Features of the Fourier Transform - Case Study Example

Task “Outline” Introduction Fourier transform is defined as an influential expository apparatus that serves to assess limitlessor semi endless differential analytics, in either Engineering or Science. It is an expansion of Fourier arrangement to a capacity g (t) of the countable period Body The various features of the Fourier transform are explained using mathematical derivations and proofs. They include linearity, frequency translation, and duality in time and frequency. Conclusion The essay concludes that, the Fourier transform is an extension of the Fourier series and it plays a central responsibility in the analysis of waves. The most applicable property of the Fourier transform is the convolution property. Fourier Transform Fourier change is an influential expository apparatus that serves to assess limitless or semi endless differential analytics, in both Engineering and Science. It is an expansion of Fourier arrangement to a capacity g (t) of the countable period (Duffy 1). Fourier change is given by the following function.  1 Hence the Fourier transform is  2 From the above, we learn that, equation 1 is the inverse Fourier transform, which changes into equation 2. A Fourier transform change back to a function f (t). For a Fourier transform to be true, the integral two must exist. The fundamental basic and necessity of a Fourier transform is that f (t) must be an integral on the interval between negative infinity to positive infinity. It is imperative to note that, the Fourier transform always involves a damping factor  (Lover 23). This follows a different definition of the Fourier transform as indicated below This translates to the inverse transform, which is as follows  The following form the foundational properties of Fourier transform Linearity Let a two function be y (t) and f (t) Therefore, the Fourier transform of the linear combination of the two functions is represented by the following series of functions. CY (t) +bf (t) = cF ( Read More