ASSIGNMENT

1. Discuss symmetrical conditions in Fourier series

2. Derive conversion formulae among different types of Fourier Series

3. State and prove any six properties of Fourier transform.

4. State and prove any four properties of Laplace transform.

5. Short notes on following

a) Energy signal- power signal b) LTI system c) Casual system d) Dirichlet's conditions e) Completeness f) Harmonic Signals g) Final value and initial value theorems in Laplace transformation h) Bandwidth of Periodic and aperiodic signals

i) ESD and PSD

6. For the given periodic signal find exponential Fourier series and trigonometric Fourier series.

7. Find the Fourier transform of the signals and also Sketch phase and magnitude spectrums

a)

b) v(t) = (Sin t) (Cos t)/t and

c) g(t) = 4 rect [(3t-2) /T]

8. Find Laplace transforms of signals

a) t² sin (at) u(t)

b) Causal Periodic signal

9. Define Impulse response? If impulse response is h(t) = 4 e^-2t +3e^-2t cos (8t), then comment on stability and causality of the system.

10. FT of periodic signals

11. The input for an LTI system with transfer function H(s) = 1/(S+1) is e^-2t u(t), then find output?

12. Draw graphs of the signals, f(t) = 5rect[(4t - 3)/-8], Sin[u(t)] and δ(sint)

13. Write short notes on
i. Parseval's theorem
ii. Region of convergence
iii. Existence conditions of FT and LT

14. Discuss the concepts of Linear, Causal, Stable and Time invariant for continuous systems

15. A continuous-time signal x(t) is shown below. Plot x(2-t) and x(t/2).

16. Prove that if x(t) ↔ x(w) then x(at) ↔ 1/|a|x(w/a)

17. Plot the amplitude spectrum of x(t) = 5 cos(2πt).

18. State whether the signal x(t) = Ae^3t is a power signal or energy signal. Justify your answer.

19. Prove that δ (at) = 1/|a|δ(t).

20. Find the Fourier transform of the signals
i) x(t) sin(wot)
ii) x(t) = δ (t + to) + δ (t-to)

21. Find the trigonometric Fourier series for the waveform shown in figure.

22. Sketch the following signal, where r(t) is ramp signal.

r(t) - 2r(t - 2) + r(t - 4).

23. Find the Laplace transform of the signal

x(t) = e^-a|t| and find its ROC

24. Find the Fourier transform of e^-at u(t).

25. Give the effect of time scaling and time shifting on the signal x(t)= cos 3w₀t.

26. Determine whether the following system is stable or not.

h(t) = 5e^-2tu(t)

27. If x(t) = δ'(t+3) -3δ(t-3) + 4δ(t+2) then sketch g(t) = _-a∫^+a x(t)dt

28. If x(t)=cos(Π/3 t) + sin(Π/4t) is x(t) periodic, if periodic find the period of x(t).

29. Find the Laplace transform of x(t) = e^-atu(-t).

30. If x(t) =  { 1      |t| < a       obtain the Fourier transform of x(t)

                    { 0      otherwise

31. If X(ω) = jd/dw {e^j2ω / (1+ jω/3) } is the Fourier transform of a signal x(t), then find the signal x(t)

32. Consider a continuous time linear time invariant system for which the input x(t) and output y(t) are related by [d²y(t)/dt²] +dy(t)/dt - 2y(t) = x(t)

(a) Find the system function

(b) Determine the impulse response

33. (a) Determine the 90% energy containment of Bandwidth of the signal

x(t) = 1/ (t² + a²)

(b) Find the initial and final values of the signal x(t) whose Laplace transform

X(s) is 10(s+4) / (s+2)(s²-2s+2)

34. A signal x(t) is shown in figure below, sketch x(t)u(1-t).

35. A periodic signal x(t) is shown in figure below. Find FT X(w)

36. If x(t)= 1/( a² + t²) Find the Fourier transform of x(t).

37. If x(s) = (s² + 6s +7) / (s² + 3s +2) Re(s)≤1, is the Laplace transform of x(t), obtain the inverse Laplace transform.

38. Prove that cos( mω₀t) and sin( nω₀t) are orthogonal to each other. Where m and n are integers (‘m' and ‘n' are not equal to zero).

39. For the periodic function shown in fig., find the Fourier series coefficients.

40. Find the Fourier transform of the waveform shown in fig.

41. The transform voltage V(s) in a network is given be the equation.

V(s) = V(s) = 2(s-1)/((s-3)(s-4))

Plot the poles and zeros and hence obtain the time domain representation if v(t) is causal signal.