Question

Prove the following shifting or translation theorems for Fourier transforms. If g(α) is the Fourier transform of f(x), then

1. The Fourier transform of $\mathbf{f}\left(x-a\right)$is ${\mathbf{e}}^{\mathbf{-}\mathbf{i\alpha a}}\mathbf{g}\left(\alpha \right)$;

2. The Fourier transform of ${\mathbf{e}}^{\mathbf{i\beta x}}\mathbf{}\mathbf{f}\left(x\right)$is $\mathbf{g}\left(\alpha -\beta \right)$.

Short answer

1. The Fourier transform of $f\left(x-a\right)$is $e^{-i\alpha a}g\left(\alpha \right)$;
2. The Fourier transform of $e^{i\beta x}f\left(x\right)$is $g\left(\alpha -\beta \right)$.

Step-by-step solution

Given Information

Fourier transform is given by,

$$\begin{gathered} f\left(t\right)={\int }_{-\infty }^{\infty }f\left(t\right)e^{-jwt}dt \\ ={\int }_{-\infty }^{\infty }f\left(t\right)e^{-iwt}dt \end{gathered}$$

Calculate the Fourier transform of$f\left(x+a\right)={\int }_{-\infty }^{\infty }f\left(x+a\right)e^{-iwx}dx$

Fourier Transformation

A Fourier transform (FT) is a mathematical transformation that decomposes functions that are spatially or temporally dependent into functions that are spatially or temporally dependent.

Fourier Transformation for a

a)

Let the equation $x+a=t$

$dx=dt$

$$\begin{gathered} ={\int }_{-\infty }^{\infty }f\left(t\right)e^{-iw\left(t-\alpha \right)}dt \\ ={\int }_{-\infty }^{\infty }f\left(t\right)e^{-ia\left(t-a\right)}dt \\ =e^{ia\alpha }{\int }_{-\infty }^{\infty }f\left(t\right)e^{-iat}dt \\ g\left(\alpha \right)={\int }_{-\infty }^{\infty }f\left(t\right)e^{-iat}dt \\ =e^{iaa}g\left(\alpha \right) \end{gathered}$$

Fourier Transformation for b

b)

The Fourier transform is given by

$$\begin{gathered} e^{i\beta x}f\left(x\right)={\int }_{-\infty }^{\infty }{e}^{i\beta x}f\left(x\right)e^{-iax}dx \\ ={\int }_{-\infty }^{\infty }f\left(x\right)e^{-i\left(a+\beta \right)x}dx \\ =g\left(\alpha +\beta \right) \end{gathered}$$