FOURIER TRANSFORM

The fourier transform, named after JOSEPH FOURIER is a mathematical transform with many applications in physics and engineering. It transforms a mathematical function of time f(t) into a new function sometimes denoted by F. which is frequency and measured in radions per second . the new function is called fourier transform.the term “Fourier transform” refers to both the transform operation and to the complex value function it produces. In case of a periodic function the fourier transform can be simplified to tha calculation of a discrete set of complex altitudes called fourier series coefficients. It is possible to recreate a version of the original fourier transform known as discrete time Fourier transform.

 The fourier transform comes from the study of Fourier series. In the study of fourier series complicated but periodic functions are written as the sum of simple waves mathematically represented by sines and cosines. Due to the properties os sine and cosine it is possible to recover the amplitude of each wave in a fourier series using an integral. In some cese it is desirable to use Euler’s formula  e^2πiθ= cos(2πθ) +isin(2πθ) . the usual interpretation of this complex number is that it gives both the amplitude of the wave present in the function and the phase of the wave.

 There is a  close  connection  between the  definition of  Fourier  series and the Fourier transform  for  functions  f  which are zero  outside of  an interval. For  such a function, we can calculate its Fourier series on any interval that includes the points where f is not identically zero. The Fourier transform is also defined for such a function. As we increase the length of the interval on which we calculate the Fourier series, then the Fourier series coefficients begin to look like the Fourier transform and the sum of the Fourier series of fbegins to look like the inverse Fourier transform.