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Harmonic analysis problem give the proper detailed answers 

To solve the problem of obtaining the first harmonics in the Fourier series expansion for the given data table of \( y = f(x) \) on the interval \( (0, 12) \), we'll proceed with the following steps:

### Steps to Obtain the First Harmonics:

1. **Fourier Series General Form**:
   The Fourier series of a periodic function \( f(x) \) with period \( T \) is given by:
   \[
   f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos\left( \frac{2n\pi x}{T} \right) + b_n \sin\left( \frac{2n\pi x}{T} \right) \right)
   \]
   Where \( a_0 \), \( a_n \), and \( b_n \) are the Fourier coefficients.

2. **Calculate the Fourier Coefficients**:
   For the first harmonics, we focus on calculating:
   - The average or DC component \( a_0 \),
   - The first cosine coefficient \( a_1 \),
   - The first sine coefficient \( b_1 \).

   These coefficients are calculated as follows:

   - **\( a_0 \) (DC component)**:
     \[
     a_0 = \frac{1}{T} \int_0^T f(x) \, dx
     \]
     For the discrete case, we can approximate this as:
     \[
     a_0 \approx \frac{1}{N} \sum_{k=0}^{N-1} f(x_k)
     \]
     Where \( N \) is the number of data points.

   - **\( a_1 \) (Cosine coefficient)**:
     \[
     a_1 = \frac{2}{T} \int_0^T f(x) \cos\left( \frac{2\pi x}{T} \right) \, dx
     \]
     Discrete approximation:
     \[
     a_1 \approx \frac{2}{N} \sum_{k=0}^{N-1} f(x_k) \cos\left( \frac{2\pi x_k}{T} \right)
     \]

   - **\( b_1 \) (Sine coefficient)**:
     \[
     b_1 = \frac{2}{T} \int_0^T f(x) \sin\left( \frac{2\pi x}{T} \right) \, dx
     \]
     Discrete approximation:
     \[
     b_1 \approx \frac{2}{N} \sum_{k=0}^{N-1} f(x_k) \sin\left( \frac{2\pi x_k}{T} \right)
     \]

3. **Given Data**:
   - Interval: \( T = 12 \) (as given in the problem)
   - \( N = 12 \) (data points corresponding to \( x = 0, 1, 2, \dots, 11 \))
   - The \( y = f(x) \) values are provided in the table.

4. **Approximating the Integrals**:
   We will use the trapezoidal rule or summation over the discrete points to approximate these Fourier coefficients.

Let me proceed with the calculations for the coefficients.

### Data:

| \( x \)  | 0  | 1  | 2  | 3  | 4  | 5  | 6  | 7  | 8  | 9  | 10 | 11 |
|---------|----|----|----|----|----|----|----|----|----|----|----|----|
| \( y \) | 1.8| 1  | 0.3| 0.1| 0.5| 1.5| 2.1| 1.8| 1.25| 1.3| 1.76| 2  |

I will now calculate the Fourier coefficients using these data points.The calculated Fourier coefficients for the first harmonics are:

- \( a_0 \) (DC component) = 1.284
- \( a_1 \) (First cosine coefficient) = -0.067
- \( b_1 \) (First sine coefficient) = -0.627

Therefore, the first harmonic in the Fourier series expansion is approximately:

\[
f(x) \approx 1.284 - 0.067 \cos\left( \frac{2\pi x}{12} \right) - 0.627 \sin\left( \frac{2\pi x}{12} \right)
\]

This represents the basic approximation of the function using the first harmonic.

Would you like to explore more details, ask any questions, or proceed with further analysis?

Here are 5 related questions you might be interested in:
1. How would the result change if more harmonics were included in the Fourier series?
2. What is the physical significance of the sine and cosine components in this context?
3. Can we visualize the original data and the first harmonic approximation graphically?
4. How do we calculate higher-order coefficients (e.g., \( a_2 \), \( b_2 \)) for more precise Fourier series expansions?
5. How do you apply Fourier series in real-world signal processing problems?

**Tip**: Fourier series is a powerful tool for breaking down complex periodic signals into simpler sinusoidal components, helping to understand the dominant frequencies within the data.

Obtain the first harmonics in the Fourier series expansion in (0, 12) for the function y=f(x) by the table given below.

Solve the Fourier series expansion problem to find the first harmonics of the function y=f(x) over the interval (0, 12) based on the given table of x and y values. This detailed explanation includes the calculation of the DC component, cosine and sine coefficients, and the resulting Fourier series.

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