Exercises and Tasks in Radar and Signal Processing

Exercises

Problem Statement: A signal emitted from a radar locator reflects off a target and returns after 200 microseconds (µs).
- Question: What is the distance from the radar locator antenna to the target?
- Solution Approach:
- Use the formula for distance based on the speed of light. The speed of light is approximately $c = 3 imes 10^8$ m/s.
- The time for the signal to return is the total time for both the onward and return journey, hence:
  - Total time = 200 µs = $200 imes 10^{-6}$ s
  - Time to reach target = $\frac{200 \times 10^{-6}}{2}$ s = $100 \times 10^{-6}$ s
  - Distance = speed x time = $c \times \text{time to target}$
  - Therefore, Distance = $3 \times 10^8 \times 100 \times 10^{-6}$ = 30,000 m (or 30 km)

Problem Statement: The capacitance in the resonant circuit of a receiver changes slowly from 50 pF to 500 pF, while the inductance remains constant at 2 µH (microhenries).
- Question: What range of wavelengths can the receiver operate in?
- Solution Approach:
- The resonant frequency $f$ of a circuit can be calculated using the formula: f = \frac{1}{2\pi\sqrt{L C}}
  - Where:
  - $L$ = inductance (2 µH = $2 \times 10^{-6}$ H)
  - $C$ = capacitance in farads (50 pF = $50 \times 10^{-12}$ F to 500 pF = $500 \times 10^{-12}$ F)
- Calculate resonant frequencies for both limits:
  - For $C = 50 pF$:
  - f_{min} = \frac{1}{2\pi\,\sqrt{2 \times 10^{-6}\, 50 \times 10^{-12}}}
  - For $C = 500 pF$:
  - f_{max} = \frac{1}{2\pi\sqrt{2 \times 10^{-6} \, 500 \times 10^{-12}}}
- Convert frequencies to wavelengths using the formula:
  \lambda = \frac{c}{f}
- Thus, calculate the resulting wavelength range.

Problem Statement: A radio station transmits signals at a wavelength of 250 m.
- Question: What frequency does the station operate at?
- Solution Approach:
- Use the formula for frequency based on wavelength:
  f = \frac{c}{\lambda}
- Given:
  - Wavelength ($\lambda$) = 250 m
  - Speed of light ($c$) = $3 \times 10^8$ m/s
- Therefore:
  f = \frac{3 \times 10^8}{250}

Problem Statement: The resonant circuit includes a capacitor with a capacitance of 0.4 µF and an inductor with an inductance of 1 mH.
- Question: Determine the wavelength of the waves emitted by this circuit.
- Solution Approach:
- First calculate the resonant frequency using the formula:
  f = \frac{1}{2\pi\sqrt{L C}}
- Given values are $C = 0.4 \times 10^{-6}$ F and $L = 1 \times 10^{-3}$ H.
- Plug in the values to find the frequency, and then calculate the wavelength using:
  \lambda = \frac{c}{f}