Fourier Transform Question || Engineering mathematics

FOURIER TRANSFORM:-

The Fourier transform (FT)

        decomposes a function of time (a signal) into its constituent frequencies. This is similar to the way a musical chord can be expressed in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. 


       The Fourier transform of a function of time is itself a complex-valued function of frequency, whose magnitude (modulus) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation.



                        Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.                               

Definition

The Fourier transform of a function f is traditionally denoted ${\displaystyle {\hat {f}}}$, by adding a circumflex to the symbol of the function. There are several common conventions for defining the Fourier transform of an integrable function ${\displaystyle f:\mathbb {R} \to \mathbb {C} }$.^[1][2] One of them is

${\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-2\pi ix\xi }\,dx,}$

(Eq.1)

for any real number ξ.

                    A reason for the negative sign in the exponent is that it is common in electrical engineering to represent by ${\displaystyle f(x)=e^{2\pi i\xi _{0}x}}$ a signal with zero initial phase and frequency ${\displaystyle \xi _{0}.}$^{[3][remark 5]} The negative sign convention causes the product ${\displaystyle e^{2\pi i\xi _{0}x}e^{-2\pi i\xi x}}$ to be 1 (frequency zero) when ${\displaystyle \xi =\xi _{0},}$ causing the integral to diverge. The result is a Dirac delta function at ${\displaystyle \xi =\xi _{0}}$, which is the only frequency component of the sinusoidal signal ${\displaystyle e^{2\pi i\xi _{0}x}.}$

When the independent variable x represents time, the transform variable ξ represents frequency (e.g. if time is measured in seconds, then frequency is in hertz). Under suitable conditions, f is determined by ${\displaystyle {\hat {f}}}$ via the inverse transform:

${\displaystyle f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )\ e^{2\pi ix\xi }\,d\xi ,}$

(Eq.2)

for any real number x.

             The statement that f can be reconstructed from ${\displaystyle {\hat {f}}}$ is known as the Fourier inversion theorem, and was first introduced in Fourier's Analytical Theory of Heat, although what would be considered a proof by modern standards was not given until much later.The functions f and ${\displaystyle {\hat {f}}}$ often are referred to as a Fourier integral pair or Fourier transform pair.

              For other common conventions and notations, including using the angular frequency ω instead of the frequency ξ, see Other conventions and Other notations below. The Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum. The conventions chosen in this article are those of harmonic analysis, and are characterized as the unique conventions such that the Fourier transform is both unitary on L² and an algebra homomorphism from L¹ to L^∞, without renormalizing the Lebesgue measure.

Many other characterizations of the Fourier transform exist. For example, one uses the Stone–von Neumann theorem: the Fourier transform is the unique unitary intertwiner for the symplectic and Euclidean Schrödinger representations of the Heisenberg group.

One Questions only important solution:-

 Q. Find the fourier transform of  pdf