Mathematical Equations of Signal Sorting of PW, PRF, PRI, Duty Cycle for AESA Radar

This PhD paper discusses the mathematical equations of signal sorting of PW, PRF, PRI, Duty Cycle for AESA Radar. It covers the radar performance, resolving the velocity ambiguity in pulse doppler radar, and the radar equation. The paper also includes MATLAB simulations and codes on the pulse train for different frequency pulses. The subject is related to radar technology and signal processing. The course code, course name, and college/university are not mentioned.

Mathematical Equations of Signal Sorting of PW, PRF, PRI, Duty Cycle for AESA Radar
Mathematical Equations of Signal Sorting of PW, PRF, PRI, Duty Cycle for AESA Radar
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Mathematical Equations of Signal Sorting of PW, PRF, PRI, Duty Cycle for AESA Radar
Introduction
The Active Electronically Scanned Array radar, AESA, detects the targets, either moving
or stationary, of reflected echoes from transmitted radio frequency pulses. Some of the
measurement metrics that coordinate the detected targets are range, doppler or velocity of
targets, and the angle. The duration a complete pulse takes is the pulse width. The parameter is
used to determine the AESA radar’s maximum and minimum detection range as well as
determining the dead zone. Consecutive pulses are transmitted at the pulse repetition frequency
which has the ability to determine the radar’s maximum detection range. It refers to all the pulses
that are transmitted by the radar per second. AESA radar is a pulse radar that transmits very large
power and high frequency pulsations to the mark. The PRF is used to determine the radar
resolution and range parameter. It mainly focusses on the doppler shifts that form the moving
target indicator radar and pulse doppler radar. There are a number of range ambiguities related to
the pulse doppler radar. The extreme pulse rate and width are determined, largely, by the
transmitter duty cycle in the velocity sphere. The wider pulses are used to improve the AESA
radar’s mode of noting the weak precipitates situated at far off ranges.
Analysis and Discussion
Radar Performance:
Pwidth ∝ Rada r sensitivity
Pwidth ∝ 1
Rang eresolution
According to the Probert-Jones equation,
Pr = Pt G2 θ2 H π3 K2 LZ
1024 ( ln 2 ) λ2 R2
H represents the pulse width which is obtained as,
H= Pr∗1024 ( ln 2 ) λ2 R2
Pt G2 θ2 π3 K2 ( loss factor , L ) ( Target Reflectivity , Z )
H is limited to a choice of 2 values as provided for by the transmitter’s duty cycle. Resolving the
velocity ambiguity in pulse doppler radar
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Mathematical Equations of Signal Sorting of PW, PRF, PRI, Duty Cycle for AESA Radar
f nd=n f r + f nd
1 … .. f nd
1 =dopper frequency
f nd= 2 V t f n
C
f r= 2 V n f n
C
V t =n V n +V nt
1
V max = ( PRF )∗λ
4
Rmax = C
2 ( PRF )
In this case, the pulse width and PRF, are constantly and dynamically adjusted to improve
system performance. In the illustration of a pulsed-RF signal, the duty cycle can be used to
compute the basic components of a pulsed radar,
The signal sorting in AESA radar [1] can be referenced as,
Period=T
PRF= 1
T
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Mathematical Equations of Signal Sorting of PW, PRF, PRI, Duty Cycle for AESA Radar
PRT = 1
PRF
pulse width=τ [ seconds ]
max range= averag epower
Pea k power
=PW∗PRF
PAve= Pi τ
T ,T − pulse period
D=
Pi τ
T
Pt
= PAve
Ppk
The radar equation is given as,
Rmax
4 = Pt G2 λ2 σ
k T0 Bn Fn ( S
N ) ( 4 π3 )
ρT = PT
As
= Pt
4 π R2
Rmax =
[ Pt G2 λ2 σ
k T 0 Bn Fn ( S
N ) ( 4 π 3 ) ] 1
4
= c0∗PRT −Pw
2
The modified radar equation based on pulsed radar performance parameters [2],
si= k T o Bn Fn S0
N o
The minimum detectable signal is given such that,
smin=k T o Bn Fn ( S0
N 0 )min
As a result,
Rmax =
[ Pt G Ae σ
k T 0 Bn Fn ( S0
N0 ) ( 4 π2 ) ]1
4
Using the radar range measurements,
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Mathematical Equations of Signal Sorting of PW, PRF, PRI, Duty Cycle for AESA Radar
D=100 ( PW
PRI )=100 ( τ
PRI )⇒ c
2 PRF =c ( PRI
2 )=[ m/ s] PRI
Runamb= c
2 ( PRF ) =c ( PRI
2 )… . signal sorting
Runamb =150[m/μs ]∗( PRI )
MATLAB Simulation and codes on the pulse train for different frequency pulses.
Frequency
pulse
Amplitude
4.7 GHz 3.45 V
4.4 GHz 3.25 V
4.2 GHz 3.10 V
10.2 GHz 3.90 V
2.4 GHz 8.40 V
2.1 GHz 15.0 V
2.9 GHz 12.1 V
The pulse width is given as 5e-5 seconds, and the propagation speed c, 3e8 m/s, and the
PRI or T or period is given as 0.0001 seconds while the PRF is the inverse of PRI, hence, it is
given as 1 MHz Using MATLAB,
x ( t ) = A ( t ) cos [ ωc t+ θ ( t ) ]
ωi ( t ) = d
dt [ ωc t+ θ ( t ) ]
ωi ( t )=ωc+ d
dt θ ( t ) … . frequency deviation
The LPE signal for the radar signal is given as,
xl ( t ) = A ( t ) e j [ wl t +θ ( t ) ]
¿ A ( t ) cos [ wl t+θ ( t ) ] + jA ( t ) sin[wl t+θ (t)]
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Mathematical Equations of Signal Sorting of PW, PRF, PRI, Duty Cycle for AESA Radar
0 0.5 1 1.5 2 2.5 3 3.5 4
10-5
-0.5
0
0.5
1
1.5 Duty cycle is 0.25
0 0.5 1 1.5 2 2.5 3 3.5 4
10-5
-0.5
0
0.5
1
1.5 Duty cycle is 0.5
0 0.5 1 1.5 2 2.5 3 3.5 4
10-5
-0.5
0
0.5
1
1.5 Duty cycle is 0.75
%% generating pulse trains at different frequencies
Fs = input('Enter Frequency(Hz):');
t = 0:1/Fs:(10*4e-6);
pulsewidth = 1e-6;
pulseperiods = [0:10]*4e-6;
x = pulstran(t,pulseperiods,@rectpuls,pulsewidth);
plot(t,x)
axis([0 4e-5 -0.5 1.5])
nwid = 3;
for nn = 1:nwid
x = pulstran(t,pulseperiods,@rectpuls,nn*pulsewidth);
subplot(nwid,1,nn)
plot(t,x)
axis([0 4e-5 -0.5 1.5])
D = dutycycle(x,Fs);
title(['Duty cycle is ' num2str(mean(D))])
end
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Mathematical Equations of Signal Sorting of PW, PRF, PRI, Duty Cycle for AESA Radar
Run the Simulink with the simulation time being 100.0 seconds
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Mathematical Equations of Signal Sorting of PW, PRF, PRI, Duty Cycle for AESA Radar
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Mathematical Equations of Signal Sorting of PW, PRF, PRI, Duty Cycle for AESA Radar
REFERENCES
[1] G. W. Stimson, "Radar Concepts in ROBOT Radar," Scitech Publications, 2011. [Online].
Available: http://home.iitk.ac.in/~nnaik/pdf/Introduction-to-Radar-Lecture-1-Material.pdf.
[2] K. Subhas, "Basics of Radar and Radar Equation," Radar Systems, 17 May 2016. [Online].
Available: https://mrcet.com/downloads/digital_notes/ECE/IV%20Year/Radar
%20Systems.pdf.
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Mathematical Equations of Signal Sorting of PW, PRF, PRI, Duty Cycle for AESA Radar
APPENDIX
Appendix A
MATLAB simulation and code
% Create a Linear FM Waveform object
h = phased.LinearFMWaveform;
h.SampleRate = 200000;
h.PulseWidth = 5e-05;
h.PRF = 10000;
% Keep changing this value to test for different frequencies
h.SweepBandwidth = 10200000000;
h.SweepDirection = 'Down';
h.SweepInterval = 'Symmetric';
h.Envelope = 'Gaussian';
h.NumPulses = 2;
%Create figure, panel, and axes
fig = figure;
panel = uipanel('Parent',fig);
hAxes = axes('Parent',panel,'Color','none');
Fs = h.SampleRate;
x = step(h);
l = (0:length(x)-1)/Fs;
hAxes1 = subplot(2,1,1);
[~, scale, Units] = engunits(l(end));
l = l*scale;
plot(l,real(x));
xlim([0 200]);
ylim([-2.1537e-11 4.229e-11]);
xlabel('Time (us)');
ylabel('Amplitude (V)');
title('Waveform: Real Part');
grid on;
hAxes2 = subplot(2,1,2);
plot(l,imag(x));
xlim([0 200]);
ylim([-2.1537e-11 4.229e-11]);
hlink = linkprop([hAxes1 hAxes2],{'XLim','YLim'});
setappdata(panel,'Link',hlink);
xlabel('Time (us)');
ylabel('Amplitude (V)');
title('Waveform: Imaginary Part');
grid on;
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Mathematical Equations of Signal Sorting of PW, PRF, PRI, Duty Cycle for AESA Radar
Appendix B
Results on figures based on different frequencies
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