Homework Answers
The MATLAB code for second question and the corresponding plots are shown below:
w=(2*pi)/6;
t=0:.1:24;
x=(8/(5*pi))*cos(10*(pi/6)*t)+(8/(3*pi))*cos(6*(pi/6)*t)+(8/(1*pi))*cos(2*(pi/6)*t)+2;
plot(t,x);
Thus option C is the correct answer
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10 points) This problem is related to Problem 9.31 in the text (a) A signal, s(t). with period T ...
(10 points) This problem is related to Problem 9.31 in the text (a) A signal, s(t), with period T 5, is approximated by using the first few terms in the frequency domain by the following non-zero (co...
(10 points) This problem is related to Problem 9.31 in the text (a) A signal, s(t), with period T 5, is approximated by using the first few terms in the frequency domain by the following non-zero (complex) Fourier coefficients (all others are zero): S(0)--, 2 6 S(1)-S(-1)-π, S(3)-S(- approximation s (t), where 5 C(n)cos(nuot + %) (See Section 9.1.2 of the text.) Write the answer as the sum of cosines with phase. There should be no complex numbers in your...
(10 points) This problem is related to Problem 9.15 in the text. Consider the periodic function...
(10 points) This problem is related to Problem 9.15 in the text.
Consider the periodic function with period 16 given by
(10 points) This problem is related to Problem 9.15 in the text Consider the periodic function with period 16 given by if 8 t<-2 if 2 t4 if 4 t<7 if 7 t8 0 -6 f(t) = 6 0 Find the first 3 coefficients of the exponential Fourier Series for f(t), i.e., F(n) for n = -2, -1,0, 1,...
Problem 32: (20 points) Consider a periodic signal f(t), with fundamental period To, that has the...
Problem 32: (20 points) Consider a periodic signal f(t), with fundamental period To, that has the exponential Fourier series representation f(t) = Σ Dnejuont . where wo 2T/To and 1. (2 points) When f(t) is a real-valued, show that DD This is known as the complex conjugate symmetry property or the Hermitian property of real signals. 2. (2 points) Show that when f(t) is an even function of time that Dn is an even function of n 3. (2 points)...
Previous Problem Problem List Next Problem (10 points) This problem is related to Problems 9.33-9.38 in the text. We have solved differential equations using the method of undetermined coefficients (...
Previous Problem Problem List Next Problem (10 points) This problem is related to Problems 9.33-9.38 in the text. We have solved differential equations using the method of undetermined coefficients (Chapter 7) and Laplace transforms (Chapter 8). We can use Fourier series to find the particular solution of an arbitrary order differential equation - as long as the driving function is periodic and can be represented by a Fourier series In the problem description and answers, all numerical angles(phases) should be...
Problem 1 The complex exponential Fourier Series of a signal over an interval 0 < t S T,-2π/wo is...
Problem 1 The complex exponential Fourier Series of a signal over an interval 0 < t S T,-2π/wo is known to be (d) Suppose x(t) is the input to a stable, continuous-time, single-input/single-output LTI system whose impulse response is given by 9sine (wot/4 2 cos (u) Determine the output y(t) for -oo<t<oo. Answer: y(t)-4m 2r(1 +9π (2r(1+9r2) tan 1(3m) cos 9T
Problem 1 The complex exponential Fourier Series of a signal over an interval 0
(10 points) This problem is related to Problem 8.3 and 8.4 in the text. Consider the...
(10 points) This problem is related to Problem 8.3 and 8.4 in the text. Consider the function (0 if 0 <t<4 5 if 4 <t<8 f(t) = 3 6 if 8 <t <10 Lo if 10 <t<o. Use the graph of this function to write it in terms of the step function. Use ut - a) for the step function shifted a units horizontally. help f(t) = (formulas) Find the Laplace transform F(s)=L{f(t)} for s + 0. F(s) = C{f(t)}...
Problem 31: (34 points) 1. (10 points) A pulse width modulated (PWM) signal fPwM(t) in Figure...
Problem 31: (34 points) 1. (10 points) A pulse width modulated (PWM) signal fPwM(t) in Figure 2. The symbol D represents a duty cycle, a number between zero and one. Determine the compact trigonometric Fourier series coefficients (Co C,11 %) of the signal f(t). 2. (10 points) One use of PWM is to generate variable DC voltages. While the PWM signal is not DC, you should be able to see from your results in part 1 that it hss a...
Problem 2 Periodic Force First Cycle The graph at the right depicts the first period of a non-harmonic periodic force (measured in Newtons). This first cycle is described by the piecewise functio...
Problem 2 Periodic Force First Cycle The graph at the right depicts the first period of a non-harmonic periodic force (measured in Newtons). This first cycle is described by the piecewise function F(t) below the graph. Per the definition of a periodic function, the function repeats every T seconds. Note that T = 1 s. 1.8 a. What is the angular frequency wT of the periodic function?2 Include units. b. What is the Fourier Series representation of this function? c....
(10 points) This problem is related to Problems 8.16-21 in the text. Consider the differential equation...
(10 points) This problem is related to Problems 8.16-21 in the text. Consider the differential equation y' (t) + 7y(t) = le 4u(t), with initial condition y(0) = 2, Find the Laplace transform of the solution Y(s). Write the solution as a single fraction in s Y(s) = help (formulas) , where cis a Find the partial fraction decomposition of Y(s). Enter all factors as first order terms in s, that is, all terms should be of the form constant...
(10 points) This problem is related to Problems 8.16-21 in the text. Consider the differential equation...
(10 points) This problem is related to Problems 8.16-21 in the text. Consider the differential equation y" (t) + 16y'(t) + 68y(t) = –20e-4t u(t), with initial conditions y(0) = -3, and y'(0) = 4. Find the Laplace transform of the solution Y(s). Write the solution as a single fraction in s Y(s) = help (formulas) Find the partial fraction decomposition of Y(s). Enter all factors as first order terms in s, that is, all terms should be of the...
(10 points) This problem is related to Problem 9.31 in the text (a) A signal, s(t), with period T 5, is approximated by using the first few terms in the frequency domain by the following non-zero (complex) Fourier coefficients (all others are zero): S(0)--, 2 6 S(1)-S(-1)-π, S(3)-S(- approximation s (t), where 5 C(n)cos(nuot + %) (See Section 9.1.2 of the text.) Write the answer as the sum of cosines with phase. There should be no complex numbers in your...
(10 points) This problem is related to Problem 9.15 in the text.
Consider the periodic function with period 16 given by
(10 points) This problem is related to Problem 9.15 in the text Consider the periodic function with period 16 given by if 8 t<-2 if 2 t4 if 4 t<7 if 7 t8 0 -6 f(t) = 6 0 Find the first 3 coefficients of the exponential Fourier Series for f(t), i.e., F(n) for n = -2, -1,0, 1,...
Problem 32: (20 points) Consider a periodic signal f(t), with fundamental period To, that has the exponential Fourier series representation f(t) = Σ Dnejuont . where wo 2T/To and 1. (2 points) When f(t) is a real-valued, show that DD This is known as the complex conjugate symmetry property or the Hermitian property of real signals. 2. (2 points) Show that when f(t) is an even function of time that Dn is an even function of n 3. (2 points)...
Previous Problem Problem List Next Problem (10 points) This problem is related to Problems 9.33-9.38 in the text. We have solved differential equations using the method of undetermined coefficients (Chapter 7) and Laplace transforms (Chapter 8). We can use Fourier series to find the particular solution of an arbitrary order differential equation - as long as the driving function is periodic and can be represented by a Fourier series In the problem description and answers, all numerical angles(phases) should be...
Problem 1 The complex exponential Fourier Series of a signal over an interval 0 < t S T,-2π/wo is known to be (d) Suppose x(t) is the input to a stable, continuous-time, single-input/single-output LTI system whose impulse response is given by 9sine (wot/4 2 cos (u) Determine the output y(t) for -oo<t<oo. Answer: y(t)-4m 2r(1 +9π (2r(1+9r2) tan 1(3m) cos 9T
Problem 1 The complex exponential Fourier Series of a signal over an interval 0
(10 points) This problem is related to Problem 8.3 and 8.4 in the text. Consider the function (0 if 0 <t<4 5 if 4 <t<8 f(t) = 3 6 if 8 <t <10 Lo if 10 <t<o. Use the graph of this function to write it in terms of the step function. Use ut - a) for the step function shifted a units horizontally. help f(t) = (formulas) Find the Laplace transform F(s)=L{f(t)} for s + 0. F(s) = C{f(t)}...
Problem 31: (34 points) 1. (10 points) A pulse width modulated (PWM) signal fPwM(t) in Figure 2. The symbol D represents a duty cycle, a number between zero and one. Determine the compact trigonometric Fourier series coefficients (Co C,11 %) of the signal f(t). 2. (10 points) One use of PWM is to generate variable DC voltages. While the PWM signal is not DC, you should be able to see from your results in part 1 that it hss a...
Problem 2 Periodic Force First Cycle The graph at the right depicts the first period of a non-harmonic periodic force (measured in Newtons). This first cycle is described by the piecewise function F(t) below the graph. Per the definition of a periodic function, the function repeats every T seconds. Note that T = 1 s. 1.8 a. What is the angular frequency wT of the periodic function?2 Include units. b. What is the Fourier Series representation of this function? c....
(10 points) This problem is related to Problems 8.16-21 in the text. Consider the differential equation y' (t) + 7y(t) = le 4u(t), with initial condition y(0) = 2, Find the Laplace transform of the solution Y(s). Write the solution as a single fraction in s Y(s) = help (formulas) , where cis a Find the partial fraction decomposition of Y(s). Enter all factors as first order terms in s, that is, all terms should be of the form constant...
(10 points) This problem is related to Problems 8.16-21 in the text. Consider the differential equation y" (t) + 16y'(t) + 68y(t) = –20e-4t u(t), with initial conditions y(0) = -3, and y'(0) = 4. Find the Laplace transform of the solution Y(s). Write the solution as a single fraction in s Y(s) = help (formulas) Find the partial fraction decomposition of Y(s). Enter all factors as first order terms in s, that is, all terms should be of the...
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