Fourier series is just a means to represent a periodic signal as an infinite sum of sine wave components. A periodic signal is just a signal that repeats its pattern at some period. The primary reason that we use Fourier series is that we can better analyze a signal in another domain rather in the original domain.
What is the purpose of Fourier series?
Fourier Series introduction. The Fourier Series allows us to model any arbitrary periodic signal with a combination of sines and cosines.
How does Fourier series work?
A Fourier series is a way of representing a periodic function as a (possibly infinite) sum of sine and cosine functions. For functions of two variables that are periodic in both variables, the trigonometric basis in the Fourier series is replaced by the spherical harmonics. …
Why do we use Fourier series and Fourier transform?
The Fourier series is used to represent a periodic function by a discrete sum of complex exponentials, while the Fourier transform is then used to represent a general, nonperiodic function by a continuous superposition or integral of complex exponentials.
How is Fourier transform used in real life?
It is used in designing electrical circuits, solving differential equations , signal processing ,signal analysis, image processing & filtering.
What are some of the most unintuitive aspects of Fourier series?
The unintuitive and surprising aspect of Fourier series is that the set of sine and cosine functions with integer frequencies actually can span the whole space of arbitrary single-variable functions.
What are Fourier approximations?
The key idea behind Fourier approximations is to think of functions themselves as vectors in some infinite-dimensional vector space. This seems weird at first, but it’s not as far out as it may seem.
What is the Fourier expansion of a function?
In short, the Fourier expansion of a single-variable real-valued function can be thought of as the real component of a more generalized expansion that makes use of Euler’s formula: In this last formula, you can see the similarity between the real component of the function and the Fourier series expansion mentioned above.
What is the best way to motivate Fourier analysis?
As opposed to taking an engineer’s approach to the problem (which may have more practical value), I like to motivate Fourier analysis using linear algebra (which, as would be expected from a math major, is more elegant). Let’s start by revisiting vector spaces.
