By using Fourier Transform show that a dc pulse would contain ac signals of frequencies mainly upto 1/pulse width

To demonstrate that a DC pulse contains AC signals with frequencies mainly up to \( 1 / \text{pulse width} \) using the **Fourier Transform**, we analyze the spectrum of a rectangular pulse in the time domain. Here is the step-by-step explanation:

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### 1. Define the Rectangular Pulse
Let the rectangular pulse be represented as:

\[
x(t) =
\begin{cases} 
1 & \text{if } -\frac{T}{2} \leq t \leq \frac{T}{2}, \\
0 & \text{otherwise}.
\end{cases}
\]

Here, \( T \) is the **pulse width**.

---

### 2. Fourier Transform of the Pulse
The Fourier Transform \( X(f) \) of the signal \( x(t) \) is defined as:

\[
X(f) = \int_{-\infty}^\infty x(t) e^{-j 2 \pi f t} dt.
\]

Substitute \( x(t) \):

\[
X(f) = \int_{-\frac{T}{2}}^{\frac{T}{2}} e^{-j 2 \pi f t} dt.
\]

---

### 3. Solve the Integral
The integral becomes:

\[
X(f) = \left[ \frac{e^{-j 2 \pi f t}}{-j 2 \pi f} \right]_{-\frac{T}{2}}^{\frac{T}{2}}.
\]

Evaluate the limits:

\[
X(f) = \frac{1}{j 2 \pi f} \left( e^{-j \pi f T} - e^{j \pi f T} \right).
\]

Using the identity \( e^{j\theta} - e^{-j\theta} = 2j \sin(\theta) \), this simplifies to:

\[
X(f) = \frac{T \sin(\pi f T)}{\pi f T}.
\]

Simplify further:

\[
X(f) = T \cdot \text{sinc}(f T),
\]

where \( \text{sinc}(u) = \frac{\sin(\pi u)}{\pi u} \).

---

### 4. Frequency Analysis
The Fourier Transform shows that the frequency spectrum of the rectangular pulse is shaped by \( \text{sinc}(fT) \), which decays as \( f \) increases. The main lobe of the \( \text{sinc} \) function spans from \( -1/T \) to \( 1/T \), beyond which the spectral components are much weaker.

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### 5. Conclusion
The rectangular pulse in the time domain contains significant AC signal components with frequencies mainly up to \( 1/T \) (where \( T \) is the pulse width). Frequencies beyond \( 1/T \) exist but are much less significant due to the rapid decay of the \( \text{sinc}(fT) \) function.

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### Relative Questions
1. How does the pulse shape affect the frequency spectrum?
2. What happens to the spectrum when the pulse width \( T \) is increased?
3. Why does the Fourier Transform of a pulse decay at higher frequencies?
4. How can this result be applied in communication systems or signal processing?
5. What role does the main lobe of the \( \text{sinc} \) function play in frequency content analysis?

### Tip
When analyzing signals in the frequency domain, remember that sharp transitions in the time domain correspond to a broader frequency spectrum.

By using Fourier Transform show that a dc pulse would contain ac signals of frequencies mainly up to 1/pulse width

Explore how the Fourier Transform reveals that a DC pulse contains AC signals with frequencies mainly up to 1/pulse width. Understand the key concepts of signal processing, the sinc function, and frequency spectrum analysis with a detailed step-by-step explanation.

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