Doppler Frequency Shift Calculation

The Doppler frequency shift is a fundamental concept in radar systems, especially for Continuous Wave (CW) radar. It refers to the change in the frequency of a wave due to the relative motion between the source (radar) and the observer (target). When a CW radar transmits an electromagnetic wave towards a moving target, the wave reflected back to the radar experiences a frequency shift. This shift, known as the Doppler frequency shift, is directly proportional to the target's speed and helps in determining how fast the target is moving.

Given Parameters for Radar System

To calculate the Doppler frequency shift, we first need to list the information provided in the question:

• Radar operating frequency ($\text{f}_\text{c}$): This is the frequency of the electromagnetic waves transmitted by the CW radar.
• Target speed ($\text{v}$): This is the speed at which the target is moving relative to the stationary radar.
• Speed of light ($\text{c}$): This is a universal constant for electromagnetic waves in a vacuum.

Unit Conversions for Accurate Calculation

For accurate calculations using the Doppler frequency shift formula, it is essential to convert all given units into a consistent system, typically the International System of Units (SI units).

• Convert Radar Frequency from GHz to Hz:

  We know that $1 \text{ GHz} = 10^9 \text{ Hz}$.

  Therefore, the radar operating frequency is: $\text{f}_\text{c} = 5 \text{ GHz} = 5 \times 10^9 \text{ Hz}$.

• Convert Target Speed from km/hr to m/s:

  We know that $1 \text{ km} = 1000 \text{ m}$ and $1 \text{ hour} = 3600 \text{ seconds}$.

  So, the target speed is: $\text{v} = 108 \text{ km/hr} = 108 \times \frac{1000 \text{ m}}{3600 \text{ s}}$.

  $\text{v} = 108 \times \frac{10}{36} \text{ m/s} = 3 \times 10 \text{ m/s} = 30 \text{ m/s}$.

Formula for Doppler Frequency Shift

For a stationary CW radar and a target moving directly towards or away from it, the Doppler frequency shift ($\text{f}_\text{d}$) is calculated using the following formula:

$$ \text{f}_\text{d} = \frac{2 \times \text{v} \times \text{f}_\text{c}}{\text{c}} $$

Where:

• $\text{f}_\text{d}$ = Doppler frequency shift (in Hz)
• $\text{v}$ = Speed of the target (in m/s)
• $\text{f}_\text{c}$ = Operating frequency of the radar (in Hz)
• $\text{c}$ = Speed of light in vacuum (approximately $3 \times 10^8 \text{ m/s}$)

Step-by-Step Calculation of Doppler Frequency Shift

Now, let's substitute the converted values into the Doppler frequency shift formula:

$$ \text{f}_\text{d} = \frac{2 \times (30 \text{ m/s}) \times (5 \times 10^9 \text{ Hz})}{3 \times 10^8 \text{ m/s}} $$

First, multiply the terms in the numerator:

$$ 2 \times 30 \times 5 \times 10^9 = 300 \times 10^9 $$

Now, substitute this back into the formula:

$$ \text{f}_\text{d} = \frac{300 \times 10^9}{3 \times 10^8} \text{ Hz} $$

Divide the numerical parts and subtract the exponents:

$$ \text{f}_\text{d} = \left(\frac{300}{3}\right) \times 10^{(9-8)} \text{ Hz} $$

$$ \text{f}_\text{d} = 100 \times 10^1 \text{ Hz} $$

$$ \text{f}_\text{d} = 1000 \text{ Hz} $$

So, the Doppler frequency shift is 1000 Hz.

Conclusion and Option Verification

After calculating the Doppler frequency shift, we compare the result with the given options:

• Option 1: 3.6 kHz (which is 3600 Hz)
• Option 2: 1000 Hz
• Option 3: 500 Hz
• Option 4: 1800 Hz

The calculated Doppler frequency shift of 1000 Hz perfectly matches Option 2.