Yes, you can use the Fourier transform pair to show that \(\text{sinc}(t) * \text{sinc}(t) = \text{sinc}(t)\). This method leverages the fact that certain functions and their Fourier transforms are well-known pairs.

### Step 1: Recognize the Fourier Transform Pair

We already know that the Fourier transform of the \(\text{sinc}(t)\) function is the rectangular function:

\[
\mathcal{F}\{\text{sinc}(t)\} = \text{rect}(f)
\]

And conversely:

\[
\mathcal{F}^{-1}\{\text{rect}(f)\} = \text{sinc}(t)
\]

### Step 2: Convolution in the Time Domain

Given that the Fourier transform of \(\text{sinc}(t)\) is \(\text{rect}(f)\), the convolution theorem tells us that the convolution of \(\text{sinc}(t)\) with itself corresponds to the multiplication of their Fourier transforms in the frequency domain.

### Step 3: Multiply in the Frequency Domain

Now consider the Fourier transform of the convolution \(\text{sinc}(t) * \text{sinc}(t)\):

\[
\mathcal{F}\{\text{sinc}(t) * \text{sinc}(t)\} = \mathcal{F}\{\text{sinc}(t)\} \cdot \mathcal{F}\{\text{sinc}(t)\} = \text{rect}(f) \cdot \text{rect}(f)
\]

Since \(\text{rect}(f) \cdot \text{rect}(f) = \text{rect}(f)\), we have:

\[
\mathcal{F}\{\text{sinc}(t) * \text{sinc}(t)\} = \text{rect}(f)
\]

### Step 4: Apply the Inverse Fourier Transform

Finally, take the inverse Fourier transform of \(\text{rect}(f)\) to return to the time domain:

\[
\mathcal{F}^{-1}\{\text{rect}(f)\} = \text{sinc}(t)
\]

Thus:

\[
\text{sinc}(t) * \text{sinc}(t) = \text{sinc}(t)
\]

### Conclusion
By using the Fourier transform pair \(\text{sinc}(t) \leftrightarrow \text{rect}(f)\) and the properties of the convolution theorem, we have shown that \(\text{sinc}(t) * \text{sinc}(t) = \text{sinc}(t)\).

Would you like any more details, or do you have other questions?

Here are five related questions for further exploration:
1. How does the width of \(\text{rect}(f)\) affect the corresponding \(\text{sinc}(t)\) in the time domain?
2. What other common Fourier transform pairs are there?
3. How would the convolution change if we used a different function, like a Gaussian, instead of \(\text{sinc}(t)\)?
4. Can you derive the Fourier transform of \(\text{sinc}^2(t)\)?
5. What is the significance of the convolution theorem in linear systems analysis?

**Tip:** Fourier transform pairs are very useful in signal processing, as they allow quick transitions between time and frequency domains, simplifying the analysis of systems and signals.

Explore how to use the Fourier transform pair to show that the convolution of \(\text{sinc}(t)\) with itself equals \(\text{sinc}(t)\). This detailed mathematical analysis covers the Fourier transform of \(\text{sinc}(t)\), the convolution theorem, and their applications in signal processing.

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