Problem 8. Use the relation zit)y"(t)dt = 1 and a known transform pa 00 sinc t h"
P7. Use the Fourier transform of the signal (t) and Parseval's theorem to: duw that「.sinc.(ka)dz-t; determine sinc(t - 2)dt
Use the Laplace transform to solve dy cost + So y(t) cos(t – t)dt, y(0) = 1 dt
Given LTI system with following input response (can use properties of the Fourier transform like, sinc(x) = sin(πx)/πx ): h(t) = 8/π sinc(8t/π) where input x(t) of the LTI system is the following continuous-time signal x(t) = cos(t) cos(8t) a) find the Fourier transform of x(t) b) find the Fourier transform of h(t) c) Is this LTI system BIBO stable? Prove d) find the output y(t) of the LTI system
Useful Formula: Fourier Transform: F[f(t)] = F(w) sof(t)e-jw dt Inverse Fourier Transform: F-1[F(w)] = f (t) = 24., F(w)ejwidw Time Transformation property of Fourier Transform: f(at – to). FC)e=itoch Laplace Transform: L[f(t)] = F(s) = $© f(t)e-st dt Shifting property: L[f(t – to)u(t – to)] = e-toSF(s) e [tuce) = 1 and c [u(e) = ) Using the convolution property of Fourier Transform to find the following convolution: sinc(t) * sinc (4t) [Hint: sinc(t) or rect(w/2)] TC .
3). Use the Laplace transform to solve for y(t) for t20. y(0 +) = 5, dt dt dt Initially relaxed dtdt
How to graph fourier transform
My question isn't on how they got the transform, my question is
how the graph was calculated, I don't know what sinc means
Note: 1) In the following problem set a notation Pa(t) is used to denote an even pulse (i.e. rectangular) shape function with the duration of a. This means that for example the function x(t) in question number one is 2P2(t-1). It is clear that using the properties of Fourier transform is greatly...
Determine the system response y(t) for h(t)=u(t)+u(t-2) and x(t)=u(t). [Hint: use Laplace Transform multiplication: L[x(t)h(t)) = x(s)H(s). Useful Formula: Fourier Transform: F[f(t)] = F(w) sof(t)e-jw dt Inverse Fourier Transform: F-1[F(w)] = f (t) = 24., F(w)ejwidw Time Transformation property of Fourier Transform: f(at – to). FC)e=itoch Laplace Transform: L[f(t)] = F(s) = $© f(t)e-st dt Shifting property: L[f(t – to)u(t – to)] = e-toSF(s) e [tuce) = 1 and c [u(e) = )
Problem 4. Linear Time-Invariant System.s A linear system has the block diagram y(t) z(t) →| Delay by 1 dt *h(t) where g(t) sinc(t Since this is a linear time invariant system, we can represent it as a convolution with a single impulse response h(t) a) Find the impulse response h(t). You don't need to explicitly differentiate. b) Find the frequency response H(j for this system.
2. Use the Laplace Transform to solve the initial value problem y"-3y'+2y=h(t), y(O)=0, y'(0)=0, where h (t) = { 0,0<t<4 2, t>4
where h is the Use the Laplace transform to solve the following initial value problem: y"+y + 2y = h(t – 5), y(0) = 2, y(0) = -1, Heaviside function. In the following parts, use h(t – c) for the shifted Heaviside function he(t) when necessary. a. First, take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation and then solve for L{y(t)}. L{y(t)}(s) = b. Express the solution y(t) as the...