Question 1: Compute graphically the convolution, f(t) fit) f2(t), of the following two time-functions (t) and...
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
Question 1: (25 marks) Compute the convolution f(t) = fi(t) + f(t) of the following two time-functions fi(t) and fu(t). Provide a complete analytical solution (formula) and also a plot of your final result f(t). You are allowed to use, if you wish, any of the properties of convolution that we have studied, but you need to explain clearly how/where you have used them. fit) f2t) +3 +4 +5 -1 0 + 1 t -1 0 +1 +2
Compute Laplace transforms of the following functions: (a) f1 = (1 + t) (b) f2 = eat sin(bt) 11, 0<t<1, (c) f3 = -1 1<t<2, | 2, t>2, Find the functions from their Laplace transforms: (a) Lyı] s(s + 1) (s +3) 2+s (b) L[42] = 52 + 2 s +5 (c) L[y3] = Solve the following initial value problems using the Laplace transform. Confirm each solution with a Matlab plot showing the function on the interval 0 <t<5. (a)...
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...
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...
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
Formal Definitions of Big-Oh, Big-Theta and Big-Omega:
1. Use the formal definition of Big-Oh to prove that if f(n) is a decreasing function, then f(n) = 0(1). A decreasing function is one in which f(x1) f(r2) if and only if xi 5 r2. You may assume that f(n) is positive evervwhere Hint: drawing a picture might make the proof for this problem more obvious 2. Use the formal definition of Big-Oh to prove that if f(n) = 0(g(n)) and g(n)...
f. The amplitude of a cosine can be observed at the origin (t=0) when there is no phase shift. Find a simplified solution for the convolution integral below for t=0. +∞ output(t) = h(t)∗ s(t) = −∞ 3 rect(3x) cos(2π f0 (t − x)) dx Hint: Set t=0, sketch the situation to help set up the integral and remember the properties of odd and even functions to simply the calculation. g. The above result gives a general expression for the...
(a) Determine the Fourier transform of x(t) 26(t-1)-6(t-3) (b) Compute the convolution sum of the following signals, (6%) [696] (c) The Fourier transform of a continuous-time signal a(t) is given below. Determine the [696] total energy of (t) 4 sin w (d) Determine the DC value and the average power of the following periodic signal. (6%) 0.5 0.5 (e) Determine the Nyquist rate for the following signal. (6%) x(t) = [1-0.78 cos(50nt + π/4)]2. (f) Sketch the frequency spectrum of...
f. The amplitude of a cosine can be observed at the origin (t=0) when there is no phase shift. Find a simplified solution for the convolution integral below for t=0. +∞ output(t) = h(t)∗ s(t) = −∞ 3 rect(3x) cos(2π f0 (t − x)) dx Hint: Set t=0, sketch the situation to help set up the integral and remember the properties of odd and even functions to simply the calculation. g. The above result gives a general expression for the...