Homework Answers
1. (a) Use Euler's identity e¡θ-cos θ + i sin θ to prove that sin θ=-(eiO , 2i (b) Use the identi...
Time series analysis
1. (a) Use Euler's identity e¡θ-cos θ + i sin θ to prove that sin θ=-(eiO , 2i (b) Use the identities above and the formula for the sum of a geometric series to prove that if n is an integer and j E 1,2,... ,n} then TL TL sin-(2Ttj/n)- n/2 so long as J关[m/2, where Laj is the greatest integer that is smaller than or equal to x (c) Show that when j 0 we have...
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...
Problem 2. (30 points) The spin states: s 1,m) and Is -2, m1) composed of spin-3/2 and spin-1/2 states are linear combi...
Problem 2. (30 points) The spin states: s 1,m) and Is -2, m1) composed of spin-3/2 and spin-1/2 states are linear combinations of s1 3/2,m-3/2;2 1/2,m2 1/2) and 81-3/2, m-1/2; 2 1/2, m2--1/2), that is 11.-1)-cos θ3/2,-3/2; 1/2, 1/2) _ sin θ|3/2.-1/2; 1/2,-1/2), 2.-1) sin θ|3/2,-3/2; 1/2, 1/2) + cos θ|3/2.-1/2: 1/2,-1/2) a) Determine the values for cos θ and sin θ b) Express |3/2,-3/2; 1/2, 1/2) and |3/2,-1/2;1/2,-1/2) as functions of |1, -1) and 2,-1) c) A system of...
The mutually coupled inductances in (Figure 1) have L1-1H. L2 2 H, and M-1H. Furthermore h(t)...
The mutually coupled inductances in (Figure 1) have L1-1H. L2 2 H, and M-1H. Furthermore h(t) sin(10t) A and 22 (t) = 0.5 sin( 10t) A. Part A Find an expression for vi(t). The arguments of the sine functions are in radians Express your answer in terms of t vec ひ Figure 1 of 1 Submit uest Ans Part B ii(t) /2(0) Find an expression for v2 (t). The arguments of the sine functions are in radians Express your answer...
solve part c Hìnt: The following identities will be useful: LWI) PVI sin e sin 2ot...
solve part c
Hìnt: The following identities will be useful: LWI) PVI sin e sin 2ot (1-59) cos a cos B = [cos (a - B) + cos (a + B)] cos (a - B) - cos a cos B + sin a sin B 1-21. A linear machine shown in Figure P1-15 has a magnetic flux density of 0.5 T di- rected into the page, a resistance of 0.25 N, a bar length I=- 1.0 m, and a battery...
5 pts D Question 1 A system has the following impulse response: .2 Sample number, n From the choices below, select the frequency response of this system. H (eju)-e(1.5 ) (2 sin( 1.5ώ) +...
5 pts D Question 1 A system has the following impulse response: .2 Sample number, n From the choices below, select the frequency response of this system. H (eju)-e(1.5 ) (2 sin( 1.5ώ) + 4 sin(0.δώ)) H (ee) = e-j(1.5e-5) (cos( 1.5 ) +2 cos(0.54)) @ H (ee)-e-n1.si) (sin( 1.54) t. 2 sin(0.δώ)) (sin(l.50) +4sin(0.0) H (ee)-e-j(1.5i) (2 cos( 1.5ώ) + 4 cos(0.5a)) H (efo)-e-n1.5u) (cos( 1.50) + 2 cos(0.50)) https://rmitinstructure.comcoursesy 5 pts DQuestion 2 A system has the following...
2) Upon starting a new position in a company, you have been assigned to work on...
2) Upon starting a new position in a company, you have been assigned to work on a project that requires you to analyze an electrical network. The project kickoff meeting is in 2 weeks and you are not sure how to complete the network analysis. While looking through the department file, you found incomplete archives that show the results for a similar network but it does not show all the components of the network. a) What would you do to...
(b) (2 pts) (t) is given as r(t) e sin(t) Find X(jw). Show that X(jw) =...
(b) (2 pts) (t) is given as r(t) e sin(t) Find X(jw). Show that X(jw) = 25 + (w- 1)225(w+1)2 (c) (4 pts) x(t) is given as x(t)-π inc(t) cos(nt). Find X(jw) (d) (4 pts) 2(t) is given as 2(t) e Áil+ 3) + e' ỗ(t-3). Find X (jw). Simplify the answer as (e) (4 pts) 2(t) is given as r(t) = rect(2(t )) reetgehj)). Hint: use Fourier Transform pair: sine(t)艹rect( ) much as possible Find X(jw). Simplify the answer...
Find the following trigonometric limit: lim sin - Hint the substitution [((u-1)E)s01 u= t-n makes life...
Find the following trigonometric limit: lim sin - Hint the substitution [((u-1)E)s01 u= t-n makes life 1. tm easier. Work inside the [...] first and then take the sine of your result, that is, use the rule that allows you to take the limit inside the sine function: lim sin(f(x)) n(Hm/s) sin x-a 2. Use the results we derived in class for power functions to find the derivative of g(x) (3x4 + v 4 atx Ans 3. When a function...
For each of the following functions indicate the matching Taylor Series centered at r=0. 1) sin(2)...
For each of the following functions indicate the matching Taylor Series centered at r=0. 1) sin(2) 2) cos(2) 3) 4) e 5) 1.2 6) D 7) 12:22 8) - In(1 - 1) 9) e--- 10) S* cos(t)dt Taylor Series Choices: a) § 3 b) (-1)=-17 c) Š(-1)" N=0 no NEO d) nr-1 e) Σα" f) 2.2 no n=0 g) 2nx2n-2 h) (-1)" (an+1)+(2n) 4+1 i) (-1)n-1 nel n=0 n=0 j) (-1)" (2n+1)! 2+1 k) § 21 k) 2ne2n-1 1) (-1)"?"...
Time series analysis
1. (a) Use Euler's identity e¡θ-cos θ + i sin θ to prove that sin θ=-(eiO , 2i (b) Use the identities above and the formula for the sum of a geometric series to prove that if n is an integer and j E 1,2,... ,n} then TL TL sin-(2Ttj/n)- n/2 so long as J关[m/2, where Laj is the greatest integer that is smaller than or equal to x (c) Show that when j 0 we have...
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...
Problem 2. (30 points) The spin states: s 1,m) and Is -2, m1) composed of spin-3/2 and spin-1/2 states are linear combinations of s1 3/2,m-3/2;2 1/2,m2 1/2) and 81-3/2, m-1/2; 2 1/2, m2--1/2), that is 11.-1)-cos θ3/2,-3/2; 1/2, 1/2) _ sin θ|3/2.-1/2; 1/2,-1/2), 2.-1) sin θ|3/2,-3/2; 1/2, 1/2) + cos θ|3/2.-1/2: 1/2,-1/2) a) Determine the values for cos θ and sin θ b) Express |3/2,-3/2; 1/2, 1/2) and |3/2,-1/2;1/2,-1/2) as functions of |1, -1) and 2,-1) c) A system of...
The mutually coupled inductances in (Figure 1) have L1-1H. L2 2 H, and M-1H. Furthermore h(t) sin(10t) A and 22 (t) = 0.5 sin( 10t) A. Part A Find an expression for vi(t). The arguments of the sine functions are in radians Express your answer in terms of t vec ひ Figure 1 of 1 Submit uest Ans Part B ii(t) /2(0) Find an expression for v2 (t). The arguments of the sine functions are in radians Express your answer...
solve part c
Hìnt: The following identities will be useful: LWI) PVI sin e sin 2ot (1-59) cos a cos B = [cos (a - B) + cos (a + B)] cos (a - B) - cos a cos B + sin a sin B 1-21. A linear machine shown in Figure P1-15 has a magnetic flux density of 0.5 T di- rected into the page, a resistance of 0.25 N, a bar length I=- 1.0 m, and a battery...
5 pts D Question 1 A system has the following impulse response: .2 Sample number, n From the choices below, select the frequency response of this system. H (eju)-e(1.5 ) (2 sin( 1.5ώ) + 4 sin(0.δώ)) H (ee) = e-j(1.5e-5) (cos( 1.5 ) +2 cos(0.54)) @ H (ee)-e-n1.si) (sin( 1.54) t. 2 sin(0.δώ)) (sin(l.50) +4sin(0.0) H (ee)-e-j(1.5i) (2 cos( 1.5ώ) + 4 cos(0.5a)) H (efo)-e-n1.5u) (cos( 1.50) + 2 cos(0.50)) https://rmitinstructure.comcoursesy 5 pts DQuestion 2 A system has the following...
2) Upon starting a new position in a company, you have been assigned to work on a project that requires you to analyze an electrical network. The project kickoff meeting is in 2 weeks and you are not sure how to complete the network analysis. While looking through the department file, you found incomplete archives that show the results for a similar network but it does not show all the components of the network. a) What would you do to...
(b) (2 pts) (t) is given as r(t) e sin(t) Find X(jw). Show that X(jw) = 25 + (w- 1)225(w+1)2 (c) (4 pts) x(t) is given as x(t)-π inc(t) cos(nt). Find X(jw) (d) (4 pts) 2(t) is given as 2(t) e Áil+ 3) + e' ỗ(t-3). Find X (jw). Simplify the answer as (e) (4 pts) 2(t) is given as r(t) = rect(2(t )) reetgehj)). Hint: use Fourier Transform pair: sine(t)艹rect( ) much as possible Find X(jw). Simplify the answer...
Find the following trigonometric limit: lim sin - Hint the substitution [((u-1)E)s01 u= t-n makes life 1. tm easier. Work inside the [...] first and then take the sine of your result, that is, use the rule that allows you to take the limit inside the sine function: lim sin(f(x)) n(Hm/s) sin x-a 2. Use the results we derived in class for power functions to find the derivative of g(x) (3x4 + v 4 atx Ans 3. When a function...
For each of the following functions indicate the matching Taylor Series centered at r=0. 1) sin(2) 2) cos(2) 3) 4) e 5) 1.2 6) D 7) 12:22 8) - In(1 - 1) 9) e--- 10) S* cos(t)dt Taylor Series Choices: a) § 3 b) (-1)=-17 c) Š(-1)" N=0 no NEO d) nr-1 e) Σα" f) 2.2 no n=0 g) 2nx2n-2 h) (-1)" (an+1)+(2n) 4+1 i) (-1)n-1 nel n=0 n=0 j) (-1)" (2n+1)! 2+1 k) § 21 k) 2ne2n-1 1) (-1)"?"...
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