Let's solve the problem step by step. ### **Step 1: Analyze the Signal** From the figure, the signal \( g_b(t) \): - Is a rectangular pulse. - Amplitude = 2 from \( t = -2 \) ms to \( t = 2 \) ms. - Amplitude = 0 otherwise. So, \[ g_b(t) = \begin{cases} 2, & -2 \leq t \leq 2 \\ 0, & \text{otherwise} \end{cases} \] ### **Step 2: Express the Signal Mathematically** This is a rectangular pulse centered at \( t = 0 \) with width \( T = 4 \) ms and height 2. We can write: \[ g_b(t) = 2 \cdot \text{rect}\left(\frac{t}{4}\right) \] Where: \[ \text{rect}\left(\frac{t}{T}\right) = \begin{cases} 1, & |t| \leq \frac{T}{2} \\ 0, & \text{otherwise} \end{cases} \] ### **Step 3: Find the Fourier Transform** The Fourier Transform of \( \text{rect}\left(\frac{t}{T}\right) \) is \( T \cdot \text{sinc}(fT) \), where: \[ \text{sinc}(x) = \frac{\sin(\pi x)}{\pi x} \] So, for our signal: - Amplitude = 2 - Width \( T = 4 \) ms = 0.004 s The Fourier Transform is: \[ G_b(f) = 2 \cdot 4 \cdot \text{sinc}(4f) \] \[ G_b(f) = 8 \cdot \text{sinc}(4f) \] But since the width is in milliseconds, \[ T = 4~\text{ms} = 0.004~\text{s} \] So, the correct argument for the sinc function is \( fT \) in Hz and seconds: \[ G_b(f) = 2 \times 0.004 \cdot \text{sinc}(0.004f) \] \[ G_b(f) = 0.008 \cdot \text{sinc}(0.004f) \] ### **Step 4: Final Answer** **The frequency spectrum of \( g_b(t) \) is:** \[ \boxed{ G_b(f) = 0.008 \cdot \text{sinc}(0.004f) } \] where \[ \text{sinc}(x) = \frac{\sin(\pi x)}{\pi x} \] --- #### **Explanation** - The spectrum is a sinc function because the time-domain signal is a rectangular pulse. - The width and amplitude of the pulse determine the scaling of the sinc function. - The main lobe width is inversely proportional to the pulse width. If you want the frequency in kHz, just substitute \( f \) with \( f \) in kHz accordingly.

Certainly! Let's go through the detailed, step-by-step solution to find the frequency spectrum of the given rectangular pulse signal. --- ### **Step 1: Understand the given signal** From the figure, the signal \( g_b(t) \): - Is a rectangular pulse centered at \( t = 0 \). - Has amplitude = 2. - Duration (width) = 4 ms, i.e., from \( t = -2 \) ms to \( t = 2 \) ms. - Zero outside this interval. **Mathematically:** \[ g_b(t) = \begin{cases} 2, & -2\, \text{ms} \leq t \leq 2\, \text{ms} \\ 0, & \text{otherwise} \end{cases} \] --- ### **Step 2: Express the signal using the rectangular function** The rectangular pulse can be expressed using the rectangular function: \[ g_b(t) = 2 \cdot \text{rect}\left( \frac{t}{T} \right) \] where: - \( T = 4\, \text{ms} = 0.004\, \text{s} \), - \( \text{rect}\left( \frac{t}{T} \right) = 1 \) for \( |t| \leq T/2 \), 0 otherwise. **Note:** The rectangular function is defined as: \[ \text{rect}\left( \frac{t}{T} \right) = \begin{cases} 1, & |t| \leq \frac{T}{2} \\ 0, & \text{otherwise} \end{cases} \] --- ### **Step 3: Recall the Fourier Transform of a rectangular pulse** The Fourier Transform of a rectangular pulse of amplitude \( A \), width \( T \), centered at zero, is well-known: \[ \mathcal{F}\left\{A \cdot \text{rect}\left( \frac{t}{T} \right) \right\} = A T \cdot \text{sinc}(f T) \] Where: \[ \text{sinc}(x) = \frac{\sin(\pi x)}{\pi x} \] This formula assumes the Fourier Transform convention: \[ G(f) = \int_{-\infty}^{\infty} g(t) e^{-j 2 \pi f t} dt \] --- ### **Step 4: Apply the Fourier Transform to our signal** Our signal: \[ g_b(t) = 2 \cdot \text{rect}\left( \frac{t}{T} \right) \] with \( T = 0.004\, \text{s} \). Using the formula: \[ G_b(f) = \text{FT}\{g_b(t)\} = 2 \times T \times \text{sinc}(f T) \] Substitute \( T = 0.004 \): \[ G_b(f) = 2 \times 0.004 \times \text{sinc}(f \times 0.004) \] Calculate the coefficient: \[ 2 \times 0.004 = 0.008 \] So, \[ \boxed{ G_b(f) = 0.008 \cdot \text{sinc}(0.004 f) } \] where: \[ \text{sinc}(x) = \frac{\sin(\pi x)}{\pi x} \] --- ### **Step 5: Final expression for the frequency spectrum** **Final answer:** \[ \boxed{ \boxed{ G_b(f) = 0.008 \cdot \frac{\sin(\pi \times 0.004 f)}{\pi \times 0.004 f} } } \] or equivalently, \[ G_b(f) = 0.008 \cdot \text{sinc}(0.004 f) \] --- ### **Summary:** | Parameter | Value | Explanation | |------------|---------|--------------| | Pulse amplitude | 2 | From the figure | | Pulse duration \( T \) | 4 ms = 0.004 s | Width of the pulse | | Fourier transform | \( G_b(f) = 0.008 \cdot \text{sinc}(0.004 f) \) | Based on properties of rectangular pulse | --- ### **Additional notes:** - The spectrum's main lobe width is inversely proportional to the pulse duration. - The sinc function determines the shape of the spectrum. - The maximum magnitude occurs at \( f = 0 \): \[ G_b(0) = 0.008 \times \text{sinc}(0) = 0.008 \times 1 = 0.008 \] --- **This completes the detailed step-by-step derivation of the frequency spectrum of the given rectangular pulse signal.**