# Questions & Answers of DFT and FFT

## Weightage of DFT and FFT

Total 11 Questions have been asked from DFT and FFT topic of Signals and Systems subject in previous GATE papers. Average marks 1.91.

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 $X\left({e}^{j\omega }\right)$ is the discrete-time Fourier transform of $x\left[\mathrm n\right],$

then $\frac1n\int_{-\mathrm\pi}^\mathrm\pi X\left(e^{j\omega}\right)\sin^2\left(2\omega\right)d\omega$ is equal to _________

A continuous-time filter with transfer function $H\left(s\right)=\frac{2s+6}{{s}^{2}+6s+8}$ is converted to a discretetime filter with transfer function so that the impulse response of the continuous-time filter, sampled at 2 Hz, is identical at the sampling instants to the impulse response of the discrete time filter. The value of k is ________

The Discrete Fourier Transform (DFT) of the 4-point sequence

$X\left[k\right]=\left\{X\left[0\right],X\left[1\right],X\left[2\right],X\left[3\right]\right\}=\left\{12,2j,0-2j\right\}.$

If${X}_{1}\left[k\right]$ is the DTF of 12-point sequence ${x}_{1}\left[n\right]=\left\{3,0,0,2,0,0,3,0,0,4,0,0\right\}$

the value of $\left|\frac{{X}_{1}\left[8\right]}{{X}_{1}\left[11\right]}\right|$ is ________

A continuous-time speech signal xa(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 __________

The direct form structure of an FIR (finite impulse response) filter is shown in the figure.

The filter can be used to approximate a

Two sequences $\lbrack a,b,c\rbrack$ and $\lbrack A,B,C\rbrack$ are related as,

If another sequence $\lbrack p,q,r\rbrack$ is derived as,

$\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\left[\begin{array}{ccc}1& 1& 1\\ 1& {W}_{3}^{1}& {W}_{3}^{2}\\ 1& {W}_{3}^{2}& {W}_{3}^{4}\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& {W}_{3}^{2}& 0\\ 0& 0& {W}_{3}^{4}\end{array}\right]\left[\begin{array}{c}A/3\\ B/3\\ C/3\end{array}\right],$

then the relationship between the sequences $\lbrack p,q,r\rbrack$ and $\lbrack a,b,c\rbrack$ is

Consider two real sequences with time – origin marked by the bold value,

x1[n] ={1,2,3,0} , x2[n] ={1,3,2,1}

Let X1(k) and X2(k) be 4-point DFTs of x1[n] and x2[n] , respectively .

Another sequence x3[n] is derived by taking 4-point inverse DFT of X3(k) =X1(k)X2(k) .

The value of x3[2] is_____.

The first six point of the 8-points 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 respectively

For an N-point FFT algorithm with N = 2m which one of the following statements is TRUE?

$y\left(n\right)=\frac1N\sum\limits_{r=0}^{N-1}x\left(r\right)x\left(n+r\right)$ is