Homework Answers
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Check the existence of the Laplace transform for the given function and hence show that -...
3 B 1. Find the third roots of 21+ Find the inverse of the Laplace transform...
3 B 1. Find the third roots of 21+ Find the inverse of the Laplace transform 2. tan" G) 3. Check the existence of the Laplace transform for the given function and hence she that -02:49 in 133+ 4 S- where LF(t)) is represent the place transform of (1) [Hint: 2 cos Acos B = (A-2).sin(A+B) + sin(A - m = sin cos sin(A + B) - Sin(A) = 0 4. Find the Fourier Sine series of f(x) <rci 5....
Q1 Write the following function in terms of unit step functions. Hence, find its Laplace transform...
Q1 Write the following function in terms of unit step functions. Hence, find its Laplace transform 10<tsI g(t) = le-3, +1 , 1<t 2 .22 Q2 Use Laplace transform to solve the following initial value problem: yty(o)-0 and y (0)-2 A function f(x) is periodic of period 2π and is defined by Q3 Sketch the graph of f(x) from x-2t to2 and prove that 2sinh π11 f(x)- Q4 Consider the function f(x)=2x, 0<x<1 Find the a Fourier cosine series b)...
Problem 1. Find the Fourier series expansion of a half-wave rectified sine wave depicted below. AS(0)...
Problem 1. Find the Fourier series expansion of a half-wave rectified sine wave depicted below. AS(0) Answer: f(t) = 1+sin at cos2nt 1 nr 15 2 Cos 4t -cost + ... 35 Problem 2. Find the Fourier series approximation of the following periodic function f(x), where the first two leading cosine and sine functions must be included. Angle sum formulas for sine / cosine functions f(x) sin(A + B) = sin A cos B + cos A sin B sin(A...
Question: Equations: My attempt (sorry for uneatness) : Tutorial questions - Sine and Cosine trans...
Question:
Equations:
My attempt (sorry for uneatness) :
Tutorial questions - Sine and Cosine transforms 9. U se the Fourier Inversion Theorem to prove for a real-valued odd function f(t) that F.(w) sin wt du at points of continuity. (Hint: first simplify the integral expression for F(w).) Fourier Inversion Theorem. At points where f0) is continuous, is that t at t=( iwt extra te Fo F(w)-マ27: J-0,f(t)e-iwt dt = F(f(t)) w) is called the Fourier cosine transform of f(t). Similarly,...
0and / is an odd function of t, find the Fourier sine sin wt d for 0<t< 1 10, (a) If f(t) = for t a 0 transfor...
0and / is an odd function of t, find the Fourier sine sin wt d for 0<t< 1 10, (a) If f(t) = for t a 0 transform of f. Deduce thato s if0<t < a. What is the value of the integral for t2 a? for 0 < t < b (b) If g(t)-{ b-t and g is an even function of t, find the Fourier 0 cosine transform of g. Deduce that foo 1-w2bw cosa t dw =...
Verify the following using MATLAB 2) (a) Consider the following function f(t)=e"" sinwt u (t (1)...
Verify the following using MATLAB
2) (a) Consider the following function f(t)=e"" sinwt u (t (1) .... Write the formula for Laplace transform. L[f)]=F(6) F(6))e"d Where f(t is the function in time domain. F(s) is the function in frequency domain Apply Laplace transform to equation 1. Le sin cot u()]F(s) Consider, f() sin wtu(t). From the frequency shifting theorem, L(e"f()F(s+a) (2) Apply Laplace transform to f(t). F,(s)sin ot u (t)e" "dt Define the step function, u(t u(t)= 1 fort >0...
Problem 3. The Fourier transform pairs of cosine and sine functions can be written as y(t)...
Problem 3. The Fourier transform pairs of cosine and sine functions can be written as y(t) = A cos 2nfot = Y(f) = 4 [86f - fo) +8(f + fo)], and y(t) = B sin 2nfot = Y(f) =-j} [8(f - fo) – 8(f + fo]. The FFT code is revised such that the resulting amplitudes in frequency domain should coincide with those in time domain after discarding the negative frequency portion of Fourier transform or the frequency domain after...
12 points) Note: The formulas for the Fourier transform on half intervals are often given in...
12 points) Note: The formulas for the Fourier transform on half intervals are often given in the form { ). f(x)cos L dx, with the Ź outside the integral. Computing these integrals will often involve u-substitutions, integration by parts, and other integration techniques that will produce all kinds of constants. With the way these problems are asked in WebWork, it would be hard to keep track of when constants should be factored out or not, so we will adopt the...
Question # 1 a) State the definition of the Laplace transform of a function f(x) b)...
Question # 1 a) State the definition of the Laplace transform of a function f(x) b) Hence determine the following transforms using the definition of Laplace transform ili. f(t) 3 ii. f(t) -sin 2t Determine the solutions of the following initial value problems using Laplace transforms c)
Expand the given function in an appropriate cosine or sine series f(x) = 1x1, -π <x<π...
Expand the given function in an appropriate cosine or sine series f(x) = 1x1, -π <x<π F(x) = sin nx cos nx + n=1L Suhmit Answor Savo Drogroso
3 B 1. Find the third roots of 21+ Find the inverse of the Laplace transform 2. tan" G) 3. Check the existence of the Laplace transform for the given function and hence she that -02:49 in 133+ 4 S- where LF(t)) is represent the place transform of (1) [Hint: 2 cos Acos B = (A-2).sin(A+B) + sin(A - m = sin cos sin(A + B) - Sin(A) = 0 4. Find the Fourier Sine series of f(x) <rci 5....
Q1 Write the following function in terms of unit step functions. Hence, find its Laplace transform 10<tsI g(t) = le-3, +1 , 1<t 2 .22 Q2 Use Laplace transform to solve the following initial value problem: yty(o)-0 and y (0)-2 A function f(x) is periodic of period 2π and is defined by Q3 Sketch the graph of f(x) from x-2t to2 and prove that 2sinh π11 f(x)- Q4 Consider the function f(x)=2x, 0<x<1 Find the a Fourier cosine series b)...
Problem 1. Find the Fourier series expansion of a half-wave rectified sine wave depicted below. AS(0) Answer: f(t) = 1+sin at cos2nt 1 nr 15 2 Cos 4t -cost + ... 35 Problem 2. Find the Fourier series approximation of the following periodic function f(x), where the first two leading cosine and sine functions must be included. Angle sum formulas for sine / cosine functions f(x) sin(A + B) = sin A cos B + cos A sin B sin(A...
Question:
Equations:
My attempt (sorry for uneatness) :
Tutorial questions - Sine and Cosine transforms 9. U se the Fourier Inversion Theorem to prove for a real-valued odd function f(t) that F.(w) sin wt du at points of continuity. (Hint: first simplify the integral expression for F(w).) Fourier Inversion Theorem. At points where f0) is continuous, is that t at t=( iwt extra te Fo F(w)-マ27: J-0,f(t)e-iwt dt = F(f(t)) w) is called the Fourier cosine transform of f(t). Similarly,...
0and / is an odd function of t, find the Fourier sine sin wt d for 0<t< 1 10, (a) If f(t) = for t a 0 transform of f. Deduce thato s if0<t < a. What is the value of the integral for t2 a? for 0 < t < b (b) If g(t)-{ b-t and g is an even function of t, find the Fourier 0 cosine transform of g. Deduce that foo 1-w2bw cosa t dw =...
Verify the following using MATLAB
2) (a) Consider the following function f(t)=e"" sinwt u (t (1) .... Write the formula for Laplace transform. L[f)]=F(6) F(6))e"d Where f(t is the function in time domain. F(s) is the function in frequency domain Apply Laplace transform to equation 1. Le sin cot u()]F(s) Consider, f() sin wtu(t). From the frequency shifting theorem, L(e"f()F(s+a) (2) Apply Laplace transform to f(t). F,(s)sin ot u (t)e" "dt Define the step function, u(t u(t)= 1 fort >0...
Problem 3. The Fourier transform pairs of cosine and sine functions can be written as y(t) = A cos 2nfot = Y(f) = 4 [86f - fo) +8(f + fo)], and y(t) = B sin 2nfot = Y(f) =-j} [8(f - fo) – 8(f + fo]. The FFT code is revised such that the resulting amplitudes in frequency domain should coincide with those in time domain after discarding the negative frequency portion of Fourier transform or the frequency domain after...
12 points) Note: The formulas for the Fourier transform on half intervals are often given in the form { ). f(x)cos L dx, with the Ź outside the integral. Computing these integrals will often involve u-substitutions, integration by parts, and other integration techniques that will produce all kinds of constants. With the way these problems are asked in WebWork, it would be hard to keep track of when constants should be factored out or not, so we will adopt the...
Question # 1 a) State the definition of the Laplace transform of a function f(x) b) Hence determine the following transforms using the definition of Laplace transform ili. f(t) 3 ii. f(t) -sin 2t Determine the solutions of the following initial value problems using Laplace transforms c)
Expand the given function in an appropriate cosine or sine series f(x) = 1x1, -π <x<π F(x) = sin nx cos nx + n=1L Suhmit Answor Savo Drogroso
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