We commonly think of mathematics as something that is not closely related to our daily lives. However, the application of mathematics is widely prevalent in our daily lives.. As an occupant of the 21st century Earth, you should, at least once in your lifetime, at least once in your lifetime, have used phones or Ipads, or any other technological innovations that impact our lives.

What makes phones work? Although phones are not completely dependent on their storage, if you want phones to work, then there should be enough of them in your phone. So, naturally, to enhance the performance of phones, what we want is to increase the size of the storage capacity—but how do we do this? Well, we can use mathematics in order to reduce the size of the data taking up space in the storage. We can maximize storage capacity by using mathematics without actually enlarging the physical size of the storage. This method has revolutionized the field of devices and goes to show that mathematics has an immense impact on our lives, whether people are aware of its influences or not..

The Fourier Transform The Fourier transform is one of the most well-known concepts and is widely used in engineering, computer science, mathematics, and many other sciences. By understanding the concept of Fourier transform, I am sure that your perception of mathematics, a “boring subject”, would deviate from the stereotype that underestimates the power and potential of mathematics.

By just description, you would not be able to, if you are not already familiar with the Fourier Transform, understand the actual functions listed below. However, you will be able to see the origin and purpose from where the formula was derived. Here, below, are the elements of the formula in which its physical properties are transcribed.

Here: ƒ stands for frequency, and t stands for space. J may not seem familiar, but it is simply the imaginary number i. 

Fourier Transform is a transformation method, in which the time domain or frequency domain is exchanged. Essentially, you are interpreting each domain in terms of frequency and time. In addition, you can exchange the time domain to frequency domain by using an inverse fourier transform. The inverse equation is not much different from the original, except that the exponential term’s sign is changed from a negative to a positive. 

Generally, the Fourier transform is a method of segregating sinusoids from a wave. In order to derive the Fourier transform, we start deriving from the Fourier Series. From now on, descriptions are brief; thus, you may skim through it.

I <- Ck stands for…(below)

II <- This equation can be applied because x(t) is a sinusoidal function

Sinusoidal function x(t) is a non-sinusoidal function transcribed into sinusoidal function. So what Fourier Series above does is to decompose functions into trigonometric form. 

 

 

Now, we are transferring the I equation which is Fourier series to the equation of Fourier 

transform by applying III to the Fourier series I and letting T → infinity.

IV

 

 

V Taking 1/T to the right

Above, integral properties are applied to equation V

VI  Simplified (V)

 

VII. Simplified(VI)

VIII.  Result of equation (VII) applied to (V)

IX.  Simplified(Application of the definition of integral)

Result of IX, Fourier Formula is derived, exp(x) means e^(x), literally exponential. If applied, now we see the Fourier transform with relatively familiar elements. F12

 

The use of Fourier transform varies. Fourier transform is mainly utilized within the engineering area. Interpreting the sinusoidal elements is very important in engineering areas, where the engineering area mostly handles the application of waves to the machines. As I mentioned earlier, the use of transform is also very handy in computer science. The software of our computers could hold the data storage of what we have due to the optimization of Fourier transform into compression.

Fourier transform granted us various useful and tangible benefits that we, to the present, enjoy. Some benefits regarding the Fourier Transform include storage augmentation by compression method, noise canceling technology, fingerprint detection, etc. Now, we are familiar with the Fourier transform and its advantages that enhanced the quality of our lives. Hopefully, the understanding of the components of Fourier formulas and its physical representations opened your eyes to the beauty of the application of mathematics in the world. There still remain, however, many real-life applications of mathematics that are prevalent in our daily lives that go without recognition.

 

 

 

 

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