Fourier Transform - Prerequisites

Prerequisites - Fourier Transform

Review of some useful Results

Multiple Choice

Consider a complex number with real part = a and imaginary part = b. Also let

 $R=\sqrt{\left(a^2+b^2\right)}$  and  $\theta=\tan^{-1}\left(\frac{b}{a}\right)$  which is the correct way to represent the complex number?

•  $a+jb$  

•  $R\angle\theta$  

•  $Re^{j\theta}$  

•  $R\left(\cos\theta+j\sin\theta\right)$  

• ALL OPTIONS are TRUE

Multiple Choice

A sinusoidal signal with peak amplitude Ac is applied across a 1 ohm resistor. The power delivered is

•  $\left(A_c\right)^2$  

•  $\frac{\left(A_c\right)^2}{2}$  

Open Ended

Significance of Fourier Transform? What information can it give you? Write only key Words.

Multiple Choice

Fourier Transform of x(t) is defined as,  $F\left\{x\left(t\right)\right\}=X\left(f\right)=$  

•  $\int_{-\infty}^{\infty}x\left(t\right)e^{-j2\pi ft}dt$  

•  $\int_{-\infty}^{\infty}x\left(t\right)e^{-j2\pi ft}df$  

•  $\int_{-\infty}^{\infty}x\left(t\right)e^{j2\pi ft}dt$  

•  $\int_{-\infty}^{\infty}x\left(t\right)e^{j2\pi ft}df$  

Multiple Choice

Inverse Fourier Transform of X(f) is defined as,  $F^{-1}\left\{X\left(f\right)\right\}=x\left(t\right)=$  

•  $\int_{-\infty}^{\infty}X\left(f\right)e^{-j2\pi ft}dt$  

•  $\int_{-\infty}^{\infty}X\left(f\right)e^{-j2\pi ft}df$  

•  $\int_{-\infty}^{\infty}X\left(f\right)e^{j2\pi ft}dt$  

•  $\int_{-\infty}^{\infty}X\left(f\right)e^{j2\pi ft}df$  

Multiple Choice

If x(t) = 1, what is X(f)?

• $\delta\left(f\right)$

• $\frac{\sin\left(\pi f\right)}{\left(\pi f\right)}$

• $1$

• $\delta\left(1\right)$

Multiple Choice

A dirac delta function in frquecy domain $\delta\left(f\right)$ is defined as 

•  $\delta\left(f\right)=0\ \forall\ f\ne0\ and\ \int_{-\infty}^{\infty}\delta\left(f\right)df=1$  

•  $\delta\left(f\right)=1\ when\ f=0,\ and\ \delta\left(f\right)\ =\ 0\ \forall\ f\ne0$  

Multiple Choice

 $e^{j\theta}=?$  

•  $\cos\left(j\theta\right)+\sin\left(j\theta\right)$  

•  $\cos\left(\theta\right)+j\sin\left(\theta\right)$  

Multiple Choice

 $\cos\left(\theta\right)=?$  

•  $\frac{e^{j\theta}+e^{-j\theta}}{2}$  

•  $\frac{e^{j\theta}-e^{-j\theta}}{2j}$  

Multiple Choice

 $\cos\left(A\right)\cos\left(B\right)=$  

•  $\frac{1}{2}\left[\cos\left(A+B\right)+\cos\left(A-B\right)\right]$  

•  $\frac{1}{2}\left[\cos\left(A-B\right)-\cos\left(A+B\right)\right]$  

Product Sum

Multiple Choice

The frequency shifting property of Fourier Transform says:

• $F\left\{x\left(t\right)e^{j2\pi f_ct}\right\}=X\left(f-f_c\right)$

• $F\left\{x\left(t\right)e^{-j2\pi f_ct}\right\}=X\left(f+f_c\right)$

• Both

• None

Multiple Choice

If M(f) is Fourier Transform of a baseband signal m(t) , then

 $F\left\{m\left(t\right).\cos\left(2\pi f_ct\right)\right\}=?$  

•  $M\left(f+f_c\right)+M\left(f-f_c\right)$  

•  $\frac{1}{2}\left[M\left(f+f_c\right)+M\left(f-f_c\right)\right]$  

•  $\frac{1}{2}\left[M\left(f+f_c\right)-M\left(f-f_c\right)\right]$  

•  $M\left(f_c\right)$  

Poll

Do we need to discuss some of the basics included in this quiz before we move ahead and apply it for obtaining spectra of AM?

• YES

• NO