Radar Systems  Quick Guide
Radar Systems  Overview
RADAR is an electromagnetic detection system that operates by radiating electromagnetic waves then by studying the echo or reflected waves.
The full form of RADAR is RA dio D etection A and R anging. Detection refers to the presence or absence of the target. The target can be fixed or mobile, that is to say not stationary. Telemetry refers to the distance between the radar and the target.
Radars can be used for applications are listed below.
 Air traffic control
 Ship safety
 Detection of distant places
 Applications military
In any Radar application, the basic principle remains the same. Let's talk nownt of the radar principle.
Radar Basics
Radar is used to detect objects and find their location. We can understand the basic principle of radar from
As shown in the figure, the radar mainly consists of a transmitter and A receiver. It uses the same antenna to transmit and receive signals. The function of the transmitter is to transmit the radar signal in the direction of the present target.
The target reflects this received signal in different directions. The signal, which is reflected back to the antenna, is received by the receiver .
Radar systems terminology
These are the basic terms, which are useful in this tutorial.
 Range
 Pulse repetition rate
 Maximum unambiguous range
 Minimum Range
Now let's talk about these basic terms one by one.
Range
The distance between the radar and the target is called the Range of the target or simply distance, R. We know that the radar transmits a signal to the target and consequently the target sends an echo signal to the radar with the speed of light, C.
Let "T" be the time required for the signal to travel from the radar to the target and go back to radar. The bidirectional distance between the radar and the target will be 2R, since the distance between the radar and the target is R.
Now here is the formula for Speed .
$$ Speed = frac {Distance} {Time} $$
$$ Rightarrow Distance = Speed times Time $$
$ $ Rightarrow 2R = C times T $$
$$ R = frac {CT} {2}::::: Equation: 1 $$
We can find the target range by replacing the values of C and T in equation 1.
Pulse repetition frequency
Radar signals must be transmitted at each clock pulsee. The duration between the two clock pulses should be suitably chosen so that the echo signal corresponding to the current clock pulse is received before the next clock pulse. A typical radar waveform is shown in the following figure.
As shown in the Figure, the radar emits a periodic signal. It has a series of narrow, rectangularshaped pulses. The time interval between successive clock pulses is called pulse repetition time , $ T_P $.
The inverse of the pulse repetition time is called pulse repetition frequency , $ f_P $. Mathematically, it can be represented by
$$ f_P = frac {1} {T_P}::::: Equation: 2 $$ Therefore, the pulse repetition frequency is nothing else frequency at which the radar transmits the signal.
Maximum unambiguous range
We know thatRadar signals must be transmitted with each clock pulse. If we select a shorter duration between the two clock pulses, then the echo signal corresponding to the current clock pulse will be received after the next clock pulse. For this reason, the target's range appears to be smaller than the actual range.
So we have to select the duration between the two clock pulses in such a way that the echo signal corresponding to the current clock pulse will be received before the start of the next one clock pulse. Then we will get the true range of the target and it is also called the maximum unambiguous range of the target or just maximum unambiguous range .
Substitute, $ R = R_ {un} $ and $ T = T_P $ in equation 1.
$$ R_ {un} = frac {CT_P} {2 }::::: Equation: 3 $$
From equation 2, we will get the pulse repetition time, $ T_P $ as the reciprocal of the pulse repetition frequency, $ f_P $. Mathematically , it can be represented by
$$ T_P = frac {1} {f_P}::::: Equation: 4 $$
Substitute , equation 4 in equation 3.
$$ R_ {un} = frac {C left (frac {1} {f_P} right)} {2} $$
$$ R_ {un} = frac {C} {2f_P}::::: Equation: 5 $$
We can use equation 3 or equation 5 to calculate the range maximum unambiguous target.

We will get the value of the target's maximum unambiguous range, $ R_ {a} $ by replacing the values of $ C $ and $ T_P $ in equation 3.

Likewise, we will get the value of the maximum unambiguous range of the target, $ R_ {a} $ by replacing the values of $ C $ and $ f_P $ in equation 5.
Minimum range
We will get the minimum range of the target, when we consider the time requirede so that the echo signal receives at the radar after the signal transmitted by the radar as pulse width. This is also called the shortest range of the target.
Substitute, $ R = R_ {min} $ and $ T = tau $ in equation 1.
$$ R_ {min} = frac {C tau} { 2}::::: Equation: 6 $$
We will get the value of the minimum range of the target, $ R_ {min} $ by replacing the values of $ C $ and $ tau $ in Equation 6.
Radar Systems  Range Equation
The radar distance equation is useful for knowing the target range theoretically . In this chapter, we will discuss the standard form of the radar distance equation, then we will discuss the two modified forms of the radar distance equation.
We will get these modified forms of the Radar Range Equation from the standard form of Radar Range Equation. Now, let's discuss the derivation of the standard form of the equation of dRadar distance.
Derivation of the Radar Range Equation
The standard form of the Radar Range Equation is also referred to as the simple form of Radar Range Equation. Now let's derive the standard form of the radar distance equation.
We know that power density is nothing more than the ratio of power to area. Thus, the power density, $ P_ {di} $ at a distance, R of the radar can be represented mathematically by 
$$ P_ {di} = frac {P_t} {4 pi R ^ 2 }::::: Equation: 1 $$
Where,
$ P_t $ is the amount of power transmitted by the radar transmitter The power density above is valid for an isotropic antenna. In general, radars use directional antennas. Therefore, the power density, $ P_ {dd} $ due to the directional antenna will be 
$$ P_ {dd} = frac {P_tG} {4 pi R ^ 2}:: ::: Equation: 2 $$
The radius targetdo the power in different directions from the received input power. The amount of power reflected back to the radar depends on its cross section. Thus, the power density $ P_ {de} $ of the radar echo signal can be represented mathematically by 
$$ P_ {de} = P_ {dd} left (frac {sigma} {4 pi R ^ 2} right)::::: Equation: 3 $$ Substitute, equation 2 in equation 3.
$$ P_ {de} = left (frac {P_tG} {4 pi R ^ 2} right) left (frac {sigma} {4 pi R ^ 2} right)::::: Equation: 4 $$
The amount of power, $ P_r $ received by the radar depends on the effective aperture, $ A_e $ of the receiving antenna.
$$ P_r = P_ {de} A_e::::: Equation: 5 $$
Substitute, Eq uation 4 in equation 5.
$$ P_r = left (frac {P_tG} {4 pi R ^ 2} right) left (frac {sigma} {4 pi R ^ 2} right) A_e $$
$$ Rightarrow P_r = frac {P_tG sigma A_e} {left (4 pi rightte) ^ 2 R ^ 4} $$
$$ Right arrow R ^ 4 = frac {P_tG sigma A_e} {left (4 pi right) ^ 2 P_r} $$
$$ Right arrow R = left [frac {P_tG sigma A_e} {left (4 pi right) ^ 2 P_r} right] ^ {1/4}::::: Equation: 6 $$
Standard form of radar distance equation
If the echo signal has a strength less than the minimum detectable signal strength, the radar cannot detect the target because it is over beyond the maximum limit of the radar range.
Therefore, we can say that the target range is said to be maximum range when the received echo signal has the power equal to that of the minimum detectable signal. We will get the following equation, replacing $ R = R_ {Max} $ and $ P_r = S_ {min} $ in equation 6.
$$ R_ {Max} = left [ frac {P_tG sigma A_e} {left (4 pi right) ^ 2 S_ {min}} right] ^ {1/4}::::: Equation: 7 $$
Equation 7 represents the standard form of the radar distance equation. Using the equation above, we can find the maximum range of the target.
Modified forms of the radar range equation
We know the following relationship between the gain of the directional antenna, $ G $, and the effective aperture, $ A_e $.
$$ G = frac {4 pi A_e} {lambda ^ 2}::::: Equation: 8 $$
Substitute, equation 8 in equation 7.
$$ R_ {Max} = left [frac {P_t sigma A_e} {left (4 pi right) ^ 2S_ {min}} left (frac {4 pi A_e} {lambda ^ 2} right) right] ^ {1/4} $$
$$ Right arrow R_ {Max} = left [frac {P_tG sigma {A_e} ^ 2} {4 pi lambda ^ 2 S_ {min}} right ] ^ {1/4}::::: Equation: 9 $$
Equation 9 represents the modified form of the radar distance equation. Using the equation above, we can find the maximum range of the target.
We will obtain the following relation between the effective opening,$ A_e $ and the gain of the directional antenna, $ G $ from equation 8.
$$ A_e = frac {G lambda ^ 2} {4 pi}::::: Equation: 10 $$
Substitute, equation 10 in equation 7.
$$ R_ {Max} = left [frac {P_tG sigma} {left (4 pi right ) ^ 2 S_ {min}} (frac {G lambda ^ 2} {4 pi}) right] ^ {1/4} $$
$$ Right arrow R_ {Max} = left [frac {P_tG ^ 2 lambda ^ 2 sigma} {left (4 pi right) ^ 2 S_ {min}} right] ^ {1/4}::::: Equation: 11 $$
L 'Equation 11 represents another modified form of the radar distance equation. Using the equation above, we can find the maximum range of the target.
Note  Based on the given data, we can find the maximum range of the target using one of these three equations namely
 Equation 7
 Equation 9
 Equation 11
Examples of problems
In the sectPrevious ion, we got the standard and modified forms of the Radar distance equation. Now let's solve some problems using these equations.
Problem 1
Calculate the maximum radar range for the following specifications 
 Peak power transmitted by the radar, $ P_t = 250KW $
 Gain of the transmitting antenna, $ G = 4000 $
 Effective opening of the receiving antenna , $ A_e = 4: m ^ 2 $
 Target radar section, $ sigma = 25: m ^ 2 $
 Minimum detectable signal strength, $ S_ {min} = 10 ^ { 12} W $
Solution
We can use the following standard form of the radar distance equation to calculate the maximum range of the radar for given specifications.
$$ R_ {Max} = left [frac {P_tG sigma A_e} {left (4 pi right) ^ 2 S_ {min}} right] ^ {1/4} $$
Replace all given elementsThese parameters in the equation above.
$$ R_ {Max} = left [frac {left (250 times 10 ^ 3 right) left (4000 right)) left (25 right) left (4 right)} {left (4 pi right ) ^ 2 left (10 ^ { 12} r ight)} right] ^ {1/4} $$
$$ Rightarrow R_ {Max} = 158: KM $$
Therefore, the maximum radar range for given specifications is $ 158: KM $.
Problem 2
Calculate the maximum radar range for the following specifications.
 Operating frequency, $ f = 10GHZ $
 Peak power transmitted by the radar, $ P_t = 400KW $
 Effective receiving antenna aperture, $ A_e = 5: m ^ 2 $
 Radar cross section of target, $ sigma = 30: m ^ 2 $
 Minimum detectable signal strength, $ S_ {min} = 10 ^ { 10} W $
Solution
We know the following formula for the wavelengthrunning , $ lambda $ in terms of running frequency, f.
$$ lambda = frac {C} {f} $$
Substitute, $ C = 3 times 10 ^ 8m / sec $ and $ f = 10GHZ $ in the equation above.
$ $ lambda = frac {3 times 10 ^ 8} {10 times 10 ^ 9} $$
$$ Rightarrow lambda = 0.03 m $$
So, the operating wavelength , $ lambda $ is equal to $ 0.03m, when the operating frequency, $ f $ is 10GHZ $.
We can use the following defined form mod of the radar range equation to calculate the maximum radar range for given specifications.
$$ R_ {Max} = left [frac {P_t sigma {A_e} ^ 2} {4 pi lambda ^ 2 S_ {min}} right] ^ {1/4} $$
Replacing , the parameters in the equation above.
$$ R_ {Max} = left [frac {left (400 times 10 ^ 3 right) left (30 right) left (5 ^ 2 right)} {4 pi left (0.003 right) ^ 2 left (10 right) ^ { 10}} right] ^ {1/4} $$
$$ Rightarrow R_ {Max} = 128KM $$
Therefore, the maximum range of the radar for one Specifications are $ 128: KM $.
Radar Systems  Performance Factors
The factors that affect the performance of the Radar are called Radar performance factors. In this chapter, let's discuss these factors. We know the following standard form of the radar range equation, which is useful for calculating the maximum radar range for given specifications.
$$ R_ {Max} = left [frac {P_tG sigma A_e} {lef t (4 pi right) ^ 2 S_ {min}} right] ^ {1/4} $$
Where,
$ P_t $ is the peak power transmitted by the radar
$ G $ is the gain of the transmitting antenna
$ sigma $ is the radar cross section of the target
$ A_e $ is the effective aperture of the receiving antenna
$ S_ {min} $ is signal strength mindetectable imum
From the above equation we can conclude that the following conditions must be taken into account to get the maximum range of the radar.
 The peak power transmitted by the radar $ P_t $ must be high.
 The gain of the transmitting antenna $ G $ must be high.
 The radar cross section of the $ sigma $ target must be high.
 Effective the opening of the receiving antenna $ A_e $ must be high.
 The minimum detectable signal strength $ S_ {min} $ must be low.
It is difficult to predict target range from the standard form of the radar range equation. This means that the degree of accuracy provided by the radar range equation on the range of the target is less. Because parameters such as radar cross section of target, $ sigma $ and minimum detectable signal, $ S_ {min} $ are of nastatistical ture .
Minimum detectable signal
If the echo signal has a minimum strength, the detection of this signal by the radar is called minimum detectable signal . This means that the radar cannot detect the echo signal if this signal has less power than the minimum power.
Usually the radar receives the echo signal in addition to the noise. If the threshold value is used to detect the presence of the target from the received signal, then this detection is called threshold detection .
We need to select the appropriate threshold value based on the strength of the signal to be detected.

A high threshold value should be chosen when the signal strength to be detected is high to remove unwanted signal noise present in it.

Likewise, a low threshold value should be chosen when the strength of the signal to be detected is low.
The following figure illustrates this concept 
A typical waveform of the radar receiver is shown in the figure above . The xaxis and the yaxis represent time and voltage respectively. RMS noise value and threshold value are indicated by dotted lines in the figure above.
We have considered three points, A, B and C in the figure above to identify valid detections and missing detections.

The signal value at point A is greater than the threshold value, so this is a valid detection .

The value of the signal at point B is equal to the threshold value, so it is valid detection .

Even if the signal value at point C is closer to the threshold value, it is missing detection . Because the signal value at point C is less than the threshold value.
So points, A and B are valid detections. While point C is a missing detection.
Receiver noise
If the receiver generates a noise component in the signal, which is received at the receiver, then this type of noise is known as receiver noise. receiver noise is an unwanted component; we should try to eliminate it with some care.
However, there is a type of noise known as thermal noise. This happens due to the thermal movement of the conduction electrons. Mathematically, we can write thermal noise power , $ N_i $ produced at the receiver as 
$$ N_i = KT_oB_n $$
Where,
$ K $ is the Boltzmann constant and it is equal to 1.38 $ times 10 ^ { 23} J / deg $
$ T_o $ is the absolute temperature and c 'equals 290 $ ^ 0K $
$ B_n $ is the receiver bandwidth
Figure of Merit
The Figure of Merit , F is nothing other than the ratio between input SNR, $ (SNR) _i $ and output SNR, $ (SNR) _o $. Mathematically, it can be represented by 
$$ F = frac {(SNR) _i} {(SNR) _o} $$
$$ Rightarrow F = frac {S_i / N_i} {S_o / N_o} $$
$$ Rightarrow F = frac {N_oS_i} {N_iS_o} $$
$$ Rightarrow S_i = frac {FN_iS_o} {N_o} $ $
Substitute, $ N_i = KT_oB_n $ in the equation above.
$$ Rightarrow S_i = FKT_oB_n left (frac {S_o} {N_o} right) $$
The power of the input signal will have a minimum value, when the SNR of output will have a minimum value.
$$ Rightarrow S_ {min} = FKT_oB_n left (frac {S_o} {N_o} right) _ {min} $$
Substitute, the $ S_ {min} $ ci above the following standard form of the radar range equation.
$$ R_ {Max} = left [frac {P_tG sigma A_e} {left (4 pi right) ^ 2 S_ {min}} right] ^ {1/4} $$
$$ Right arrow R_ {Max} = leftche [frac {P_tG sigma A_e} {left (4 pi right) ^ 2 FKT_oB_n left (frac {S_o} {N_o} right) _ {min}} right] ^ {1/4} $$
From the equation above, we can conclude that the following conditions must be considered in order to get the radar run as maximum.
 Peak power transmitted by radar, $ P_t $ must be high.
 Gain of transmitting antenna $ G $ must be high.
 The radar cross section of the $ sigma $ target must be high.
 The effective opening of the receiving antenna $ A_e $ must be high.
 The merit value F must be low.
 The bandwidth of receiver $ B_n $ must be low.
Radar systems  Radar types
In this chapter we briefly have the different types of radar. This chapter briefly provides information about the types of speed cameras. Radars can be classified under deux types following depending on the type of signal with which the radar can be used.
 Pulse Radar
 Continuous wave radar
Now let's talk about these two types of Ra dars one by one.
Pulse Radar
Radar, which works with a pulse signal, is called Pulse Radar . Pulse radars can be classified into the following two types depending on the type of target it detects.
 Basic pulse radar
 Moving target indication Radar
Now let's talk briefly about the two radars.
Basic Pulse Radar
Radar, which works with a pulse signal to detect stationary targets, is called the Basic Pulse Radar or simply, Pulse Radar. It uses a single antenna to transmit and receive signals using the duplexer.
The antenna transmitsra a pulse signal with each clock pulse. The duration between the two clock pulses should be chosen so that the echo signal corresponding to the current clock pulse is received before the next clock pulse.
Moving Target Indication Radar
Radar, which works with a pulse signal to detect nonstationary targets, is called Target Indication Radar mobile or simply MTI Radar . It uses a single antenna for transmitting and receiving signals using the duplexer.
MTI Radar uses the principle of the Doppler effect to distinguish nonstationary targets from stationary objects.
Continuous wave radar
Radar, which works with a continuous wave or signal, is called Continuous wave radar . They use the Doppler effect to detect nonstationary targets. Continuous wave radars can be classés in the following two types.
 Unmodulated continuous wave radar
 Frequency modulated continuous wave radar
Now , let's talk briefly about the two radars.
Unmodulated continuous wave radar
Radar, which works with a continuous signal (wave) to detect nonstationary targets is called unmodulated continuous wave radar or simply CW radar . It is also called CW Doppler radar.
This radar requires two antennas. Of these two antennas, one antenna is used to transmit the signal and the other antenna is used to receive the signal. It only measures the target's speed but not the target's distance from the radar.
Frequencymodulated continuous wave radar
If CW Doppler radar uses frequency modulation, then this radar is called frequencymodulated continuous wave radar (FMCW ) or FMC Doppler radarW. It is also called continuous wave frequency modulated radar or CWFM radar.
This radar requires two antennas. Among which, one antenna is used to transmit the signal and the other antenna is used to receive the signal. It measures not only the speed of the target but also the distance of the target from the radar.
In our next chapters we will discuss in detail how all of these radars work.
Radar Systems  Pulse Radar
Radar, which works with a pulse signal to detect stationary targets, is called Basic Pulse Radar or simply Pulse radar . In this chapter, let's talk about how pulse radar works.
Pulse radar block diagram
Pulse radar uses a single antenna to transmit and receive signals using the duplexer. Here is the block diagram of Pulse Radar 
Now let's see the function of each Pulse Radar block 

Modulator of pulses  It produces a pulse modulated signal and is applied to the transmitter.

Transmitter  It transmits the pulse modulated signal, which is a repetitive pulse train.

Duplexer  It 'sa microwave switch, which alternately connects the antenna to the transmitter section and to the receiver section . The antenna transmits the pulse modulated signal, when the duplexer connects the antenna to the transmitter. Likewise, the signal, which is received by the antenna will be transmitted to the low noise RF amplifier, when the duplexer connects the antenna to the low noise RF amplifier.

Low noise RF Amplifier  It amplifies the weak RF signal, which is received by the antenna. The output of this amplifier is connected to the mixer.

Local oscillator  It produces a signal with a stable frequency. The output of Local Oscillator is connected to Mixer.

Mixer  We know that Mixer can produce both the sum and the difference of the frequencies applied to it. Among which, the difference in frequencies will be of the Intermediate Frequency (IF) type.

IF Amplifier  The IF amplifier amplifies the Intermediate Frequency (IF) signal. The IF amplifier shown in the figure allows only the intermediate frequency, which is obtained from the mixer and amplifies it. It improves the signal / noise ratio at the output.

Detector  It demodulates the signal, which is obtained at the output of the IF amplifier.

Video amplifier  As the name suggests, it amplifies the video signal, which is obtained at the output of the detector.

Display  Usually displays the emptied signalo Amplified on CRT screen.
In this chapter, we have discussed how pulse radar works and how useful it is for detecting stationary targets. In our following chapters, we will discuss Radars, which are useful for detecting nonstationary targets.
Radar Systems  Doppler Effect
In this chapter we will learn more about the Doppler Effect in radar systems.
If the target is not stationary, then there will be a change in the frequency of the signal that is emitted by the radar and that is received by the radar. This effect is known as the Doppler effect .
Depending on the Doppler effect, we will get the following two possible cases 

The frequency of the received signal will increase , when the target moves towards the direction of the radar.

The frequency of the received signal decreases when the target ison the radar.
Now let's derive the formula for Doppler frequency.
Doppler frequency derivation
The distance between the radar and the target is nothing other than the Range of the target or simply the distance, R. Therefore, the total distance between radar and target in a twoway communication path will be 2R, since the radar transmits a signal to the target and as a result the target sends an echo signal to the Radar .
If $ lambda $ is a wavelength, then the number of N wavelengths present in a twoway communication path between the radar and the target will be equal to $ 2R / lambda $ .
We know that a wavelength $ lambda $ corresponds to an angular excursion of 2 $ pi $ radians. Therefore, the total angle of excursion made by the electromagnetic wave during the twoway communication path between the radar and the target will be equal to 4 $ pi R /lambda $ radians.
Here is the mathematical formula for angular frequency , $ omega $ 
$$ omega = 2 pi f::::: Equation: 1 $$
The following equation shows the mathematical relationship between the angular frequency $ omega $ and the phase angle $ phi $ 
$$ omega = frac {d phi} {dt}::::: Equation: 2 $$
Equation the terms on the right side of equation 1 and equation 2 since the terms of left side of these two equations are identical.
$$ 2 pi f = frac {d phi} {dt} $$
$$ Rightarrow f = frac {1} {2 pi} frac {d phi} {dt }::::: Equation: 3 $$
Substitute , $ f = f_d $ and $ phi = 4 pi R / lambda $ in Equation 3.
$$ f_d = frac {1} {2 pi} frac {d} {dt} left (frac {4 pi R} {lambda} right) $$
$$ Rightarrow f_d = frac {1} {2 pi} frac {4 pi} {lambda} frac {dR} {dt} $$
$$ Rightarrow f_d = frac {2V_r} {lambda}::::: Equation:4 $$
Where,
$ f_d $ is the Doppler frequency
$ V_r $ is the relative speed
We can find the value of Doppler Frequency $ f_d $ by replacing the values of $ V_r $ and $ lambda $ in equation 4.
Substitute , $ lambda = C / f $ in equation 4.
$$ f_d = frac {2V_r} {C / f} $$
$$ Rightarrow f_d = frac {2V_rf} {C}:: ::: Equation: 5 $$
Where,
$ f $ is the frequency of the transmitted signal
$ C $ is the speed of light and it is equal to 3 $ times 10 ^ 8m / sec $
We can find the value of the Doppler frequency, $ f_d $ by substituting the values of $ V_r, f $ and $ C $ in l 'Equation 5.
Note  Equation 4 and Equation 5 show the formulas for the Doppler frequency, $ f_d $. We can use Equation 4 or Equation 5 to find the Doppler frequency , $ f_d $ based on the given data.
Example problem
If the radar is operating at a frequency of 5GHZ $, then find the Doppler frequency of an airplane moving at a speed of 100KMph.
Solution
Given,
The frequency of the transmitted signal, $ f = 5GHZ $
Airplane speed ( target), $ V_r = 100KMph $
$$ Rightarrow V_r = frac {100 times 10 ^ 3} {3600} m / sec $$
$$ Rightarrow V_r = 27.78m / sec $$
We have converted the speed data of the aircraft (target), which is present in KMph in its m / sec equivalent.
We know that, the speed of light, $ C = 3 times 10 ^ 8m / sec $
Now here is the Doppler frequency formula 
$$ f_d = frac {2Vrf} {C} $$
Replace the values of ð '‰ ð ' Ÿ, $ V_r , f $ and $ C $ in the equation above.
$$ Rightarrow f_d = frac {2 left (27.78 right) left (5 times 10 ^ 9 right)} {3 times 10 ^ 8} $$
$$ Right arrow f_d = 926HZ $$
By cTherefore, the value of Doppler frequency , $ f_d $ is 926 $ HZ $ for the given specifications.
Radar Systems  CW Radar
The basic radar uses the same antenna for both transmission and reception of signals. We can use this type of radar, when the target is stationary, that is to say immobile and / or when this radar can be used with a pulse signal.
Radar, which works with a continuous signal (wave) for detection of nonstationary targets, is called continuous wave radar or simply CW radar . This radar requires two antennas. Among which, one antenna is used to transmit the signal and the other antenna is used to receive the signal.
Functional diagram of CW radar
We know that CW Doppler radar contains two antennas  Transmitting antenna and receiving antenna. The following figure shows the block diagram of CW Radar 
LThe CW Doppler Radar diagram contains a set of blocks and the function of each block is mentioned below.

CW transmitter  It produces an analog signal having a frequency of $ f_o $. The output of the CW transmitter is connected to both the transmit antenna and the Imixer.

Local Oscillator  It produces a signal having a frequency of $ f_l $. The output of the Local Oscillator is connected to MixerI.

MixerI  The Mixer can produce both the sum and the difference of the frequencies applied to it. Signals with frequencies of $ f_o $ and $ f_l $ are applied to MixerI. So the MixerI will produce the output having the frequencies $ f_o + f_l $ or $ f_o  f_l $.

Side Band Filter  As the name suggests, the sideband filter allows for particular sideband frequencies  i.e. frequencies of side bandhigher, or lower sideband frequencies. The sideband filter shown in the figure above only produces the upper sideband frequency, i.e. $ f_o + f_l $.

MixerII  Mixer can produce both the sum and the difference of the frequencies applied to it. Signals with frequencies of $ f_o + f_l $ and $ f_o pm f_d $ are applied to MixerII. So the MixerII will produce the output with frequencies of 2 $ f_o + f_l pm f_d $ or $ f_l pm f_d $.

IF Amplifier  The IF amplifier amplifies the intermediate frequency (IF) signal. The IF amplifier shown in the figure allows only the intermediate frequency, $ f_l pm f_d $ and amplifies it.

Detector  It detects the signal, which has a Doppler frequency, $ f_d $.

Doppler Amplifier  As the name suggests, Doppler Amplifier amplifies the signall, which has Doppler frequency, $ f_d $.

Indicator  It indicates relative speed information and whether the target is entering or exiting.
CW Doppler radars provide an accurate measurement of relative speeds . Therefore, these are mainly used, where the speed information is more important than the actual range.
Radar systems  FMCW Radar
If CW Doppler Radar uses frequency modulation, then this Radar is called FMCW Doppler Radar or just FMCW Radar . It is also called continuous wave frequency modulated radar or CWFM radar. It not only measures the speed of the target, but also the distance of the target from the radar.
Functional diagram of FMCW radar
The FMCW radar is mainly used as a radar altimeter in order to measure the exact height during the landing of the aircraft. The figure followednte shows the block diagram of the FMCW radar 
The FMCW radar contains two antennas  a transmitting antenna and an antenna reception as shown in the figure. The transmitting antenna transmits the signal and the receiving antenna receives the echo signal.
The block diagram of the FMCW radar looks like the block diagram of the CW radar. It contains some modified blocks and some other blocks besides th blocks are present in the CW radar block diagram. The function of each FMCW Radar block is mentioned below.

FM modulator  It produces a frequency modulated (FM) signal having a variable frequency, $ f_o left ( t right) $ and it is applied to the FM transmitter.

FM Transmitter  It transmits the FM signal using the transmitting antenna. The output of the FM transmitter is also connected to the MixerI.

Local Oscillator  In general, the Local Oscillator is used to produce an RF signal. But, here it is used to produce a signal having an intermediate frequency, $ f_ {IF} $. The output of the Local Oscillator is connected to both MixerI and Balanced Detector.

MixerI  The Mixer can produce both the sum and the difference of the frequencies applied to it. Signals with frequencies of $ f_o left (t right) $ and $ f_ {IF} $ are applied to MixerI. So the MixerI will produce the output having a frequency either $ f_o left (t right) + f_ {IF} $ or $ f_o left (t right) f_ {IF} $.

Sideband filter  It only allows one sideband frequency, i.e. higher sideband frequencies or lower sideband frequencies. The sideband filter shown in the figure produces only a lower sideband frequency. ie, $ f_o left (t right) f_ {IF} $.

MixerII  The mixer can produce both sums and the difference of frequencies applied to it. Signals with frequencies of $ f_o left (t right) f_ {IF} $ and $ f_o left (tT right) $ are applied to MixerII. Thus, the MixerII will produce the output having a frequency either $ f_o left (tT right) + f_o left (t right) f_ {IF} $ or $ f_o left (tT right) f_o left (t right) + f_ {IF} $.

IF Amplifier  The IF Amplifier amplifies the Intermediate Frequency Signal (IF). The IF amplifier shown in the figure amplifies the signal having a frequency of $ f_o left (tT right) f_o left (t right) + f_ {IF} $. This amplified signal is applied as an input to the balanced detector.

Balanced detector  It is used to produce the output signal having a frequency of $ f_o left (tT right) f_o left (t right ) $ from dthem applied input signals, which have frequencies of $ f_o left (tT right) f_o left (t right) + f_ {IF} $ and $ f_ {IF} $. The output of the balanced detector is applied as an input to the low frequency amplifier.

Low Frequency Amplifier  It amplifies the output of the balanced detector to the required level. The output of the low frequency amplifier is applied to both the switched frequency counter and the medium frequency counter.

Switched frequency counter  It is useful to get the value of Doppler speed.

Average frequency counter  It is useful to get the value of Range.
Radar Systems  MTI Radar
If the Radar is used to detect the moving target, then the Radar should only receive the signal from echo due to this moving target. This echo signal is the desired one. However, in practical applications theradar receives echo signals due to stationary objects in addition to the echo signal due to this moving target.
Echo signals due to fixed objects (places) such as land and sea are called congestion because they are unwanted signals. Therefore, we have to choose the radar in such a way that it only considers the echo signal due to the moving target but not the congestion.
For this, Radar uses the principle of the Doppler effect to distinguish nonstationary targets from fixed objects. This type of radar is referred to as a moving target indicating radar or simply MTI Radar .
According to the Doppler effect , the frequency of the received signal will increase if the target moves towards the direction of the radar. Likewise, the frequency of the received signal will be folded if the target moves away from the radar.
Types of MTI radars
We can classify MTI radars into the following two types on the basis of type of transmitter used.
 MTI Radar with Power Amplifier Transmitter
 MTI Radar with Power Oscillator Transmitter
Now let's talk about these two MTI radars one by one.
MTI Radar with Power Amplifier Transmitter
MTI Radar uses a single antenna for transmitting and receiving signals using Duplexer. The functional diagram of the MTI radar with transmitter power amplifier is shown in the following figure.
The function of each MTI radar block with transmitter power amplifier is mentioned below.

Pulse modulator  It produces a pulse modulated signal and it is applied to the power amplifier.

Power amplifier  It amplifies the power levels of the pulse modulated signal.

Local oscillator  It produces a signal with a stable frequency $ f_l $. Hence, it is also called a stable local oscillator. The output of Local Oscillator is applied to both MixerI and MixerII.

Coherent Oscillator  It produces a signal with an intermediate frequency, $ f_c $. This signal is used as a reference signal. The output of the Coherent Oscillator is applied to both the MixerI and the Phase Detector.

MixerI  The Mixer can produce the sum or the difference of the frequencies applied to it. Signals with frequencies of $ f_l $ and $ f_c $ are applied to MixerI. Here, the MixerI is used to produce the output, which has the frequency $ f_l + f_c $.

Duplexer  It 'sa microwave switch, which connects the antenna to the transmitter section or to the ion receiver sect based on the requirement. The antenna transmits the signal with the frequency $ f_l + f_c $ whenthe duplexer connects the antenna to the power amplifier. Likewise, Antenna receives the signal having a frequency of $ f_l + f_c pm f_d $ when the duplexer connects the antenna to MixerII.

MixerII  The mixer can produce the sum or difference of frequencies applied to it. The signals of frequencies $ f_l + f_c pm f_d $ and $ f_l $ are applied to MixerII. Here, the MixerII is used to produce the output, which has the frequency $ f_c pm f_d $.

IF Amplifier  The IF amplifier amplifies the intermediate frequency (IF) signal. The amplifier IF shown in the figure amplifies the signal of frequency $ f_c + f_d $. This amplified signal is applied as an input to the phase detector.
Phase detector  It is used to produce the output signal having frequency $ f_d $ of the two signals of input applied, which have the frequencies of $ f_c + f_d $ and $ f_c $. The exitof the phase detector can be connected to the delay line canceller.
MTI Radar with Power Oscillator Transmitter
The schematic diagram of MTI Radar with power oscillator transmitter looks like the schematic diagram of MTI Radar with amplifier transmitter power. The blocks corresponding to the receiver section will be the same in both block diagrams. While the blocks corresponding to the section of the transmitter may differ in the two block diagrams.
The block diagram of the MTI radar with power oscillator transmitter is shown in the following figure.
As shown in the figure, MTI Radar uses the single antenna for transmitting and receiving signals using the duplexer. operation of MTI radar with power oscillator transmitter is mentioned below.

The o The output of the magnetron oscillator and theThe output of the local oscillator is applied to MixerI. This will further produce an IF signal , the phase of which is directly related to the phase of the transmitted signal.

The output of MixerI is applied to the coherent oscillator. Therefore, the phase of the coherent oscillator output will be locked to the phase of the IF signal. This means that the phase of the Coherent Oscillator output will also be directly related to the phase of the transmitted signal.

Thus, the output of Coherent Oscillator can be used as a reference signal to compare the received echo signal with the corresponding transmitted signal using the phase detector .
The above tasks will be repeated for each newly transmitted signal.
Radar systems  Line delay cancellers
In this chapter, we will learn about delay line cancellers in radar systems. As the name suggests, thedelay line introduces a certain delay. Thus, the delay line is mainly used in the delay line canceller in order to introduce a delay of pulse repetition time.
The delay line canceller is a filter, which eliminates DC components from echo signals received from fixed targets. This means that it allows the CA components of echo signals received from nonstationary targets, i.e. moving targets.
Types of delay line cancellers
Delay line cancellers can be classified into the following two types according to the number of lines delay that are present there.
 Single Delay Line Canceller
 Double Delay Line Canceller
In our following sections, we will discuss these two delay line cancellers in more detail.
Single delay line canceller
The combination of a delay line anda subtracter is known as a delay line canceller. It is also referred to as a single delay line canceller. The block diagram of the MTI receiver with single delay line canceller is shown in figure bel ow.
We can write the mathematical equation of the echo signal received after the Doppler effect as 
$$ V_1 = A sin left [2 pi f_dt  phi_0 right]::::: Equation: 1 $$
Where,
A is the amplitude of the video signal
$ f_d $ is the Doppler frequency
$ phi_o $ is phase shift and it is equal to $ 4 pi f_tR_o / C $
We will get the delay line cancellation output , replacing $ t $ with $ t T_P $ in equation 1.
$$ V_2 = Left side [2 pi f_d left (tT_P right)  phi_0 right]::::: Equation: 2 $$
Where,
$ T_P $ is the pulse repetition time
We will get the output of the subtractside by subtracting Equation 2 from equation 1.
$$ V_1V_2 = A sin left [2 pi f_dt  phi_0 right] A sin left [2 pi f_d gauche ( tT_P right)  phi_0 right] $$
$$ Right tarrow V_1V_2 = 2A sin left [frac {2 pi f_dt  phi_0  left [2 pi f_d left (tT_P right)  phi_0 right]} {2} right] cos left [frac {2 pi f_dt  phi_o + 2 pi f_d left (tT_P right)  phi_0} {2} right] $$
$$ V_1V_2 = 2A sin left [frac {2 pi f_dT_P} {2} right] cos left [frac {2 pi f_d left (2tT_P right) 2 phi_0} {2} right] $$
$$ Right arrow V_1V_2 = 2A sin left [pi f_dT_p right] cos left [2 pi f_d left (t  frac {T_P} {2} right)  phi_0 right]::::: Equation: 3 $ $
The output of the subtractor is applied as the input of the Full Wave Rectifier. Therefore, the output of Full Wave Rectifier looks like the one shown in the following figure. This is just the answer in frsingle delay line canceller equence .
From equation 3, we can observe that the frequency response of the line canceller delay becomes zero, when $ pi f_dT_P $ equals integer multiples of $ pi $ This means that $ pi f_dT_P $ equals $ n pi $ Mathematically, it can be written
$$ pi f_dT_P = n pi $$
$$ Rightarrow f_dT_P = n $$
$$ Rightarrow f_d = frac {n} {T_P}: :::: Equation: 4 $$
From equation 4 we can conclude that the frequency response of the single delay line canceller becomes zero, when the Doppler frequency $ f_d $ is integer multiples of the inverse of the pulse repetition time $ T_P $.
We know the following relationship between the pulse repetition time and the repetition frequency impulses.
$$ f_d = frac {1} {T_P} $$
$$ Rightarrow frac {1} {T_P} = f_P::::: Equation: 5 $$
We will get the following equation, by replacing equation 5 in equation 4.
$$ Rightarrow f_d = nf_P:: ::: Equation: 6 $$
From Equation 6, we can conclude that the frequency response of the single delay line canceller becomes zero, when the Doppler frequency, $ f_d $ is equal to integer multiples of the pulse repetition frequency $ f_P $.
Blind speeds
From what we have learned so far, the unique delay line canceller eliminates the DC components from the echo signals received from fixed targets, when $ n $ is zero. In addition to this, it also removes the AC components from echo signals received from nonstationary targets, when the Doppler frequency $ f_d $ is equal to integer multiples (other than zero) of the pulse repetition frequency $ f_P $.
Thus, the relative speeds for which the freq responseUence of the single delay line canceller becomes zero are called blind speeds . Mathematically, we can write the expression for the blind speed $ v_n $ as 
$$ v_n = frac {n lambda} {2T_P}::::: Equation: 7 $$
$$ Rightarrow v_n = frac {n lambda f_P} {2}::::: Equation: 8 $$
Where,
$ n $ is an integer and it is equal to 1, 2, 3 and so on
$ lambda $ is the working wave length
Example problem
An MTI radar operates at a frequency of $ 6GHZ $ with a pulse repetition frequency of $ 1KHZ $. Find the first, second and third blind speeds of this radar.
Solution
Given,
The operating frequency of the MTI radar, $ f = 6GHZ $
Pulse repetition frequency , $ f_P = 1KHZ $.
Here is the formula for operating wavelength $ lambda $ in terms of frequencyoperating, f.
$$ lambda = frac {C} {f} $$
Substitute, $ C = 3 times10 ^ 8m / sec $ and $ f = 6GHZ $ in the equation above.
$$ lambda = frac {3 times10 ^ 8} {6 times10 ^ 9} $$
$$ Rightarrow lambda = 0.05m $$
So , the operating wavelength $ lambda $ is equal to 0.05m $, when the operating frequency f is 6GHZ $.
We know the following formula for blind speed .
$$ v_n = frac {n lambda f_p} {2} $$
By replacing, $ n $ = 1,2 & 3 in the equation above, we will obtain the following equations for the first, second and third blind speeds respectively.
$$ v_1 = frac {1 times lambda f_p} {2} = frac {lambda f_p} {2} $$
$$ v_2 = frac {2 times lambda f_p} {2} = 2 left (fra c {lambda f_p} {2} right) = 2v_1 $$
$$ v_3 = frac {3 times lambda f_p} {2} = 3 left (frac {lambda f_p} {2} right) =3v_1 $$
Replace the values of $ lambda $ and $ f_P $ in the equation for the first blind speed.
$$ v_1 = frac {0.05 times 10 ^ 3} {2} $$
$$ Right arrow v_1 = 25m / sec $$
Therefore, the first blind speed $ v_1 $ is equal to $ 25m / sec $ for the given specifications.
We will get the values of second and third blind speeds like 50m $ / sec $ and 75m $ / sec $ respectively by replacing the value of ð '£ 1 in the equations second and third blind speeds.
Double delay line canceller
We know that a single delay line canceller consists of a delay line and a subtractor. If two of these delay line cancellers are cascaded together, then this combination is called a double delay line canceller. The block diagram of the double delay line canceller is shown in the following figure.
Let $ p left (t right) $ and $ q left (t right) $ be the input and the output of the first canceller delay line. We will get the following mathematical relation from first line delay canceller .
$$ q left (t right) = p left (t right) p left (tT_P right)::::: Equation: 9 $$
The output of the first delay line canceller is applied as input to the second delay line canceller. Therefore, $ q left (t right) $ will be the input of the second delay row canceller. Let $ r left (t right) $ be the output of the second delay row canceller. We will get the following mathematical relation from the second delay row canceller .
$$ r left (t right) = q left (t right) q left (tT_P right)::::: Equation: 10 $$
Replace $ t $ by $ t T_P $ in equation 9.
$$ q left (tT_P right) = p left (tT_P right) p left (tT_PT_P right) $$
$$ q left (tT_P right) = p left (tT_P right) p left (t 2T_P right)::::: Equation: $ 11 $
Substitute , equation 9 and equation 11 in equation 10.
$$ r left (t right) = p left (t right ) p left (tT_P right)  left [p left (tT_P right) p left (t2T_P right) right] $$
$$ Right arrow r left (t right ) = p left (t right) 2p left (tT_P right) + p left (t2T_P right)::::: Equation: 12 $$
The advantage of the double delay line canceller is that it largely rejects clutter. The output of two delay line cancellers, which are cascaded, will be equal to the square of the output of the single delay line canceller.
So the amplitude of the output of the double delay line canceller, which is present The MTI radar receiver will be equal to $ 4A ^ 2 left (sin left [pi f_dT_P right] right) ^ $ 2.
The frequency response characteristics of the double delay line canceller and the cascade combination of two delay line cancellers are the same. The advantage of the time domain delay line canceller is that it can be used for all frequency ranges.
Radar Systems  Tracking Radar
Radar, which is used to track the trajectory of one or more targets is known as tracking radar . Typically, it performs the following functions before starting the tracking activity.
 Detect target
 Target range
 Find elevation and angle angles 'azimuth
 Doppler frequency offset search
Tracking radar therefore follows the target by following one of three parameters  distance, angle, frequency Doppler shift. Most tracking radars use the principle of angle tracking . MaintenLet 's talk about what angular tracking is.
Angular tracking
The pencil beams of the radar antenna are tracking at an angle. The axis of the radar antenna is considered as the reference direction. If the direction of the target and the reference direction are not the same, then there will be an angular error , which is nothing more than the difference between the two directions.
If the angular error signal is applied to a servo control system, then it will shift the axis of the radar antenna towards the direction of the target. The axis of the radar antenna and the direction of the target will coincide when the angular error is zero. There is a feedback mechanism in the tracking radar, which works until the angular error becomes zero.
Here are the two techniques , which are used in Angular tracking.
 Sequential lobbing
 Baltaper ayage
Now let's talk about these two techniques one by one.
S Equential lobing
If the antenna beams are switched alternately between two models to follow the target, then this is called sequential lobing . It is also called sequential switching and lobe switching. This technique is used to find the angular error in a coordinate. It gives details of the magnitude and direction of the angular error.
The following figure shows an example of a sequential lob in polar coordinates .
As shown in the figure, the antenna beams switch between position 1 and position 2 alternately. The angular error θ is shown in the figure above. Sequential lobing gives the position of the target with great precision. This is the main advantage of sequential lobing.
Conical sweep
If the beam of the anten does not rotate continuously to follow a target, so it is called a conical sweep . Taper sweep modulation is used to find the position of the target. The following figure shows an example of a conical scan.
The angle of strabismus is the angle between the beam axis and the axis of rotation and it is shown in the figure above.The echo signal obtained from the target is modulated at a frequency equal to the frequency at which the beam of the 'antenna rotates.
The angle between the direction of the target and the axis of rotation determines the amplitude of the modulated signal . Thus, the modulation of Conical sweep must be extracted from the echo signal, then it must be applied to the servo control system, which shifts the beam axis of the antenna towards the direction of the target.
Radar Systems  Antenna Parameters
An Antenna or Antenna is a transducer, which converts electrical energy into wavess electromagnetic and vice versa.
An Antenna has the following f Parameters 
 Directivity
 Opening efficiency
 Antenna efficiency
 Gain
Now let's talk about these parameters in detail 
Directivity
According to the standard definition, "The ratio of maximum radiation intensity of the antenna in question to the radiation intensity of an isotropic or reference antenna, radiating the same total power is called the Directivity . of great importance. The antenna under study is called the subject antenna . Its radiation intensity is focused in a particular direction, while it is emitting or receiving. Therefore, the antenna is said to have its directivity in that particular direction.

The ratio of the radiation intensity in a given direction of an antenna to the average radiation intensity in all directions, is called Directivity .

If this particular direction is not specified, then the direction in which the maximum intensity is observed , can be considered as the directivity of this antenna.

The directivity of a nonisotropic antenna is equal to the ratio of the radiation intensity in a given direction to the radiation intensity of the isotropic source.
Mathematically , we can write the expression for directivity as 
$$ Directivity = frac {U_ {Max} left (theta, phi right)} {U_0} $$
Where,
$ U_ {Max} left (theta, phi right) $ is the intensity subject's antenna maximum radiation
$ U_0 $ is the radiation intensity of an isotropic antenna.
Aperture efficiency
Accordin g to standard definition, "The aperture efficiency of an antenna is the ratio of the effective radiating area(or effective area) to the physical area of the opening. "
An antenna emits power through an opening. This radiation must be effective with minimal loss. The physical area of the aperture should also be considered, as the efficiency of the radiation depends on the area of the aperture, physically on the antenna.
Mathematically , we can write the expression for the efficiency of Aperture $ epsilon_A $ as
$$ epsilon _A = frac { A_ {eff}} {A_p} $$
Where,
$ A_ {eff} $ is the effective area
$ A_P $ is the physical area
Antenna efficiency
According to standard definition, "antenna efficiency is the ratio of the radiated power of the antenna and the input power accepted by the antenna. "
Any antenna is designed to emit power with a minimum of loss, for a given input. The efficiency of an antenna explains howan antenna is able to deliver its output efficiently with minimal losses in the transmission line. It is also called Radiation efficiency factor of the antenna.
Mathematically , we can write the expression for antenna efficiency 𝜂𝑒 like 
$$ eta _e = frac { P_ {Rad}} {P_ {in}} $$
Where,
$ P_ {Rad} $ is the amount of radiated power
$ P_ {in} $ is the input power of the antenna
Gain
According to the standard definition, "Gain of a antenna is the ratio of the intensity of radiation in a given direction to the intensity of radiation that would be obtained if the power accepted by the antenna were radiated isotropically. ”
Simply, The gain of an antenna takes into account the directivity of the antenna as well as its effective performance.If the power accepted by the antenna was radiated isotropically (that is, to sayin all directions), then the radiation intensity we get can be taken as a benchmark.

The term Antenna gain describes the amount of power transmitted in the direction of peak radiation to that of an isotropic source.

Gain is generally measured in dB.

Unlike directivity, gain antenna takes the losses that also occur in into account and therefore focuses on efficiency.
Mathematically , we can write the expression for Antenna Gain $ G $ as 
$$ G = eta_eD $$
Where,
$ eta_e $ is the efficiency of the antenna
$ D $ is the directivity of the antenna
Radar Systems  Radar Antennas
In this chapter, let's learn about Antennas, which are useful in Radar communication. We can classify radar antennas into the following two types based ontion of the physical structure.
 Parabolic reflector antennas
 Objective antennas
In our following sections we will discuss in detail the two types of antennas.
Parabolic reflector antennas
Parabolic reflector antennas are microwave antennas. Knowledge of the parabolic reflector is essential to understand how antennas work in depth.
Principle of Operation
Parabola is nothing other than the Locus of points, which move in such a way that its distance by compared to the fixed point (called focus) plus its distance from a straight line (called directrix) is constant.
The following figure shows the geometry of the parabolic reflector . Points F and V are respectively the focus (power is given) and the vertex. The line joining F and V is the axis of symmetry. $ P_1Q_1, P_2Q_2 $ and $ P_3Q_3 $are the reflected rays. The line L represents the directrix on which the reflected points rest (to say that they are collinear).
As shown in the figure, the distance between F and L is constant with respect to focused waves. The reflected wave forms a collimated wavefront, out of the parabolic shape. of focal length to aperture size (i.e. $ f / D $) is called "f over D ratio" . This is an important parameter of the parabolic reflector and its value varies from 0.25 to 0.50 .
The law of reflection states that the angle of incidence and angle of reflection are equal. This law, when used with a parabola, helps in focusing the beam. The shape of the parabola when used for wave reflection purposes, shows some properties of the parabola, which are useful for constructing an antenna, using the reflected waves.
Propertyés of the parabola
Here are the different properties of Parabola 

All the waves coming from the focus are reflected on the parabolic axis. Therefore, all waves reaching the aperture are in phase.

As the waves are in phase, the beam of radiation along the parabolic axis will be strong and focused.
By following these points, parabolic reflectors help produce high directivity with a narrower beamwidth.
Construction and operation of a parabolic reflector
If a parabolic reflector antenna is used to transmit a signal , the signal from the power supply goes out. 'a dipole antenna or a horn antenna, to focus the wave on the parabola. This means that the waves come out of the focal point and hit the paraboloid reflector. This wave is now reflected as a collimated wavefront, as indpreviously mentioned, to be transmitted.
The same antenna is used as receiver . When the electromagnetic wave reaches the shape of the parabola, the wave is reflected on the feed point. The dipole antenna or horn antenna, which acts as the antenna of the receiver to its feed receives this signal, to convert it into an electrical signal and transmits it to the receiver circuit.
The gain of the paraboloid is a function of the opening ratio $ D / lambda $. The effective radiated power (ERP) of an antenna is the multiplication of the input power supplied to the antenna and its power gain.
Usually, a waveguide horn antenna is used as the power radiator for the dish reflector antenna. Along with this technique, we have the following two types of flux given to the paraboloid reflector antenna.
 Cassegrain feed
 Grego feednothing
Cassegrain Flux
In this type, the flux is located at the top of the paraboloid, unlike the parabolic reflector. A convex shaped reflector, which acts like a hyperboloid, is placed opposite to the antenna feed. It is also called a secondary hyperboloid reflector or subreflector. It is placed in such a way that one of its focal points coincides with the focal point of the paraboloid. Thus, the wave is reflected twice.
The figure above shows the working model of the cassegrain feed.
Gregorian Feed
The type of feed where a pair of certain configurations are present and where the width of the feed beam is gradually increased while the dimensions of the antenna are kept fixed, this is referred to as feed Gregorian . Here, the convex shaped hyperboloid of Cassegrain is replaced by a concave shaped paraboloid reflector,which is of course smaller in size.
These Gregorian type reflectors can be used in the following four ways 

Gregorian systems using ellipsoidal reflector subreflector at focal points F1.

Gregorian systems using ellipsoidal reflector subreflector at focal points F2.

Cassegrain systems using a hyperboloid (convex) subreflector.

Cassegrain systems using a reflective hyperboloid subreflector (concave but the feed being very close).
Among the different types of reflector antennas, simple parabolic reflectors and Cassegrain powered parabolic reflectors are the most common
'lens
Lens antennas use the curved surface for signal transmission and reception. These antennas are made of glass, where the convergent properties and f The lens antenna usage requirement range starts at 1 GHz but its usage is greatest at 3 GHz and more .
A knowledge of the lens is required to fully understand the operation of the lens antenna. Remember that a normal glass lens works on the principle of refraction .
Construction and Function of the Lens Antenna
If a light source is assumed to be present at a focal point of a lens, which is at a focal length of the lens, then the rays pass through the lens as collimated or parallel rays on the plane wavefront.
There are two phenomena that occur when rays fall from different sides of a lens. They are given here 

The rays passing through the center of the lens are lessrefracted as rays passing through the edges of the lens. All rays are sent parallel to the plane wavefront. This Lens phenomenon is called Divergence .

The same procedure is reversed if a light beam is sent from the right side to the left side of the same goal. Then the beam is refracted and meets at a point called the focal point, at a focal length of the lens. This phenomenon is called Convergence.
The following diagram will help us better understand the phenomenon.
The ray diagram represents the focal point and focal length from source to lens. The parallel rays obtained are also called collimated rays.
In the figure above, the source at the focal point, at a focal length of the lens, is collimated in the plane wave front. This phenomenon can be reversed, which means that the light, if it is sent from theleft side, converges to the right side of the lens.
It is because of this reciprocity that the lens can be used as an antenna, because the same phenomenon helps to use the same antenna for both transmission and reception .
To obtain the focusing properties at higher frequencies, the refractive index must be less than unity. Regardless of the refractive index, Lens's goal is to straighten out the waveform. On this basis, Eplan and Hplane objectives are developed, which also delay or accelerate the wavefront.
Radar Systems  Matched Filter Receiver
If a filter produces an output in such a way as to maximize the ratio of the output peak power to the average noise power in its frequency response , then this filter is called Matched filter .
This is an important criterion, which is taken into account when designing any radar receiver. InIn this chapter, let's talk about the frequency response function of the matched filter and the impulse response of the corresponding filter.
Matched filter frequency response function
The frequency response of the corresponding filter will be proportional to the complex conjugate of the spectrum of the input signal. Mathematically, we can write the expression for frequency response function , $ H left (f right) $ of the Matched filter as 
$$ H left (f right ) = G_aS ^ ast left (f right) e ^ { j2 pi ft_1}::::: Equation: 1 $$
Where,
$ G_a $ is the gain maximum of the corresponding filter
$ S left (f right) $ is the Fourier transform of the input signal, $ s left (t right) $
$ S ^ ast left (f right) $ is the complex conjugate of $ S left (f right) $
$ t_1 $ is the instant at which the observed signal is maximum
In general, the value of $ G_a $ is considered one. We will get thefollowing equation by replacing $ G_a = 1 $ in equation 1.
$$ H left (f right) = S ^ ast left (f right)) e ^ { j2 pi ft_1}: :::: Equation: 2 $$
The frequency response function, $ H left (f right) $ of the filter corresponding to the magnitude of $ S ^ ast left (f right) $ and phase angle of $ e ^ { j2 pi ft_1} $, which varies uniformly with frequency.
Im Impulse response of the matched filter
In the time domain , we will get the output, $ h (t) $ of the matched filter receiver by applying the inverse Fourier transform of the frequency response function, $ H (f) $.
$$ h left (t right) = int _ { infty} ^ {infty} H left (f right) e ^ { j2 pi ft_1} df::::: Equation: $ 3 $
Substitute , Equation 1 in equation 3.
$$ h left (t right) = int _ { infty} ^ { infty} lbrace G_aS ^ ast left (f right) e ^ { j2 pi ft_1} rbracee ^ {j2 pi ft} df $$
$$ Right arrow h left (t right) = int _ { infty} ^ {infty} G_aS ^ ast left (f right) e ^ { j2 pi f left (t_1t right)} df::::: Equation: 4 $$
We know the following relation.
$$ S ^ ast left (f right) = S left (f right)::::: Equation: $ 5
Substitute , equation 5 in equation 4.
$$ h left (t right) = int _ { infty} ^ {infty} G_aS (f) e ^ { j2 pi f left ( t_1t right)} df $$
$$ Right arrow h left (t right) = int _ { infty} ^ {infty} G_aS ^ left (f right) e ^ {j2 pi f left (t_1t right)} df $$
$$ Right arrow h left (t right) = G_as (t_1  t)::::: Equation: 6 $$
In general, the value of $ G_a $ is considered to be one. We will get the following equation by replacing $ G_a = 1 $ in equation 6.
$$ h (t) = s left (t_1t right) $$
The above equation proves that the Matched filter impulse response is the mirror image of the signal received at approximately $ t_1 $. The following figures illustrate this concept.
The received signal, $ s left (t right) $ and the impulse response, $ h left (t right) $ of the corresponding filter corresponding to the signal, $ s left (t right) $ are shown in the figures above.
Radar Systems  Radar Displays
An electronic instrument, which is used to display data visually, is called a display. So, the electronic instrument which visually displays the information about the radar target is called radar display . It visually shows the echo signal information on the screen.
Types of radar displays
In this section we will learn about the different types of radar displays. Radar displays can be classified into the following types.
A Scope
These are 'a twodimensional radar display. The horizontal and vertical coordinates represent the range and amplitude of the echo of the target, respectively. In AScope, deflection modulation takes place. It is more suitable for manually following the radar .
BScope
This is a twodimensional radar display. The horizontal and vertical coordinates represent the azimuthal angle and range of the target, respectively. In BScope, intensity modulation takes place. It is more suitable for military radars .
CScope
This is a twodimensional radar display. The horizontal and vertical coordinates represent the azimuthal angle and the elevation angle, respectively. In CScope, intensity modulation takes place.
DScope
If the electron beam is deflected or if the intensity modulated spot appears on the radar screen due to the presence of a target , then it is known as nom from blip. CScope becomes DScope, when the blips extend vertically in order to provide distance.
EScope
This is a two dimensional radar display. Horizontal and vertical coordinates represent distance and elevation angle respectively. In EScope, intensity modulation takes place.
FScope
If the radar antenna is pointed at the target, then FScope displays the target as a central blip. Thus, the horizontal and vertical movements of the blip represent the horizontal and vertical sighting errors, respectively.
GScope
If the Ant enna radar is aiming at the target, then GScope displays the target as a centralized blip laterally. The horizontal and vertical movements of the blip represent the horizontal and vertical aiming errors, respectively.
HScope
This is the modified version of BScope in order to provide information on the elevation angle of the target. It displaysthe target as two spots, which are closely spaced. This can be approximated by a short bright line and the slope of that line will be proportional to the sine of the elevation angle.
IScope
If the radar antenna is pointed at the target, then IScope displays the target as a circle . The radius of this circle will be proportional to the distance from the target. If the radar antenna is not aiming the target correctly, the IScope displays the target as a segment instead of a circle. The length of the arc of this segment will be inversely proportional to the magnitude of the pointing error.
JScope
This is the modified version of AScope. It displays the target as a radial deviation from the time base.
KScope
This is the modified version of AScope. If the radar antenna is pointed at the target, then KScope displays the target as a pair of vertical deflections, which have the same height. Yesthe radar antenna is not aiming the target correctly, there will be a pointing error. Thus, the magnitude and direction of the pointing error depends on the difference between the two vertical deflections.
LScope
If the radar antenna is pointed at the target, then L Scope displays the target as two horizontal blips of equal amplitude. One horizontal blip is to the right of the central vertical time base and the other to the left of the central vertical time base.
MScope
This is the modified version of A Scope. An adjustable pedestal signal should be moved along the baseline until it coincides with the signal deviations, which come from the horizontal position of the target. In this way, the distance to the target can be determined.
NScope
This is the modified version of KScope. An adjustable pedestal signal is used to measure the distance.
OScope
This is the modified version of AScope. We will get OScope, including an adjustable notch in AScope to measure distance.
PScope
This is a radar display, which uses intensity modulation. It displays the echo signal information as a plan view. Distance and azimuth angle are displayed in polar coordinates. Therefore, it is called a plane position indicator or PPI display .
RScope
This is a radar display, which uses intensity modulation. The horizontal and vertical coordinates represent the range and height of the target, respectively. Therefore, it is referred to as a distanceheight indicator or RHI display .
Radar Systems  Duplexers
In twoway communication, if we are supposed to use the same antenna for transmitting and receiving signals, then we need a duplexer. Duplexer is a microwave switch, which connects the antenna to the transmitter section for signal transmission. Therefore, the radar cannot receive the signal during the transmission time.
Likewise, it connects the antenna to the reception section for signal reception. Radar cannot transmit signal during reception time. In this way, Duplexer isolates both the transmitter and receiver sections.
Types of duplexers
In this section, we will discover the different types of duplexers. We can classify duplexers into the following three types .
 Branch type duplexer
 Balanced duplexer
 Circulator as duplexer
In our following sections we will discuss in detail the types of duplexers.
Branch type duplexer
The branch type duplexer consists of two switches: the t switchtransmissionreception (TR) and the antitransmissionreception switch (ATR). The following figure shows the block diagram of the Branch type duplexer 
As shown in the figure, the two switches TR and ATR are placed at a distance of $ lambda / 4 $ from the transmission line and the two switches are separated by a distance from $ lambda / 4 $. operation of the branch type duplexer is mentioned below.

During transmission , both TR & ATR will look like an open circuit of the transmission line. Therefore, the antenna will be connected to the transmitter through the transmission line.

During reception , the ATR will look like a short circuit on the transmission line, therefore, the antenna will be connected to the receiver via a transmission line. transmission.
Branch type duplexer is only suitable for low cost radar,because it has less power handling capability.
Symmetrical duplexer
We know that a twohole directional coupler is a 4port waveguide junction made up of a primary waveguide and a secondary waveguide. There are two small holes, which will be common to these two waveguides.
The balanced duplexer consists of two TR tubes. The configuration of the balanced duplexer for transmission purposes is shown in the following figure.
The signal, which is produced by the transmitter must reach the antenna in order for the antenna to transmit this signal during transmission time. The solid lines with arrows shown in the figure above represent how the signal reaches the antenna from the transmitter.
The dotted lines with arrows shown in the figure above represent the signal, leaking from Dual TR tubes; this will only reachthe corresponding load. So no signal was reached to the iver receivers.
The configuration of the balanced duplexer for receive purposes is shown in the figure below.
We know that Antenna is receiving signal while receiving. The signal received by the antenna must reach the receiver. The solid lines with arrows shown in the figure above represent how the signal reaches the receiver from the antenna. In this case, Dual TR tubes transmit the signal from the first section of the waveguide to the next section of the waveguide.
Symmetrical duplexer has high power handling capability and high bandwidth compared to branch type duplexer.
Circulator as a duplexer
We know that the functionality of the circulator is that if we apply an entry to a port, it will be produced at the port, which for it is adjacentclockwise. There is no outlet to the remaining ports of the circulator.
So, consider a 4 port circulator and connect the transmitter, antenna, receiver and the corresponding load to port1, port2, port3 and port4 respectively. Now let's understand how the 4 port circulator works as a duplexer.
The signal, which is produced by the transmitter must reach the antenna in order for the antenna to transmit this signal during the transmission time. This will be achieved when the transmitter generates a signal at port 1.
The signal, which is received by the antenna, must reach the receiver during the reception time. . This objective will be achieved when the antenna present at port2 receives an external signal.
The following figure shows the block diagram of the duplexer circulator 
The figure above is made up of a 4  port circulator  The emitsur, the antenna and the corresponding load are connected respectively to port1, port2 and port4 of the circulator as shown at the beginning of the section.
The receiver is not directly connected to port3. Instead, the blocks corresponding to the passive TR limiter are placed between port 3 of the circulator and the receiver. The blocks, the TR tube and the diode limiter are the blocks corresponding to the passive TR limiter.
In fact, the circulator itself acts as a duplexer. It does not require any additional block. However, it will not give any protection to the receiver. Thus, the blocks corresponding to the passive TR limiter are used to provide receiver protection .
Radar Systems  Phased Array Antennas
Only One Antenna can emit a certain amount of power in a particular direction. Obviously, the amount of radiating power will be increased when we use a group of antennas inseems. The group of antennas is called Antenna array .
An Antenna array is a radiating system comprising radiators and elements. Each of these radiators has its own induction field. The elements are placed so close that each is within the neighbor's induction field. Therefore, the radiation pattern produced by them would be the vector sum of each.
The antennas radiate individually and when they are in an array, the radiation of all elements is upwards, to form the radiation beam, which has high gain, high directivity and better performance, with minimal losses.
An array of antennas is said to be a array of phased antennas if the shape and direction of the radiation pattern depends on the relative phases and amplitudes of the currents present at each antenna of this network.
Radiation diagram
Consider "n" elements ofisotropic radiation, which when combined form an array . The figure below will help you understand the same. Let 's the spacing between successive elements of units "d".
As shown in the figure, all radiation elements receive the same incoming signal. So each element produces an output voltage equal to $ sin left (omega t right) $. However, there will be an equal phase difference $ Psi $ between successive elements. Mathematically, it can be written 
$$ Psi = frac {2 pi d sin theta} {lambda}::::: Equation: 1 $$
Where,
$ theta $ is the angle at which the incoming signal is incident on each radiation element.
Mathematically, we can write the expressions for the output voltages of 'n ' radiation elements individually as
$$ E_1 = sin left [ omega t right] $$
$$ E_2 = sin left [omega t + Psi right] $$
$$ E_3 = sin left [omega t + 2 Psi right] $$
$$. $$
$$. $$
$$. $$
$$ E_n = sin left [omega t + left (N1 right) Psi right] $$
Where,
$ E_1, E_2, E_3,…, E_n $ are t The output voltages of the first, second, third,…, n ^{ th } radiation elements respectively.
$ omega $ is the angular frequency of the signal.
We will get the global output voltage $ E_a $ from the table by adding the output voltages of each element present in this table, since all these radiation elements are networked linear. Mathematically, it can be represented by 
$$ E_a = E_1 + E_2 + E_3 +… + E_n::: Equation: 2 $$
Replace , the values of $ E_1, E_2, E_3,…, E_n $ in equation 2.
$$ E_a = sin left [omega t right] + sin left [omega t + Psi right ] + sin left [omega t + 2 Psi right] +left sin [omega t + left (n1 right) Psi right] $$
$$ Rightarrow E_a = sin left [omega t + frac {(n1) Psi)} {2} right ] frac {sin left [frac {n Psi} {2} right]} {sin left [frac {Psi} {2} right]}::::: Equation: 3 $$
In l Equation 3, there are two terms. From the first trimester, we can observe that the whole output voltage $ E_a $ is a sine wave having an angular frequency $ omega $. But, it has a phase shift of $ left (n  1 right) Psi / 2 $. The second term in Equation 3 is an amplitude factor .
The amplitude of equation 3 will be
$$ left  E_a right  = left  frac {left sin [frac {n Psi} {2} right]} {left sin [frac {Psi} {2} right]} right  ::::: Equation: 4 $$
We will obtain the following equation by replacing equation 1 in equation 4.
$$ left  E_a right  = left  frac {sin left [frac {n pi dsin theta} {lambda} right]} {sin left [frac {pi d sin theta} {lambda} right]} right  ::::: Equation: 5 $$
Equation 5 is called field strength model . The field strength model will have the values of zeros when the numerator of equation 5 is zero
$$ sin left [frac {n pi d sin theta} {lambda} right] = 0 $$
$$ Rightarrow frac {n pi d sin theta} {lambda} = pm m pi $$
$$ Rightarrow nd sin theta = pm m lambda $$
$$ Rightarrow sin theta = pm frac {m lambda} {nd} $$
Where,
$ m $ is an integer and is equal to 1, 2 , 3 and so on.
We can find the maximum values of the field strength model using the LHospital rule when the numerator and denominator of equation 5 are equal to zero. We can observe that if the denominator of equation 5 becomes zero, then the numerator of equation 5 also becomes zero.
Now, let's get the condition for which the denominator of equation 5 becomes zero.
$$ sin left [frac {pi d sin theta} {lambda} right] = 0 $$
$$ Rightarrow frac {pi d sin theta} {lambda} = pm p pi $$
$$ Rightarrow d sin theta = pm p lambda $$
$$ Rightarrow sin theta = pm frac {p lambda} {d} $ $
Where,
$ p $ is an integer and it is equal to 0, 1, 2, 3 and so on.
If we consider $ p $ to be zero, then we will get the value of $ sin theta $ as zero. For this, we will obtain the maximum value of the field intensity model corresponding to the main lobe . We will get the maximum values of the field strength model corresponding to the side lobes , when we consider other values of $ p $.
The direction of the radiation pattern of the phased array can be directed by varying the relative phases of the current present to each antenna. This is the advantage of phased electronic scanning.