Consider an analog pulse 1 0 ≤ t ≤ 1 x(t) = { 0 Otℎerwise Find mathematical expression for x(t)delayed by 2, advanced by 2, and the reflected signal x(−t).
[3 marks]Determine whether or not the following signals is periodic. If a signal is periodic, determine its fundamental period. i. x(t) = cost+sin√2t 𝜋 ii. x[n] = e j( 4 )n
[4 marks]Evaluate y[n] = x[n]∗ℎ[n], by graphical method. where x[n] and ℎ[n] are shown figure below.
[7 marks]Determine the energy and power of a unit step signal.
[3 marks]Consider a discrete-time LTI system with impulse response ℎ[n] given by ℎ[n] = 𝛼nu[n] i. Is this system causal? ii. Is this system BIBO stable?
[4 marks]Determine natural response of the first order system governed by the equation, dy(t) +3y(t) = x(t);y(0) = dt OR1
[2 marks]Find the overall impulse response of the system shown in figure below. Take, ℎ (t) = tu(t); ℎ (t) = 3u(t); ℎ (t) = 2u(t);123
[7 marks]Find the Laplace transform of x(t) = sin2t.
[3 marks]Determine the complex exponential Fourier series representation for the 𝜋 signals x(t) = cos(2t + ).4
[4 marks]Determine the trigonometric Fourier series of periodic impulse train ∞ 𝛿 (t) = ∑ 𝛿(t−kT ) T0 0 k=−∞
[7 marks]State and prove the frequency differentiation property of Fourier transform.
[3 marks]Find the Fourier transform of x(n) = { 2, 1, 2 }.
[4 marks]Determine the frequency response of the LTI system defined by, y(n) = x(n)+by(n−1)
[7 marks]Determine the z-transform of x(n) = (n−3)u(n)
[3 marks]State and prove shifting property for one sided z-transform.
[4 marks]Determine the inverse z-transform of1 X(z) = for ROC, |z| > 0.6. 1−0.8z−1+0.12z−2
[7 marks]Find the even part of signal x(n) = u(n)+u(−n).
[3 marks]Determine the inverse z-transform of X(z) = log(1+az−1) ; |z| > |a|.
[4 marks]Determine the impulse response h(n) for the system described by the second order difference equation, y(n)−4y(n−1)+4y(n−2) = x(n−1)
[7 marks]Test the following systems for linearity. dx(t) y(t) = 4x(t)+2 . dt
[3 marks]State and prove the time scaling property of Laplace transform.
[4 marks]Asystem has impulse response h(n) given by,2 ℎ(n) = −0.25𝛿(n+1)+0.5𝛿(n)−0.25𝛿(n−1). i. Is the system BIBO stable? ii. Is the system causal? Justify your answer.
[7 marks]i. Define Fourier transform. ii. State the condition for existence of Fourier integral.
[3 marks]Calculate the DFT of the sequence, x(n) = {1,1,−2,−2}
[4 marks]Define ROC for z-transform. List the property of ROC.
[7 marks]