How do you mathematically represent a Fourier transform?

In the study of Fourier series, complicated but periodic functions are written as the sum of simple waves mathematically represented by sines and cosines. The Fourier transform is an extension of the Fourier series that results when the period of the represented function is lengthened and allowed to approach infinity.

What is the basic difference between a Fourier series and a Fourier transform?

Fourier series is an expansion of periodic signal as a linear combination of sines and cosines while Fourier transform is the process or function used to convert signals from time domain in to frequency domain.

Why Fourier Transform is used in signal processing?

There are multiple Fourier methods that are used in signal processing. The Fourier transform is used to analyze problems involving continuous-time signals or mixtures of continuous- and discrete-time signals. The discrete-time Fourier transform is used to analyze problems involving discrete-time signals or systems.

Why we need Fourier transform even we have Fourier Series?

Fourier transforms is an extremely powerful mathematical tool that allows you to view your signals in a different domain, inside which several difficult problems become very simple to analyze.

What is the difference between Fourier integrals and Fourier transforms?

Fourier integral is any integral of the form ∫∞−∞y(ω)eiωtdω . Fourier integral of a function f is any Fourier integral, that satisfies x(t)=∫∞−∞y(ω)eiωtdω . You can choose y=Fx to find a suitable y. The Fourier transform is usually defined with an expression such that it has to exist everywhere.

Why we need Fourier Transform even we have Fourier series?

How do you calculate the Fourier transform of a function?

For a general real function, the Fourier transform will have both real and imaginary parts. We can write f˜(k)=f˜c(k)+if˜ s(k) (18) where f˜ s(k) is the Fourier sine transform and f˜c(k) the Fourier cosine transform. One hardly ever uses Fourier sine and cosine transforms. We practically always talk about the complex Fourier transform.

Is the Fourier transform real or imaginary?

Fourier transform is purely imaginary. For a general real function, the Fourier transform will have both real and imaginary parts. We can write f˜(k)=f˜c(k)+if˜ s(k) (18) where f˜ s(k) is the Fourier sine transform and f˜c(k) the Fourier cosine transform. One hardly ever uses Fourier sine and cosine transforms.

What are the applications of the fastfourier transform?

Fourier transform has many applications in physics and engineering such as analysis of LTI systems, RADAR, astronomy, signal processing etc.

What are the applications of Fourier series in physics?

A Fourier Series has many applications in mathematical analysis as it is defined as the sum of multiple sines and cosines. Thus, it can be easily differentiated and integrated, which usually analyses the functions such as saw waves which are periodic signals in experimentation.