1
GATE ECE 2022
+1
-0.33

The Fourier transform X(j$$\omega$$) of the signal $$x(t) = {t \over {{{(1 + {t^2})}^2}}}$$ is ____________.

A
$${\pi \over {2j}}w{e^{ - |\omega |}}$$
B
$${\pi \over 2}w{e^{ - |\omega |}}$$
C
$${\pi \over {2j}}{e^{ - |\omega |}}$$
D
$${\pi \over 2}{e^{ - |\omega |}}$$
2
GATE ECE 2022
Numerical
+1
-0.33

For a vector $$\overline x$$ = [x[0], x[1], ....., x[7]], the 8-point discrete Fourier transform (DFT) is denoted by $$\overline X$$ = DFT($$\overline x$$) = [X[0], X[1], ....., X[7]], where

$$X[k] = \sum\limits_{n = 0}^7 {x[n]\exp \left( { - j{{2\pi } \over 8}nk} \right)}$$.

Here, $$j = \sqrt { - 1}$$. If $$\overline x$$ = [1, 0, 0, 0, 2, 0, 0, 0] and $$\overline y$$ = DFT (DFT($$\overline x$$)), then the value of y[0] is __________ (rounded off to one decimal place).

3
GATE ECE 2021
+1
-0.33
Consider two 16-point sequences x[n] and h[n]. Let the linear convolution of x[n] and h[n] be denoted by y[n], while z[n] denotes the 16-point inverse discrete Fourier transform (IDFT) of the product of the 16-point DFTs of x[n] and h[n]. The value(s) of k for which z[k] = y[k] is/are
A
k = 0, 1, 2, ....., 15
B
k = 0 and k = 15
C
k = 0
D
k = 15
4
GATE ECE 2010
+1
-0.3
For an N-point FFT algorithm with N = $${2^m}$$ which one of the following statements is TRUE?
A
It is not possible to construct a signal flow graph with both input and output in normal order.
B
The number of butterflies in the $${m^{th}}$$ stage is N/m.
C
In-place computation requires storage of only 2N node data.
D
Computation of a butterfly requires only one complex multiplication.
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