Fourier Series

Let $f(t)$ be a periodic signal with fundamental period $T_0>0$. Consider the signal $y(t)=f(\alpha t)$, where $\alpha>1$. The Fourier series expansions of $f(t)$ and $y(t)$ are given by $$ f(t)=\sum\limits_{k = - \infty...

Consider a discrete-time periodic signal with period N = 5. Let the discrete-time Fourier series (DTFS) representation be $$x[n] = \sum\limits_{k = 0}^4 {{a_k}{e^{{{jk2\pi m} \over 5}}}} $$, where $${a_0} = 1,{a_1} = 3j,...

The exponential Fourier series representation of a continu-ous-time periodic signal $X(t)$ is defined as $$ x(t)=\sum\limits_{k=-\infty}^{\infty} a_k e^{j k w_0 t} $$ Where $\omega_0$ is the fundamental angular frequency...

Let 𝑥(𝑡) be a periodic function with period 𝑇 = 10. The Fourier series coefficients for this series are denoted by 𝑎 𝑘 , that is $$x\left( t \right) = \sum\limits_{k = - \infty }^\infty {{a_k}} {e^{jk{{2\pi } \over...

A periodic signal x(t) has a trigonometric Fourier series expansion $$$x\left(t\right)=a_0\;+\;\sum_{n=1}^\infty\left(a_n\cos\;n\omega_0t\;+\;b_n\sin\;n\omega_0t\right)$$$ If $$x\left(t\right)=-x\left(-t\right)=-x\left(t...

Let x(t) be a continuous time periodic signal with fundamental period T = 1 seconds. Let {a k } be the complex Fourier series coefficients of x(t), where k is integer valued. Consider the following statements about x(3t)...

Let $$\widetilde x\left[ n \right]\, = \,1 + \cos \left[ {{{\pi n} \over 8}} \right]$$ be a periodic signal with period 16. Its DFS coefficients are defined by $${a_k}$$ = $${1 \over {16}}\sum\limits_{n = 0}^{15} {\widet...

Consider a discrete time periodic signal x$$\left[ n \right]$$= $$\sin \left( {{{\pi n} \over 5}} \right)$$. Let a k be the complex Fourier serier coefficients of x$$\left[ n \right]$$. The coefficients $$\left\{ {{a_k}}...

A function is given by f(t) = sin 2 t +cos2t . Which of the following is true?

The Fourier series of a real periodic function has only P. Cosine terms if it is even Q. Sine terms if it is even R. Cosine terms if it odd S. Sine terms if it is odd Which of the above statement are correct?

The trigonometric Fourier series of a periodic time function can have only

The trigonometric Fourier series of an even function of time does not have the

The Fourier Series of an odd periodic function, contains only