# JWSP-Lec20 ```System properties: linearity
✗
→
☒
-51=17&times;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
]&yuml;
^ &yuml;
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&times;111&times;-11-1
+
FEW
'
I on the other hand
–Tsq {x}(t) + –
˜ Tsq {x̃}(t) =
0&times;11-1
&quot;
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≠&middot; )/RC
e
vs (&middot; ) d&middot;
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≠&middot; ) (–vc (&middot; ) + –
˜ ṽSc (&middot; )) d&middot;
0
⁄ t
⁄ t
1 ≠(t≠&middot; )/RC
1 ≠(t≠&middot; )/RC
=–
e
v
(&middot;
)
d&middot;
+
–
˜
ṽSc (&middot; ) d&middot;
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&middot; (t) = x(t ≠ &middot; ) 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&middot; = T {x&middot; } for all time shifts &middot; 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&middot; (t) =
I# it
1&times;(1--4)
'
I if we instead shift the input signal to be x&middot; (t) = x(t ≠ &middot; ), we compute
I # 21
Tsq {x&middot; }(t) =
( ✗ 11-15=1&times;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) &middot; A sin(&Ecirc;0 t)
I if we simply shift the obtained output, we obtain
1&yen;11
y&middot; (t) =
ylt-i-1-xlt-c-1.Asinlw.lt -4 )
I if we instead shift the input signal to be x&middot; (t) = x(t ≠ &middot; ), we compute
(# 2)
Tmod {x&middot; }(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 &middot; œ R, we have
⁄ t≠&middot;
1 ≠(t≠&middot; ≠‡)/RC
11
#
y&middot; (t) = y(t ≠ &middot; ) =
x(‡) d‡.
I
RC e
0
I if we instead shift the input signal by &middot; , the corresponding output is
I # 2)
TRC {x&middot; }(t) =
=
⁄
t≠&middot;
⁄
0
t
1 ≠(t≠›)/RC
e
x(›
RC
≠ &middot; ) d›
1 ≠(t≠&middot; ≠‡)/RC
e
x(‡) d‡
RC
≠&middot;
change
=
⁄
→
because
g-
t≠&middot;
0
✗
(g)
d- variables
=3
-
t
1 ≠(t≠&middot; ≠‡)/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&times;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(&middot; )}&middot; &AElig;t .
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≠&middot; )/RC
I the RC circuit y(t) = 0t RC
e
x(&middot; ) d&middot; is causal, because the
integral uses only the values {x(&middot; )} for 0 &AElig; &middot; &AElig; t
I the differentiating system y(t) =
y(t) = lim
h&aelig;0
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 &gt; 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 &AElig; t0 , the corresponding
outputs y = T {x} and ỹ = T {x̃} satisfy y(t) = ỹ(t) for all t &AElig; 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
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