GATE ECE
Signals and Systems
Discrete Fourier Transform and Fast Fourier Transform
Previous Years Questions

## Marks 1

The Fourier transform X(j$$\omega$$) of the signal $$x(t) = {t \over {{{(1 + {t^2})}^2}}}$$ is ____________.
For a vector $$\overline x$$ = [x[0], x[1], ....., x[7]], the 8-point discrete Fourier transform (DFT) is denoted by $$\overline X$$ = DFT($$\overli... Consider two 16-point sequences x[n] and h[n]. Let the linear convolution of x[n] and h[n] be denoted by y[n], while z[n] denotes the 16-point inverse... For an N-point FFT algorithm with N =$${2^m}$$which one of the following statements is TRUE? ## Marks 2 A continuous-time speech signal$${x_a}(t)$$is sampled at a rate of 8 kHz and the samples are subsequently grouped in blocks, each of size N. The D... The Discrete Fourier Transform (DFT) of the 4-point sequence$$x\left[ n \right]$$= {x[0], x[1], x[2], x[3]} = {3, 2, 3, 4 } is x[k] = {X[0], X[1... Two sequences [a, b, c ] and [A, B, C ] are related as,$$\left[ {\matrix{ A \cr B \cr C \cr } } \right] = \left[ {\matrix{ 1 ...
Consider two real sequences with time- origin marked by the bold value, $${x_1}\left[ n \right] = \left\{ {1,\,2,\,3,\,0} \right\}\,,\,{x_2}\left[ n \... The N-point DFT X of a sequence x[n] 0 ≤ n ≤ N − 1 is given by$$X\left[ k \right] = {1 \over {\sqrt N }}\,\,\sum\limits_{n = 0}^{N - 1} x \,[n\,]e...
Consider a discrete time periodic signal x$$\left[ n \right]$$= $$\sin \left( {{{\pi n} \over 5}} \right)$$. Let ak be the complex Fourier serier coe...
The DFT of a vector [a b c d] is the vector [α β γ δ ]. Consider the product The DFT of the vector [ p q r s] is a scaled version of ...
The first six points of the 8-point DFT of a real valued sequence are 5, 1 - j3, 0, 3- j4, 0 and 3+ j4. The last two points of the DFT are respectiv...
The 4-point Discrete Fourier Transform (DFT) of a discrete time sequence $$\left\{ {1,\,0,\,2,\,3} \right\}$$ is
{x(n)} is a real-valued periodic sequence with a period N. x(n) and X(k) form N-point. Discrete Fourier Transform (DFT) pairs. The DFT Y(k) of the se...
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