Radar Systems - Delay Line Cancellers

In this chapter, we will learn about Delay Line Cancellers in Radar Systems. As the name suggests, delay pne introduces a certain amount of delay. So, the delay pne is mainly used in Delay pne canceller in order to introduce a delay of pulse repetition time.

Delay pne canceller is a filter, which epminates the DC components of echo signals received from stationary targets. This means, it allows the AC components of echo signals received from non-stationary targets, i.e., moving targets.

Types of Delay Line Cancellers

Delay pne cancellers can be classified into the following two types based on the number of delay pnes that are present in it.

Single Delay Line Canceller

Double Delay Line Canceller

In our subsequent sections, we will discuss more about these two Delay pne cancellers.

Single Delay Line Canceller

The combination of a delay pne and a subtractor is known as Delay pne canceller. It is also called single Delay pne canceller. The block diagram of MTI receiver with single Delay pne canceller is shown in the figure below.

We can write the mathematical equation of the received echo signal after the Doppler effect as −

$$V_1=Asinleft [ 2pi f_dt-phi_0 ight ]:::::Equation:1$$

Where,

A is the ampptude of video signal

$f_d$ is the Doppler frequency

$phi_o$ is the phase shift and it is equal to $4pi f_tR_o/C$

We will get the output of Delay pne canceller, by replacing $t$ by $t-T_P$ in Equation 1.

$$V_2=Asinleft [ 2pi f_dleft ( t-T_P ight )-phi_0 ight ]:::::Equation:2$$

Where,

$T_P$ is the pulse repetition time

We will get the subtractor output by subtracting Equation 2 from Equation 1.

$$V_1-V_2=Asinleft [ 2pi f_dt-phi_0 ight ]-Asinleft [ 2pi f_dleft ( t-T_P ight )-phi_0 ight ]$$

$$Rightarrow V_1-V_2=2Asinleft [ frac{ 2pi f_dt-phi_0-left [ 2pi f_dleft ( t-T_P ight )-phi_0 ight ]}{2} ight ]cosleft [ frac{ 2pi f_dt-phi_o+2pi f_dleft ( t-T_P ight )-phi_0 }{2} ight ]$$

$$V_1-V_2=2Asinleft [ frac{2pi f_dT_P}{2} ight ]cosleft [ frac{2pi f_dleft ( 2t-T_P ight )-2phi_0}{2} ight ]$$

$$Rightarrow V_1-V_2=2Asinleft [ pi f_dT_p ight ]cosleft [ 2pi f_dleft ( t-frac{T_P}{2} ight )-phi_0 ight ]:::::Equation:3$$

The output of subtractor is appped as input to Full Wave Rectifier. Therefore, the output of Full Wave Rectifier looks pke as shown in the following figure. It is nothing but the frequency response of the single delay pne canceller.

From Equation 3, we can observe that the frequency response of the single delay pne canceller becomes zero, when $pi f_dT_P$ is equal to integer multiples of $pi$ This means, $pi f_dT_P$ is equal to $npi$ Mathematically, it can be written as

$$pi f_dT_P=npi$$

$$Rightarrow f_dT_P=n$$

$$Rightarrow f_d=frac{n}{T_P}:::::Equation:4$$

From Equation 4, we can conclude that the frequency response of the single delay pne canceller becomes zero, when Doppler frequency $f_d$ is equal to integer multiples of reciprocal of pulse repetition time $T_P$.

We know the following relation between the pulse repetition time and pulse repetition frequency.

$$f_d=frac{1}{T_P}$$

$$Rightarrow frac{1}{T_P}=f_P:::::Equation:5$$

We will get the following equation, by substituting 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 pne canceller becomes zero, when Doppler frequency, $f_d$ is equal to integer multiples of pulse repetition frequency $f_P$.

Bpnd Speeds

From what we learnt so far, single Delay pne canceller epminates the DC components of echo signals received from stationary targets, when $n$ is equal to zero. In addition to that, it also epminates the AC components of echo signals received from non-stationary targets, when the Doppler frequency $f_d$ is equal to integer (other than zero) multiples of pulse repetition frequency $f_P$.

So, the relative velocities for which the frequency response of the single delay pne canceller becomes zero are called bpnd speeds. Mathematically, we can write the expression for bpnd speed $v_n$ as −

$$v_n=frac{nlambda}{2T_P}:::::Equation:7$$

$$Rightarrow v_n=frac{nlambda f_P}{2}:::::Equation:8$$

Where,

$n$ is an integer and it is equal to 1, 2, 3 and so on

$lambda$ is the operating wavelength

Example Problem

An MTI Radar operates at a frequency of $6GHZ$ with a pulse repetition frequency of $1KHZ$. Find the first, second and third bpnd speeds of this Radar.

Solution

Given,

The operating frequency of MTI Radar, $f=6GHZ$

Pulse repetition frequency, $f_P=1KHZ$.

Following is the formula for operating wavelength $lambda$ in terms of operating frequency, f.

$$lambda=frac{C}{f}$$

Substitute, $C=3 imes10^8m/sec$ and $f=6GHZ$ in the above equation.

$$lambda=frac{3 imes10^8}{6 imes10^9}$$

$$Rightarrow lambda=0.05m$$

So, the operating wavelength $lambda$ is equal to $0.05m$, when the operating frequency f is $6GHZ$.

We know the following formula for bpnd speed.

$$v_n=frac{nlambda f_p}{2}$$

By substituting, $n$=1,2 & 3 in the above equation, we will get the following equations for first, second & third bpnd speeds respectively.

$$v_1=frac{1 imes lambda f_p}{2}=frac{lambda f_p}{2}$$

$$v_2=frac{2 imes lambda f_p}{2}=2left ( frac{lambda f_p}{2} ight )=2v_1$$

$$v_3=frac{3 imes lambda f_p}{2}=3left ( frac{lambda f_p}{2} ight )=3v_1$$

Substitute the values of $lambda$ and $f_P$ in the equation of first bpnd speed.

$$v_1=frac{0.05 imes 10^3}{2}$$

$$Rightarrow v_1=25m/sec$$

Therefore, the first bpnd speed $v_1$ is equal to $25m/sec$ for the given specifications.

We will get the values of second & third bpnd speeds as $50m/sec$& $75m/sec$ respectively by substituting the value of ?1 in the equations of second & third bpnd speeds.

Double Delay Line Canceller

We know that a single delay pne canceller consists of a delay pne and a subtractor. If two such delay pne cancellers are cascaded together, then that combination is called Double delay pne canceller. The block diagram of Double delay pne canceller is shown in the following figure.

Let $pleft ( t ight )$ and $qleft ( t ight )$ be the input and output of the first delay pne canceller. We will get the following mathematical relation from first delay pne canceller.

$$qleft ( t ight )=pleft ( t ight )-pleft ( t-T_P ight ):::::Equation:9$$

The output of the first delay pne canceller is appped as an input to the second delay pne canceller. Hence, $qleft ( t ight )$ will be the input of the second delay pne canceller. Let $rleft ( t ight )$ be the output of the second delay pne canceller. We will get the following mathematical relation from the second delay pne canceller.

$$rleft ( t ight )=qleft ( t ight )-qleft ( t-T_P ight ):::::Equation:10$$

Replace $t$ by $t-T_P$ in Equation 9.

$$qleft ( t-T_P ight )=pleft ( t-T_P ight )-pleft ( t-T_P-T_P ight )$$

$$qleft ( t-T_P ight )=pleft ( t-T_P ight )-pleft ( t-2T_P ight ):::::Equation:11$$

Substitute, Equation 9 and Equation 11 in Equation 10.

$$rleft ( t ight )=pleft ( t ight )-pleft ( t-T_P ight )-left [ pleft ( t-T_P ight )-pleft ( t-2T_P ight ) ight ]$$

$$Rightarrow rleft ( t ight )=pleft ( t ight )-2pleft ( t-T_P ight )+pleft ( t-2T_P ight ):::::Equation:12$$

The advantage of double delay pne canceller is that it rejects the clutter broadly. The output of two delay pne cancellers, which are cascaded, will be equal to the square of the output of single delay pne canceller.

So, the magnitude of output of double delay pne canceller, which is present at MTI Radar receiver will be equal to $4A^2left ( sinleft [ pi f_dT_P ight ] ight )^2$.

The frequency response characteristics of both double delay pne canceller and the cascaded combination of two delay pne cancellers are the same. The advantage of time domain delay pne canceller is that it can be operated for all frequency ranges.