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Question 1 Suppose m-2 kg, k- 32 N/m and F() is an odd periodic force with a period of 2 s given in one period by ON if O<t<I Construct the steady periodic motion, Xsn) using the Fourier series...
The Fourier series of a periodic signal s(2) of period T can be expressed as k...
The Fourier series of a periodic signal s(2) of period T can be expressed as k s(x) = cxexp ( 21 - where the coefficients Ck are given by 7/2 CR 1 T -T/2 | $(z) exp (-27 k -27=cdc T (i) Consider s(2) of period T = 6 and amplitude A= 2: 8(z) = 2 * |< T 2 Compute the Fourier coefficients ok. (ii) Use the identities exp(Trik) + exp(-rik) cos(Tk) = 2 sin(Tk) exp(Trik) – exp(-rik) 2i...
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)...
2, For the following periodic series, f(t) = t^2, ( -1<t<1), period = 2 (a) graph...
2, For the following periodic series, f(t) = t^2, ( -1<t<1), period = 2 (a) graph this periodic function, range is ( -8<t<8), and find a key point (b)Even function or odd function? (c ) find a Fourier series
The sketch of the following periodic function f (t) given in one period f(t) t2 -1,...
The sketch of the following periodic function f (t) given in one period f(t) t2 -1, 0s t s 2 is given as follows f(t) 2 -1 We proceed as follows to find the Fourier series representation of f (t) (Note:Jt2 cos at dt = 2t as at + (a--)sina:Jt2 sin at dt = 2t sin at + sin at. Г t2 sin at dt-tsi. )cos at.) Please scroll to the bottom of page for END of question a) The...
1. Find the Fourier series for the following 1-periodic function f(t) = t, t < -- 2. Find the sum 24 3444 (Hint: Consider the Fourier series for the function f(t)-t2 on [- integer k.) 1) and f(t-k...
1. Find the Fourier series for the following 1-periodic function f(t) = t, t < -- 2. Find the sum 24 3444 (Hint: Consider the Fourier series for the function f(t)-t2 on [- integer k.) 1) and f(t-k)-f(t) for all
1. Find the Fourier series for the following 1-periodic function f(t) = t, t
l, f) is a periodic signal with period f(t)-n(t)-u(t-t/2 ) for 0 2π a.) Find the exponential Fourier series of f() and sketchf). What is the fundamental radian frequency. b.) Evaluate and sketch...
l, f) is a periodic signal with period f(t)-n(t)-u(t-t/2 ) for 0 2π a.) Find the exponential Fourier series of f() and sketchf). What is the fundamental radian frequency. b.) Evaluate and sketch |Dml, the magnitude of the Fourier series coefficients vs.o in the range of -4s n S4 c.) Evaluate and sketch the phase angle of D, vs. co in the same range (-4S n S4) d.) Find the signal average power e) Find the approximate average power of...
F(N) 2. A 15 kg oscillator with a stiffness of k = 960 N/m and damping...
F(N) 2. A 15 kg oscillator with a stiffness of k = 960 N/m and damping coefficient c = 60 Ns/m is driven by a square- wave excitation F(t) shown in the figure. Determine and plot the steady state response for 12 s using 100 terms in the Fourier series solution. 100 -100
Consider the periodic function defined by 1<t0, 0<t<1, f(t)= f(t+2) f(), and its Fourier series F(t): Σ A,...
Consider the periodic function defined by 1<t0, 0<t<1, f(t)= f(t+2) f(), and its Fourier series F(t): Σ A, cos(nmi) +ΣB, sin (nπί), F(t)= Ao+ n1 n=1 (a) Sketch the function f(t) the function is even, odd or neither even nor odd. over the range -3<t< 3 and hence state whether (b) Calculate the constant term Ao
Consider the periodic function defined by 1
The periodic function so(t) with period 28 given by if 14 t<-0.5 1 if _ 0.5 〈 t < 0.5 so(t)- 0if 0.5 t< 14. has the Fourier series defined by So(0)-0.0357143 and for n 0 0.0357143 * sin(nTl...
The periodic function so(t) with period 28 given by if 14 t<-0.5 1 if _ 0.5 〈 t < 0.5 so(t)- 0if 0.5 t< 14. has the Fourier series defined by So(0)-0.0357143 and for n 0 0.0357143 * sin(nTl/28) nT1/28 Use linearity and the shifting property to find the Fourier Series for s(t), defined by f -14 t <4.5 -5 if 4.5 t< 5.5 3 if 5.5 t<6.5 if 6.5 < t < 14. s(t) S(0) and for n S(n)...
F Fosin t m k 2 Figure Qla: System is subjected to a periodic force excitation (a) Derive the equation of motion of the...
F Fosin t m k 2 Figure Qla: System is subjected to a periodic force excitation (a) Derive the equation of motion of the system (state the concepts you use) (b) Write the characteristic equation of the system [4 marks 12 marks (c) What is the category of differential equations does the characteristic equation [2 marks fall into? (d) Prove that the steady state amplitude of vibration of the system is Xk Fo 25 + 0 marks (e) Prove that...
The Fourier series of a periodic signal s(2) of period T can be expressed as k s(x) = cxexp ( 21 - where the coefficients Ck are given by 7/2 CR 1 T -T/2 | $(z) exp (-27 k -27=cdc T (i) Consider s(2) of period T = 6 and amplitude A= 2: 8(z) = 2 * |< T 2 Compute the Fourier coefficients ok. (ii) Use the identities exp(Trik) + exp(-rik) cos(Tk) = 2 sin(Tk) exp(Trik) – exp(-rik) 2i...
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)...
The sketch of the following periodic function f (t) given in one period f(t) t2 -1, 0s t s 2 is given as follows f(t) 2 -1 We proceed as follows to find the Fourier series representation of f (t) (Note:Jt2 cos at dt = 2t as at + (a--)sina:Jt2 sin at dt = 2t sin at + sin at. Г t2 sin at dt-tsi. )cos at.) Please scroll to the bottom of page for END of question a) The...
1. Find the Fourier series for the following 1-periodic function f(t) = t, t < -- 2. Find the sum 24 3444 (Hint: Consider the Fourier series for the function f(t)-t2 on [- integer k.) 1) and f(t-k)-f(t) for all
1. Find the Fourier series for the following 1-periodic function f(t) = t, t
l, f) is a periodic signal with period f(t)-n(t)-u(t-t/2 ) for 0 2π a.) Find the exponential Fourier series of f() and sketchf). What is the fundamental radian frequency. b.) Evaluate and sketch |Dml, the magnitude of the Fourier series coefficients vs.o in the range of -4s n S4 c.) Evaluate and sketch the phase angle of D, vs. co in the same range (-4S n S4) d.) Find the signal average power e) Find the approximate average power of...
F(N) 2. A 15 kg oscillator with a stiffness of k = 960 N/m and damping coefficient c = 60 Ns/m is driven by a square- wave excitation F(t) shown in the figure. Determine and plot the steady state response for 12 s using 100 terms in the Fourier series solution. 100 -100
Consider the periodic function defined by 1<t0, 0<t<1, f(t)= f(t+2) f(), and its Fourier series F(t): Σ A, cos(nmi) +ΣB, sin (nπί), F(t)= Ao+ n1 n=1 (a) Sketch the function f(t) the function is even, odd or neither even nor odd. over the range -3<t< 3 and hence state whether (b) Calculate the constant term Ao
Consider the periodic function defined by 1
The periodic function so(t) with period 28 given by if 14 t<-0.5 1 if _ 0.5 〈 t < 0.5 so(t)- 0if 0.5 t< 14. has the Fourier series defined by So(0)-0.0357143 and for n 0 0.0357143 * sin(nTl/28) nT1/28 Use linearity and the shifting property to find the Fourier Series for s(t), defined by f -14 t <4.5 -5 if 4.5 t< 5.5 3 if 5.5 t<6.5 if 6.5 < t < 14. s(t) S(0) and for n S(n)...
F Fosin t m k 2 Figure Qla: System is subjected to a periodic force excitation (a) Derive the equation of motion of the system (state the concepts you use) (b) Write the characteristic equation of the system [4 marks 12 marks (c) What is the category of differential equations does the characteristic equation [2 marks fall into? (d) Prove that the steady state amplitude of vibration of the system is Xk Fo 25 + 0 marks (e) Prove that...
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Amplitude=3;
fs=8000;
n=0:399;
t=0:1/fs: n*1/fs-1/fs;
signal=3+3*cos(2*pi*1100*t)+3*cos(2*pi*2200*t)+3*cos(2*pi*3300*t);
fftSignal= fft(signal);
fftSignal=f ftshift (fftSignal);
f=fs/2*linspace(-1,1,fs);
plot(f,abs(fftsignal);
xlabel('Frequency(Hz)’)
yla
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