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LIB
x=
[
1
R →
)
Continuous
Time
Signals
:
Y
: [
R
Discrete - Time
Signals
:
∅
·
i
(x
: [
((,
,))
y
[
a
→
]
Vector Spaces
at of
objects
(that
satisfies the
following
ten
axioms
Closure Axionis
:
Al :
Closur
under Vector
addition
:
Va,
Y EW,
there exists an
unique
element
z
in I
denoted
by
E
: x
y.
A2 :
Closer under Scalar
multiplication
: VEW
,
UNER ,
the
existen
uniqu
IEU devoted
by
z
= &x
Axioms of Vector addition
A
: Xx ,
y
x+y
=
y
A
:
fx,
y ,
ztY , x
(y
z)
=
(x
y)
A5: Existence of a
Zero element 0
:
Then exists
an
element
devoted
by
0
St
. VxEY
,
x
O
= x
A6 :
Existence of a
Negation
:
VEW ,
the
element (-1) exist S.
t
. (+ 1-1(x
Axioms of
Scalar
Multiplication
AZ
:
UuE γ
. VL
BER
,
α ( BR
) = (α
B
] x
Ag :
foscy
trad FaER
.
xty
sceay
A
: Fat
V .
Fa
. BElR ,
(x
p)x
= xx+
Bx
Al0 : Unit
Multiplication
XxtV
,
tER
.
+ x
= x
L2A
Subspace
A
Subset S of
a vector
space
(SE2)
that is itself
a vector
space
Ex
:
S
= SVER" /v
=
(i)
=
0
S.
t .
Only
cheek
: a)
is
O an
elemet of
S? (is
S now
empty
?
b) is S
closed under vector addition?
C)
is S
closed
under scalar multiplication
Norm
:
A function
11
. 11
:
V-R is
a
norm if F,
YE]
and
all HER
,
following properties
hold.
· Ikill I 0
w /
equality
itf
cc =
0
. I / abll
3 all
·
Ikty1I[(kII
11y
w/
equality
ift a
By
are colinar.
t-
Norm
: 1K/
K
,
(kn
:
Kl
t Norm
for
DT
Signals
:
1kll
:Kil
,
If 1111 ,
>.
Signals
is
absolutely
summable.
n
= c
Norm
: 1klla
:
(1,
+...
k) :
(*
III
:
Norm for
DT
Signals
:
1/
:G
:illa
energy
of the
Signal
co
:
1l xIIas
= max (1x
...,
Kn)
L 23
Distance ((x
,
y)
= 1/x-
y1)
3 * x complex
conjugate
:
changes
the Sign
of the
imaginary part
Inner
Product
:
J...
>: VxV +
(
=
> (x,
y)
y
,
x)
,
(x,
y
+ z) : (x ,
y
z)
,
(ax.
y)
= 2(x .
y)
(x. x) 28
w/ equality
iff x
= 0
(i
Inver
Product of Continous
Signals
: K.
Y
:
R
(x .
y
:
S@xHt(y
(t)d+
Carchy
.
Schwarz Inequalities
:
,
y
1kIIIIy
~
angle
between-vectors my
Cosie Law
Angle
Between Vectors
: -
IIIIIII(y)/IIy
=
, Losing
:
<
:
<)
: (i
. j) =>
orthgouliff
(x,
=
Linear
Dependence
:
S
:
Ea
,.. .,
a
3 in
W
is
dependent
if
a
set
Sx
. ,.. ., 13 of Scalars (not all zero)
exists
S.
t. G
,
a +... + apar
=
.
Linea
Independence
:
S
= Sa....., ap
is
lively
independent
if a
,
a
, ... than =O is tore
iff
x
... xk
= 0
An
infinite
fut
of
vectors
J
:
Sa ,,
an
,
... 3 is
Linearly
independent
if
every
finite Subset
of S
is
limely
independent
Span
:
a vector
b
is
in the
Span
of
a Set
S
= Sa,... ach of vectors
if
b
= a ,
a , +..
dai
·
Span
of Set S
: The
Get
of
all
live combinations
of a ......
ai
Span(S)
:
Se (QE
Basis
:
A
Linerly
independent
at S
. 20 ,
...
of Vectors
Spans
a vector
space
L
3 A
Complex
exponentials
:
x(t)
: eit
,
: 1 ,
35/t)
: < (t) : jeit
isint >
xt)
: eit
: cost+
is
int.
ict
Euler's
formula
:
eit
= cost + isint
,
ect
:
cost
=
cosSut) +
·Sin(nt)
w
frequency
in
rad/s,
clockwise WO counter
clockwise
ettect : Zcost ,
cost
:
eteit
krt
%
u
Stationary
w(g
clockwite
e
et 225 t
tourier
,
confizie.t.
L 3 B
Los
contains two fraquencias
.
DT Frequencies
CT
Periodicity
: x
: R-D => x
is
p-periodic
if
the
exists a
post
.
xlt
p)
:
(t)
WLER
=> The smallest
p
for
which
the
above holds is called the
fadamtal
period
od
.
8
:
,
W
=
28 lad/s)
DT
Periodicity
-D
is
p-periodic
if <(n
p)
= <(n) Une
For DT
Signals
, first
determin
the
fundamental
period p
, and
only
the look at
w
For
CT-Signals
,
not an
issue
: w
= p
:
L4A Forrier
Analysis
· Time Forrier
Series
(DTFS)
Only
need to
study
for
periodic
DT
Signals
. (DFTS)
·
Decomposition
of a
signal
into
a
Liver Combination of Individual Funquencies
.
DFT
: Discute Forrier
Transform
A
change
of basis. The
new
basic is
directly
related to
the consistent
frequencies
.
…
2
Peridic
sign.in
!“
! 1
11
1
:
”
Pucmy
B
, eay
....
…
,
:
"
℃
Y
la
] : 1 - cion
. :..
.
(n)
: (
= ei
=
c0)(in)
isin(iu) =
((u] : X
.
e
o"
+ X
,
e
** "
for
any
p
Signals
.
X: X , are
forrier
coefficients
(DTFS Loeffs)
& x : (2)
: Xo (i)
. (.
i)
{
?
for p-periodic
, wo :*, <(n)
: Xoeor + X
,
eirorXeizwou
+...
Xp- ,
eW
3 ( [
n ]
:
Xkeiknoon x(n) : 3econ +
e
ih
X
eipwon
is
not needed becaude Wo :,wop
:
IT
"PWon
intcos
(2πn)
lu Geroral ,to :
:
=
Subspace
of DT
Signals
:
[x
: D-) [(x(n
,
2 ,
... 3
. VEL)
fr
p
: c
,
ω o
=π
: 3
x
=
(
""
]
,
[u
}
ek
"
,
K
:
[ →
40 :
(
:
1 ,
.
:
{
}
=
(
]
poif
Aoy
.
Ca
]
yOsyCuy
,
bupelre
Respone
:Letscln
} :
lu}
h
( @
L
Mntnd
I :lelw
=
aghonse
'
wn,sIkw
]:
kGo
}e
: woeh
{
aew
!
t ☆
w ,
H : R
→ 4
Mellad
I:
eigo
property
, afu
] :
ciwen
:rkw
ew^
,
ylu}
lef {
u]:
ec ω^= 3 xlu
lJ:
e :
wln
'
:
e"
eawn
4
lwleiw
^
:
e
ω^
x
e:
^
elω^
HKw
}
_
This
filter
favors low frog (around
even
multiples
of
TT)
Muguslole
:flCw )
:
(
)
e
cos e
Hb ):
| Kos )e \
cos
(
'\e 1
kws ) |
disfavors
high fre (aord odd multiples
of
T
...
Low-pass
filter
☆
Firrt Ouder 1 IR
Fooz
Respoute
:
ySu
= α
ySu
. l
]
talbn]
,
him]
: α
"
n
[n]
,
Y
"
) cO
MoShdI: tHlw
}:Y
hlnd
e '
aws
MithdI
:Lats (a
] : eawn
,
y
(
u
}, “
'
"
miu e
' w "
: (
Re '"w
)^
,
B
=
Le
' "
,
IBl= lalle
' vg
Chubeivo ,
n
= = C
Bounded
.
Input
Bounded
Output
Stability
: BIBOstable if for
any
bounded import
Signal c
,
the
corresponding
output signal
y
is also bonded .
VEX
St. (n)1[Ba
UNE
,
the 70?
By as
s.
t .
→
Bounded
Signal
:
ADT-Signal
is bonded if
the
exists a
non-negative
woul number
Bir St
.
k(n)/[Bic
FREE
IyKnyl
E 1 ☆
Z
Un tR
A
DILT
1 iS B 113 O STuble i 88
lheuylEs
15
InCayl
so Eurf
bounded
impefprodvvesabunded
ortpot
IfHis
B( BOStusk
=
Incn
1
508 Proof
n=
(n
- :
AqSn)
Evolution
3
: C-R scalar impt
to C
EIR'
**
Ortput
Matrin
LCDEs &State-
Span
Reps
.
Setop
:
y(ul
:
(q(n)
Ortpot
Equ
y
: c + RScalarortput
:
/@nl]EIR" Statevestor
Ai
State
.
Trancitio-Miti
is
□ E
R
L14BLCCDEs
defines
how
you
relates to
its
past
values & x(U)
DA 4
:
Delay
Bloks :
sx
{ur- t
3
Adders
→
ty
,
Gain/nult '
iplier
0 r
