f using

Rice. 16.the formula    f / F... Compare the deviation with the measured spectrum width.

V. Measure the relative amplitudes C (f) / C max of the first three to four components of the spectrum and compare them with the theoretical values determined by the Bessel functions
... Pay attention to the phases of the spectral components.

In contrast to the spectrum of the bell pack, the spectra of rectangular packs have a different lobe shape, namely.

Spectra of packets of rectangular radio pulses

· The shape of the ASF arches is determined by the shape of the ASF impulses.

· The shape of the ASF petals is determined by the shape of the ASF pack.

· Spectra of bursts of video pulses are located on the frequency axis in the vicinity of the lower frequencies, and the spectra of bursts of radio pulses are located in the vicinity of the carrier frequency.

The numerical value of the spectral density of a burst of pulses is determined by its energy, which, in turn, is directly proportional to the amplitude of pulses in a burst of pulse duration and the number of pulses in a burst TO(burst duration) and is inversely proportional to the pulse repetition period

With the number of pulses in a burst, the signal base (bandwidth factor) =

1.5.2. Intra-pulse modulated signals

In the theory of radar, it has been proven that in order to increase the range of the radar, it is necessary to increase the duration of the sounding pulses, and to improve the resolution, to expand the spectrum of these pulses.

Radio signals without intrapulse modulation (“smooth”), used as sounding signals, cannot simultaneously satisfy these requirements, because their duration and spectrum width are inversely proportional to each other.

Therefore, at present, in radar, sounding radio pulses with intra-pulse modulation are increasingly used.

Chirp radio pulse

An analytical expression for such a radio signal will be:

where is the amplitude of the radio pulse,

Pulse duration,

Average carrier frequency,

rate of change of frequency;

Frequency variation law.

The graph of a radio signal with a chirp and the law of changing the frequency of the signal inside the pulse (shown in Figure 1.63 a radio pulse with a frequency increasing in time) are shown in Figure 1.63

The amplitude-frequency spectrum of such a radio pulse has an approximately rectangular shape (Fig. 1.64)

For comparison, the AFR of a single rectangular radio pulse without intra-pulse frequency modulation is shown below. Due to the fact that the duration of a radio pulse with a chirp is long, it can be conditionally divided into a set of radio pulses without a chirp, the frequencies of which change according to the stepwise law shown in Figure 1.65

The spectra of each of the radio pulses without JICHM will each be at its own frequency: .

signal. It is easy to show that the shape of the AFC will coincide with the shape of the original signal.

Phase-code-manipulated impulses (PCM)

PCM radio pulses are characterized by a jump-like phase change within the pulse according to a certain law, for example (Fig. 1.66):

three-element signal code

phase change law

three-element signal

or seven-element signal (fig. 1.67)

Thus, we can draw conclusions:

· AShS signals with chirp is continuous.

· The AFR envelope is determined by the shape of the signal envelope.

· The maximum value of AFC is determined by the signal energy, which, in turn, is directly proportional to the amplitude and duration of the signal.

The spectrum width is where is the frequency deviation and does not depend on the signal duration.

Signal base (bandwidth ratio) may be n>> 1. Therefore, chirp signals are called broadband.

PCM radio pulses with a duration are a set of elementary radio pulses following one after another without intervals, the duration of each of them is the same and is equal to ... The amplitudes and frequencies of elementary pulses are the same, and the initial phases may differ by (or some other value). The law (code) of the alternation of the initial phases is determined by the purpose of the signal. For FKM radio pulses used in radar, the corresponding codes have been developed, for example:

1, +1, -1 - three-element codes

- two variants of the four-element code

1 +1 +1, -1, -1, +1, -2 - seven-element code

The spectral density of the encoded pulses is determined using the additivity property of the Fourier transforms, in the form of the sum of the spectral densities of elementary radio pulses.

ASF graphs for three-element and seven-element impulses are shown in Figure 1.68

As can be seen from the above figures, the spectrum width of the PCM radio signals is determined by the duration of an elementary radio pulse

or .

Broadband ratio , where N- the number of elementary radio pulses.

2. Analysis of processes by temporary methods. General information about transient processes in electrical circuits and the classical method of their analysis

2.1. The concept of a transient regime. Commutation laws and initial conditions

Processes in electrical circuits can be stationary and non-stationary (transient). A transient process in an electrical circuit is a process in which currents and voltages are not constant or periodic functions of time. Transient processes can occur in circuits containing reactive elements when connecting or disconnecting energy sources, abrupt changes in the circuit or parameters of incoming elements (switching), as well as when signals pass through the circuits. In the diagrams, switching is denoted in the form of a key (Fig. 2.1), it is assumed that switching occurs instantly. The moment of switching is conventionally taken as the origin of time. In circuits that do not contain energy-intensive elements L and C during switching, transitional

there are no processes. In circuits with energy-consuming elements, transient processes continue for some time, because capacitor stored energy or inductance cannot change abruptly, because this would require an energy source of infinite power. In this regard, the voltage across the capacitor and the current through the inductance cannot change abruptly. Denoting

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Objective

Study of the temporal and spectral characteristics of pulsed radio signals used in radar, radio navigation, radio telemetry and related fields;

Acquisition of skills in calculating and analyzing the correlation and spectral characteristics of deterministic signals: autocorrelation functions, amplitude spectra, phase spectra and energy spectra;

Study of methods for optimal matched filtering of signals of a known shape against the background of noise such as white noise;

Acquisition of skills in performing engineering calculations to determine the spectral characteristics of signals on a PC

All calculations performed in the work were performed using the Mathcad 14 program.

List of symbols, units and terms

u - carrier frequency, Hz

F S - repetition frequency, Hz

f - pulse duration, s

N is the number of pulses in a burst

T n - distance between two pulses (period), s

U1 (t) - envelope of one radio pulse

S1 (t) - single radio pulse

S (t) - burst of radio pulses

S11 (u) - spectral density of the amplitude of one video pulse

Sw (u) - spectral density of a burst of radio pulses

W (u) - energy spectrum

W (f1) - ACF signal

A - an arbitrary constant coefficient

h (t) - impulse response of the matched filter

Coursework assignment

Preset signal type:

Rectangular coherent bundle of rectangular radio pulses. In the middle of each pulse, the phase changes abruptly by 180 °.

Sub-option number - 3:

Carrier frequency - u = 2.02 MHz,

Pulse duration - f = 55 μs,

Repetition frequency -Fs = 40kHz,

The number of pulses in a pack - N = 7

1) The mathematical model of the signal.

2) Calculation of the ACF.

3) Calculation of the spectrum of amplitudes and energy spectrum.

4) Calculation of the impulse response of the matched filter.

Chapter 1.Calculation of signal parameters

1.1 Calculation of the mathematical model of the signal

A single rectangular pulse, in the middle of which the phase changes abruptly by 180є, can be described by the expression:

The graph of a single radio pulse is shown in Fig. 1.

Fig. 1. Single radio pulse graph

In Fig. 2, let us consider in more detail the middle of the pulse, where the phase changes by 180є

Fig. 2. Detailed graph of a single radio pulse.

The envelope of one radio pulse is shown in Fig. 3.

Fig. 3 Envelope of one radio pulse

Since all pulses in a pack have the same shape, then when constructing a coherent pack, you can use the formula:

where T n is the pulse repetition period, N is the number of pulses in the burst, U1 (t) is the envelope of the first pulse

Figure 4 shows a view of a coherent rectangular burst of radio pulses.

Fig. 4-Coherent burst of radio pulses

1.2 Calculation of the amplitude spectrum

The modulus of the spectral density characterizes the density of the distribution of the amplitudes of the components of the continuous spectrum of the signal in frequency, and the argument of the spectral density characterizes the distribution of the phases of the components.

In this case, there is no need to integrate over these limits, since a single signal is in the range from (0; f), and outside that limit is identically equal to zero.

For a given signal, the spectral density of the amplitudes of a single video pulse is shown in Fig. 5

Fig. 5-Spectral density of a single radio pulse

The spectrum of the amplitudes of a burst of radio pulses is the product of the spectrum of the amplitudes of a single pulse and a function of the form | sin (Nx) / sin (x) | called the "lattice factor". This function is periodic.

The spectrum of the amplitudes of a burst of radio pulses is shown in Fig. 7.

Fig. 6 Spectral density of the packet

1.3 Calculation of the energy spectrum

spectrum pulsed radio signal amplitude

The energy spectrum is calculated using a simple ratio

The energy spectrum is shown in Fig. 11. Figure 12 shows an enlarged fragment of the energy spectrum.

Fig. 7 - Energy spectrum of the signal

1.4 Calculation of the autocorrelation function

The autocorrelation function (ACF) of the signal is used to quantify the degree of difference between the signal and its time-shifted copy s (t-) and is their scalar product on an infinite interval

The ACF for the envelope of one pulse is shown in Fig. 13.

Fig. 13 ACF for one pulse envelope

The autocorrelation function for a given signal is shown in Fig. 14.

Fig. 14 ACF of a given signal

Chapter 2... Calculation of the parameters of the matched filter

2.1 Impulse response calculation

The impulse response of the matched filter is a scaled copy of the mirror image of the input signal shifted by a certain time interval. Otherwise, the condition of the physical realizability of the filter is not met, since the signal must have time to be "processed" by the filter during this time.

We build the impulse response for the envelope of a given signal.

The envelope of the pack is shown in Fig. 15.

Fig. 15 Bundle envelope

The impulse response is shown in Fig. 16.

Fig. 16 Impulse response of the matched filter

The block diagram of the matched filter for a given signal is shown in Fig. 18.

In this course work, the signal parameters were calculated for a rectangular coherent bundle of rectangular radio pulses, in which the phase changes by 180є in the middle of the pulse.

Also in the Mathcad 14 program, graphs of the signal envelope, spectral density, energy spectrum, autocorrelation function were built.

The impulse response of the matched filter was also plotted.

Bibliography

1) Baskakov S.I., Radio circuits and signals: Textbook. for universities on specials. "Radiotekhnika" .- 2nd ed .., revised. and additional-M: Higher school .., 1988.

2) Kobernichenko V.G., Methodical instructions for term paper.

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A single radio pulse is given by the amplitude U= 1V, frequency f and pulse duration τ specified in table 1.

1. Determine the spectrum of amplitudes and phases for the variant of a single radio pulse indicated in the table. Provide tables and graphs, give an analysis of the results

2. To study changes in the spectrum of amplitudes and phases when changing τ them . (τ them =0,5τ , τ them =τ , τ them =1,5τ ). Provide tables and graphs, give an analysis of the results.

3. To study the changes in the spectrum of amplitudes and phases at a pulse shift Δt relative to t = 0Δt = 0.5 τ themΔt = 1.5 τ them... Provide tables and graphs to provide an analysis of the results.

4. Determine the signal spectrum width in accordance with

the criteria used.

5. Determine the width of the signal spectrum that provides the transfer of 0.9 signal energy at various signal durations.

using the programs provided in the appendix

I... Periodic pulse train

The calculation of the spectral characteristics of a periodic rectangular signal can be carried out using programs developed by students, using spreadsheets or the program "Spectrum_1.xls" given in the electronic

versions of this manual. The program "Spectrum_1.xls" uses a numerical method for finding the spectral components.

Formulas used to calculate the spectrum for

periodic signals

The method is based on the formulas below

(2)

(3)

(4)

where C 0 - constant component,

ω 1 = 2π / T is the angular frequency of the first harmonic,

T - function repetition period,

k – harmonic number

C k- amplitude k- th harmonic,

φ k- phase k- th harmonic.

The calculation of the harmonic components is reduced to the calculation by the approximate integration formulas

(5)

(6)

where N- the number of discrete samples per period

the function under study f(t)

Δ t = T/ N- the step with which the function counts are located f(.).

The constant component is found by the formula C 0 = a 0

The transition to a complex form of presentation is carried out according to the formulas given below:

;
; (7)

For periodic signals with a limited spectrum, the power is found by the formula:

(8)

where P – spectrum limited signal power n harmonics.

To solve the problem of spectral analysis according to the above formulas, the appendix contains programs for calculating spectral characteristics. The programs are executed in the VBAMicrosoftExcel environment.

The program is launched from the “Spectrum” folder by double-clicking the left mouse button on the program name. The window with the name of the program is shown in Fig. 1. After the appearance of the image shown in Fig. 2, you should enter the initial data for the calculation in the corresponding fields highlighted in color

Fig 1. Launching the program

Fig. 2. Periodic signal with a period of 1000 μs and

duration 500 μs

After the appearance of the image shown in Fig. 2, you should enter the initial data for the calculation in the corresponding fields highlighted in color. In accordance with the specification for a variant of a sequence of rectangular pulses with a period of 1000 μs and a duration of 500 μs, a spectrum of amplitudes and phases is found. After entering data in each field, press the "Enter" key. To start the program, move the cursor to the "Calculate spectrum" button and press the left mouse button.

Tables and graphs of the dependence of the modulus of amplitudes and phases on the harmonic number and frequency are shown in Fig. 3 - 5

Rice. 3. Table with calculation results

In fig. 3 shows the results of the calculation, collected in a table on sheet 3. The following results are displayed in the columns: 1 - harmonic number, 2 - frequency of the harmonic component, 3 - amplitude of the cosine component of the spectrum, 4 - amplitude of the sine component of the spectrum, 5 - amplitude modulus, 6 - phase spectral component. The table in Fig. 3 shows an example of calculation for a pulse repetition period T = 1000 μs and a pulse duration τ = 500 μs. The number of points per period is selected depending on the required accuracy of the calculation and must be at least twice the number of calculated harmonics.

Rice. 4. The module of the spectral components of the signal with a period of 1000 μs and a duration of 500 μs

Rice. 5. Phases of the spectral components of the signal with a period of 1000 μs and a duration of 500 μs

Fig. 6. The sum of the powers of the harmonic components.

The reconstructed signal is shown in Fig. 7. The shape of the reconstructed signal is determined by formula (1) and depends on the number of harmonics