Fourier Transform Basics-1

The Fourier Transform of f(x) is  $F[f(x)]=$  

•  $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ dx\ =\ F\left[s\right]$  

•  $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)e^{isx}\ dx\ =\ F\left[s\right]$  

•  $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ ds\ =\ F\left[s\right]$  

•  $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ \ \cos sx\ dx\ =\ F\left[s\right]$  

The  inverse Fourier Transform of   $F[f(x)]\$ is  $f\left(x\right)\ =$   

•  $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\ F\left[s\right]\ dx\ =\ f\left(x\right)$  

•  $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F\left[s\right]e^{isx}\ ds\ =f\left(x\right)$  

•  $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F\left[s\right]\ e^{-isx}ds\ =f\left(x\right)$  

•  $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F\left[s\right]\ \ \cos sx\ ds\ =\ f\left(x\right)$  

The complex form  $e^{isx\ }$  and  $e^{-isx}$   is 

•  $e^{isx\ }\ =\ \cos sx\ +\ \sin sx\ \ and\ e^{-isx\ }\ =\ \cos sx\ -\ \sin sx\ \$  

•  $e^{isx\ }\ =\ \cos sx\ +i\sin sx\ \ and\ e^{-isx\ }\ =\ \cos sx\ -i\sin sx\ \$  

•  $e^{isx\ }\ =\ \cos sx\ \ and\ e^{-isx\ }\ =\ \ \sin sx\ \$  

• None of the above

The Real part of  $e^{isx\ }$  is

• sinsx

• i sinsx

• cossx

• cosx

The Imaginary part of  $e^{isx\ }$  is

• sinsx

• i sinsx

• cossx

• cosx

The Fourier sine Transform of f(x) is  $F_s[f(x)]=$  

•  $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ dx\ =\ F_s\left[s\right]$  

•  $\frac{1}{\sqrt{2\pi}}\int_0^{\infty}f\left(x\right)\ \sin sx\ dx\ =\ F_s\left[s\right]$  

•  $\sqrt{\frac{2}{\pi}}\int_0^{\infty}f\left(x\right)\ \sin sx\ dx\ =\ F_s\left[s\right]$  

•  $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ \ \cos sx\ dx\ =\ F_s\left[s\right]$  

The inverse Fourier sine Transform of   $F^{-1}\left\{F_s[f(x)]\right\}\ =\ f\left(x\right)$  

•  $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ ds=\ F_s\left[s\right]$  

•  $\frac{1}{\sqrt{2\pi}}\int_0^{\infty}\ F_s\left[s\right]\sin sx\ ds=f\left(x\right)\$  

•  $\sqrt{\frac{2}{\pi}}\int_0^{\infty}F_s\left[s\right]\sin sx\ ds\ =\ f\left(x\right)\$  

•  $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\ F_s\left[s\right]\ \ \cos sx\ dx\ =f\left(x\right)$  

The inverse Fourier cosine Transform of   $F^{-1}\left\{F_c[f(x)]\right\}\ =\ f\left(x\right)$  

•  $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ \cos sx\ ds=\ F_c\left[s\right]$  

•  $\sqrt{\frac{2}{\pi}}\int_0^{\infty}\ F_c\left[s\right]\ \cos sx\ \ ds=f\left(x\right)\$  

•  $\sqrt{\frac{2}{\pi}}\int_0^{\infty}F_s\left[s\right]\sin sx\ ds\ =\ f\left(x\right)\$  

•  $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\ F_s\left[s\right]\ \ \cos sx\ dx\ =f\left(x\right)$  

The Fourier cosine Transform of f(x) is  $F_c[f(x)]=$  

•  $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ \cos sx\ dx\ =\ F_c\left[s\right]$  

•  $\frac{1}{\sqrt{2\pi}}\int_0^{\infty}f\left(x\right)\ \cos sx\ dx\ =\ F_c\left[s\right]$  

•  $\sqrt{\frac{2}{\pi}}\int_0^{\infty}f\left(x\right)\ \cos sx\ dx\ =\ F_c\left[s\right]$  

•  $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ \ \cos sx\ dx\ =\ F_s\left[s\right]$  

Infinite Fourier Transform pair is also called

• complex form

• Complex pair

• complex transform pair

• none of the above