System properties: linearity
✗
→
☒
-51=17×3
Definition 6.1. A CT system T is linear if for any two input signals x, x̃
and any two constants –, –
˜ œ C it holds that
T {–x + –
˜ x̃} =
rT{ }
a
+
IT { I }
I in words: if the input is a linear combination of different signals, you
can apply T to each piece and add the results (“superposition”)
I for a linear system, zero input always produces zero output (why?)
I superposition immediately extends to more general sums
Y
Z
n
n
]ÿ
^ ÿ
T
–j xj =
–j T {xj }.
[
\
j=1
Section 6: Fundamentals of Continuous-Time Systems
j=1
6-210
Example: the squaring system is not linear
I recall: the squaring system Tsq defined by y(t) = (x(t))2
I let x(t) and x̃(t) be two inputs with corresponding outputs
y(t) = Tsq {x}(t) = (x(t))2 ,
ỹ(t) = Tsq {x̃}(t) = (x̃(t))2 .
I for constants –, –
˜ , we have
Tsq {–x + –
˜ x̃}(t) =
( 981+1+2 Elt) )
= a' ✗ Hi
+
'
248×111×-11-1
+
FEW
'
I on the other hand
–Tsq {x}(t) + –
˜ Tsq {x̃}(t) =
0×11-1
"
t
IE Iti
Therefore, the system Tsq is not linear
Section 6: Fundamentals of Continuous-Time Systems
6-211
Example: the RC circuit w/ zero I.C. is linear
I recall: the RC circuit TRC {vs }(t) =
st
1 ≠(t≠· )/RC
e
vs (· ) d·
0 RC
I let vSc , ṽcS be two input signals and let –, –
˜ be constants
I we calculate that
c-
-
⁄
t
Kc
A
TRC {–vSc + –
˜ ṽcS }(t) =
ce≠c(t≠· ) (–vc (· ) + –
˜ ṽSc (· )) d·
0
⁄ t
⁄ t
1 ≠(t≠· )/RC
1 ≠(t≠· )/RC
=–
e
v
(·
)
d·
+
–
˜
ṽSc (· ) d·
c
S
RC
RC e
S
0
=
0
oTRc{ us }lH
+
IT,zc{ Is } / t )
Therefore, the system TRC is linear.
Section 6: Fundamentals of Continuous-Time Systems
6-212
System properties: time-invariance
I notation: let x· (t) = x(t ≠ · ) be short form notation for a
time-delayed signal
Definition 6.2. A CT system T is time-invariant if for any input x with
output y = T {x}, it holds that y· = T {x· } for all time shifts · œ R.
physical meaning: an
experiment on the system
tomorrow will produce the
same results as an
experiment on the system
today.
Section 6: Fundamentals of Continuous-Time Systems
6-213
Example: the squaring system is time-invariant
I for the squaring system, let x be an input with corresponding output
y(t) = Tsq {x}(t) = (x(t))2
I if we simply shift the obtained output, we obtain
y· (t) =
I# it
1×(1--4)
'
I if we instead shift the input signal to be x· (t) = x(t ≠ · ), we compute
I # 21
Tsq {x· }(t) =
( ✗ 11-15=1×11--1-15
*
These two calculations agree, so Tsq is time-invariant.
Section 6: Fundamentals of Continuous-Time Systems
6-214
Example: the system Tmod is not time-invariant
I for the system Tmod let x be an input with output
y(t) = Tmod {x}(t) = x(t) · A sin(Ê0 t)
I if we simply shift the obtained output, we obtain
1¥11
y· (t) =
ylt-i-1-xlt-c-1.Asinlw.lt -4 )
I if we instead shift the input signal to be x· (t) = x(t ≠ · ), we compute
(# 2)
Tmod {x· }(t) = XHt-A-simlwotl-xlt-i-l.A-sinlw.tl
These two calculations disagree, so Tmod is not time-invariant.
Section 6: Fundamentals of Continuous-Time Systems
6-215
w/ right-sided input
Example: the RC circuit w/ zero I.C. is time-invariant
s 1 ≠(t≠‡)/RC
I for TRC we have y(t) = 0t RC
e
x(‡) d‡
I if we simply shift the obtained output by · œ R, we have
⁄ t≠·
1 ≠(t≠· ≠‡)/RC
11
#
y· (t) = y(t ≠ · ) =
x(‡) d‡.
I
RC e
0
I if we instead shift the input signal by · , the corresponding output is
I # 2)
TRC {x· }(t) =
=
⁄
t≠·
⁄
0
t
1 ≠(t≠›)/RC
e
x(›
RC
≠ · ) d›
1 ≠(t≠· ≠‡)/RC
e
x(‡) d‡
RC
≠·
change
=
⁄
→
because
g-
t≠·
0
✗
(g)
d- variables
=3
-
t
1 ≠(t≠· ≠‡)/RC
e
x(‡) d‡.
RC
=
o
for
set
0
These two calculations agree, so TRC is time-invariant. Physically,
the circuit is time-invariant because R and C are constant.
Section 6: Fundamentals of Continuous-Time Systems
6-216
yltl
is
=
1×11-15
causal
System properties: causality
Definition 6.3. A CT system T is causal if for all t œ R, the output
value y(t) depends only on the past input values {x(· )}· Æt .
Tard
present
I physical meaning: the system does not “look into the future”
You can often check causality by just inspecting the formula for y(t).
s 1 ≠(t≠· )/RC
I the RC circuit y(t) = 0t RC
e
x(· ) d· is causal, because the
integral uses only the values {x(· )} for 0 Æ · Æ t
I the differentiating system y(t) =
y(t) = lim
hæ0
d
dt x(t)
is not causal, because
x(t + h/2) ≠ x(t ≠ h/2)
,
h
and therefore y(t) depends on x(t + h/2) for (very) small h > 0.
Section 6: Fundamentals of Continuous-Time Systems
6-217
System properties: causality (equivalent definition)
I the following equivalent definition is easy to visualize
Definition 6.4. A CT system T is causal if for any time t0 and any two
inputs x, x̃ that satisfy x(t) = x̃(t) for all t Æ t0 , the corresponding
outputs y = T {x} and ỹ = T {x̃} satisfy y(t) = ỹ(t) for all t Æ t0 .
interpretation: if two
inputs agree up to some
time, then the
corresponding outputs
must also agree up to
that time.
Section 6: Fundamentals of Continuous-Time Systems
6-218