Exploring Fourier Series and Signal Processing

In this assignment, the primary objective is to understand how Fourier series can be employed to

approximate a given signal, and to explore its practical applications in real$-$world scenarios.

Task $1$: Implementation in a Programming Language

Problem Statement:

Implement a Fourier series for a given periodic function using a programming language of your

choice $($e$.$g$.,$ Python, MATLAB$).$ Follow these steps:

Write a function to calculate Fourier coefficients for a given function.

Implement the Fourier series using the calculated coefficients.

Visualize the original function and the Fourier series approximation.

Consider the signal $f(x)=3e^{5x}-5x$ defined on the interval $-,.$ Your task is to

approximate this signal using a Fourier series. Follow these steps:

$3\mathrm{.}1$ Express the given function as a periodic function.

$3\mathrm{.}2$ Determine the constituent sinusoidal coefficients $($Fourier coefficients$)$ using the

appropriate formulas.

$3\mathrm{.}3$ Construct the Fourier series representation of $f(x).$

$3\mathrm{.}4$ Discuss the convergence properties of the Fourier series for the given signal.

Discuss any challenges faced during implementation and how they were overcome.

Submission Instructions

Code Implementation: Implement the Fourier series analysis and modification for the real$-$world signal in MATLAB or in Python $($e$.$g$.,$ using

Jupyter Notebooks, Python scripts$).$ Implement the Fourier series approximation for the mathematical function.

Visualizations: Include visualizations of the frequency spectrum for the real$-$world signal and the Fourier series approximation for the

mathematical function.

Discussion: Provide a clear discussion of the impact of modifications on the real$-$world signal. Discuss the convergence properties of the

Fourier series for the mathematical function.

Conclusion: Conclude your assignment with a summary of your findings and insights gained from the Fourier series analysis and

approximation.