The Fourier transform $$x(\omega )$$ of $$x(t) = {e^{ - {t^2}}}$$ is
Note : $$\int\limits_{ - \infty }^\infty {{e^{ - {y^2}}}dy = \sqrt \pi } $$
In the table shown below, match the signal type with its spectral characteristics.
| Signal type | Spectral characteristics | ||
|---|---|---|---|
| (i) | Continuous, aperiodic | (a) | Continuous, aperiodic |
| (ii) | Continuous, periodic | (b) | Continuous, periodic |
| (iii) | Discrete, aperiodic | (c) | Discrete, aperiodic |
| (iv) | Discrete, periodic | (d) | Discrete, periodic |
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 $${a_0} = 1,{a_1} = 3j,{a_2} = 2j,{a_3} = - 2j$$ and $${a_4} = - 3j$$. The value of the sum $$\sum\limits_{n = 0}^4 {x[n]\sin {{4\pi n} \over 5}} $$ is
Let an input $$x[n]$$ having discrete time Fourier transform $$x({e^{j\Omega }}) = 1 - {e^{ - j\Omega }} + 2{e^{ - 3j\Omega }}$$ be passed through an LTI system. The frequency response of the LTI system is $$H({e^{j\Omega }}) = 1 - {1 \over 2}{e^{ - j2\Omega }}$$. The output $$y[n]$$ of the system is