The final signal recovery using the process of modulation

Question

To build on the work of Level 1 and extend those aspects of Mathematics required in this and later stages of the degree Programme.

To make sure that students achieve the following learning outcomes.

- to emphasize on mathematical notations, concepts and problem solving.
- to develop competence in relevant applied mathematics concepts and application to engineering problems

1. The final signal recovery using the process of modulation, demodulation and frequency domain filtering was found to be periodic with period and is defined by

              -x       -5 ≤ x< 0.

f(x) = 

             1 + x²   0 ≤ x < 0.

Find the Fourier series expansion for f(x)

2. Evaluate ∫∫√(4 - x² - y²) over the region bounded by the semi-circle x² + y² -2x = 0 lying in the first quadrant

3. Find the volume bounded by the elliptic paraboloids x² + 3y² = z and -y² -x² + 8 = z

4. Prove that the Fourier series expansion of the function Π/2 -t can be expressed as ∑_n=1^∞ 1/n.sin2nt; 0 < t < Π.

5. Evaluate the double integral by changing the order of integration in

∫∫xy(x² + y²)^n/2 dxdy

over the positive quadrant of x² + y² = 4

6. In a two dimensional fluid flow, the stream lines is given by

φ = 1/2ln(x² + y²)

Find the corresponding stream function and its complex potential. Also verify whether stream lines and the stream function are harmonic.

7. Determine the analytic function

f(z) = u + jv +, if u -v = (cos x + sinx - e^-y)/(2(cosx -coxhy))

and f(Π/2) = 0

8. Find the Laurent's series expansion of

1/(z² + 1)(z² +2) for

(a)  0 < |z| < 1

(b)  1 < |z| < √2

(c) |z| > 2