# Questions & Answers of DFT and FFT

Question No. 45

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[n],

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

Question No. 142

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 ________

Question No. 143

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 ________

Question No. 245

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 __________

Question No. 246

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

The filter can be used to approximate a

Question No. 53

Two sequences [a,b,c] and [A,B,C] are related as,

If another sequence [pqr] 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 [p, q, r] and [a,b,c] is

Question No. 154

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_____.

Question No. 31

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

Question No. 16

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