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...