Simulation of radar emitter modulated signal

Radar emitter modulation signal simulation

explain ： adopt Matlab Carry out single carrier frequency （CW）, LFM （LFM）、 Nonlinear frequency modulation （NLFM）、 Two phase code （BPSK）、 Four phase code （QPSK）、 Two frequency coding （BFSK）、 Quad frequency coding （QFSK） Square simulation of seven kinds of radar pulse modulated signals .

Environmental Science ：Matlab

Radar emitter modulation signal simulation （ Code ）

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1. Single carrier frequency signal

The frequency and phase of a single carrier frequency conventional pulse signal do not change with time, which is the most basic radar signal , At that time, the broadband product was about 1. The mathematical expression is as follows ：
$$s\left(t\right)=u\left(t\right)e^{j(2\pi f_{0}t+{\phi }_0)}$$
among $u\left(t\right)$ Is the normalized envelope of the conventional rectangular pulse signal ：
$$\{\begin{array}{l}\frac{1}{\sqrt{T}},-\frac{T}{2}<t<\frac{T}{2}\\ 0,\end{array}$$

$T$ Single carrier frequency pulse width , $f_0$ Carrier frequency , ${\phi }_0$ It's the first phase . Time domain of single carrier frequency pulse signal 、 The frequency domain waveform is shown in the figure below ：

Time domain ：

frequency domain ：

Frequency domain plus noise ：

As can be seen from the above figure , Single carrier frequency pulse signal is a signal whose frequency and phase do not change with time , Its spectrum has only a single frequency point .

2. Linear frequency modulation signal

Linear frequency modulation based on adaptive filter theory （LFM） The signal , When the signal has a large Doppler shift , The matched filter can still play the role of pulse compression , Its mathematical expression is as follows ：

$$s(t)=u(t)e^{j(2\pi f_{0}t+{\phi }_0)}$$
among $u(t)$ Is the complex envelope of linear frequency modulated rectangular pulse ：

$$u(t)=\frac{1}{\sqrt{T}}rect(\frac{t}{T})e^{j\pi \mu t^2},rect(\frac{t}{T})=\{\begin{array}{l}1,|t|\le T/2\\ 0,|t|>T/2\end{array}$$

$T$ Is the pulse width of LFM pulse signal , $f_0$ Is the carrier frequency , ${\phi }_0$ Is the initial phase , $\mu =B/T$ It's the FM slope , $B$ Is frequency offset , That is, frequency modulation bandwidth . Time domain of LFM pulse signal 、 The frequency domain diagram is as follows ：

Time domain ：

frequency domain ：

Frequency domain plus noise ：

The signal intensity varies with t change , That is, the frequency of the signal changes linearly with time , From the frequency domain diagram, it can be seen that the frequency spectrum also changes as a linear function .

3. Nonlinear FM signal

The requirement of large time and wide bandwidth of modern radar in linear frequency modulation can be adjusted by means of pulse compression , However, the ratio of main and side valves is still low . After that, nonlinear frequency modulation technology is derived , It is famous for its comprehensive variety and wide application range , In this paper, a Nonlinear FM signal is solved approximately by using the principle of sojourn phase , First, choose one hamming window , The expression is ：
$$W(f)=\{\begin{array}{l}0.54-0.46\cos (\frac{2\pi f}{B}),|f|\le \frac{B}{2}\\ 0,|f|>\frac{B}{2}\end{array}$$

For a given function $W(f)$ Derive the group function $T(f)$ ：
$$T(f)=K_1{\int }_{-\infty }^{f}W(x)dx,-\frac{B}{2}\le f\le \frac{B}{2}$$

among $K_1$ Constant coefficient , It can be determined by specific time delay and frequency offset , Through the iterative numerical calculation method, the $T(f)$ The inverse function of $f(t)$ ：
$$f(t)=T^{-1}(f),0\le t\le T$$

Frequency modulation function , For functions $f(t)$ Integrate to get the phase $\theta (t)$ ：
$$\theta (t)=2\pi {\int }_0^{t}f(x)dx,0\le t\le T$$
Finally, the Nonlinear FM signal is obtained ：
$$\begin{array}{l}s(t)=Arect(\frac{t}{\sqrt{T}})e^{j[2\pi f_{0}t+\theta (t)+{\phi }_0]},0\le t\le T\\ Arect(\frac{t}{\sqrt{T}})=\{\begin{array}{l}1,|t|\le T/2\\ 0,|t|>T/2\end{array}\end{array}$$

among $T$ Is the pulse width of the nonlinear frequency modulated pulse signal , $f_0$ Is the loading rate , ${\phi }_0$ Is the initial phase of the signal , $\theta (t)$ It is a Nonlinear FM signal . Nonlinear frequency modulation time domain 、 The frequency domain diagram is as follows ：

Time domain ：

frequency domain ：

Frequency domain plus noise ：

The density of the time domain diagram of the Nonlinear FM signal varies nonlinearly , In the frequency domain diagram, the spectrum is also nonlinear .

4. Two phase coded signal

The phase coded signal has the advantage of good main sidelobe ratio under the condition of hourly broadband product , Therefore, phase coded modulation is widely used in modern radar . Phase coded pulse signal , The internal sub pulse phase is encoded by a specified sequence , The sub pulses are closely connected . The phase difference between the sub pulses of the two-phase coded signal is only 0 and π Two values , Commonly used codes are Barker code 、M Sequence sum L Sequence code, etc . Its mathematical expression is as follows ：
$$s(t)=u(t)e^{j(2\pi f_0+{\phi }_0)}$$
among ：
$$u(t)=\{\begin{array}{l}\frac{1}{\sqrt{P}}\sum _{K=0}^{P-1}{\mathrm{c}}_{K}v(t-KT),0<t<PT\\ 0,\end{array}$$

It is the complex envelope of two-phase coded rectangular pulse , $v(t)$ Is an internal pulse function , $T$ Is the internal pulse width , $P$ Is the length of the symbol , $PT$ Is the duration of the signal , The value of the sequence corresponding to the two-phase coded signal is { $c_K=1,-1$}. The coding sequence used in the biphasic coding in this paper is Barker code with ideal aperiodic autocorrelation function . Time domain of two-phase coded signal 、 The frequency domain is as follows ：

Time domain ：

frequency domain ：

Frequency domain plus noise ：

It can be seen from the above two-phase coded signal time domain diagram , The phase between two sub pulses appears according to the law of coding sequence 0 and π The migration , The signal intensity has not changed , That is, the frequency does not change . In the frequency domain diagram of two-phase coded signal, the spectrum is represented as a single frequency point .

5. Four phase coded signal

The four phase coded signal is , Phase difference between sub pulses 0,$\pi /2$ ,π as well as $3\pi /2$ Four possibilities . Because the known Barker code sequence has limited length , The optimal sidelobe attenuation that can be provided is -22.2dB, In order to meet the needs of intra pulse modulation , Taylor is often used for four phase coded signals （Taylor）, Frank （Frank） Polyphase code, etc , This paper adopts Frank Code to construct four phase coded signal . Its mathematical expression is as follows ：
$$s(t)=u(t)e^{j(2\pi f_0+{\phi }_0)}$$

among ：
$$u(t)=\{\begin{array}{l}\frac{1}{\sqrt{P}}\sum _{K=0}^{P-1}{\mathrm{c}}_{K}v(t-KT),0<t<PT\\ 0,\end{array}$$

It is the complex envelope of four phase coded rectangular pulse , $v(t)$ Is an internal pulse function , $T$ Is the internal pulse width , $P$ Is the symbol length , $PT$ Is the duration of the signal , The corresponding sequence value of four phase coded signal is { $c_K=1,j,-1,-j$}. Time domain of four phase coded signal 、 The frequency domain is as follows ：

Time domain ：

frequency domain ：

Frequency domain plus noise ：

It can be seen from the above four phase coded signal time domain diagram , The phases between some two sub pulses appear respectively according to the law of coding sequence 0,$\pi /2$ ,π as well as $3\pi /2$ The migration , The signal intensity has not changed , That is, the frequency does not change . In the frequency domain diagram of the four phase coded signal, the spectrum is represented as a single frequency point .

6. Two frequency coded signal

Frequency coded signals are also widely used in modern radar , The sub pulses in the rectangular pulse are encoded by the encoding sequence , It shows that the internal sub pulse signal frequency is within the specified frequency set { $f_1...f_M$ } Change in , This kind of coded signal has good range resolution and instantaneous bandwidth . The mathematical expression of two frequency coding is as follows ：
$$s(t)=u(t)e^{j(2\pi f_i+2\pi f_0+{\phi }_0)}$$

among $u(t)$
$$\begin{array}{l}u(t)=\sum _{i=0}^{N_c-1}Arect[t-i\Delta L,\Delta L]\\ Arect[t-i\Delta L,\Delta L]=\{\begin{array}{l}1,0\le n\Delta T\le \Delta L\\ 0\end{array}\end{array}$$
It is the complex envelope of two frequency coded pulse signal , $f_0$ Is the carrier frequency , $f_i$ Is the frequency code group , ${\phi }_0$ Is the initial phase , $\Delta L$ Is the subcode width . The frequency code group of two frequency coding has only { $f_1,f_2$} Two values , That is, inside the rectangular pulse , The sub pulse frequency depends on the coding sequence , Take the frequency in the frequency code group respectively . Time domain of two frequency coded signal 、 The frequency domain is as follows ：

Time domain ：

frequency domain ：

Frequency domain plus noise ：

From the above two frequency coding time domain diagram, we can see , The density of the signal, that is, the frequency of the signal, is in accordance with the coding law { $f_1,f_2$} Change in . In the frequency domain diagram, the spectrum is represented by two frequency points .

7. Four frequency coded signal

The mathematical expression of the four frequency coded signal is as follows ：

$s(t)=u(t)e^{j(2\pi f_i+2\pi f_0+{\phi }_0)}$

among $u(t)$：

$\begin{array}{l}u(t)=\sum _{i=0}^{N_c-1}Arect[t-i\Delta L,\Delta L]\\ Arect[t-i\Delta L,\Delta L]=\{\begin{array}{l}1,0\le n\Delta T\le \Delta L\\ 0\end{array}\end{array}$

It is the complex envelope of the four frequency coded pulse signal , $f_0$ Is the carrier frequency , $f_i$ Is the frequency code group , ${\phi }_0$ Is the initial phase , $\Delta L$ Is the subcode width . The frequency code group of four frequency coding has { $f_1,f_2,f_3,f_4$} Four values , That is, inside the rectangular pulse , The sub pulse frequency is determined according to Frank The coding sequence respectively takes the frequency in the frequency code group . Time domain of four frequency coded signal 、 The frequency domain is as follows ：

Time domain ：

From the above time domain diagram of four frequency coding, we can see , The density of the signal, that is, the frequency of the signal, is in accordance with the coding law { $f_1,f_2,f_3,f_4$ } Change in . In the frequency domain diagram, the spectrum is represented by four frequency points .