Unveiling the Mysteries of Fourier Transform: A Practical Series — Part 2

Time-domain representation of the signal in Question 1

Welcome back to Part 2 of our intriguing journey into the world of Fourier Transform! If you’ve just joined us, consider [catching up with Part 1](https://medium.com/@oldseedling1992/unveiling-the-mysteries-of-fourier-transform-a-practical-series-part-1-89ed6da9b388), where we explored the Fourier Transform of a complex signal. Today, we are venturing into the realm of signal filtering, with a specific focus on a filter characterized by a real and even transfer function H(s).

## The Puzzle: Filters and Frequencies

The question for today’s exploration is a tantalizing one: Given a filter with a real and even transfer function H(s), what happens when the input to this filter is \cos(2 \pi a t)? A hint to navigate this puzzle lies in the properties of H(s) and the Fourier Transform of the cosine function.

### The Fourier Transform of Cosine: A Brief Detour

Before solving our question, let’s revisit the Fourier Transform of \cos(2 \pi a t). Employing Euler’s formula to expand the cosine term, we can rewrite the Fourier Transform as:

    \[ \mathcal{F}(\cos(2 \pi a t)) = \frac{1}{2} (\delta(a+s) + \delta(a-s)) \quad \text{(1)} \]

### Filtering the Signal: A Tale of Convolution

In the world of signal processing, the output of a filter is the convolution of the input signal and the transfer function of the filter. Let’s denote this output as Y(s):

    \[ Y(s) = H(s) \odot F(s) \quad \text{(2)} \]

Because H(s) is both real and even, which means H(a) = H(-a), the filtered output Y(s) becomes:

    \[ Y(s) = \frac{H(a)}{2} (\delta(a+s) + \delta(a-s)) \quad \text{(3)} \]

### The Final Revelation

To find the time-domain representation of the filtered signal, we can simply take the inverse Fourier Transform of Y(s):

    \[ \mathcal{F}^{-1}(Y(s)) = H(a) \cos(2 \pi a t) \quad \text{(4)} \]

And there you have it! The output of the filter, when the input is \cos(2 \pi a t), is H(a) \cos(2 \pi a t), just as we set out to show.

## Concluding Thoughts: Filters and Fourier Unite

Today’s exploration has led us through the beauty of Fourier Transforms, from the simple cosine function to the complex behavior of filters. The result is as elegant as it is powerful: the output of a real and even filter can be precisely determined for a cosine input.

This revelation not only underscores the versatility of Fourier Transforms but also highlights the immense applications in signal processing, telecommunications, and beyond.

As we gear up for more complex questions and applications in future installments, remember that each Fourier Transform we tackle uncovers another layer of our complex, frequency-filled world. Until next time, keep transforming!

And if you have any thoughts or questions, feel free to drop them in the comments section below. Your curiosity is the fuel for this academic adventure!

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