Questions & Answers of DFT and FFT

Consider the signal


If Xejω 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 _________

A continuous-time filter with transfer function Hs=2s+6s2+6s+8 is converted to a discretetime filter with transfer function Gz=2z2-0.5032 zz2-0.5032 z+k 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

xn=0,x1,x2,x3=3,2,3,4 is


IfX1k is the DTF of 12-point sequence x1n=3,0,0,2,0,0,3,0,0,4,0,0

the value of X18X111 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 [a,b,c] and [A,B,C] are related as,

ABC=1111W3-1W3-21W3-2W3-4abcWhere W3=ej2π3

If another sequence [pqr] is derived as,


then the relationship between the sequences [p, q, r] and [a,b,c] 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?

The 4 point Discrete Fourier Transform (DFT) of a discrete time sequence {1, 0, 2, 3} 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 sequence

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