Discrete Time Signal Fourier Series Fourier Transform · Signals and Systems · GATE ECE
Marks 1
1
The value of $$\sum\limits_{n = 0}^\infty n {\left( {{1 \over 2}} \right)^n}$$ is ________________.
GATE ECE 2015 Set 3
2
A Fourier transform pair is given by $${\left( {{2 \over 3}} \right)^n}$$ u $$\left[ {n + 3} \right]\,\mathop \Leftrightarrow \limits^{FT} \,{{A{e^{ - j6\pi f}}} \over {1 - \left( {{2 \over 3}} \right){e^{ - j2\pi f}}}}$$ ,
where u(n) donotes the unit step sequence. The values of A is_______________.
where u(n) donotes the unit step sequence. The values of A is_______________.
GATE ECE 2014 Set 4
3
Let x(n) = $${\left( {{1 \over 2}} \right)^n}$$ u(n), y(n) = $${x^2}$$, and Y ($$({e^{j\omega }})\,$$ be the Fourier transform of y(n). Then Y ($$({e^{jo}})$$ is
GATE ECE 2005
Marks 2
1
The radian frequency value(s) for which the discrete time sinusoidal signal $x[n] = A \cos(\Omega n + \pi/3)$ has a period of 40 is/are __.
GATE ECE 2024
2
Consider a discrete-time periodic signal with period N = 5. Let the discrete-time Fourier series (DTFS) representation be $$x[n] = \sum\limits_{k = 0}^4 {{a_k}{e^{{{jk2\pi m} \over 5}}}} $$, where $${a_0} = 1,{a_1} = 3j,{a_2} = 2j,{a_3} = - 2j$$ and $${a_4} = - 3j$$. The value of the sum $$\sum\limits_{n = 0}^4 {x[n]\sin {{4\pi n} \over 5}} $$ is
GATE ECE 2023
3
Let X[k] = k + 1, 0 ≤ k ≤ 7 be 8-point DFT of a sequence x[n],
where X[k] = $$\sum\limits_{n = 0}^{N - 1} {x\left[ n \right]{e^{ - j2\pi nk/N}}} $$.
The value (correct to two decimal places) of $$\sum\limits_{n = 0}^3 {x\left[ {2n} \right]} $$ is ___________.
where X[k] = $$\sum\limits_{n = 0}^{N - 1} {x\left[ n \right]{e^{ - j2\pi nk/N}}} $$.
The value (correct to two decimal places) of $$\sum\limits_{n = 0}^3 {x\left[ {2n} \right]} $$ is ___________.
GATE ECE 2018
4
Let h[n] be the impulse response of a discrete time linear time invariant (LTI) filter. The impulse response is given by h(0)= $${1 \over 3};h\left[ 1 \right] = {1 \over 3};h\left[ 2 \right] = {1 \over 3};\,and\,h\,\left[ n \right]$$ =0 for n < 0 and n > 2. Let H ($$\omega $$) be the Discrete- time Fourier transform (DTFT) of h[n], where $$\omega $$ is the normalized angular frequency in radians. Given that ($${\omega _o}$$) = 0 and 0 < $${\omega _0}$$ < $$\pi $$, the value of $${\omega _o}$$ (in ratians ) is equal to ____________.
GATE ECE 2017 Set 1
5
Two discrete-time signals x [n] and h [n] are both non-zero only for n = 0, 1, 2 and are zero otherwise. It is given that x(0)=1, x[1] = 2, x [2] =1, h[0] = 1, let y [n] be the linear convolution of x[n] and h [n]. Given that y[1]= 3 and y [2] = 4, the value of the expression (10y[3] +y[4]) is _____________________.
GATE ECE 2017 Set 1
6
Consider the signal $$x\left[ n \right] = 6\delta \left[ {n + 2} \right] + 3\delta \left[ {n + 1} \right] + 8\delta \left[ n \right] + 7\delta \left[ {n - 1} \right] + 4\delta \left[ {n - 2} \right]$$.
If X$$({e^{t\omega }})$$is the discrete-time Fourier transform of x[n],
then $${1 \over \pi }\int\limits_{ - \pi }^\pi X ({e^{j\omega }}){\sin ^2}(2\omega )d\omega $$ is equal to ____________.
GATE ECE 2016 Set 1
7
Let $$\widetilde x\left[ n \right]\, = \,1 + \cos \left[ {{{\pi n} \over 8}} \right]$$ be a periodic signal with period 16. Its DFS coefficients are defined by
$${a_k}$$ = $${1 \over {16}}\sum\limits_{n = 0}^{15} {\widetilde x} \left[ n \right]\exp \left( { - j{\pi \over 8}kn} \right)$$ for all k. The value of the coeffcients $${a_{31}}$$ is _____________________.
$${a_k}$$ = $${1 \over {16}}\sum\limits_{n = 0}^{15} {\widetilde x} \left[ n \right]\exp \left( { - j{\pi \over 8}kn} \right)$$ for all k. The value of the coeffcients $${a_{31}}$$ is _____________________.
GATE ECE 2015 Set 3
8
A 5-point sequence x [n] is given as x$$\left[ { - 3} \right]$$ =1, x$$\left[ { - 2} \right]$$ =1, x$$\left[ { - 1} \right]$$ =0, x$$\left[ { - 0} \right]$$ = 5, x$$\left[ { - 1} \right]$$ = 1. Let X$$({e^{j\omega }})\,$$ denote the discrete - time Fourier transform of x(n). The value of $$\int\limits_{ - \pi }^\pi x $$
($$({e^{j\omega }})\,$$ d$$\omega $$ is
GATE ECE 2007
9
A sequence x(n) has non-zero values as shown in Fig.
The sequence $$$y(n)=\left\{\begin{array}{l}x\left(\frac n2-1\right)\;\;\;for\;n\;even\\0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;for\;n\;odd\end{array}\right.$$$
will be
The sequence $$$y(n)=\left\{\begin{array}{l}x\left(\frac n2-1\right)\;\;\;for\;n\;even\\0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;for\;n\;odd\end{array}\right.$$$
will be
GATE ECE 2005
10
A sequence x(n) has non-zero values as shown in figure. 1
The Fourier transform of y(2n) will be
The Fourier transform of y(2n) will be
GATE ECE 2005