The problem asks to find the Fourier series coefficients for the signal:

$x(t) = 3 + 8 \cos(5\pi t) + 6 \sin(15\pi t)$

To determine the Fourier series coefficients, we need to express the signal in the form:

$x(t) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t) \right)$

Where:

• $a_0$ is the average (or DC) term,
• $a_n$ and $b_n$ are the cosine and sine Fourier coefficients respectively,
• $\omega_0$ is the fundamental angular frequency, which is related to the period of the signal.

Let's break down the signal:

1. The constant term $3$ contributes to $a_0$.
2. The cosine term $8 \cos(5\pi t)$ contributes to the $a_n$ coefficient for $n = 5$.
3. The sine term $6 \sin(15\pi t)$ contributes to the $b_n$ coefficient for $n = 15$.

Next, we will compute the Fourier series coefficients.

Steps:

1. DC coefficient ($a_0$): It is simply the constant term: $a_0 = 3$

2. Cosine coefficients ($a_n$): The general form of the coefficient $a_n$ is given by: $a_n = \frac{2}{T} \int_{0}^{T} x(t) \cos(n\omega_0 t) dt$ In the case of this signal, we already see that for the term $8 \cos(5\pi t)$, we will get a non-zero $a_n$ only for $n = 5$. Hence: a_5 = 8 \quad \text{(from the cosine term with frequency 5\pi)} For all other $n$, $a_n = 0$.

3. Sine coefficients ($b_n$): The general form of the coefficient $b_n$ is given by: $b_n = \frac{2}{T} \int_{0}^{T} x(t) \sin(n\omega_0 t) dt$ In this case, the term $6 \sin(15\pi t)$ contributes a non-zero value only for $n = 15$: b_{15} = 6 \quad \text{(from the sine term with frequency 15\pi)} For all other $n$, $b_n = 0$.

Conclusion:

The Fourier series coefficients for the given signal are:

• $a_0 = 3$,
• $a_5 = 8$,
• $b_{15} = 6$,
• All other $a_n = 0$ and $b_n = 0$ for $n \neq 5, 15$.

Let me know if you need further explanation or a detailed breakdown of any steps!

Here are some related questions you may find helpful:

1. How do I compute the fundamental frequency for a periodic signal?
2. What does it mean when Fourier coefficients are zero for certain frequencies?
3. How can the Fourier series be used in signal processing applications?
4. What is the significance of the DC term in the Fourier series?
5. How do I handle complex exponentials in Fourier series analysis?

Tip: Always ensure that the frequency components in the signal are clearly identified before starting your Fourier analysis!