Welcome to the first installment of this practical series on Fourier Transform and its applications. As someone deeply engrossed in a course on this subject, I find it profoundly fascinating to explore the mathematical foundations and real-world applications of Fourier Transform. Let’s embark on this intellectual journey, where today we will delve into an intriguing problem that beckons us to make creative use of the 
function.
Table of Contents
## The Enigma: Transforming a Unique Signal
Our first question is an enticing one: What is the Fourier transform of a particular signal? The time-domain representation of this signal is illustrated in the figure below:
The signal in question is a sum of two scaled and shifted 
functions:
![\[2\Lambda\left(\frac{x - 2}{2}\right) + 2.5\Lambda\left(\frac{x - 4}{2}\right)\]](https://lazying.art/wp-content/ql-cache/quicklatex.com-b0f6260f1b6510207251aa53f96b2c73_l3.png "Rendered by QuickLaTeX.com")
### An Exploration into
Before diving headlong into solving our central problem, let’s take a detour through the mathematical landscape of 
. We begin by deriving the Fourier transform of the scaled function 
.
The Fourier transform of this scaled function can be expressed as:
![\[\mathcal{F}\left(\Lambda\left(\frac{x}{c}\right)\right) = \int_{-c}^{c} \Lambda\left(\frac{x}{c}\right) e^{-i 2 \pi s x} dx \quad \text{(1)}\]](https://lazying.art/wp-content/ql-cache/quicklatex.com-05b79106320954fb7eef89c3f87aaaf5_l3.png "Rendered by QuickLaTeX.com")
Through a substitution 
, we can elegantly rewrite the integral:
![\[\mathcal{F}\left(\Lambda(y)\right) = c \mathbf{sinc}^2(cs) \quad \text{(2)}\]](https://lazying.art/wp-content/ql-cache/quicklatex.com-ecb2d36ab9f5f25c720e90f49837e3e8_l3.png "Rendered by QuickLaTeX.com")
### Generalizing the Fourier Transform
With the foundation laid, we introduce the shift relation in Fourier transform to broaden our discussion:
![\[g(t-b) \xleftrightarrow{} e^{-j 2 \pi b s} G(s)\]](https://lazying.art/wp-content/ql-cache/quicklatex.com-0678d84a6b349d115d898146cb781849_l3.png "Rendered by QuickLaTeX.com")
From this, we can now generalize for 
:
![\[\mathcal{F}\left(a\Lambda\left(\frac{x-b}{c}\right)\right) = a c \, e^{-j 2 \pi b s} \mathbf{sinc}^2(cs) \quad \text{(3)}\]](https://lazying.art/wp-content/ql-cache/quicklatex.com-f5d841f9702b9b603e9c647345c29bac_l3.png "Rendered by QuickLaTeX.com")
### Solving the Enigma
Armed with these tools, we find the Fourier Transform of our signal as:
![\[\mathcal{F}\left(2\Lambda\left(\frac{x-2}{2}\right) + 2.5\Lambda\left(\frac{x-4}{2}\right)\right) = 4 e^{-j 4 \pi s} \mathbf{sinc}^2(2s) + 5 e^{-j 8 \pi s} \mathbf{sinc}^2(2s) \quad \text{(4)}\]](https://lazying.art/wp-content/ql-cache/quicklatex.com-546cfaf2c99a9b6a1325fce56d5f98b1_l3.png "Rendered by QuickLaTeX.com")
Below is a stunning visual representation of the Fourier Transform of this signal, displaying both its real and imaginary components:
## Conclusion: The Transformative Journey
Today, we’ve cracked the code of a complex signal using the Fourier Transform and unearthed the role of the function in this process. As we journey deeper into this series, more mathematical enigmas await us. The Fourier Transform, it turns out, is not just a mathematical technique; it’s a lens through which we can view the world—distilled into its constituent frequencies.
Join me in the next part of this series, where we’ll explore more complex questions and applications of Fourier Transform. Until then, keep transforming!
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