1. For the systems given below, investigate for the given properties. Explain your answers. i. y(t) = exp(x(1)) ii. y(t)=x(t)x(t-1) linearity, time-invariance linearity, stability dx(t) iii. y(t) = sin(t)x(t) + dt time-invariance, causality

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1. For the systems given below, investigate for the given properties. Explain your
answers.
i. у() — еxp(x())
ii. y(1)=x(t)x(t-1)
dx(t)
iii. y(t) = sin(t)x(t) +
linearity, time-invariance
linearity, stability
time-invariance, causality
dt
2. The output of a linear system is y(t)=In(kt+1) to the input signals of the form a“ (a>1,
k are constants). Answer the following:
i. Find the output for x(1) = 1 + 6(2²ª) – 4(3* ).(Hint: Express 2 and 3 as
exponents of a!)
ii. Show that the output of the system is identically zero to a constant
input.
iii. Is this system a LTI system? Why or why not?
3. Two discrete-time LTI systems, whose impulse responses are h(n) and h,(n), are
cascade connected as in the figure (below right).
a. Find w(n) for x(n)=8(n)-0.58(n-1).
b. Find the response y(n) to the input x(n)={(n)-0.58(n-1).
c. Find the overall impulse response of the cascade system.
4. The step (i.e. the input is u(t)) response of a LTI system is given by y(1) = sin(-21)u(t).
Find the response to the input signal x(1) given in the figure (below left).
5. Sketch the discrete time signals x1(n) and x2(n) corresponding to the sampling periods
T,1=0.5s and T,=0.3s, respectively, for the signal x(t) given below left. Assume that
the sampler takes the right limit of the signal at nT, (i.e. x,(nT,) = lim x(t).
x(n)
w(n)
y(n)
h,(n)=0.5u(n)
h{n}=2*u(n)
For question 3
For question 4 and 5
Transcribed Image Text:1. For the systems given below, investigate for the given properties. Explain your answers. i. у() — еxp(x()) ii. y(1)=x(t)x(t-1) dx(t) iii. y(t) = sin(t)x(t) + linearity, time-invariance linearity, stability time-invariance, causality dt 2. The output of a linear system is y(t)=In(kt+1) to the input signals of the form a“ (a>1, k are constants). Answer the following: i. Find the output for x(1) = 1 + 6(2²ª) – 4(3* ).(Hint: Express 2 and 3 as exponents of a!) ii. Show that the output of the system is identically zero to a constant input. iii. Is this system a LTI system? Why or why not? 3. Two discrete-time LTI systems, whose impulse responses are h(n) and h,(n), are cascade connected as in the figure (below right). a. Find w(n) for x(n)=8(n)-0.58(n-1). b. Find the response y(n) to the input x(n)={(n)-0.58(n-1). c. Find the overall impulse response of the cascade system. 4. The step (i.e. the input is u(t)) response of a LTI system is given by y(1) = sin(-21)u(t). Find the response to the input signal x(1) given in the figure (below left). 5. Sketch the discrete time signals x1(n) and x2(n) corresponding to the sampling periods T,1=0.5s and T,=0.3s, respectively, for the signal x(t) given below left. Assume that the sampler takes the right limit of the signal at nT, (i.e. x,(nT,) = lim x(t). x(n) w(n) y(n) h,(n)=0.5u(n) h{n}=2*u(n) For question 3 For question 4 and 5
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