Question 1: (25 marks) Compute the convolution f(t) = fi(t) + f(t) of the following two...
Question 1: Compute graphically the convolution, f(t) fit) f2(t), of the following two time-functions (t) and f2(t). Sketch your final result f(t)· (Hint: To avoid having to do twice as many calculations, you may want to use these properties of convolution: the distributive and the shift properties.) fi(t) f2(t) +3 +4 +5 -1 0 +1 t -1 0
Compute graphically the convolution, f (t)-fi(t f2(t), of the following two time-functions (t) and f3(t). Sketch your final result f(t)· (Hint: To avoid having to do twice as many calculations, you may want to use these properties of convolution: the distributive and the shift properties.) fi(t) f2(t) -1 0 +1t -1 -1
..3-1 In Fig. P1.3-1, the signal fi(t) = f(-). Express signals fa(t), f(t), fu(t), and f(t) in terms of signals f(t), fi(t), and their time-shifted, time-scaled or time-inverted versions. For instance f2(t) = f(t-T) + fit - T). f (1) f (1) 1,(0) f(t) 0 - 0 1 Fig. P1.3-1
Please show full solution and explanation Consider the following two functions h (t) and f (t). and (a) Plot h(t) and f(t). (b)Use the convolution integral to calculate the convolution g (t) of the function h (t) with f (t) and plot. So if t > 0 h(t) = 1 et if t > 0 Ji if 0 <t<T f(t) = 10 if otherwise
1. Consider density functions f and g. The convolution formula we have in the class is (c) Suppose t(f * g) (t)dt 10 and J tf (t)dt 4. What is Jtg(t)dt?
Question 2 (3 marks) Function f(t) is described as a sudden change of 1 SI unit at t-0 followed by a sudden change of -1 SI unit after 0.2 seconds. Assume f(t)-0 for t<0. (a) Write the analytical form of f(t). Sketch its graph indicating clearly all relevant values. (b) Find the Laplace transform of f(t) (c) Derive the expression for F, (s) if f(t) f(t)dt (d) Please see over Question 2 (3 marks) Function f(t) is described as a...
Question 1: (2 marks) Find the zero-input response yz(t) for a linear time-invariant (LTI) system described by the following differential equation: j(t) + 5y(t) + 6y(t) = f(t) + 2x(t) with the initial conditions yz (0) = 0 and jz (0) = 10. Question 2: (4 marks) The impulse response of an LTI system is given by: h(t) = 3e?'u(t) Find the zero-state response yzs (t) of the system for each the following input signals using convolution with direct integration....
1. Consider the two waveforms f(t) and g(t) shown in the figures below. (a) Characterize both functions by expressing each in a suitable mathematical functional form. Write the resultant equation next to the equal sign for each function (b) Using direct integration, compute the convolution integral using the functions you defined in part (a). (c) Sketch the resultant function or use a plotting package of your choice to plot your result for h(t). What do you observe about the relative...
Vv = 8 V Ii = 5 A fv = 19 Hz fi = 11 Hz theta_v = 120 degrees theta_i = 25 degrees t_0 = -500 mseconds w_0 = 6 Joules p_w from vi Given 1. The voltage and current are given by v(t) = V, cos (27fyt + O,) (Volts) and i(t) = I; cos (2Tf;t + 0;) (Amps) for t to seconds, respectively. Note that neither v(t) or i(t) are given for t <to, and that to...
Problem 4: Evaluation of the convolution integral too y(t) = (f * h)(t) = f(t)h(t – 7)dt is greatly simplified when either the input f(t) or impulse response h(t) is the sum of weighted impulse functions. This fact will be used later in the semester when we study the operation of communication systems using Fourier analysis methods. a) Use the convolution integral to prove that f(t) *8(t – T) = f(t – T) and 8(t – T) *h(t) = h(t...