(10 points) This problem is related to Problem 9.15 in the text. Consider the periodic function with period 16 given by
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(10 points) This problem is related to Problem 9.15 in the text. Consider the periodic function...
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)...
10 points) This problem is related to Problem 9.31 in the text (a) A signal, s(t). with period T 6, is approximated by using the first few terms in the frequency domain by the following non-zero (complex) Fourier coefficients (all others are zerol S(0)S(1) 1)) 3)5)-.nd the epproximation (t), where See Section 9.1.2 of the text) Write the answer es the sum of cosines with phese. There should be no complex numbers in your formula Ib) Using a MATLAB, graph...
(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 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)}...
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...
The periodic function so(t) with period 16 given by so(t) = 0 1 10 if – 8<t< -1 if –1<t<1 if 1 <t<8. has the Fourier series defined by S.(0) = 0.125 and for n +0 So(n) = 0.125 * sin(n72/16) 772/16 Use linearity and the shifting property to find the Fourier Series for s(t), defined by s(t) = O 1-5 4 lo if – 8<t<1 if i<t<3 if 3 <t<5 if 5 <t<8. S(0) = and for n 70...
(20 points) 1. (8 points) Suppose that f(t) is a periodic signal with exponential Fourier series coefficients Dn. Show that the power P of f(t) is This is Parseval's theorem for the exponential Fourier series. 2. (12 points) If f(t) is real-valued, Parseval's theorem can be as a) (3 points) Find the power of the PWM signal shown in figure 1. Hint: for this part don't use Parseval's theorem b) (9 points) Use Parseval's theorem for a real-valued signal to...
(20 points) Consider the function f(z) z in the interval [0, 2π). (a) Derive the Fourier coefficients ck fork = 0, 1,土2, (b) Derive the Fourier coefficients ao, ak, bk for k 1,2,... .. (c) Plot the partial Fourier series, along with the function f, by retaining 1, 10, 50, 100 terms in the summation (use the second form involving cosines and sines) d) Comment on the convergence of the partial Fourier series. Note: you should submit only 1 plot...
[4 Mar (c) Consider the following periodic function, defined as: fO) = 7? - ?, - <t<T and f(t) = f(t + 27) (0 ) State the period, P. [1 Marks) ( 11) Sketch a graph of f(t). [2 marks] State if f(t) is either even or odd, or neither. (1 Marks) (iv) Which Fourier coefficients are zero and why? [1 Marks) (v) Compute do [2 marks] (vi) Compute the non-zero Fourier coefficients. [5 Marks) (vii) Write down the Fourier...
11. (10 points) Let f(t) be a 27-periodic function defined by f(t) = -{ 2 if – <t<0, -2 if 0 <t<, f(t + 2) = f(t). a) Find the Fourier series of f(t). b) What is the sum of the Fourier series of f at t = /2.