GATE ECE
Signals and Systems
Discrete Time Signal Fourier Series Fourier Transform
Previous Years Questions

## Marks 1

The value of $$\sum\limits_{n = 0}^\infty n {\left( {{1 \over 2}} \right)^n}$$ is ________________.
A Fourier transform pair is given by $${\left( {{2 \over 3}} \right)^n}$$ u $$\left[ {n + 3} \right]\,\mathop \Leftrightarrow \limits^{FT} \,{{A{e^{ ... Let x(n) =$${\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^...

## Marks 2

Let X[k] = k + 1, 0 ≤ k ≤ 7 be 8-point DFT of a sequence x[n], where X[k] = $$\sum\limits_{n = 0}^{N - 1} {x\left[ n \right]{e^{ - j2\pi nk/N}}}$$. T...
Let h[n] be the impulse response of a discrete time linear time invariant (LTI) filter. The impulse response is given by h(0)= $${1 \over 3};h\left[ 1... Two discrete-time signals x [n] and h [n] are both non-zero only for n = 0, 1, 2 and are zero otherwise. It is given that x(0)=1, x = 2, x  =1, ... Consider the signal$$x\left[ n \right] = 6\delta \left[ {n + 2} \right] + 3\delta \left[ {n + 1} \right] + 8\delta \left[ n \right] + 7\delta \left[ ...
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 d...
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$$\left[ { - 0} \r... A sequence x(n) has non-zero values as shown in Fig. The sequence$$\$y(n)=\left\{\begin{array}{l}x\left(\frac n2-1\right)\;\;\;for\;n\;even\\0\;\;\...
A sequence x(n) has non-zero values as shown in figure. 1 The Fourier transform of y(2n) will be ...
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