Fourier Transform · Signals and Systems · GATE ECE

Consider a continuous-time, real-valued signal $f(t)$ whose Fourier transform $F(\omega)=$$\mathop f\limits_{ - \infty }^\infty $$ f(t) \exp (-j \omega t) d t$ exists.

Which one of the following statements is always TRUE?

Let $$m(t)$$ be a strictly band-limited signal with bandwidth B and energy E. Assuming $${\omega _0} = 10B$$, the energy in the signal $$m(t)\cos {\omega _0}t$$ is

The Fourier transform $$x(\omega )$$ of $$x(t) = {e^{ - {t^2}}}$$ is

Note : $$\int\limits_{ - \infty }^\infty {{e^{ - {y^2}}}dy = \sqrt \pi } $$

Consider a real-valued base-band signal $x(t)$. band limited to 10 kHz . The Nyquist rate for the signal $y(t)=x(t) \times \left(1+\frac{t}{2}\right)$ is

Consider a real baseband signal $x(t)=e^{-2 t}$, for $t$ (in seconds) $\geq 0$. If $99 \%$ of energy of $x(t)$ lies within $B \mathrm{~Hz}$, then which of the following options is TRUE for the value of $B$ ?

Consider two continuous time signals $x(t)$ and $y(t)$ as shown below

If $X(f)$ denotes the Fourier transform of $x(t)$, then the Fourier transform of $y(t)$ is ______.

$X(\omega)$ is the Fourier transform of $x(t)$ shown below. The value of $\int\limits_{-\infty}^{\infty}|X(\omega)|^2 d \omega$ (rounded off to two decimal places) is $\_\_\_\_$ .

Two of the angular frequencies at which its Fourier transform becomes zero are

Then Y(f) is

Item-1
(1) The Fourier transform of g (t - 2) is
(2) The Fourier transform of g (t/2) is

Item - 2
(A) G(f) $$e^{-j\left(4\mathrm{πf}\right)}$$
(B) G(2f)
(C) 2G(2f)
(D) G(f-2)

Using this information, and the time-shifting and time-scaling properties, determine and Fourier transform of signals in Fig (2), (3) and (4).