lincar time invariant systcm (LTI), Determine whcther cach of the following systems is a wherc uis thc input and y is t...
7. Show that the following functions u(x, y) monic functions v(x, y) and determine f(z) = u(x,y) + iv(x, y) are harmonic, find their conjugate har- as functions of 2. 2x2 2лу — 5х — 22. Зл? — 8ху — Зу? + 2у, (а) и(х, у) (b) и(х, у) (с) и(х, у) (d) u(a, y) 2e cos y 3e" sin y, = 3e-* cos y + 5e-" sin y, = elx cos y - e y sin y, (e) u(x,...
Linear time-invariant sys tems. Consider a SISO, LTI system: t(t) Az(t) + bu(t) y(t) cr (t) with IC x(0) 3o. If the nominal input is a non-zero constant, i.e. if u(t)= u, under what conditions does there exist a constant nominal solu- tion (t) nominal output zero? Under what conditions do there exist constant nom- inal solutions that satisfy g = u for all u? To for some axo? Under what conditions is the corresponding
3. (10 points) Two linear time invariant (LTI) systems with impulse response hi(k) and h2(k) are connected in cascade as shown in Figure 1. Let x(k) be the input, yı(k) be the output of the first LTI, and y2(k) be the output of the second LTI. Let hi(k) = k(0.7)k u(k), h2(k) = ku(k), and x(k) = (0.3)k u(k). Use z-transform to (a) find yı(k). (b) find y2(k). x(k) yi(k) y2(k) hi(k) h2(k)
4. LTI Systems and Erponential Response. (12 pts) (a) (2 pts) Suppose an LTI system has input-output relationship y(t) 2r(t+3). What is the transfer function H(jw) of the given system. Show that H(jw)2. Hint: H(jw(tejdt (b) (5 pts) Suppose an LTI system has input-output relationship y(t)2r(t+3) as Problem 4-(a). Find the output y(t) using the complex exponential response method as discussed in lecture for the input r(t) = ej2t + 2 cos2(t). Hint: cos2(0) 1 (20 cos(26) an d 1-ejot...
(a) LTI Systems. Consider two LTI subsystems that are connected in series, where system Tl has step response s1(t)=u(t-1)-u(t-5) and system T2 has impulse response h2t = e-3tu(t). Find the overall impulse response h(t). Hint: you will need to find h1(t) first (b)Fourier Series. The input signal r(t) and impulse response h(t) of an LTI system are as follows:x(t) = sin(2t)cos(t)-ej3t +2 and h(t) = sin(2t)/t Use the Fourier Series method to find the output y(t) (c)Parseval's Identity and Theorem. Consider the system in the...
Q8) Consider the following causal linear time-invariant (LTI) discrete-time filter with input x[n] and output y[n] described by bx[n-21- ax[n-3 for n 2 0, where a and b are real-valued positive coefficients. A) Is this a finite impulse response (FIR) or infinite impulse response (IIR) filter? Why? B) What are the initial conditions and their values? Why? C) Draw the block diagram of the filter relating input x[n] and output y[n] D) Derive a formula for the transfer function in...
(c) If the impulse response function of a linear time invariant (LTI) system is h0)-Se u(), compute the output of this system due to an input ) which is a 4 second pulse of height 3, as shown in Fig.1 below. x(t) t(sec) 0 Fig.1 Input signal 10 marks/
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...
Linear Time Invariant Systems 4] For each of the following continuous-time systems xt) is a real input. Determine whether the system is (1) stable, (2) causal, (3) linear and (4) time invariant (5% each): (a) T(x(t)] = sin(2π) x(t + 2%)-cos(2π) x(t-ro), where τ。> (b)T(x(t)] = x(4) (c) T(x(t)]Ξ14- AxzQ A is a complex constant.
2.10. Window/modulator Consider the system where for an input x(t) the output is y(t) = x(oft) for some function f(t). (a) Letf(t)=u(t)-11(t-10). Determine whether the system with input x(t) and output y(t)is linear, time invariant, and causal, Suppose x(t) = 4 cos(T/2), and f(t)=cos(67t/7) periodic? What frequencies are present in the output? Is this system linear? Is it time invariant? Explain. (b) and both are periodic. Is the output y(t) also (c) Let f(t) = u(t)-u (t-2) and the input...