fourier quiz second quiz model

A square wave time signal is defined as: $f\left(t\right)\left\lbrace-1,\right\rbrace if,-\frac{\pi}{3}\le t\le0,+1,if,0\le t\le\frac{\pi}{3}.$

Given this information, select the most accurate statement from the list of choices below.

a) one can determine that f(0) = 0 without determining its entire Fourier series.

b) one simply cannot evaluate f(0), because we need to find the Fourier series

c) one can accurately find f(0) only by determining the complex form of Fourier series.

d) it is evident f(0) does not exist

a) one can determine that f(0) = 0 without determining its entire Fourier series.

Select the most correct statement from the list of choices below

a) Functions that are neither f(-x) = -f(x) nor f(-x) = +f(x), must contain both sine and cosine terms

b) Fourier series for odd functions are difficult to calculate

c) Fourier series for even functions are easy to calculate

d) Fourier series for odd and even functions are tedious to calculate

a) Functions that are neither f(-x) = -f(x) nor f(-x) = +f(x), must contain both sine and cosine terms

Select the most correct statement from the list of choices below

a) Application of Parseval’s theorem can often lead to expressions facilitating the calculation of the irrational number $\pi$

b) Parseval’s theorem applied to even functions only may lead to calculation of rapidly convergent series for the irrational number $\pi$

c) Parseval’s theorem cannot be employed to facilitate the calculation of the irrational number $\pi$

a) Application of Parseval’s theorem can often lead to expressions facilitating the calculation of the irrational number $\pi$

The Fourier series for $\sin^3\theta=\sum_{n=0}^{+N}a_{n}\cos\left(n\theta\right)+\sum_{n=0}^{+N}b_{n}\sin\left(n\theta\right)$ . Given this information, and the fact $\sin\left(3\theta\right)=3\sin\theta-4\sin^3\theta$ , select the best statement from the list of choices below:

a) the coefficients $a_{n}\ne0$ and $b_1=\frac34,b_2=\frac{-1}{4}$

b) the coefficients $a_{n}=0$ and $b_1=\frac34,b_2=\frac{-1}{4}$

c) the coefficients $a_{n}\ne0$ and $b_1=-\frac14,b_2=\frac34$

d) the coefficients $a_{n}=0$ and $b_1=0,b_{n\ge2}=-\frac14$ , because of symmetry.

b) the coefficients $a_{n}=0$ and $b_1=\frac34,b_2=\frac{-1}{4}$

Select the single most correct answer from the list of statements below

a) The complex Fourier series always converges faster than the corresponding trigonometric form

b) The trigonometric Fourier series can be derived from the complex Fourier series

c) The trigonometric Fourier series can be derived from the complex Fourier series, and vice-versa

d) The Parseval’s theorem does not apply to complex Fourier series

c) The trigonometric Fourier series can be derived from the complex Fourier series, and vice-versa