Derive the following results, where H(x) is the Heaviside unit step function.
![[sin x H(x)]*2 =-(sin x-x cos x)H(x) e)](http://img.homeworklib.com/questions/86401bf0-a7ce-11eb-90dd-4f4460b542b1.png?x-oss-process=image/resize,w_560)
Do the 1st, 2nd, and 4th only. In each case, sketch the input functions and their CONVOLUTION.
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Derive the following results, where H(x) is the Heaviside unit
step function.
Do the 1st, 2nd,...
The Heaviside step function is a mathematical function defined by h(x) := ( 1 : x ≥ 0 0 : x < ...
The Heaviside step function is a mathematical function defined by h(x) := ( 1 : x ≥ 0 0 : x < 0 that is to say, h(x) is 1 for non-negative x and 0 for negative x. In this question we will prove that the following code computes h(x) if (x >= 0): y = 1 else: y = 0 a) Define a predicate p(x, y), written solely in terms of logical ANDs and ORs, inequalities and equalities, which...
e) Let n] denote the discrete-time unit-step function. Derive the discrete-time Fourier transform of the following...
e) Let n] denote the discrete-time unit-step function. Derive the discrete-time Fourier transform of the following signals ytn) - 5 0.25" cos 2xm 5 marks]
Derive the following equation and explain each step in words and sentences: y'(x,t)=[2ymcos1/2phi...
Derive the following equation and explain each step in words and
sentences:
y'(x,t)=[2ymcos1/2phi]sin(kx-wt+1/2phi)
Derive the following equation and explain each step in words and
sentences:
y'(x,t)=[2ymsinkx] cos(wt)
Derive the following equation and explain each step in words and sentences: Derive the following equation and explain each step in words and sentences:
Derive the following equation and explain each step in words and sentences: Derive the following equation and explain each step in words and sentences:
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...
Question 5 Using the graphical method (i.e., the method used during the lectures), compute x* h(t),...
Question 5 Using the graphical method (i.e., the method used during the lectures), compute x* h(t), where x(t) = e-and h is as shown in the figure. (You must compute x* h, not hx.) For each separate case in your solution, you must state the convolution result and the corresponding range of t as well as show the fully-labelled graph from which this result is derived. Each convolution result may be stated in the form of an integral, but the...
9. (a) Find the inverse Fourier transform of the following function 1 (2 iw)(5 iw) (b)...
9. (a) Find the inverse Fourier transform of the following function 1 (2 iw)(5 iw) (b) The displacement of a particular mechanical system is governed by the following ordinary differential equation dy 10y f(t) 7 dt where y(t) is the displacement and f(t) is the applied load Page 2 of 4 MATH2124 SaMplE EXAM IV i Use the Fourier transform to obtain the impulse response h(t) of the mechanical system (ii) If the applied load is f(t) = H (t1)-...
1. Evaluate and sketch the convolution integral (the output y(t)) for a system with input x(t)...
1. Evaluate and sketch the convolution integral (the output y(t)) for a system with input x(t) and impulse response h(t), where x(t) = u(1-2) and h(t)= "u(t) u(t) is the unit step function. Please show clearly all the necessary steps of convolution. Determine the values of the output y(t) at 1 = 0,1 = 3,1 = 00. (3 pts)
Can someone explain each part of this solution I don’t understand Example 1 square wave Derive...
Can someone explain each part of this solution I don’t
understand
Example 1 square wave Derive the Fourier series (FS) representation of a square wave of period T with duty cycle τ-AT, where 0< B<1. The square wave is symmetrically defined over one period by a Heaviside unit-step function, as in Eq. (28) It! <汁 (77) The ordinary unit-step could also be used, but the Heaviside is more natural here because the FS representation will pass through the 1/2 point...
Please do b and d. The result for 26.5 a is below 26.6. Using the results...
Please do b and d. The result for 26.5 a is below
26.6. Using the results from exercise 26.5 a, find the solution to y" + 4y fo with y(0) = 0 and y'(0) = 0 for each of the following choices of /: a. fo = 1 b. - 1 d. f(1) = sin(21) e. f(1) = sin(a) where a #2 c. / (0) = 1 26.5 a. C[SOL {[y" + 4y]. [Y'). + 4L[y]1, → = F(s) [s?Y(s)...
1. Auto- and Cross-Correlation. For each of the following, compute the cross correlation T/2 Rry(,) = E[drpd, + n-linx...
1. Auto- and Cross-Correlation. For each of the following, compute the cross correlation T/2 Rry(,) = E[drpd, + n-linx t-Tax(ry(, + rdr . Hint: Use trigonometric identities (see HW 1), 27T such as sin a sin b-2 [cos(a-b)-cos(a + b)] . Also use the fact that j cos(ont-б unless co-0 x(t) = sin(2n/r), y(t)-sin(2nft) (here x and y are the same, so Rry-Rrr is the a. autocorrelation of x). x(t) = sin(2nft), y(t) = sin(2nf(t-to)) c. x(t)-n(), y()2x(t) +n2(t) where...
e) Let n] denote the discrete-time unit-step function. Derive the discrete-time Fourier transform of the following signals ytn) - 5 0.25" cos 2xm 5 marks]
Derive the following equation and explain each step in words and
sentences:
y'(x,t)=[2ymcos1/2phi]sin(kx-wt+1/2phi)
Derive the following equation and explain each step in words and
sentences:
y'(x,t)=[2ymsinkx] cos(wt)
Derive the following equation and explain each step in words and sentences: Derive the following equation and explain each step in words and sentences:
Derive the following equation and explain each step in words and sentences: Derive the following equation and explain each step in words and sentences:
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...
Question 5 Using the graphical method (i.e., the method used during the lectures), compute x* h(t), where x(t) = e-and h is as shown in the figure. (You must compute x* h, not hx.) For each separate case in your solution, you must state the convolution result and the corresponding range of t as well as show the fully-labelled graph from which this result is derived. Each convolution result may be stated in the form of an integral, but the...
9. (a) Find the inverse Fourier transform of the following function 1 (2 iw)(5 iw) (b) The displacement of a particular mechanical system is governed by the following ordinary differential equation dy 10y f(t) 7 dt where y(t) is the displacement and f(t) is the applied load Page 2 of 4 MATH2124 SaMplE EXAM IV i Use the Fourier transform to obtain the impulse response h(t) of the mechanical system (ii) If the applied load is f(t) = H (t1)-...
1. Evaluate and sketch the convolution integral (the output y(t)) for a system with input x(t) and impulse response h(t), where x(t) = u(1-2) and h(t)= "u(t) u(t) is the unit step function. Please show clearly all the necessary steps of convolution. Determine the values of the output y(t) at 1 = 0,1 = 3,1 = 00. (3 pts)
Can someone explain each part of this solution I don’t
understand
Example 1 square wave Derive the Fourier series (FS) representation of a square wave of period T with duty cycle τ-AT, where 0< B<1. The square wave is symmetrically defined over one period by a Heaviside unit-step function, as in Eq. (28) It! <汁 (77) The ordinary unit-step could also be used, but the Heaviside is more natural here because the FS representation will pass through the 1/2 point...
Please do b and d. The result for 26.5 a is below
26.6. Using the results from exercise 26.5 a, find the solution to y" + 4y fo with y(0) = 0 and y'(0) = 0 for each of the following choices of /: a. fo = 1 b. - 1 d. f(1) = sin(21) e. f(1) = sin(a) where a #2 c. / (0) = 1 26.5 a. C[SOL {[y" + 4y]. [Y'). + 4L[y]1, → = F(s) [s?Y(s)...
1. Auto- and Cross-Correlation. For each of the following, compute the cross correlation T/2 Rry(,) = E[drpd, + n-linx t-Tax(ry(, + rdr . Hint: Use trigonometric identities (see HW 1), 27T such as sin a sin b-2 [cos(a-b)-cos(a + b)] . Also use the fact that j cos(ont-б unless co-0 x(t) = sin(2n/r), y(t)-sin(2nft) (here x and y are the same, so Rry-Rrr is the a. autocorrelation of x). x(t) = sin(2nft), y(t) = sin(2nf(t-to)) c. x(t)-n(), y()2x(t) +n2(t) where...
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