# Radar systems - Delay line cancellers

Tutorial on radar systems 2020-11-20 02:30:37# Radar Systems - Delay Line Cancellers

In this chapter, we will learn more about Delay Line Cancellers in Radar Systems. As the name suggests, the delay 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 non-stationary 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 next sections, we will discuss these two delay line cancellers in more detail.

## Single delay line canceller

The combination of a delay line and a 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 below.

We can write the ** mathematical equation ** from echo signal received after Doppler effect like -

$$ V_1 = A sin left [2 pi f_dt - phi_0 right]::::: Equation: 1 $$

Where,

A is the amplitude of the video signal

$ f_d $ is the Doppler frequency

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

We will get the ** delay line canceller output **, by replacing $ t $ by $ t-T_P $ in equation 1.

$$ V_2 = A sin left [2 pi f_d left (t-T_P right) - phi_0 right]:: ::: Equation: 2 $$

Where,

$ T_P $ is the repetition of the ition time pulse

We will get the ** output of the subtractor ** by subtracting equation 2 from equation 1.

$$ V_1-V_2 = A sin left [2 pi f_dt - phi_0 right] -A sin left [ 2 pi f_d left (t-T_P right) - phi_0 right] $$

$$ Right arrow V_1-V_2 = 2A sin left [frac {2 pi f_dt - phi_0 - left [2 pi f_d left ( t-T_P right) - phi_0 right]} {2} right] cos left [frac {2 pi f_dt - phi_o + 2 pi f_d left (t-T_P right) - phi_0} {2} right] $$

$$ V_1-V_2 = 2A sin left [frac {2 pi f_dT_P} {2} right] cos left [frac {2 pi f_d left (2t-T_P right) -2 phi_0} {2} right] $$

$$ Right arrow V_1-V_2 = 2A sin left [pi f_dT_p right] cos left [2 pi f_d left (t - frac {T_P} {2} right) - phi_0 right]::::: Equation: 3 $$

The output of the subtractor is applied as input to the Full Wave Rectifier. Therefore, the output of Full Wave Rectifier looks like the one shown in the following figure. This is just the ** frequency response ** of the single-delay line canceller.

From equation 3, we can observe that the frequency response of the single delay line canceller 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 canceller of linesingle delay becomes zero, when the Doppler frequency $ f_d $ is equal to integer multiples of the inverse of the pulse repetition time $ T_P $.

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

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

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

We w ill 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 single delay line canceller eliminates DC components from echo signals received fromstationary targets, when $ n $ is zero. In addition to this, it also removes the AC components from echo signals received from non-stationary 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 at which the frequency response 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 operating wavelength

### Example problem

An MTI radar operates at a frequency of $ 6GHZ $ with a repetition frequencypulse 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 operating frequency, 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, wes will obtain the following equations for the first, second and third blind speeds respectively.

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

$$ v_2 = frac {2 times lambda f_p} {2} = 2 left (frac {lambda f_p} {2} right) = 2v_1 $$

$$ v_3 = frac {3 times lambda f_p} {2} = 3 left (frac {lambda f_p } {2} right) = 3v_1 $$

** 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 spec.

We will obtain the values of the ** second and third blind speeds ** as respectively 50 m / s $ and 75 m $ / s $ by substituting the value of 𝑣1 in the equations of second and third blind gears.

## Double Delay Line Cance ller

We know that a single delay line canceler consists of a delay line and a subtractor. If two such 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 delay row canceller. We will get the following mathematical relation from ** first line delay canceller **.

$$ q left (t right) = p left (t right) -p left (t-T_P right)::::: Equation: 9 $$

The output of the first delay line canceller is applied as input to the second delay line canceller. Therefore, $ q left (t right) $ will bethe input of the second delay line 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 **.

$$ r left (t right) = q left (t right) -q left (t-T_P right)::::: Equation: 10 $$

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

$$ q left (t-T_P right) = p left (t-T_P right) -p left (t-T_P -T_P right) $$

$$ q left (t-T_P right) = p left (t-T_P right) -p left (t- 2T_P right)::::: Equation: 11 $$

** Substitute **, equation 9 and equation 11 in equation 10.

$$ r left (t right) = p left (t right) - p left (t-T_P right) - left [p left (t-T_P right) -p left (t-2T_P right) right] $$

$$ Right arrow r left (t right) = p left (t right) -2p left (t-T_P right) + p left (t-2T_P right)::::: Equation: 12 $$

The ** advantage ** of the double delay line canceler 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 at 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 two double delays l The line canceller and the cascading 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.