x() — А() x() + B()u(t) Consider the following time-varying system У() %3D С()x() + D()u(t)...
() At)x()B(f)u() Consider the following time-varying system y(t) C(f)x(t) where x) R", u(t)E R R 1 1) Derive the state transition matrix D(t,r) when A(f) = 0 0 sint 2) Assume that x(to) = x0 is given and u(f) is known in the interval [to, 4] Based on these assumptions, derive the complete solution by using the state transition matrix D(f, r). Also show that the solution is unique in the interval [to, 4]. 3) Let x(1) 0 and u(f)...
x(t)-A()x(t) + B(t)u(t) 2. Consider the following system y(t)-C(t)x(t) where A(t)- 0 1) Derive the state transition matrix Φ(t, τ). 2) Derive the impulse response function g(t, z). x(t)-A()x(t) + B(t)u(t) 2. Consider the following system y(t)-C(t)x(t) where A(t)- 0 1) Derive the state transition matrix Φ(t, τ). 2) Derive the impulse response function g(t, z).
2. Given two initial value problems, у" — р(г)у +q()у +r(x) with a <I<b,y(a) — с,1 (а) —0 (1) and у" — р(г)у + g(х)у with a < r <ь,y(a) — 0, and / (а) — 1 (2) [a, b) where p(x), q(z) and r(x) Show that given two solutions yı(x), y2(x) to the linear value problems above, (1) and (2), respectively, then there exists a solution y(x) to a linear boundary value problem above where y(a) %3D 0, у...
2.6.1-2.6.62.6.1 Consider a causal contimuous-time LTI system described by the differential equation$$ y^{\prime \prime}(t)+y(t)=x(t) $$(a) Find the transfer function \(H(s)\), its \(R O C\), and its poles.(b) Find the impulse response \(h(t)\).(c) Classify the system as stable/unstable.(d) Find the step response of the system.2.6.2 Given the impulse response of a continuous-time LTI system, find the transfer function \(H(s),\) the \(\mathrm{ROC}\) of \(H(s)\), and the poles of the system. Also find the differential equation describing each system.(a) \(h(t)=\sin (3 t) u(t)\)(b)...
Consider a continuous-time LTI system S with impulse response h(t) = 2(u(t + 1)-u(t 1)). Determine the values of the amplitude scaling and the tme shifting that takes place when each of the following input signals is provided to the system S. Don't use the convolution integral, instead use the result about how LTI systems respond to complex exponential signals. (a) x(t) 2 (b) x(t) ej0.5Tt (c) x(t) = e-j0.5πt (d) x(t) = e-jmt (e) x(t) = cos (0.5t) (f)...
Consider an LTI system with the impulse response h(t) = e- . Is the system casual? Explain. Find and plot the output s(t) given that the system input is x(t) = u(t). Note that s(t) in this case is commonly known as the step response of the system. If the input is x(t) = u(t)-u(t-T). Express the output y(t) as a function of s(t). Also, explicitly write the output y(t) as a function of t. a) b) c)
For a continuous time linear time-invariant system, the input-output relation is the following (x(t) the input, y(t) the output): , where h(t) is the impulse response function of the system. Please explain why a signal like e/“* is always an eigenvector of this linear map for any w. Also, if ¥(w),X(w),and H(w) are the Fourier transforms of y(t),x(t),and h(t), respectively. Please derive in detail the relation between Y(w),X(w),and H(w), which means to reproduce the proof of the basic convolution property...
Consider the following zero-sum game. a b с e X у u V W What conditions do x, y, u, v, and w need to satisfy for the alpha-beta pruning algorithm to cause a cutoff, such that node ε does not need to be evaluated? Explain how you arrived at your condition or conditions. [6 marks]
please show complete answers for a,b,c,d Find the solution to the given system that satisfies the given initial condition. -5 1 |x(t) 10 3 х'() 3 -3 1 (b) х(т) %—D (d) x(T/6) (с) x(- 2т)%3 (а) x(0) %3 -1 1 3 (а) x() 3D| | (Use parentheses to clearly denote the argument of each function.) Find the solution to the given system that satisfies the given initial condition. -5 1 |x(t) 10 3 х'() 3 -3 1 (b) х(т)...
lincar time invariant systcm (LTI), Determine whcther cach of the following systems is a wherc uis thc input and y is the output (а) у — —и + (3 sin t cos t)u S 3 (b) y =t+ u (с) ў — —у +Зи (d) y = tsinu + e (с) у — (е 'sin t)u lincar time invariant systcm (LTI), Determine whcther cach of the following systems is a wherc uis thc input and y is the output (а)...