1
GATE ECE 2022
Numerical
+2
-0
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).
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2
GATE ECE 2016 Set 3
Numerical
+2
-0
A continuous-time speech signal $${x_a}(t)$$ is sampled at a rate of 8 kHz and the samples are subsequently grouped in
blocks, each of size N. The DFT of each block is to be computed in real time using the radix-2 decimation-in-frequency
FFT algorithm. If the processor performs all operations sequentially, and takes 20 µs for computing each complex multiplication (including multiplications by 1 and −1) and the time required for addition/ subtraction is negligible, then the
maximum value of N is __________.
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3
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-----------------------.
$$x\left[ n \right]$$= {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-----------------------.
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4
GATE ECE 2015 Set 1
MCQ (Single Correct Answer)
+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
$$\left[ {\matrix{ A \cr B \cr C \cr } } \right] = \left[ {\matrix{ 1 \cr 1 \cr 1 \cr } {\mkern 1mu} \,\matrix{ 1 \cr {W_3^{ - 1}} \cr {W_3^{ - 2}} \cr } \,\matrix{ 1 \cr {W_3^{ - 2}} \cr {W_3^{ - 4}} \cr } } \right]{\mkern 1mu} \left[ {\matrix{ a \cr b \cr c \cr } } \right]$$ 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
Questions Asked from Discrete Fourier Transform and Fast Fourier Transform (Marks 2)
Number in Brackets after Paper Indicates No. of Questions
GATE ECE Subjects
Signals and Systems
Representation of Continuous Time Signal Fourier Series Fourier Transform Continuous Time Signal Laplace Transform Discrete Time Signal Fourier Series Fourier Transform Discrete Fourier Transform and Fast Fourier Transform Discrete Time Signal Z Transform Continuous Time Linear Invariant System Discrete Time Linear Time Invariant Systems Transmission of Signal Through Continuous Time LTI Systems Sampling Transmission of Signal Through Discrete Time Lti Systems Miscellaneous
Network Theory
Control Systems
Digital Circuits
General Aptitude
Electronic Devices and VLSI
Analog Circuits
Engineering Mathematics
Microprocessors
Communications
Electromagnetics