1
GATE ECE 2016 Set 2
Numerical
+2
-0
The Discrete Fourier Transform (DFT) of the 4-point sequence
$$x\left[ n \right]$$= {x[0], x[1], x[2], x[3]}
= {3, 2, 3, 4 } is
x[k] = {X[0], X[1], X[2], X[3]}
= {12, 2j, 0, -2j }
If $${X_1}$$ [k] is the DFT of the 12- point sequence$${X_1}$$[n] = {3, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 0 },
The value of $$\left| {{{{X_1}[8]} \over {{X_1}[11]}}} \right|$$ is-----------------------.
2
GATE ECE 2015 Set 1
+2
-0.6
Two sequences [a, b, c ] and [A, B, C ] are related as,
$$\left[ {\matrix{ A \cr B \cr C \cr } } \right] = \left[ {\matrix{ 1 \cr 1 \cr 1 \cr } {\mkern 1mu} \,\matrix{ 1 \cr {W_3^{ - 1}} \cr {W_3^{ - 2}} \cr } \,\matrix{ 1 \cr {W_3^{ - 2}} \cr {W_3^{ - 4}} \cr } } \right]{\mkern 1mu} \left[ {\matrix{ a \cr b \cr c \cr } } \right]$$ Where
$${W_3}$$ = $${e^{j{{2\pi } \over 3}}}$$ .
if another sequence $$\left[ {p,\,q,\,r} \right]$$ is derived as,
$$\left[ {\matrix{ p \cr q \cr r \cr } } \right] = \left[ {\matrix{ 1 \cr 1 \cr 1 \cr } \,\,\matrix{ 1 \cr {W_3^1} \cr {W_3^2} \cr } \,\matrix{ 1 \cr {W_3^2} \cr {W_3^4} \cr } } \right]\,\left[ {\matrix{ 1 \cr 0 \cr 0 \cr } \,\matrix{ 0 \cr {W_3^2} \cr {0\,} \cr } \,\matrix{ 0 \cr 0 \cr {W_3^4} \cr } } \right]\,\left[ {\matrix{ {A/3} \cr {B/3} \cr {C/3} \cr } } \right]$$ ,
Then the relationship between the sequences $$\left[ {p,\,q,\,r} \right]$$ and $$\left[ {a,\,b,\,c} \right]$$ is
A
$$\left[ {p,\,q,\,r} \right]$$= $$\left[ {b,\,a,\,c} \right]$$
B
$$\left[ {p,\,q,\,r} \right]$$ = $$\left[ {b,\,c,\,a} \right]$$
C
$$\left[ {p,\,q,\,r} \right]$$= $$\left[ {c,\,a,\,b} \right]$$
D
$$\left[ {p,\,q,\,r} \right]$$ = $$\left[ {c,\,b,\,a} \right]$$
3
GATE ECE 2015 Set 1
Numerical
+2
-0
Consider two real sequences with time- origin marked by the bold value, $${x_1}\left[ n \right] = \left\{ {1,\,2,\,3,\,0} \right\}\,,\,{x_2}\left[ n \right] = \left\{ {1,\,3,\,2,\,1} \right\}$$ Let $${X_1}(k)$$ and $${X_2}(k)$$ be 4-point DFTs of $${x_1}\left[ n \right]$$ and $${x_2}\left[ n \right]$$, respectively. Another sequence $${X_3}(n)$$ is derived by taking 4-ponit inverse DFT of $${X_3}(k)$$= $${X_1}(k)$$$${X_2}(k)$$. The value of $${x_3}\left[ 2 \right]$$
4
GATE ECE 2014 Set 4
+2
-0.6
The N-point DFT X of a sequence x[n] 0 ≤ n ≤ N − 1 is given by
$$X\left[ k \right] = {1 \over {\sqrt N }}\,\,\sum\limits_{n = 0}^{N - 1} x \,[n\,]e{\,^{ - j{{2\pi } \over N}nk}}$$, 0$$\le k \le N - 1$$
Denote this relation as X = DFT(x). For N= 4 which one of the following sequences satisfies DFT (DFT(x) ) = ___________.
A
x = $$\left[ {1\,2\,3\,4} \right]$$
B
x = $$\left[ {1\,2\,3\,2} \right]$$
C
x = $$\left[ {1\,3\,2\,2} \right]$$
D
x = $$\left[ {1\,2\,2\,3} \right]$$
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