The task involves finding the numerical Fourier series of a signal and matching it to the corresponding graphs. Based on the given images, let's analyze the problem step-by-step:

Problem Analysis:

1. Given Graph (First Image): The task is to find the numerical Fourier series of a signal, where the provided graph likely shows a periodic signal over a time period, denoted as $T$.
2. Answer Graphs (Second Image): The solution graphs display the results of Fourier series calculations. These include:
   • A magnitude spectrum $|X(k)|$
   • A phase spectrum $\angle X(k)$
   • The Fourier series representation $\sum_{k=-\infty}^{\infty} X(k)e^{j\omega k t}$

Fourier Series Representation:

The Fourier series decomposes a periodic signal into a sum of sinusoids with different frequencies. The formula given in the answer sheet shows that:

$X_k = \frac{2}{\pi} \cdot \frac{1 - e^{-j \pi/2}}{k(1 - j6k)}$

Solution Explanation:

• The first graph represents the magnitude $|X(k)|$, which shows the amplitudes of each harmonic component.
• The second graph displays the phase $\angle X(k)$, indicating the phase shift of each component.
• The third graph shows the reconstructed signal $x(t)$, representing the signal as a sum of its Fourier components.

The graphs align with the expected characteristics of a Fourier series:

• Magnitude and phase spectra help identify the harmonics contributing to the signal.
• The time-domain plot verifies that the series accurately reconstructs the original periodic behavior.

Would you like me to go into more detail about the calculations, analyze the graphs more thoroughly, or clarify a specific part of this problem?

Follow-Up Questions:

1. How do you determine the Fourier coefficients for a given periodic signal?
2. What are the effects of changing the signal period $T$ on the Fourier series?
3. How do phase shifts affect the reconstructed signal in the time domain?
4. Why is the magnitude of some frequencies higher than others in the magnitude spectrum?
5. What role does symmetry play in simplifying the Fourier series calculation?

Tip:

When interpreting Fourier series graphs, always check the frequency domain plots first (magnitude and phase) to understand the contributions of each harmonic before analyzing the reconstructed signal in the time domain.