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Compute graphically the convolution, f (t)-fi(t f2(t), of the following two time-functions (t) and f3(t). Sketch...
Question 1: Compute graphically the convolution, f(t) fit) f2(t), of the following two time-functions (t) and...
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
Question 1: (25 marks) Compute the convolution f(t) = fi(t) + f(t) of the following two...
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 =...
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 –...
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...
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...
f. The amplitude of a cosine can be observed at the origin (t=0) when there is...
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 bel...
(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 solutio...
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...
3. Let z,-34J4 and z2-10e'. Find the results of: zit 22,21 * 22,21/22, ะ?, and å . Find the Laplace transforms of the following functions: note that t2 0 1). f(t) =e®.4tcos 12t. f) sin(t 3),...
3. Let z,-34J4 and z2-10e'. Find the results of: zit 22,21 * 22,21/22, ะ?, and å . Find the Laplace transforms of the following functions: note that t2 0 1). f(t) =e®.4tcos 12t. f) sin(t 3), f(t) =cos2wtcos3wt. Hint: You may want to use the following identities: cos(o + β)-cosoosß-sino sind, sin θ = , 2 2j
3. Let z,-34J4 and z2-10e'. Find the results of: zit 22,21 * 22,21/22, ะ?, and å . Find the Laplace transforms of the...
C++. Need some help getting started. We will also have the following two functions: 1. A...
C++. Need some help getting started. We will also have the following two functions: 1. A mutate function that randomly modifies a chromosome. 2. A crossover function that takes two chromosomes and splits each one at the same spot, then combines them together. Our genetic algorithm works by iterating over generations of chromosomes via the following process: 1. Generate random population. 2. Until we get an answer that is good enough, do the next steps in a loop: (a) Do...
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
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...
(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...
3. Let z,-34J4 and z2-10e'. Find the results of: zit 22,21 * 22,21/22, ะ?, and å . Find the Laplace transforms of the following functions: note that t2 0 1). f(t) =e®.4tcos 12t. f) sin(t 3), f(t) =cos2wtcos3wt. Hint: You may want to use the following identities: cos(o + β)-cosoosß-sino sind, sin θ = , 2 2j
3. Let z,-34J4 and z2-10e'. Find the results of: zit 22,21 * 22,21/22, ะ?, and å . Find the Laplace transforms of the...
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