Problem 1. (B01) Characterize the systems below, i.e. see what properties they have or do not have. The properties you should be investigate for are (i) homogeneity, (ii) superposition property, (iii) linearity, (iv) time-invariance, (v) causality, (vi) memoryless property and (vii) Bounded- Input/Bounded-Output (BIBO) stability. Assume that their inputs r is an arbitrary, complex- valued signal. T1 {r(t)} = |r(t)I (la)

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Problem 1. (B01) Characterize the systems below, i.e. see what properties they have or do not
us-00 JT=-
have. The properties you should be investigate for are (i) homogeneity, (ii) superposition property,
(iii) linearity, (iv) time-invariance, (v) causality, (vi) memoryless property and (vii) Bounded-
Input/Bounded-Output (BIBO) stability. Assume that their inputs z is an arbitrary, complex-
valued signal.
T {x(t)} = |x(t)|
T2 {r(t)} = Re {x(t))}
(la)
(1b)
T3 {r(t)} = x*(t)
TA {r(t)} = x²(t)
x² (t-1)
(lc)
(ld)
when r(t) + 0
T3 {x(t)} =
r(t)
(le)
otherwise
T6 {r(t)} = x(t) cos (wt), w> 0
T7 {r(t)} = x(t)u(t)
(1f)
(1g)
1
Ts {r(t)} =
T
I(T)dr, T> 0
(1h)
dr(t)
T9 {r(t)} = 2.
dt2
t
rtoo
| v(T)x(t' –T)dr, where v is a given signal
(li)
Comments: T; is not BIBO stable, but it is quite hard to show. So, if you don't manage to tackle
it, don't worry; just review its solution, when it is posted.
Transcribed Image Text:Problem 1. (B01) Characterize the systems below, i.e. see what properties they have or do not us-00 JT=- have. The properties you should be investigate for are (i) homogeneity, (ii) superposition property, (iii) linearity, (iv) time-invariance, (v) causality, (vi) memoryless property and (vii) Bounded- Input/Bounded-Output (BIBO) stability. Assume that their inputs z is an arbitrary, complex- valued signal. T {x(t)} = |x(t)| T2 {r(t)} = Re {x(t))} (la) (1b) T3 {r(t)} = x*(t) TA {r(t)} = x²(t) x² (t-1) (lc) (ld) when r(t) + 0 T3 {x(t)} = r(t) (le) otherwise T6 {r(t)} = x(t) cos (wt), w> 0 T7 {r(t)} = x(t)u(t) (1f) (1g) 1 Ts {r(t)} = T I(T)dr, T> 0 (1h) dr(t) T9 {r(t)} = 2. dt2 t rtoo | v(T)x(t' –T)dr, where v is a given signal (li) Comments: T; is not BIBO stable, but it is quite hard to show. So, if you don't manage to tackle it, don't worry; just review its solution, when it is posted.
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