Talk:Video/The imaginary number i and the Fourier Transform

After watching the video:The imaginary number i and the Fourier Transform.

Trigonometry

Fourier Transform and the notion of $i$ (imaginary number), cannot be separated from Trigonometry.

Fourier Series(Q179467)

Some functions are hard to work with, for example, discontinuous and fractal functions. On the other hand, some functions have wonderful properties for example sin, cos, and the linear function. In the Fourier series, we can use functions with wonderful properties to approximate functions that are hard to work with, for example, we use sin and cos to approximate fractal functions . Moreover, we can also approximate the functions by adding up functions together and the Fourier series will tell us what coefficient to use in our combination. However, you can only approximate a function on an interval.

Convolution(Q210857)

Convolution is a mathematical operation that expresses the product of two functions. It refers to both the result function and to the process of computing it. After one function is reversed and shifted it could be seen as the integral of the product of the two functions. The following equations is copied from the video: " The Convolution of Two Functions Definition & Properties"

The equation of convolution of $f$ of $t$ and $g$ of $t$: $f(t)*g(t)=\int _{0}^{t}f(\tau )f(t-\tau )d\tau$

In this equation, the star between $f(t)$ and $g(t)$ is not multiplication. The star operator:($*$), takes two different functions and combines them into one function. The $t$ in the equation is just a variable, and the tau $\tau$ were just the dummy variable of integration.

Fourier transform

Fourier transform is the next level of the Fourier Series, it comes up with a way to approximate hole functions by using exponentials $e^{ikx}$ that means, unlike Fourier Series can only approximate a function on an interval, now we can approximate functions that are infinitely long.

Fourier transforms integral equation : ${\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-i2\pi \xi x}\,dx,\quad \forall \ \xi \in \mathbb {R} .$

Example for Fourier transform: We have a signal called $x(t)$ we will represent it in terms of the time domain. We also can represent it in another way which is called $x(f)$ we will represent it in terms of the frequency domain and This is why we called transformation. Fourier Transform is an equivalent representation of the signal.

Convolution equation : $(f*g)(t):=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .$

Possible Works to be done for this page

1. Learn to use Blockquote and Cite!
2. What did you learn from the video?
3. What are the links of Fourier Transform with Sine/Cosine, and Exponential functions?
4. What other sources related to Sine/Cosine/Exponent and Fourier Transform you have found and what are their Links?
5. Does it relate to Polynomials, Infinite Series, and the eventual implementation of Fast Fourier Transform?
6. What did you learn that changed the views about certain beliefs that you hold before watching this video?
7. Who can you ask to learn more about what you wish to know.
8. Can you ask other people, such as your English teacher to look for help in documenting your observations and thoughts?
9. Since Fourier Transform relates to adding many waves together, What kind of arithmetic does Fourier Transform can do? How does it relate to numerical arithmetic, or how does the arithmetics of waves differ from the arithmetics of numbers?

--Benkoo (talk) 03:04, 28 July 2022 (UTC)