1. Given below are the input/output relations of four systems, where x(t) is the input and y(t) is the system output. Classify each system as (i) Linear/ Non-linear (ii) Time variant/ Time invariant (iii) Causal/ Non-causal with proper justification.
Also find the output y(t) for each system, when input x(t) = u(t − 2) − u(t − 4) is applied.
(a) y(t) = −2t∞ x(τ + 3)dτ
(b) y(t) = x(t) sin(πt)
(c) y(t) = dx(t)
dt
(d) y(t) = x(2 − t) + x(2 + t)
2. Consider the following input/output relationship for a system S :
y(t) = x(t)
t+1
−
e|t−τ|x(τ)dτ
t−1
(a) Rewrite y(t) in the form y(t) =
∞
[?]dτ where [?] is a function to be determined.
−∞
(b) Classify S as Linear/ Non-linear, Time varying/ Time invariant, Causal/ Non-causal. Justify your answer.
(c) Find the output y(t) given: x(t) = e−tu(t + 2)
3. Consider an LTI system whose response to the signal x1(t) in Figure 1(a) is the signal y1(t) illustrated in Figure 1(b).
Figure 1
1
(a) Determine and sketch carefully the response of the system to the input x2(t) depicted in Figure 1(c).
(b) Determine and sketch the response of the system considered in part (a) to the input x3(t) shown in Figure 1(d).
(c) Consider the LTI system with input/output relation y(t) = tt−2 x(τ)dτ. Find the impulse
response h(t). Sketch the output when an input of x(t) = u(t+2)+u(t)−2u(t−1)+δ(t−1) is applied to the system.
4. Given the following input-output relation (IPOP) of a system:
∞
y(t) = e−t(t − τ)2u(τ + t)x(τ − 2)dτ, t ∈ (−∞, ∞).
−∞
a) Find impulse response of the system h(t, τ). Is the system time variant (TV) or time invariant (TI)? Is it causal (C) or non-causal (NC)?
b) Find the corresponding output, y(t), given an input of:
x(t) = δ(t − 2) − e−tu(t + 1), t ∈ (−∞, ∞)
2
