Exercises and Tasks in Radar and Signal Processing

Exercises

Exercise 1

  • 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)

Exercise 2

  • 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=12πLCf = \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 $C = 50 pF$:
      • fmin=12π 2×10−6 50×10−12f_{min} = \frac{1}{2\pi\,\sqrt{2 \times 10^{-6}\, 50 \times 10^{-12}}}
      • For $C = 500 pF$:
      • fmax=12π2×10−6 500×10−12f_{max} = \frac{1}{2\pi\sqrt{2 \times 10^{-6} \, 500 \times 10^{-12}}}
    • Convert frequencies to wavelengths using the formula:
      λ=cf\lambda = \frac{c}{f}
    • Thus, calculate the resulting wavelength range.

Exercise 3

  • 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=cλf = \frac{c}{\lambda}
    • Given:
      • Wavelength ($\lambda$) = 250 m
      • Speed of light ($c$) = $3 \times 10^8$ m/s
    • Therefore:
      f=3×108250f = \frac{3 \times 10^8}{250}

Exercise 4

  • 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=12πLCf = \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:
      λ=cf\lambda = \frac{c}{f}

Creative Task

  • Instruction: Prepare a report on one of the given topics.