Consider a causal, linear and time-invariant system of continuous time, with an input-output relation that obeys...
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...
2. Let y(t)(e')u(t) represent the output of a causal, linear and time-invariant continuous-time system with unit impulse response h[nu(t) for some input signal z(t). Find r(t) Hint: Use the Laplace transform of y(t) and h(t) to first find the Laplace transform of r(t), and then find r(t) using inverse Laplace transform. 25 points
Determine which of these properties (Memoryless, Time invariant, Linear, Causal, and Stable) hold and which do not hold for each of the continuous-time system, y[n] = x [4n + 1]. Justify your answers. y(t) denotes the system output and x(t) is the system input
4. Let S be a linear, time-invariant, and causal system whose input x(t) and corresponding output y(t) are shown below: r(t) Page 1 of 2 Please go to next page... y(t) ? (a) Find the impulse response function h(t) of ? (b) Find the output of S when its input is e*, t<0, t2, t20
6. (15) Consider the following causal linear time-invariant (LTT) discrete-time filter with input in and output yn described by y[n] = x[n] – rn - 2 for n 20 . Is this a finite impulse response (FIR) or infinite impulse response (IIR) filter? Why? • What are the initial conditions and their values for this causal and linear time-invariant system? Why? • Draw the block diagram of the filter relating input x[n) and output y[n] • Derive a formula for...
If of the input signal X(t) and output signal yl to a continuous - time linear time - invariant system - Y(t) = 5 to xc ej elt- el de o find the impluse response of the system hat) o If the input signal is x (t) = f(t) - f (t-1) find the output signal y ct) o Find the impulse response of the system hit) @ If the input signal Is x (t) = f(t)-f(t-1) find the -...
2. (Chapter 2). A linear, time-invariant, continuous-time (LTIC) system with input 1(1) and output y(t) is specified by the differential equation D(D? + 1)y(t) = Df(t). a. Find the characteristic polynomial, characteristic equation, characteristic root(s), and characteristic mode(s) of this system. b. Is this system asymptotically stable, marginally stable, or unstable? Justify your answer.
solve all 22. The input-output relationship for a linear, time-invariant system is described by differential equation y") +5y'()+6y(1)=2x'()+x(1) This system is excited from rest by a unit-strength impulse, i.e., X(t) = 8(t). Find the corresponding response y(t) using Fourier transform methods. 23. A signal x(1) = 2 + cos (215001)+cos (210001)+cos (2.15001). a) Sketch the Fourier transform X b) Signal x() is input to a filter with impulse response (1) given below. In each case, sketch the associated frequency response...
Consider a discrete-time system that is linear (but not necessarily time-invariant), and where: - if the input x[n] is even, then the output is y[n]= x[n-1] - if the input x[n] is odd, then the output y[n]= x [n=1] Find the ouput of this system if the inpus is (a) δ [n] (b) u[n]. (Do not use la place transform) Hint: if a signal is neither even nor odd, then you can write it as a sum of an even...
uestion A causal, linear time-invariant system is excited with an input x (n) described as x(n) 3u(n) with the output y(n) of the system as follows: 7l n) -2"u(n) y(n)- a) Determine z-transform X(z) and Y (z) (4 marks) b) Determine the transfer function H(z). (3 marks) Based on (b), determine the impulse response h(n). Based on (b), sketch the z-plane for the transfer function of the system Based on (d), determine the stability of the system and discuss the...