https://file.aiquickdraw.com/mathbot/file/1726710669708y411hsjf.jpg

Let's break down the problem step by step.

### The Problem:
You are asked to calculate the Fourier series representation of the given function \( f(t) \) from the graph, up to at least the 5th harmonic of its fundamental frequency. You are required to integrate over the whole cycle to determine the Fourier coefficients.

### Understanding the Graph:
From the graph:
- The function appears to be periodic with a period \( T = 2\pi \).
- \( f(t) \) has specific values at critical points:
  - \( f(0) = 0 \)
  - \( f(\pi) = 1 \)
  - \( f(2\pi) = 0 \)
  - \( f(3\pi) = -1 \)
  
### Fourier Series:
The Fourier series for a periodic function \( f(t) \) with period \( T = 2\pi \) is given by:

\[
f(t) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos\left( n \omega_0 t \right) + b_n \sin\left( n \omega_0 t \right) \right)
\]
where \( \omega_0 \) is the fundamental frequency, \( \omega_0 = \frac{2\pi}{T} \).

In this case, \( T = 2\pi \), so \( \omega_0 = 1 \).

The Fourier coefficients \( a_0 \), \( a_n \), and \( b_n \) are given

Given the figure shown below calculate the Fourier series representation to at least 5th harmonics of its fundamental frequency. Take note please integrate over the whole cycle and show your detailed solution.

Detailed step-by-step solution to find the Fourier series representation of a periodic function, integrating over the full cycle and calculating up to the 5th harmonic. This problem involves key concepts in digital signal processing and Fourier series representation.

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