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If x1[n] ↔ X1(Ω) and x2[n] ↔ X2(Ω) are two Fourier transform pairs, then prove that


x1[n] * x2[n] = X1(Ω)X2(Ω)


If x[n] ↔ X(Ω) are Fourier transform pairs, find the Fourier transform of nx[n].


Find the inverse Fourier transform of X(Ω), which is a rectangular pulse:


X(Ω) = 1, −W0 < Ω < W0

0 elsewhere


Prove the Parseval’s relation for discrete-time aperiodic signals:


Comment on the stability and causality of a system h(t) which has a frequency response:


H(ω) = 1, −W0 < ω < W0

0, elsewhere


Comment on the stability and causality of a system h[n] which has a frequency response:



H(Ω) = 1, −Ω0 < Ω < Ω0


0, elsewhere

Construct a high pass filter using a low-pass filter having an impulse response:



h[n] =(a^n)(u[n]) 0 < a < 1

  1. How does an electric field and magnetic field affect the kinetic energy of a charged particle moving in free space?
  2. What is the relationship between magnetic torque and force?
  3. Differentiate the magnetic moments: orbital spin moment, electron spin moment, and nuclear spin moment.

The turning moment diagram for a four stroke gas engine may be assumed for simplicity to be repre-




sented by four triangles, the areas of which from the line of zero pressure are as follows:




Expansion stroke = 3550 mm2; exhaust stroke = 500 mm2; suction stroke = 350 mm2; and compres- sion stroke = 1400 mm2. Each mm2 represents 3 N-m.




Assuming the resisting moment to be uniform, find the mass of the rim of a flywheel required to keep the mean speed 200 r.p.m. within ± 2%. The mean radius of the rim may be taken as 0.75 m. Also determine the crank positions for the maximum and minimum speeds

An ice-skating rink is located in an indoor shopping mall with an environmental air temperature of 22◦C and radiation surrounding walls of about 25◦C. The convection heat-transfer coefficient between the ice and air is about 10 W/m2 · ◦C because of air movement and the skaters’ motion. The emissivity of the ice is about 0.95. Calculate the cooling required to maintain the ice at 0◦C for an ice rink having dimensions of 12 by 40 m. Furthermore, the rink is located on the 2nd floor with which the room air temperature of the 1st floor is 24◦C with convection heat-transfer coefficient between the concrete floor (see appendix Table A3 for normal weight concrete) of the 2nd floor is about 12 W/m2. ◦C. Thickness of the concrete is 4 inches. The Obtain a value for the heat of fusion of ice and estimate how long it would take to melt 3 mm of ice from the surface of the rink if no cooling is supplied.

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