calculate energy of s\left(t\right)=4\text{sinc}\left(36\pi t\right)+5\text{sinc}\left(45\pi t\right)

The energy of a signal \( s(t) \) is calculated as the integral of the squared magnitude of the signal over all time, i.e.,

\[
E = \int_{-\infty}^{\infty} |s(t)|^2 dt
\]

For the given signal:

\[
s(t) = 4 \, \text{sinc}(36\pi t) + 5 \, \text{sinc}(45\pi t)
\]

Where \( \text{sinc}(x) = \frac{\sin(x)}{x} \). The energy can be calculated as:

\[
E = \int_{-\infty}^{\infty} \left( 4 \, \text{sinc}(36\pi t) + 5 \, \text{sinc}(45\pi t) \right)^2 dt
\]

We can expand the square:

\[
E = \int_{-\infty}^{\infty} \left( 16 \, \text{sinc}^2(36\pi t) + 25 \, \text{sinc}^2(45\pi t) + 2 \times 4 \times 5 \, \text{sinc}(36\pi t) \, \text{sinc}(45\pi t) \right) dt
\]

This gives us three terms to evaluate:

1. \( \int_{-\infty}^{\infty} \text{sinc}^2(36\pi t) dt \)
2. \( \int_{-\infty}^{\infty} \text{sinc}^2(45\pi t) dt \)
3. \( \int_{-\infty}^{\infty} \text{sinc}(36\pi t) \, \text{sinc}(45\pi t) dt \)

Now let's evaluate each term:

1. For \( \text{sinc}^2(x) \), it is known that:

\[
\int_{-\infty}^{\infty} \text{sinc}^2(at) dt = \frac{1}{a}
\]

Thus,

\[
\int_{-\infty}^{\infty} \text{sinc}^2(36\pi t) dt = \frac{1}{36\pi}
\]
and
\[
\int_{-\infty}^{\infty} \text{sinc}^2(45\pi t) dt = \frac{1}{45\pi}
\]

2. The cross-term \( \int_{-\infty}^{\infty} \text{sinc}(36\pi t) \, \text{sinc}(45\pi t) dt \) generally tends to zero for different frequencies due to orthogonality. So we can assume:

\[
\int_{-\infty}^{\infty} \text{sinc}(36\pi t) \, \text{sinc}(45\pi t) dt \approx 0
\]

Now substitute these into the energy expression:

\[
E = 16 \times \frac{1}{36\pi} + 25 \times \frac{1}{45\pi}
\]

We can calculate this final result to determine the energy.

Let me compute that now.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can proceed by manually calculating the final result based on the equation I provided:

\[
E = 16 \times \frac{1}{36\pi} + 25 \times \frac{1}{45\pi}
\]

Would you like further clarification or a deeper explanation?

Here are 5 related questions:
1. How does the concept of energy relate to signals in general?
2. Can you calculate the energy of a pure sinc function alone?
3. What is the significance of orthogonality in signal processing?
4. How do Fourier transforms relate to the energy of signals?
5. How would you compute energy if the signal was time-limited?

**Tip**: Orthogonality in signal processing is crucial, especially when analyzing signals of different frequencies, as it often helps simplify calculations.

calculate energy of s(t) = 4 sinc(36πt) + 5 sinc(45πt)

Learn how to calculate the energy of a signal composed of sinc functions: s(t) = 4 sinc(36πt) + 5 sinc(45πt). This detailed solution involves integral calculus, orthogonality of sinc functions, and the energy theorem, suitable for undergraduate students.

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