complet numbers
2 = X + it
RL8o
j= F
R =++2
-
o=
Rose
ran" (
z = 2
X YIRSM g
grad angle Tan
sin LOS
I OCOC Ye t T
T
Z
< +
-
3
-
+
TheT
-
-
~ t -
<o
-
-
je
2=
Reje e = lose +isine
=
Taylor : f(x) Scos() iSm+
E
=l
macurin fl
@lox.
Ficos =
-
=
fetti
j0
2
!
~ 1 + ix -- ... cos() + isin
,Trig identities erlers identity
e
je
= cos(o) + ism(0) eiπ = - 1
58
isin (0)
-
e = cos(e) -
orejo = zcose
20 =o _ = I sne
Complex Conjugate , swap all Imaginary variables
Re neYuiNe
2= a + jb >
- 2= a -
1)
z = fee >
- rese
functions line the domain (infuil to The
codomain (CPUT)
if f(z) : 22+ ·
and Zo : Hj2 ,
what is
F(20) ?
2( jz) +
+ 12+ j4 + 4
Olerations on functions-
1. Additionbo a funcion >
- G(t) = fil + X- Vertical Swift
2
. multiplication to function-- G(i) : xf(T) Magnitude
-
a
Shift
.
3 Add/supiraci functions >
- Glitt fil
,Multiply functions >
-
His . G(i)
Time Snifi >
-
Gl = fi-io) left/right Sniff
-
Time reversal - GH) = -
fli)-mirror across y X
Time Scaling >
- G(T) = f(ai)-Streich/Compress X
M
( I
function :
S
Unot star T
1/ -
ULT)
40
=
Continuous ; 0 .
even function : symmetry across Y
odd function : Origin Symmetry
Dirac Peta function :
↑
area is I
G(n) =
)! -
S(i)
(ai 0)
=
=
8 + 10
Properties
f(i)(i) = EG) fol
G
FLO)
SFLT) &(T)dT =
Sifting
-
es
Lil-Linear Time Inveriant (sysical
Memery-Dependent on other man the resent
, yn = 10x (n 1) -
memory
100 x (n)
no memory
Yn= cos(2 +
-
memory
Yn =
x)in)
Y (R + 5) Xca) no Memery
Yet X(k) Memor
k= =
and 1-1 function , an inverse exists
Y(i) = f(x(T)) =
Y( + 1 = 1x (3i)/
Gail
memory,
not
not
Invertable
invertible
-
8188 sinuity : If ((6)1 < 8 for all +, then
T C exists where X(c)( /
Y(i1 = not 8 180
Xd + X Chees lands
-
80
Casuality : if a system Is dependent Past er Present
In Purs (no futures
Y(T) = x (5)
.
Past ,
cause
xH) = X (ii) future ,
non causal
2 = X + it
RL8o
j= F
R =++2
-
o=
Rose
ran" (
z = 2
X YIRSM g
grad angle Tan
sin LOS
I OCOC Ye t T
T
Z
< +
-
3
-
+
TheT
-
-
~ t -
<o
-
-
je
2=
Reje e = lose +isine
=
Taylor : f(x) Scos() iSm+
E
=l
macurin fl
@lox.
Ficos =
-
=
fetti
j0
2
!
~ 1 + ix -- ... cos() + isin
,Trig identities erlers identity
e
je
= cos(o) + ism(0) eiπ = - 1
58
isin (0)
-
e = cos(e) -
orejo = zcose
20 =o _ = I sne
Complex Conjugate , swap all Imaginary variables
Re neYuiNe
2= a + jb >
- 2= a -
1)
z = fee >
- rese
functions line the domain (infuil to The
codomain (CPUT)
if f(z) : 22+ ·
and Zo : Hj2 ,
what is
F(20) ?
2( jz) +
+ 12+ j4 + 4
Olerations on functions-
1. Additionbo a funcion >
- G(t) = fil + X- Vertical Swift
2
. multiplication to function-- G(i) : xf(T) Magnitude
-
a
Shift
.
3 Add/supiraci functions >
- Glitt fil
,Multiply functions >
-
His . G(i)
Time Snifi >
-
Gl = fi-io) left/right Sniff
-
Time reversal - GH) = -
fli)-mirror across y X
Time Scaling >
- G(T) = f(ai)-Streich/Compress X
M
( I
function :
S
Unot star T
1/ -
ULT)
40
=
Continuous ; 0 .
even function : symmetry across Y
odd function : Origin Symmetry
Dirac Peta function :
↑
area is I
G(n) =
)! -
S(i)
(ai 0)
=
=
8 + 10
Properties
f(i)(i) = EG) fol
G
FLO)
SFLT) &(T)dT =
Sifting
-
es
Lil-Linear Time Inveriant (sysical
Memery-Dependent on other man the resent
, yn = 10x (n 1) -
memory
100 x (n)
no memory
Yn= cos(2 +
-
memory
Yn =
x)in)
Y (R + 5) Xca) no Memery
Yet X(k) Memor
k= =
and 1-1 function , an inverse exists
Y(i) = f(x(T)) =
Y( + 1 = 1x (3i)/
Gail
memory,
not
not
Invertable
invertible
-
8188 sinuity : If ((6)1 < 8 for all +, then
T C exists where X(c)( /
Y(i1 = not 8 180
Xd + X Chees lands
-
80
Casuality : if a system Is dependent Past er Present
In Purs (no futures
Y(T) = x (5)
.
Past ,
cause
xH) = X (ii) future ,
non causal
