Discrete Fourier Transform and Fast Fourier Transform | Signals and Systems | GATE ECE Previous Year Questions

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

GATE ECE 2023

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).

GATE ECE 2022

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 __________.

GATE ECE 2016 Set 3

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-----------------------.

GATE ECE 2016 Set 2

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

GATE ECE 2015 Set 1

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 $${x_3}\left[ 2 \right]$$

GATE ECE 2015 Set 1

Consider a discrete time periodic signal x$$\left[ n \right]$$= $$\sin \left( {{{\pi n} \over 5}} \right)$$. Let a_k be the complex Fourier serier coefficients of x$$\left[ n \right]$$. The coefficients $$\left\{ {{a_k}} \right\}$$ are non- zero when k = Bm $$ \pm $$ 1, where m is any integer. The value of B is _________________.

GATE ECE 2014 Set 1

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) ) = ___________.