Signals And Systems

    Quicktest 1answers. Signals and systems. collection editor: richard baraniuk. authors: thanos antoulas. richard baraniuk. steven cox. benjamin fite. roy ha. michael haag. ufdcimages.uflib.ufl.edu.

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Signals and Systems
Collection Editor:
Richard Baraniuk
Signals and Systems
Collection Editor:
Richard Baraniuk
Authors:
Thanos Antoulas
Richard Baraniuk
Steven Cox
Benjamin Fite
Roy Ha
Michael Haag
Matthew Hutchinson
Don Johnson
Ricardo Radaelli-Sanchez
Justin Romberg
Phil Schniter
Melissa Selik
JP Slavinsky
Online:
<http://cnx.org/content/col10064/1.11/ >
C O N N E X I O N S
Rice University, Houston, Texas
T
t
T
f (t) = f (T + t)
f (t) T
T
0
f f (t) = f (−t)
f
f (t) = −(f (−t))
f (t) =
1
2
(f (t) + f (−t)) +
1
2
(f (t) − f (−t))
f (t) + f (−t) f (t) −f (−t)
e (t) =
1
2
(f (t) + f (−t)) o (t) =
1
2
(f (t) − f (−t)) e (t) + o (t) = f (t)
f (t) = 0 t < t
1
< ∞ f (t) = 0
t > t
1
> −∞ t
1
f (t)
t
1
< f (t) < t
2
t
1
> −∞ t
2
< ∞
f (t)
∞ ≤ f (t) ≤ −∞
(|f (t) |)
2
E
f
=
Z
∞
−∞
(|f (t) |)
2
dt
E
f
L
p
k f k
p
=

Z
(|f (t) |)
p
dt

1
p
1 ≤ p < ∞
E
f
= (k f k
2
)
2
L
p
k f k
p
=

Z
(|f (t) |)
p
dt

1
p
p → ∞
L
∞
k f k
∞
= |f (t) |
E
f
=
∞
X
n=−∞

(|f [n] |)
2

k f k
p
=

Z
(|f (t) |)
p
dt

1
p
1 ≤ p < ∞
k f k
∞
= max
n
{|f [n] |}

k f k
p

p
=
N−1
X
n=0
((|f [n] |)
p
)
1 ≤ p < ∞
k f k
∞
= max
n=
{|f [n] |}
k f k
p
< ∞
N
f =










f [0]
f [1]
f [2]
f [...]
f [N −1]










=










f
0
f
1
f
2
f
N1










f ∈ C
N
f ∈
N
N
N = 3 f
l
p
[0, N − 1] =
n
f ∈ C
N
, k f k
p
< ∞
o
l
p
[0, N − 1] = C
N
f
f =












f [...]
f [−1]
f [0]
f [1]
f [2]
f [...]












C
∞
R
∞
l
p
(z) =
n
f, k f k
p
< ∞
o

k f k
p

p
=
∞
X
n=−∞
((|f [n] |)
p
)
1 ≤ p < ∞
k f k
∞
= max
n∈z
{|f [n] |}
f l
p
(z)
f ∈ l
p
(z) f /∈ l
p
(z)
f ∈ l
p
(z) p
f (t)
L
p
[T
1
, T
2
] =
n
f [T
1
, T
2
] , k f k
p
< ∞
o
k f k
p
=
Z
T
2
T
1
(|f (t) |)
p
dt
!
1
p
1 ≤ p < ∞
k f k
p
= |f (t) |
T
1
≤ t ≤ T
2
f L
p
[T
1
, T
2
]
f (t)
L
p
(R) =
n
f, k f k
p
< ∞
o
k f k
p
=

Z
∞
−∞
(|f [n] |)
p
dt

1
p
1 ≤ p < ∞
k f k
∞
= |f (t) |
−∞ < t < ∞
f ∈ L
p
(R)
f ∈ L
p
(R) f /∈ L
p
(R)
k f k
p
= ∞
•
•
kfk
p
T
p = 2
L
2
=
• E
f
= ∞
•
P
f
= lim
T →∞
Z
T
2
−
(
T
2
)
(|f (t) |)
2
dt
T
=
(
kfk
2
)
2
T
lim
T →∞
T
=
(
kfk
2
)
2
T
P
f
f
p
P
f
f
E
f
< ∞
P
f
< ∞ P
f
6= 0 → E
f
= ∞
f (t) = t
f (t − T ) f T
f (at) f a
f (t) f (−(at))
f (t) t at f (at) t
t −
b
a
f
`
a
`
t −
b
a
´´
= f (at − b)
x (t) = Acos (ωt + φ)
A ω φ ωt
2Ï€ft
T =
2Ï€
ω
A = 2 w = 2 φ = 0
f (t) = Be
st
s σ ω
s = σ + jω
f (t) = Be
αt
B α
α
• α < 0
• α > 0
t = 0
δ (t)
Z
∞
−∞
δ (t) dt = 1
u (t) =



0 t < 0
1 t ≥ 0
t
1
t
1
r (t) =







0 t < 0
t
t
0
0 ≤ t ≤ t
0
1 t > t
0
t
1
t
0
a −

2
a +

2
1

lim
→0
0
δ (t)
Z
∞
−∞
δ (t) dt = 1
f (t) δ (t) = f (0) δ (t)
R
∞
−∞
f (t) δ (t) dt =
R
∞
−∞
f (0) δ (t) dt
= f (0)
R
∞
−∞
δ (t) dt
= f (0)
δ (t − T ) δ (t) f (T )
• δ (αt) =
1
|α|
δ (t)
• δ (t) = δ (−t)
• δ (t) =
d
dt
u (t) u (t)
δ [n] =



1 n = 0
0
k
s [k] k δ [n − k]
s [n] =
∞
X
k=−∞
(s [k] δ [n − k])
δ (t) δ [n]
x
e
x
= 1 +
x
1
1!
+
x
2
2!
+
x
3
3!
+ . . .
e
x
=
∞
X
k=0

1
k!
x
k

e
x
1
+x
2
= (e
x
1
) (e
x
2
)
s
f (t) = Ae
st
= Ae
jωt
A t s s = jω
Ae
jωt
= Acos (ωt) + j (Asin (ωt))
s s = σ + jω
f (t) = Ae
st
= Ae
(σ+jω)t
= Ae
σt
e
jωt
S
S = Ae
σt
f (t) = Ae
σt
(cos (ωt) + jsin (ωt))
Re (f (t)) = Ae
σt
cos (ωt)
Im (f (t)) = Ae
σt
sin (ωt)
f [n] = Be
snT
= Be
jωnT
nT
∼
cos (ωt) =
e
jwt
+ e
−(jwt)
2
sin (ωt) =
e
jwt
− e
−(jwt)
2j
e
jwt
= cos (ωt) + jsin (ωt)
t = 0 σ
σ
σ
σ ω
ω
s
s
s (n) n = {. . . , −1, 0, 1, . . . }
δ (n − m) n = m
s (n) = e
j2Ï€f n
s (n) = Acos (2πfn + φ)
f

−

1
2

,
1
2

e
j2Ï€(f +m)n
= e
j2Ï€f n
e
j2Ï€mn
= e
j2Ï€f n
2Ï€
δ (n) =



1 n = 0
0
1
n
δ
n
m s (m) m
δ (n − m)
s (n) =
∞
X
m=−∞
(s (m) δ (n − m))
s (n) {a
1
, . . . , a
K
}
A
t = 0
H
H (kf (t)) = kH (f (t))
H
H (f
1
(t) + f
2
(t)) = H (f
1
(t)) + H (f
2
(t))
H (k
1
f
1
(t) + k
2
f
2
(t)) = k
2
H (f
1
(t)) + k
2
H (f
2
(t))
H H (f (t)) = y (t) H
T
H (f (t − T )) = y (t − T )
t
0
y (t
0
)
t
0
x (t)
y (t)
|y (t) | ≤ M
y
< ∞
|x (t) | ≤ M
x
< ∞
M
x
M
y
t
x L y x α
α
α β
t
t
0
t
0
x (t) x (t − t
0
)
x (t) x (t − t
0
)
x (t − t
0
) t
0
t
0
h (t) t < 0
f (t)
y (t)
d
n
dt
n
y (t) + a
n−1
d
n−1
dt
n−1
y (t) + ··· + a
1
d
dt
y (t) + a
0
y (t) = b
m
d
m
dt
m
f (t) + ··· + b
1
d
dt
f (t) + b
0
f (t)
n
X
i=0

a
i
d
i
dt
i
y (t)

=
m
X
i=0

b
i
d
i
dt
i
f (t)

a
n
= 1
y (t) f (t)
y (t) = y
i
(t) + y
s
(t)
y
i
(t)
y
s
(t)
f (t)
y
i
(t)
y
0
(t)
n
X
i=0

a
i
d
i
dt
i
y
0
(t)

= 0 , a
n
= 1
D

D
n
+ a
n−1
D
n−1
+ ··· + a
0

y
0
(t) = 0
y
0
(t) 0 t
n
y
0
(t) ,
d
dt
y
0
(t) ,
d
2
dt
2
y
0
(t) , . . .
o
e
st
s ∈ C
y
0
(t) = ce
st
, c 6= 0
c s
d
dt
y
0
(t) = cse
st
d
2
dt
2
y
0
(t) = cs
2
e
st
. . .

D
n
+ a
n−1
D
n−1
+ ··· + a
0

y
0
(t) = 0
c

s
n
+ a
n−1
s
n−1
+ ··· + a
1
s + a
0

e
st
= 0
t
s
n
+ a
n−1
s
n−1
+ ··· + a
1
s + a
0
= 0
s
{s
1
, s
2
, . . . , s
n
}
(s − s
1
) (s − s
2
) (s − s
3
) . . . (s − s
n
) = 0
c
1
e
s
1
t
c
2
e
s
2
t
. . . c
n
e
s
n
t
y
0
(t) = c
1
e
s
1
t
+ c
2
e
s
2
t
+ ··· + c
n
e
s
n
t
{c
1
, . . . , c
n
}
y
i
(t) = 0
y
i
(t)
n
X
i=0

a
i
d
i
dt
i
y (t)

=
m
X
i=0

b
i
d
i
dt
i
f (t)

f (t)
f (t)
f (t)
h (t)
m < n
d
n
dt
n
y (t) + a
n−1
d
n−1
dt
n−1
y (t) + ··· + a
1
d
dt
y (t) + a
0
y (t) = b
m
d
m
dt
m
f (t) + ··· + b
1
d
dt
f (t) + b
0
f (t)
Q
D
[y (t)] = P
D
[f (t)]
Q
D
[·] y (t)
Q
D
[y (t)] =
d
n
dt
n
y (t) + a
n−1
d
n−1
dt
n−1
y (t) + ··· + a
1
d
dt
y (t) + a
0
y (t)
P
D
[·] f (t)
h (t) = b
n
δ (t) + P
D
[y
n
(t)] µ (t)
m < n b
n
= 0 y
n

y
n−1
(0) = 1, y
n−2
(0) = 1, . . . , y (0) = 0

x (t) h (t)
y (t) =
Z
∞
−∞
x (τ ) h (t − τ ) dτ
∗
y (t) = x (t) ∗ h (t)
τ = t − τ
x (t) ∗ h (t) = h (t) ∗ x (t)
∼
f (t)
h (t)
f (Ï„)
t
Ï„
f (t)
δ
∆
(t) =



1
∆
−

∆
2

< t <
∆
2
0
lim
∆→0
δ
∆
(t) → h → lim
∆→0
h (t)
lim
∆→0
δ
∆
(t − n∆) → h → lim
∆→0
h (t − n∆)
lim
∆→0
f (n∆) δ
∆
(t − n∆) ∆ → h → lim
∆→0
f (n∆) h (t − n∆) ∆
lim
∆→0
P
n
(f (n∆) δ
∆
(t − n∆) ∆) → h → lim
∆→0
P
n
(f (n∆) h (t − n∆) ∆)
R
∞
−∞
f (τ ) δ (t −τ) dτ → h →
R
∞
−∞
f (τ ) h (t − τ ) dτ
f (t) → h → y (t) =
R
∞
−∞
f (τ ) h (t − τ ) dτ
t
t t
x (t) h (t)
Ï„
h (t − τ )
h (t − τ )
h (t − τ ) x (t)
t h (t − τ )
t
t < 0
0 ≤ t < 1
1 ≤ t < 2
t ≥ 2
x (t) h (t − τ) 0
0 ≤ t < 1
y (t) =
R
t
0
1dτ
= t
1 ≤ t < 2
h (t − τ ) t − 1
y (t) =
R
1
t−1
1dτ
= 1 − (t − 1)
= 2 − t
y (t) =













0 t < 0
t 0 ≤ t < 1
2 − t 1 ≤ t < 2
0 t ≥ 2
x (t) ∗ h (t)
x (t)
f
1
(t) ∗ (f
2
(t) ∗ f
3
(t)) = (f
1
(t) ∗ f
2
(t)) ∗ f
3
(t)
y (t) = f (t) ∗ h (t)
= h (t) ∗ f (t)
y (t) =
Z
∞
−∞
f (τ ) h (t − τ ) dτ
τ = t − τ
y (t) =
R
∞
−∞
f (t − τ ) h (τ ) dτ
=
R
∞
−∞
h (τ ) f (t − τ ) dτ
f (t) ∗ h (t) = h (t) ∗ f (t)
f
1
(t) ∗ (f
2
(t) + f
3
(t)) = f
1
(t) ∗ f
2
(t) + f
1
(t) ∗ f
3
(t)
c (t) = f (t) ∗ h (t)
c (t − T ) = f (t − T ) ∗ h (t)
c (t − T ) = f (t) ∗ h (t − T )
f (t) ∗ δ (t) = f (t)
δ (t)
f (t) ∗ δ (t) =
Z
∞
−∞
δ (τ ) f (t − τ) dτ
δ (τ ) = 0 τ 6= 0
f (t) ∗ δ (t) =
R
∞
−∞
δ (τ ) f (t) dτ
= f (t)
R
∞
−∞
(δ (τ )) dτ
δ (τ ) τ = 0
f (t) ∗ δ (t) = f (t)
Duration (f
1
) = T
1
Duration (f
2
) = T
2
Duration (f
1
∗ f
2
) = T
1
+ T
2
Duration (f
1
) = N
1
Duration (f
2
) = N
2
Duration (f
1
∗ f
2
) = N
1
+ N
2
− 1
f h f ∗ h
A |f (t) | < A
h (t)
Z
∞
−∞
|h (t) |dt < ∞
h (n)
∞
X
n=−∞
(|h (n) |) < ∞
jω
jω
s (n) n
0
s (n − n
0
) n
0
> 0 n
0

s (n − n
0
) ↔ e
−(j2πf n
0
)
S

e
j2Ï€f

S (a
1
x
1
(n) + a
2
x
2
(n)) = a
1
S (x
1
(n)) + a
2
S (x
2
(n))
S (x (n)) = y (n)
S (x (n − n
0
)) = y (n − n
0
)
y (n) = a
1
y (n − 1) + ··· + a
p
y (n − p) + b
0
x (n) + b
1
x (n − 1) + ··· + b
q
x (n − q)
y (n) y (n − l) l = {1, . . . , p}
x (n)
p q {a
1
, . . . , a
p
} {b
0
, b
1
, . . . , b
q
}
a
0
y (n)
a
0
y (1) y (0)
y (−1)
p
p = 1 q = 0
y (n) = ay (n − 1) + bx (n)
y (n − 1)
x (n) = δ (n)
n = 0
y (0) = ay (−1) + b
y (−1)
y (−1) = 0 y (0) = b n > 0
y (n) = ay (n − 1) , n > 0
y (n) = ay (n − 1) + bδ (n)
n x (n) y (n)
−1 0 0
0 1 b
1 0 ba
2 0 ba
2
0
n 0 ba
n
b
a
b a
a
a = 1 a
−1 a = −1
b −b |a| > 1
1
n
y(n)
a = 0.5, b = 1
n
-1
1
y(n)
a = –0.5, b = 1
n
0
2
4
y(n)
a = 1.1, b = 1
x(n)
n
n
a
n
n
a
b = 1
|a| = 1
y (n) = a
1
y (n − 1) + ··· + a
p
y (n − p) + b
0
x (n) + b
1
x (n − 1) + ··· + b
q
x (n − q)
y (n + 1) x (n + 1)
y(n)
n
1
5
a
y (n) =
1
q
(x (n) + ··· + x (n − q + 1))
1
q
n = {0, . . . , q − 1}
q
1
q
q
q = 7
y [n] =
∞
X
k=−∞
(x [k] h [n − k])
y [n] = x [n] ∗ h [n]
k = n − k
x [n] ∗ h [n] = h [n] ∗ x [n]
H
y [n] = H[x [n]]
= H

P
∞
k=−∞
(x [k] δ [n − k])

=
P
∞
k=−∞
(H[x [k] δ [n −k]])
=
P
∞
k=−∞
(x [k] H[δ [n −k]])
=
P
∞
k=−∞
(x [k] h [n − k])
x [n]
H
H[·] x [k]
H[·] H[·]
∼
x m h k
y [n] =
m−1
X
l=0
(x [l] h [n − l])
n x
l
h
−l
x h
h 0
1
n = 4
x [n] h [n]
y [n] = x [n] ∗ h [n]
=
P
∞
k=−∞
(x [k] h [n − k])
N
n ≥ N
n ≥ N
n = {0, 1, . . . , N − 1}
Y [k] y [n]
Y [k] = F [k] H [k]
0 ≤ k ≤ N − 1
y [n]
y [n] =
1
N
N−1
X
k=0

F [k] H [k] e
j
2Ï€
N
kn

F [k] =
P
N−1
m=0

f [m] e
(−j)
2Ï€
N
kn

y [n] =
1
N
P
N−1
k=0

P
N−1
m=0

f [m] e
(−j)
2Ï€
N
kn

H [k] e
j
2Ï€
N
kn

=
P
N−1
m=0

f [m]

1
N
P
N−1
k=0

H [k] e
j
2Ï€
N
k(n−m)

h [((n − m))
N
] =
1
N
P
N−1
k=0

H [k] e
j
2Ï€
N
k(n−m)

y [n] =
N−1
X
m=0
(f [m] h [((n − m))
N
])
0 ≤ n ≤ N − 1
y [n] ≡ (f [n] ~ h [n])
~
• f [m] h [((−m))
N
]
• h [((−m))
N
] n h [((n − m))
N
]
h [n] n
• f [m] h [((n − m))
N
]
sum = y [n]
• 0 ≤ n ≤ N − 1
• h [((−m))
N
]
f [m] sum y [0] = 3
• h [((1 − m))
N
]
f [m] sum y [1] = 5
• h [((2 − m))
N
]
f [m] sum y [2] = 3
• h [((3 − m))
N
]
f [m] sum y [3] = 1
• f [n] F [k] h [n] H [k]
• Y [k] = F [k] H [k]
• Y [k] y [n]
2 N
y [n] =
N−1
X
m=0
(f [m] h [((n − m))
N
])
n N N − 1
N N
2
N (N −1) O

N
2

•
d
dt
y (t) − y (t) = x (t)
•
x [n] − x [n − 1]
d
dt
x (t)
• x [n] x [n]
N
N
X
K=0
(a
K
y [n − K]) =
M
X
K=0
(b
K
x [n − K])
y [n] =
1
M
1
+ M
2
+ 1
M
2
X
K=M
1
(x [n − K])
M
1
= 0 M
2
= M a
K
=



1 K = 0
0
b
K
=



1
M+1
0 ≤ K ≤ M
0
M = 2 y [n] =
1
3
P
2
K=0
(x [n − K])
y [n] =
N
X
K=1
(α
K
y [n − K]) + x [n]
N
X
K=0
(a
K
y [n − K]) = x [n]
a
K
=







1 K = 0
−α
K
1 ≤ K ≤ N
0
N = 2 y [n] =
P
2
K=1
(α
K
y [n − K]) + x [n]
∼ ∼
∼ ∼
•
• x [n] y
p
[n]
N
X
k=0
(a
k
y
p
[n − k]) =
M
X
k=0
(b
k
x [n − k])
P
N
k=0
(a
k
y
h
[n − k]) = 0
N
X
k=0
(a
k
(y
p
[n − k] + y
h
[n − k])) =
M
X
k=0
(b
k
x [n − k])
y
p
[n] y
h
[n]
y (n) = y
p
(n) + y
h
(n)
• x [n]
N
X
k=0
(a
k
y
p
[n − k]) =
M
X
k=0
(b
k
x [n − k])
!
y
p
[n]
•
N
X
k=0
(a
k
y
h
[n − k]) = 0
y
h
[n]
•
y [n] = y
p
[n] + y
h
[n]
•
• y
h
[n] T
·
·
·
• T
→
y [n] − ay [n − 1] = x [n]
|a| < 1 x [n] = δ [n]
• n ≥ 0
y
p
[0] = (δ [0] → 1) + a (y
p
[−1] → 0) = 1
y
p
[1] = (δ [1] → 0) + a (y
p
[0] → 1) = a
y
p
[2] = (δ [2] → 0) + a (y
p
[1] → a) = a
2
y
p
[n] = a
n
n ≥ 0
• x [n] = 0 y
h
[n] − ay
h
[n − 1] = 0
y
h
[n] = ay
h
[n − 1]
y
h
[n] = ca
n
n
•
y [n] = y
n
[p] + y
h
[n]
= a
n
u [n] + ca
n
c
c = 0
y [n] = a
n
u [n]
c = −1
y [n] = −(a
n
u [−n])
x = δ
0
= 0 y
p
y
p
= h
H = F
H
δ
0
=
1
√
N
N−1
X
n=0

h [n] e
−
(
2Ï€
N
kn
)

=
1
√
N
N−1
X
n=0

δ
0
[n] e
−
(
2Ï€
N
kn
)

δ
0
[n] =



1 n = 0
0
=
1
√
N
{x
1
, x
2
, . . . , x
k
} , x
i
∈ C
n
{x
1
, x
2
, . . . , x
n
}
c
1
x
1
+ c
2
x
2
+ ··· + c
n
x
n
= 0
c
1
= c
2
= ··· = c
n
= 0
x
1
=


3
2


x
2
=


−6
−4


x
2
= −2x
1
⇒ 2x
1
+ x
2
= 0
3
-6
2
4
x
1
=


3
2


x
2
=


1
2


c
1
x
1
= −(c
2
x
2
)
c
1
= c
2
= 0
3
2
1
{x
1
, x
2
, x
3
}
x
1
=


3
2


x
2
=


1
2


x
3
=


−1
0


3
2
1
-1
m C
n
m > n
{x
1
, x
2
, . . . , x
k
}
{x
1
, x
2
, . . . , x
k
}
span ({x
1
, . . . , x
k
}) = {α
1
x
1
+ α
2
x
2
+ ··· + α
k
x
k
, α
i
∈ C
n
}
x
1
=


3
2


x
1
x
1
=


3
2


x
2
=


1
2


C
2
C
n
C
n
n C
n
e
i
=
















0
0
1
0
0
















1 i C
n
{e
i
, i = [1, 2, . . . , n] }
{e
i
, i = [1, 2, . . . , n] }
h
1
=


1
1


h
2
=


1
−1


{h
1
, h
2
} C
2
C
2
{b
1
, . . . , b
2
} C
n
x ∈ C
n
b
i
x = α
1
b
1
+ α
2
b
2
+ ··· + α
n
b
n
, α
i
∈ C
x =


1
2


x {e
1
, e
2
}
x = e
1
+ 2e
2
x {h
1
, h
2
}
x
f (t) e
jω
0
nt
×
A × A C
n
Ax = b
x b ×
A v ∈ C
n
Av = λv
λ A v
Ax = b
v A
λ
Av = λv
A
v
1
v
2
A =


3 0
0 −1


λ
1
λ
2
v
1
=


1
1


v
2
=


1
−1


A A =


3 −1
−1 3


λ ∈ C v 6= 0 0
λ
Av = λv
Av − λv = 0
(A − λI) v = 0
λv = λIv
I
I =







1 0 . . . 0
0 1 . . . 0
0 0
0 . . . . . . 1







A − λI
A A − λI
A =


a
11
a
12
a
21
a
22


A − λI =


a
11
− λ a
12
a
21
a
22
− λ


(A − λI) v = 0 v 6= 0 A − λI
det (A − λI) = 0
n
A
A =


3 −1
−1 3


A − λI =


3 − λ −1
−1 3 − λ


det (A − λI) = (3 − λ)
2
− (−1)
2
= λ
2
− 6λ + 8
λ = {2, 4}
A
A =


a
11
a
12
a
21
a
22


A − λI =


a
11
− λ a
12
a
21
a
22
− λ


det (A − λI) = λ
2
− (a
11
+ a
22
) λ − a
21
a
12
+ a
11
a
22
det (A − λI) = c
n
λ
n
+ c
n−1
λ
n−1
+ ··· + c
1
λ + c
0
= 0
λ
i
Av = λ
i
v
A





v
1
v
n





=





λ
1
v
1
λ
n
v
n





n n n
A {v
1
, v
2
, . . . , v
n
} C
n
{v
1
, v
2
, . . . , v
n
}
x ∈ C
n
x = α
1
v
1
+ α
2
v
2
+ ··· + α
n
v
n
{α
1
, α
2
, . . . , α
n
} ∈ C x A
Ax = A (α
1
v
1
+ α
2
v
2
+ ··· + α
n
v
n
)
Ax = α
1
Av
1
+ α
2
Av
2
+ ··· + α
n
Av
n
Ax = α
1
λ
1
v
1
+ α
2
λ
2
v
2
+ ··· + α
n
λ
n
v
n
= b
x =
X
i
(α
i
v
i
)
x
b =
X
i
(α
i
λ
i
v
i
)
x Ax
A x
A =


3 −1
−1 3


x =


5
3


A A C
n
x {v
1
, v
2
, . . . , v
n
} A A x
A
{v
1
, v
2
, . . . , v
n
} A C
n
{v
1
, v
2
, . . . , v
n
}
x {v
1
, v
2
, . . . , v
n
}
{v
1
, v
2
, . . . , v
n
} A C
n
A n
λ
i
6= λ
j
, i 6= j
i j A n {v
1
, v
2
, . . . , v
n
}
C
n
n
det (A − λI) = c
n
λ
n
+ c
n−1
λ
n−1
+ ··· + c
1
λ + c
0
= 0
n
x {v
1
, v
2
, . . . , v
n
}
{α
1
, α
2
, . . . , α
n
} ∈ C
x = α
1
v
1
+ α
2
v
2
+ ··· + α
n
v
n
{v
1
, v
2
, . . . , v
n
}
× V
V =





v
1
v
2
. . . v
n





x =





v
1
v
2
. . . v
n










α
1
α
n





x = V α
α
α = V
−1
x
V n
V
x = V α





x
1
x
n





= V





α
1
α
n





α x
x = x
1







1
0
0







+ x
2







0
1
0







+ ··· + x
n







0
0
1







x = α
1





v
1





+ α
2





v
2





+ ··· + α
n





v
n





V x {v
1
, v
2
, . . . , v
n
}
{v
1
, v
2
, . . . , v
n
} b
b = Ax = A (α
1
v
1
+ α
2
v
2
+ ··· + α
n
v
n
)
Ax = α
1
λ
1
v
1
+ α
2
λ
2
v
2
+ ··· + α
n
λ
n
v
n
= b
Ax =





v
1
v
2
. . . v
n










λ
1
α
1
λ
1
α
n





Ax = V Λα
Ax = V ΛV
−1
x
Λ
Λ =







λ
1
0 . . . 0
0 λ
2
. . . 0
0 0 . . . λ
n







x A
A = V ΛV
−1
α = V
−1
x
b =
X
i
(α
i
λ
i
v
i
)
x A





x
1
x
n





→





α
1
α
n





→





λ
1
α
1
λ
1
α
n





→





b
1
b
n





x V
−1
α
Λ
V b
A v
Av = λv
v λ
A v
A
λ C
n
x ∈ C
n
{α
1
, α
2
, α
n
} ∈ C
x = α
1
v
1
+ α
2
v
2
+ ··· + α
n
v
n
Ax = b
x =
X
i
(α
i
v
i
)
b =
X
i
(α
i
λ
i
v
i
)
x b
x →

α = V
−1
x

→

ΛV
−1
x

→ V ΛV
−1
x = b
x V
−1
α
A Λ
V b
H f (t)
y (t)
H[f (t)] = y (t)
H [f (t)] = y (t) f t H
N N A x ∈ C
N
b ∈ C
N
Ax = b
Ax = b x b C
N
A N N
A v ∈ C
N
Av = λv λ ∈ C
Av = λv v ∈ C
N
A
H f (t)
H[f (t)] = λf (t) , λ ∈ C
H [f (t)] = λf (t) f H
H
λ
e
st
s ∈ C
H

e
st

= λ
s
e
st
H
ˆ
e
st
˜
= λ
s
e
st
H
{e
st
, s ∈ C }
e
st
h (t) H
H[e
st
] =
R
∞
−∞
h (Ï„ ) e
s(t−τ)
dτ
=
R
∞
−∞
h (Ï„ ) e
st
e
−(sτ)
dτ
= e
st
R
∞
−∞
h (Ï„ ) e
−(sτ)
dτ
t λ
s
H

e
st

= λ
s
e
st
λ
s
s H
H (s) ≡ λ
s
e
st
H (s)
e
st
f (t)
e
−(at)
u (t)
1
j2Ï€f +a
e
(−a)|t|
2a
4Ï€
2
f
2
+a
2
p (t) =



1 |t| <
∆
2
0 |t| >
∆
2
sin(πf∆)
Ï€f
sin(2Ï€W t)
Ï€t
S (f) =



1 |f| < W
0 |f| > W
a
1
s
1
(t) + a
2
s
2
(t) a
1
S
1
(f) + a
2
S
2
(f)
s (t) ∈ R S (f ) = S (−f )
∗
s (t) = s (−t) S (f) = S (−f)
s (t) = −(s (−t)) S (f ) = −(S (−f))
s (at)
1
|a|
S

f
a

s (t − τ ) e
−(j2πf τ )
S (f)
e
j2Ï€f
0
t
s (t) S (f − f
0
)
s (t) cos (2Ï€f
0
t)
S(f−f
0
)+S(f+f
0
)
2
s (t) sin (2Ï€f
0
t)
S(f−f
0
)−S(f+f
0
)
2j
d
dt
s (t) j2Ï€fS (f )
R
t
−∞
s (α) dα
1
j2Ï€f
S (f) S (0) = 0
t ts (t)
1
−(j2π)
d
df
S (f)
R
∞
−∞
s (t) dt S (0)
s (0)
R
∞
−∞
S (f) df
R
∞
−∞
(|s (t) |)
2
dt
R
∞
−∞
(|S (f) |)
2
df
x
1
− x
2
+ 2x
3
= 0
x =
3
2
h
1
+
−1
2
h
2
v
1
=


1
0


v
2
=


0
1


λ
1
= 3
λ
2
= −1
Av
1
=


3 −1
−1 3




1
1


=


2
2


Av
2
=


3 −1
−1 3




1
−1


=


4
−4


A
λ
1
= 2
λ
2
= 4
A v
1
v
2
Ax =


3 −1
−1 3




5
3


=


12
4


v
1
=


1
1


v
2
=


1
−1


λ
1
= 2
λ
2
= 4
x
x = 4v
1
+ v
2
x =


5
3


= 4


1
1


+


1
−1


Ax = A (4v
1
+ v
2
) = λ
i
(4v
1
+ v
2
)
Ax = 4 × 2


1
1


+ 4


1
−1


=


12
4


A
f (t) = f (t + mT ) m ∈ Z
T > 0
T
R
R f (t
0
) = f (t
0
+ T )
[0, T ]
[a, a + T ]
f (t) [0, T ]
f (t) T ∈ R T
H e
st
H (s) ∈ C
y (t) = H (s) e
st
H y (t)
H y (t)
c
1
e
s
1
t
+ c
2
e
s
2
t
→ c
1
H (s
1
) e
s
1
t
+ c
2
H (s
2
) e
s
2
t
X
n

c
n
e
s
n
t

→
X
n

c
n
H (s
n
) e
s
n
t

H H
e
s
n
t
H (s
n
) ∈ C
f (t)
• H f (t) H (s)
• H f (t)
f (t)
f (t) =
∞
X
n=−∞

c
n
e
jω
0
nt

ω
0
=
2Ï€
T
f (t) c
n
f (t) f (t) ∈ L
2
[0, T ]
f (t) f (t)
c
n
e
jω
0
nt
f (t)
f (t)

e
jω
0
nt
, n ∈ Z

f (t)
f (t) = cos (ω
0
t)
f (t) = sin (2ω
0
t)
f (t) = 3 + 4cos (ω
0
t) + 2cos (2ω
0
t)
∼
f (t) c
n
c
n
=
1
T
Z
T
0
f (t) e
−(jω
0
nt)
dt
{c
n
, n ∈ Z } f (t)
c
n
f (t)
c
n
f (t)
f (t) f (t)
f (t) =



1 |t| ≤ T
0
f (t)
f (t) =
∞
X
n=−∞

c
n
e
jω
0
nt

f (t)
c
n
=
1
T
Z
T
0
f (t) e
−(jω
0
nt)
dt
ω
0
=
2Ï€
T
[0, T ]
[a, a + T ] T
f (t) =
∞
X
n=−∞

c
n
e
jω
0
nt

c
n
f (t)
c
n
e
−(jω
0
kt)
k ∈ Z
f (t) e
−(jω
0
kt)
=
∞
X
n=−∞

c
n
e
jω
0
nt
e
−(jω
0
kt)

T
Z
T
0
f (t) e
−(jω
0
kt)
dt =
Z
T
0
∞
X
n=−∞

c
n
e
jω
0
nt
e
−(jω
0
kt)

dt
Z
T
0
f (t) e
−(jω
0
kt)
dt =
∞
X
n=−∞
c
n
Z
T
0
e
jω
0
(n−k)t
dt
!
R
T
0
e
jω
0
(n−k)t
dt
n = k
n 6= k n = k
Z
T
0
e
jω
0
(n−k)t
dt = T , n = k
n 6= k
Z
T
0
e
jω
0
(n−k)t
dt =
Z
T
0
cos (ω
0
(n − k) t) dt + j
Z
T
0
sin (ω
0
(n − k) t) dt , n 6= k
cos (ω
0
(n − k) t) n − k 0 T
sin (ω
0
(n − k) t)
Z
T
0
cos (ω
0
(n − k) t) dt = 0
Z
T
0
e
jω
0
(n−k)t
dt =



T n = k
0
k n
Z
T
0
f (t) e
−(jω
0
nt)
dt = T c
n
, n = k
c
n
=
1
T
Z
T
0
f (t) e
−(jω
0
nt)
dt
f (t)
k f (t) e
−(jω
0
kt)
T
k ∈ Z
f (t)
H[. . . ] e
st
H[st] = H [s] e
st
H[. . . ] e
st
H[. . . ]
f (t)
{c
n
} n ∈ Z c
i
∈ C
f (t) =
∞
X
n=−∞

c
n
e
jω
0
nt

H[t] = y (t)
f (t) =
X
n

c
n
e
jω
0
nt

y (t) =
X
n

c
n
H (jω
0
n) e
jω
0
nt

f (t) y (t)
f (t) → {c
n
} → {c
n
H (jω
0
n)} → y (t)
c
n
=
1
T
Z
T
0
f (t) e
−(jω
0
nt)
dt
H H (jω
0
n)
y (t) =
∞
X
n=−∞

c
n
e
jω
0
nt

{c
n
} f (t)
c
n
ω
0
n
c
n
n
f (t)
c
n
n
f (t)
f (t)

−

T
2

,
T
2

f (t)
c
n
=
1
T
Z
T
0
f (t) e
−(jω
0
nt)
dt
c
n
f (t)
c
n
=



2T
1
T
n = 0
2sin(ω
0
nT
1
)
nπ
n 6= 0
T
1
=
T
8
T
1
=
T
8
f (t) c
n
n = 0
c
n
n
c
n
= 0 n = {. . . , −4, 4, 8, 16, . . . } e
−(jω
0
nt)
n
f (t) =
∞
X
n=−∞

c
n
e
jω
0
nt

c
n
=
1
T
Z
T
0
f (t) e
−(jω
0
nt)
dt
F {·} f (t)
F {f (t)} = c
n
, n ∈ Z
F {·}
F {·}
F {f (t)} = c
n
F {g (t)} = d
n
F {αf (t)} = αc
n
, α ∈ C
F {f (t) + g (t)} = c
n
+ d
n
F {f (t) + g (t)} =
R
T
0
(f (t) + g (t)) e
−(jω
0
nt)
dt , n ∈ Z
=
1
T
R
T
0
f (t) e
−(jω
0
nt)
dt +
1
T
R
T
0
g (t) e
−(jω
0
nt)
dt , n ∈ Z
= c
n
+ d
n
, n ∈ Z
= c
n
+ d
n
F {f (t − t
0
)} = e
−(jω
0
nt
0
)
c
n
c
n
= |c
n
|e
j∠c
n
|e
−(jω
0
nt
0
)
c
n
| = |e
−(jω
0
nt
0
)
||c
n
| = |c
n
|
∠e
−(jω
0
t
0
n)
= ∠c
n
− ω
0
t
0
n
F {f (t − t
0
)} =
1
T
R
T
0
f (t − t
0
) e
−(jω
0
nt)
dt , n ∈ Z
=
1
T
R
T −t
0
−t
0
f (t − t
0
) e
−(jω
0
n(t−t
0
))
e
−(jω
0
nt
0
)
dt , n ∈ Z
=
1
T
R
T −t
0
−t
0
f

∼
t

e
−
“
jω
0
n
∼
t
”
e
−(jω
0
nt
0
)
dt , n ∈ Z
= e
−
“
jω
0
n
∼
t
”
c
n
, n ∈ Z
Z
T
0
(|f (t) |)
2
dt = T
∞
X
n=−∞

(|c
n
|)
2

L
2
([0, T ]) l
2
(Z)
f (t) c
n
n → ∞
c
n
=
1
n
, |n| > 0 f ∈ L
2
([0, T ])
c
n
=
1
√
n
, |n| > 0 f ∈ L
2
([0, T ])
f (t)
F {f (t)} = c
n
⇒ F

d
dt
f (t)

= jnω
0
c
n
f (t) =
∞
X
n=−∞

c
n
e
jω
0
nt

d
dt
f (t) =
P
∞
n=−∞

c
n
d
dt

e
jω
0
nt

=
P
∞
n=−∞

c
n
jω
0
ne
jω
0
nt

f (t)
f (t)
n → ∞
F {f (t)} = c
n
|c
n
|
1
n
k
F

d
m
dt
m
f (t)

= (jnω
0
)
m
c
n
n
m
n
k
m
th
X

|
n
m
n
k
|

2
!
< ∞
n
m
n
k
1
n
2k − 2m > 1
k >
2m + 1
2
F {f (t)} = c
n
F

Z
t
−∞
f (τ ) dτ

=
1
jω
0
n
c
n
c
0
6= 0
f (t) c
n
g (t) d
n
y (t) y (t) = f (t) g (t) y (t) e
n
e
n
=
P
∞
k=−∞
(c
k
d
n−k
)
e
n
=
1
T
R
T
0
f (t) g (t) e
−(jω
0
nt)
dt
=
1
T
R
T
0

P
∞
k=−∞

c
k
e
jω
0
kt

g (t) e
−(jω
0
nt)
dt
=
P
∞
k=−∞

c
k

1
T
R
T
0
g (t) e
−(jω
0
(n−k)t)
dt

=
P
∞
k=−∞
(c
k
d
n−k
)
f (t) f (t) = f (t)
∗
f (t)
∗
f (t) c
n
= c
−n
∗
Re (c
n
) = Re (c
−n
) c
n
Im (c
n
) = −(Im (c
−n
))
c
n
|c
n
| = |c
−n
|
∠c
n
= (∠ − c
−n
)
c
−n
=
1
T
R
T
0
f (t) e
jω
0
nt
dt
=
1
T
R
T
0
f (t)
∗
e
−(jω
0
nt)
dt
∗
, f (t) = f (t)
∗
=
1
T
R
T
0
f (t) e
−(jω
0
nt)
dt
∗
= c
n
∗
Re (c
n
) = Re (c
−n
) Im (c
n
) = − (Im (c
−n
))
|c
n
| = |c
−n
| ∠c
n
= (∠ − c
−n
)
f (t) = f (t)
∗
f (t) = (f (−t)) c
n
= c
−n
c
n
= c
n
∗
c
n
=
1
T
R
T
2
−
(
T
2
)
f (t) e
−(jω
0
nt)
dt
=
1
T
R
0
−
(
T
2
)
f (t) e
−(jω
0
nt)
dt +
1
T
R
T
2
0
f (t) e
−(jω
0
nt)
dt
=
1
T
R
T
2
0
f (−t) e
jω
0
nt
dt +
1
T
R
T
2
0
f (t) e
−(jω
0
nt)
dt
=
2
T
R
T
2
0
f (t) cos (ω
0
nt) dt
f (t) cos (ω
0
nt) c
n
cos (ω
0
nt) = cos (−(ω
0
nt))
c
n
= c
−n
f (t) = 2
P
∞
n=0
(c
n
cos (ω
0
nt)) f (t) c
n
cos (ω
0
nt)
f (t) = −(f (−t)) f (t) = f (t)
∗
c
n
= −c
−n
c
n
= −(c
n
∗
) c
n
f (t) sin (ω
0
nt)
f (t) =
∞
X
n=1
(2c
n
sin (ω
0
nt))
f
e
(t) f
o
(t)
f (t) = f
e
(t) + f
o
(t)
f (t) {a
n
} {b
n
}
f (t) =
∞
X
n=0
(a
n
cos (ω
0
nt)) +
∞
X
n=1
(b
n
sin (ω
0
nt))
T = 1 ω
0
= 2Ï€
f (t)
c
n
=







4A
jπ
2
n
2
n = {. . . , −11, −7, −3, 1, 5, 9, . . . }
−

4A
jπ
2
n
2

n = {. . . , −9, −5, −1, 3, 7, 11, . . . }
0 n = {. . . , −4, −2, 0, 2, 4, . . . }
c
n
= −c
−n
1
n
1
n
2
f (t) c
n
g (t) d
n
v (t) v (t) = (f (t) ~ g (t))
y (t) a
n
a
n
= c
n
d
n
(f (t) ~ g (t))
(f (t) ~ g (t)) =
R
T
0
R
T
0
f (τ ) g (t − τ ) dτdt
a
n
=
1
T
R
T
0
v (t) e
−(jω
0
nt)
dt
=
1
T
2
R
T
0
R
T
0
f (τ ) g (t − τ ) dτe
−(jω
0
nt)
dt
=
1
T
R
T
0
f (Ï„ )

1
T
R
T
0
g (t − τ ) e
−(jω
0
nt)
dt

dτ
=
1
T
R
T
0
f (Ï„ )

1
T
R
T −τ
−τ
g (ν) e
−(jω
0
(ν+τ ))
dν

dτ , ν = t − τ
=
1
T
R
T
0
f (Ï„ )

1
T
R
T −τ
−τ
g (ν) e
−(jω
0
nν)
dν

e
−(jω
0
nτ)
dτ
=
1
T
R
T
0
f (Ï„ ) d
n
e
−(jω
0
nτ)
dτ
= d
n

1
T
R
T
0
f (Ï„ ) e
−(jω
0
nτ)
dτ

= c
n
d
n
T
1
=
T
4
c
n
=



1
T
n = 0
1
2
sin
(
Ï€
2
n
)
Ï€
2
n
a
n
= c
n
2
=
1
4
sin
2
(
Ï€
2
n
)
(
Ï€
2
n
)
2
H
e
st
H H (s)
h (t)
H (s) =
Z
∞
−∞
h (Ï„ ) e
−(sτ)
dτ
f (t)
f (t) =
X
n

c
n
e
jω
0
nt

y (t)
y (t) =
X
n

H (jω
0
n) c
n
e
jω
0
nt

f (t) c
n
y (t)
H
{c
n
} {H (jw
0
n)}
{H (jw
0
n) c
n
}
H (jw
0
n)
T = 2Ï€w
0
h (t) =
1
RC
e
−t
RC
u (t)
f (t)
H (s) =
R
∞
−∞
h (Ï„ ) e
−(sτ)
dτ
=
R
∞
0
1
RC
e
−τ
RC
e
−(sτ)
dτ
=
1
RC
R
∞
0
e
(−τ)
(
1
RC
+s
)
dτ
=
1
RC
1
1
RC
+s
e
(−τ)
(
1
RC
+s
)
|
∞
Ï„=0
=
1
1+RCs
T = 2Ï€w
0
f (t)
s = jw
0
n
|H (jw
0
n) | =
1
|1 + RCjw
0
n|
=
1
√
1 + R
2
C
2
w
0
2
n
2
n 0
n
• f (t)
c
n
=
1
2
sin

Ï€
2
n

Ï€
2
n
1
t
n = 0
•
H (jw
0
n) =
1
1 + jRCw
0
n
• y (t)
d
n
= H (jw
0
n) c
n
=
1
1 + jRCw
0
n
1
2
sin

Ï€
2
n

Ï€
2
n
d
n
=
1
1 + jRCw
0
n
1
2
sin

Ï€
2
n

Ï€
2
n
y (t) =
X

d
n
e
jw
0
nt

y (t) {d
n
}
y (t)
y (t)
y (t) f (t) d
n
c
n
d
n
=
1
1 + jRCw
0
n
1
2
sin

Ï€
2
n

Ï€
2
n
|d
n
| =
1
q
1 + (RCw
0
)
2
n
2
1
2
sin

Ï€
2
n

Ï€
2
n
f (t)
f (t) =
X
n

c
n
e
jω
0
nt

c
n
=
1
T
Z
T
0
f (t) e
−(jω
0
nt)
dt
∼
f
N
0
(t) =
N
X
n=−N

c
n
e
jω
0
nt

c
n
f (t)
f
N
0
(t) f (t) 2N + 1 f
N
0
(t)
f (t) N

f
N
0
(t) , N = {0, 1, . . . }

f (t)
f (t) f
N
0
(t) → f (t) N → ∞
e
N
(t) = f (t) − f
N
0
(t)
lim
N→∞
f
N
0
(t) f (t)
e
N
(t) f (t) f
N
0
(t)
e
N
(t) = f (t) − f
N
0
(t)
f (t) ∈ L
2
([0, T ]) e
N
(t) → 0 N → ∞
Z
T
0
(|e
N
(t) |)
2
dt =
Z
T
0

f (t) − f
N
0
(t)

2
dt → 0
lim
N→∞
Z
T
0

|f (t) − f
N
0
(t) |

2
dt = lim
N→∞
∞
X
N=−∞


|F
n
f (t) − F
n
f
N
0
(t) |

2

= lim
N→∞
X
|n|>N

(|c
n
|)
2

= 0
f (t) ∈
L
2
([0, T ])
f (t) ∈ L
2
([0, T ])
e
N
→ 0 f (t) lim
N→∞
f
N
0
(t)
f (t) g (t) t f (t) 6= g (t)
Z
T
0
(|f (t) − g (t) |)
2
dt = 0
energy convergence 6= pointwise convergence
pointwise convergence ⇒ convergence in L
2
([0, T ])
f (t) g (t) t
0
f (t
0
) 6= lim
N→∞
f
N
0
(t
0
)
f (t)
f (t
0
) = lim
N→∞
f
N
0
(t
0
)
f (t) t = t
0
Z
T
0
|f (t) |dt < ∞
|c
n
| = |
1
T
Z
T
0
f (t) e
−(jω
0
nt)
dt| ≤
1
T
Z
T
0
|f (t) ||e
−(jω
0
nt)
|dt
|e
−(jω
0
nt)
| = 1
1
T
Z
T
0
|f (t) |dt =
1
T
Z
T
0
|f (t) |dt
< ∞
f (t) =
1
t
, 0 < t ≤ T
Z
∞
−∞
|f (t) |dt < ∞
f (t)
f (t)
sin (logt)
f
0
(t) = lim
N→∞
f
N
0
(t)
f (t)
f (Ï„ ) = f
0
(Ï„)
Ï„ f (t) f (t) f
0
(t)
f (t)
f
N
0
(t) =
N
X
n=−N

c
n
e
jω
0
nt

c
n
n = ±79
f
79
0
(t) =
79
X
n=−79

c
n
e
jω
0
nt

s (t) = a
0
+
∞
X
k=1

a
k
cos

2Ï€kt
T

+
∞
X
k=1

b
k
sin

2Ï€kt
T

K
sq (t)
t
∼
t
lim
K→∞
rms (
K
) = 0
s
1
(t) s
2
(t) rms (s
1
− s
2
) = 0
s
1
(t) = s
2
(t) t
f (t)
f (t) =
∞
X
n=−∞

c
n
e
jω
0
nt

c
n
=
1
T
Z
T
0
f (t) e
−(jω
0
nt)
dt
c
n
ω
0
n
e
jω
0
nt
H (jω
0
n) =
Z
∞
−∞
h (t) e
−(jω
0
nt)
dt
• |H (jω
0
n) | ⇒ ω
0
n
• |H (jω
0
n) | ⇒ ω
0
n
{c
n
} f (t)
• f (t)
• f (t) |c
n
|
f
N
0
(t) =
X
n≤|N|

c
n
e
jω
0
nt

f (t) f (t)
e
jω
0
nt
f (t) =
1
2

e
jω
0
t
+ e
−(jω
0
t)

c
n
=



1
2
|n| = 1
0
f (t) =
1
2j

e
jω
0
t
− e
−(jω
0
t)

c
n
=







−j
2
n = −1
j
2
n = 1
0
f (t) = 3 + 4

1
2


e
jω
0
t
+ e
−(jω
0
t)

+ 2

1
2


e
j2ω
0
t
+ e
−(j2ω
0
t)

c
n
=













3 n = 0
2 |n| = 1
1 |n| = 2
0
f (t)
c
n
=
1
T
Z
T
1
−T
1
(1) e
−(jω
0
nt)
dt
n = 0 n 6= 0 n = 0
c
n
=
2T
1
T
, n = 0
n 6= 0
c
n
=
1
T

1
jω
0
n

e
−(jω
0
nt)
|
T
1
t=−T
1
c
n
=
1
T

1
−(jω
0
n)


e
−(jω
0
nT
1
)
− e
jω
0
nT
1

2j
2j
c
n
=
1
T

2j
jω
0
n

sin (ω
0
nT
1
)
ω
0
=
2Ï€
T
T
c
n
=
2jω
0
jω
0
n2Ï€
sin (ω
0
nT
1
)
f (t)
c
n
=
sin (ω
0
nT
1
)
nπ
, n 6= 0
(|c
n
|)
2
< ∞ f (t)
(|c
n
|)
2
=
1
n
2
(|c
n
|)
2
=
1
n
T
4
L
2
([0, T )) l
2
(Z)
L
2
(R) L
2
(R)
l
2
(Z) L
2
([0, 2Ï€))
l
2
([0, N − 1]) l
2
([0, N − 1])
f (t) =
X
n

c
n
e
jω
0
nt

c
n
=
1
T
R
n
f (t) e
−(jω
0
nt)
dt
=
1
T
< f, e
jω
0
nt
>
c
n
ω
0
n f (t)
[0, T ]
T
f(t)
T
[0, T ]
N
N = 4
{. . . , 3, 2, −2, 1, 3, . . . }
f[n]
n
n
C
4
[0, T ]
N C
N
e
jωn
ω = 0
ω =
Ï€
4
|e
jωn
| = 1 , n ∈ R
∠e
jωn
= wn
n e
jωn
n e
jωn
n = 0 n = 1 n = 2
e
jωn
e
jωN
= 1 N
ω =
2Ï€
7
N = 7
N = 7 Re
“
e
j
2Ï€
7
n
”
ω = 1 N = 7
N = 7 Re
`
e
jn
´
e
jωn
= e
j(ω+2π)n
ω + 2π
ω
e
jωn
ω ∈ [0, 2π)
π < ω < 2π
Ï€
2π − ω
w 2π − ω
e
jωn
ω =
5Ï€
4
n = 1
ω = −
`
3Ï€
4
´
−

3Ï€
4

[−π, π) ω
e
jωn
e
−(jωn)
e
jωn
∗
= e
−(jωn)
N
b
k
= e
jω
k
n
, n = {0, . . . , N − 1} {b
k
} C
n
ω
0
=
2Ï€
N
e
j
2Ï€
N
kn
k = 0
Im
“
e
j
2Ï€
N
1n
”
k = 1 Im
“
e
j
2Ï€
N
2n
”
k = 2
e
j
2Ï€
N
kn
N k n = 0 n = N − 1
b
k
[n] =
1
√
N
e
j
2Ï€
N
kn
, n = {0, . . . , N − 1}
C
N
{b
k
}|
k={0,...,N −1}
C
N
{b
k
} < b
k
, b
l
>= δ
kl
< b
k
, b
l
>=
N−1
X
n=0

b
k
[n] b
l
[n]
∗

=
1
N
N−1
X
n=0

e
j
2Ï€
N
kn
e
−
(
j
2Ï€
N
ln
)

< b
k
, b
l
>=
1
N
N−1
X
n=0

e
j
2Ï€
N
(l−k)n

l = k
< b
k
, b
l
> =
1
N
P
N−1
n=0
(1)
= 1
l 6= k
N−1
X
n=0
(α
n
) =
∞
X
n=0
(α
n
) −
∞
X
n=N
(α
n
) =
1
1 − α
−
α
N
1 − α
=
1 − α
N
1 − α
< b
k
, b
l
>=
1
N
N−1
X
n=0

e
j
2Ï€
N
(l−k)n

α = e
j
2Ï€
N
(l−k)
< b
k
, b
l
>=
1
N
1 − e
j
2Ï€
N
(l−k)N
1 − e
j
2Ï€
N
(l−k)
!
=
1
N

1 − 1
1 − e
j
2Ï€
N
(l−k)

= 0
< b
k
, b
l
>=



1 k = l
0 k 6= l
{b
k
} {b
k
} N
n
1
√
N
e
j
2Ï€
N
kn
o
C
n
C
n
f [n]
f [n] =
1
√
N
N−1
X
k=0

c
k
e
j
2Ï€
N
kn

c
k
=
1
√
N
N−1
X
n=0

f [n] e
−
(
j
2Ï€
N
kn
)

1
√
N
c
k
f [n] =
N−1
X
k=0

c
k
e
j
2Ï€
N
kn

c
k
=
1
N
N−1
X
n=0

f [n] e
−
(
j
2Ï€
N
kn
)

C
N
f [n]
f [n] = f [0] e
0
+ f [1] e
1
+ ··· + f [N − 1] e
N−1
=
P
n−1
k=0
(f [k] δ [k − n])










f [0]
f [1]
f [2]
f [N −1]










=










f [0]
0
0
0










+










0
f [1]
0
0










+










0
0
f [2]
0










+ ··· +










0
0
0
f [N −1]










f [n]
f [n] =
N−1
X
k=0

c
k
e
j
2Ï€
N
kn











f [0]
f [1]
f [2]
f [N −1]










= c
0










1
1
1
1










+ c
1










1
e
j
2Ï€
N
e
j
4Ï€
N
e
j
2Ï€
N
(N−1)










+ c
2










1
e
j
4Ï€
N
e
j
8Ï€
N
e
j
4Ï€
N
(N−1)










+ . . .
W B
W =

b
0
[n] b
1
[n] . . . b
N−1
[n]

=










1 1 1 . . . 1
1 e
j
2Ï€
N
e
j
4Ï€
N
. . . e
j
2Ï€
N
(N−1)
1 e
j
4Ï€
N
e
j
8Ï€
N
. . . e
j
2Ï€
N
2(N−1)
1 e
j
2Ï€
N
(N−1)
e
j
2Ï€
N
2(N−1)
. . . e
j
2Ï€
N
(N−1)(N−1)










b
k
[n] = e
j
2Ï€
N
kn
W
j,k
= e
j
2Ï€
N
kn
= W
n,k
W = W
T
⇒ W
T
∗
= W
∗
=
1
N
W
−1
{b
k
[n]}
• f C
N
• c C
N
f = W c f [n] =< c, b
n
∗
>
c = W
T
∗
f = W
∗
f c [k] =< f , b
k
>
C
N
c [k]
c [k]
c [k]
c [k] =
1
N
N−1
X
n=0

f [n] e
−
(
j
2Ï€
N
kn
)

c
k
=
1
N
N
1
X
n=−N
1

e
−
(
j
2Ï€
N
kn
)

m = n + N
1
c
k
=
1
N
P
2N
1
m=0

e
−
(
j
2Ï€
N
(m−N
1
)k
)

=
1
N
e
j
2Ï€
N
k
P
2N
1
m=0

e
−
(
j
2Ï€
N
mk
)

M
X
n=0
(a
n
) =
1 − a
M+1
1 − a
c
k
=
1
N
e
j
2Ï€
N
N
1
k
P
2N
1
m=0

e
−
(
j
2Ï€
N
k
)

m

=
1
N
e
j
2Ï€
N
N
1
k
1−e
−
(
j
2Ï€
N
(2N
1
+1)
)
1−e
−
(
jk
2Ï€
N
)
c
k
=
1
N
e
−
(
jk
2Ï€
2N
)
„
e
jk
2Ï€
N
(
N
1
+
1
2
)
−e
−
(
jk
2Ï€
N
(
N
1
+
1
2
))
«
e
−
(
jk
2Ï€
2N
)
„
e
jk
2Ï€
N
1
2
−e
−
(
jk
2Ï€
N
1
2
)
«
=
1
N
sin
2Ï€k
(
N
1
+
1
2
)
N
!
sin
(
Ï€k
N
)
=
N
1
N
1
= 1 f [n] c [k]
N
1
= 3 f [n] c [k]
N
1
= 7 f [n] c [k]
m f (n + m) m
m = −2
m = −2
f (n)
N = 8
m
(f [n] → f [((n + m))
N
])
m = −3
f (n) = [0, 1, 2, 3, 4, 5, 6, 7] f (((n − 3))
N
) = [3, 4, 5, 6, 7, 0, 1, 2]
f [((n + N ))
N
] = f [n]
N
f [((n + N ))
N
] = f [((n − (N −m)))
N
]
m N − m
f [((−n))
N
]
f [n]
f [n] f
ˆ
((−n))
N
˜
f [n] ↔ F [k]
f [((n − m))
N
] ↔ e
−
(
j
2Ï€
N
km
)
F [k]
f [n] =
1
N
N−1
X
k=0

F [k] e
j
2Ï€
N
kn

f [n] =
1
N
P
N−1
k=0

F [k] e
−
(
j
2Ï€
N
kn
)
e
j
2Ï€
N
kn

=
1
N
P
N−1
k=0

F [k] e
j
2Ï€
N
k(n−m)

= f [((n − m))
N
]
x [n] h [n]
y [n] = x [n] ∗ h [n]
=
P
∞
k=−∞
(x [k] h [n − k])
N
n ≥ N
n ≥ N
n = {0, 1, . . . , N − 1}
Y [k] y [n]
Y [k] = F [k] H [k]
0 ≤ k ≤ N − 1
y [n]
y [n] =
1
N
N−1
X
k=0

F [k] H [k] e
j
2Ï€
N
kn

F [k] =
P
N−1
m=0

f [m] e
(−j)
2Ï€
N
kn

y [n] =
1
N
P
N−1
k=0

P
N−1
m=0

f [m] e
(−j)
2Ï€
N
kn

H [k] e
j
2Ï€
N
kn

=
P
N−1
m=0

f [m]

1
N
P
N−1
k=0

H [k] e
j
2Ï€
N
k(n−m)

h [((n − m))
N
] =
1
N
P
N−1
k=0

H [k] e
j
2Ï€
N
k(n−m)

y [n] =
N−1
X
m=0
(f [m] h [((n − m))
N
])
0 ≤ n ≤ N − 1
y [n] ≡ (f [n] ~ h [n])
~
• f [m] h [((−m))
N
]
• h [((−m))
N
] n h [((n − m))
N
]
h [n] n
• f [m] h [((n − m))
N
]
sum = y [n]
• 0 ≤ n ≤ N − 1
• h [((−m))
N
]
f [m] sum y [0] = 3
• h [((1 − m))
N
]
f [m] sum y [1] = 5
• h [((2 − m))
N
]
f [m] sum y [2] = 3
• h [((3 − m))
N
]
f [m] sum y [3] = 1
• f [n] F [k] h [n] H [k]
• Y [k] = F [k] H [k]
• Y [k] y [n]
2 N
y [n] =
N−1
X
m=0
(f [m] h [((n − m))
N
])
n N N − 1
N N
2
N (N −1) O

N
2

b
k
= e
j
2Ï€
N
kn
b
k+N
= e
j
2Ï€
N
(k+N )n
= e
j
2Ï€
N
kn
e
j2Ï€n
= e
j
2Ï€
N
n
= b
k
→ N
N N
2N
N
N − 1 2N + 2 (N − 1) = 4N − 2
N N (4N −2)
4N
2
O

N
2

N
k =

N
2
+ 1, ..., N + 1

K
N
•
•
•
W
N
2
N 10 100 1000 10
6
10
9
N
2
10
4
10
6
10
12
10
18
NlogN 1 200 3000 6 × 10
6
9 × 10
9
N = 1million =
10
6
N
2
10
12
→ 10
6
'
NlogN 6 × 10
6
→
N = 1million
3×10
6
N
f [n] h [n] (f [n] ~ h [n]) N
2
N = 2
l
S (k) = s (0) + s (2) e
(−j)
2Ï€2k
N
+ ··· + s (N − 2) e
(−j)
2π(N −2)k
N
+
s (1) e
(−j)
2Ï€k
N
+ s (3) e
(−j)
2Ï€(2+1)k
N
+ ··· + s (N − 1) e
(−j)
2π(N −2+1)k
N
=
s (0) + s (2) e
(−j)
2Ï€k
N
2
+ ··· + s (N − 2) e
(−j)
2Ï€
(
N
2
−1
)
k
N
2
+


s (1) + s (3) e
(−j)
2Ï€k
N
2
+ ··· + s (N − 1) e
(−j)
2Ï€
(
N
2
−1
)
k
N
2


e
−(j2πk)
N
N
2
e
−(j2πk)
N
k ∈ {0, . . . , N − 1}
e
−(j2πk)
N
N
2
N
2
2O

N
2
4

O (N)
O (N)
N = 2
l
N
2
8
N
4
=
N
2
log
2
N O (NlogN)
e
−(jπ)
N
2
= 4 2N = 16 log
2
N = 3
3N
2
log
2
N
2
4
3
4
2
1
3
2
K O (KN)
g
n
n
{g
n
}|
∞
n=1
g
n
=
1
n
g
n
=


sin

nπ
2

cos

nπ
2



g
n
(t) =



1 0 ≤ t <
1
n
0
t
{g
n
}|
∞
n=1
g ∈ R  > 0 N
|g
i
− g| <  , i ≥ N
lim
i→∞
g
i
= g
g
i
→ g
 g
i
 g
g
n
=
1
n
lim
n→∞
g
n
= 0
 > 0 N = d
1

e
dxe x n ≥ N
|g
n
− 0| =
1
n
≤
1
N
< 
lim
n→∞
g
n
= 0
g
n
=



1 n = even
−1 n = odd
g
n
= n
g
n
=



1
n
n = even
−1
n
n = odd
g
n
=



1
n
n 6= power of 10
1
g
n
=



n n < 10
5
1
n
n ≥ 10
5
g
n
= sin

Ï€
n

g
n
= j
n
{g
n
}|
∞
n=1
g g
n
g
g
n
=


g
n
[1]
g
n
[2]


=


1 +
1
n
2 −
1
n


g
n
lim
n→∞
(g
n
[1]) = 1
lim
n→∞
(g
n
[2]) = 2
lim
n→∞
g
n
= g
g =


1
2


g
n
(t) =
t
n
, t ∈ R
lim
n→∞
g
n
(t
0
) = lim
n→∞
t
0
n
= 0
t
0
∈ R lim
n→∞
g
n
= g g (t) = 0 t ∈ R
{g
n
}|
∞
n=1
g lim
n→∞
k g
n
− g k= 0 k · k
g
n
g
n
g 0
g
n
=


1 +
1
n
2 −
1
n


g =


1
2


k g
n
− g k =
q

1 +
1
n
− 1

2
+

2 −
1
n

2
=
q
1
n
2
+
1
n
2
=
√
2
n
lim
n→∞
k g
n
− g k= 0 g
n
→ g
g
n
(t) =



t
n
0 ≤ t ≤ 1
0
g (t) = 0 t
k g
n
(t) − g (t) k =
R
1
0
t
2
n
2
dt
=
t
3
3n
2
|
1
n=0
=
1
3n
2
lim
n→∞
k g
n
(t) − g (t) k= 0 g
n
(t) → g (t)
R
m
⇒
g
n
[i] → g [i]
(k g
n
− g k)
2
=
m
X
i=1

(g
n
[i] − g [i])
2

lim
n→∞
(k g
n
− g k)
2
= lim
n→∞
P
m
i=1
2
=
P
m
i=1

lim
n→∞
2

= 0
⇒
k g
n
− g k→ 0
lim
n→∞
P
m
i=1
2 =
P
m
i=1

lim
n→∞
2

= 0
m
lim
n→∞
2 = 0
i
g
n
→ g
[U+21CF]
g
n
(t) =



n 0 < t <
1
n
0
lim
n→∞
g
n
(t) = 0
g
n
(t) → g (t)
t g (t) = 0
(k g
n
k)
2
=
R
∞
−∞
(|g
n
(t) |)
2
dt
=
R
1
n
0
n
2
dt
= n → ∞
[U+21CF]
g
n
(t) =



1 0 < t <
1
n
0
g
n
(t) =



−1 0 < t <
1
n
0
k g
n
− g k=
Z
1
n
0
1dt =
1
n
→ 0
g (t) = 0 t
g
n
→ g
t = 0 g
n
(t) g
n
(t)
g
n
(t) =



1
nt
0 < t
0 t ≤ 0
g
n
(t) =



e
−(nt)
t ≥ 0
0 t < 0
g
n
R → R
{g
n
}|
∞
n=1
g  > 0
N n ≥ N
|g
n
(t) − g (t) | ≤ 
t ∈ R
{g
n
} g  > 0
t ∈ R N  t n ≥ N {g
n
}
g  > 0 N t ∈ R
g
n
(t) =
1
n
, t ∈ R
 > 0 N = d
1

e
|g
n
(t) − 0| ≤  , n ≥ N
t g
n
(t) 0
g
n
(t) =
t
n
, t ∈ R
 > 0 g
n
(t) g (t) = 0
t g
n
g
n
(t) → g (t)
g
n
(t) =
sin(t)
n
g
n
(t) = e
t
n
g
n
(t) =



1
nt
t > 0
0 t ≤ 0
X [k] =
N−1
X
n=0

x [n] e
(−j)
2Ï€
n
kn

, k = {0, . . . , N − 1}
x [n] =
1
N
N−1
X
k=0

X [k] e
j
2Ï€
n
kn

, n = {0, . . . , N − 1}
• X [k] ω =
2Ï€
N
k , k = {0, . . . , N − 1}
• x [n] M M
X

e
j
2Ï€
M
k

=
N−1
X
n=0

x [n] e
(−j)
2Ï€
M
k

X

e
j
2Ï€
M
k

=
N−1
X
n=0

x
zp
[n] e
(−j)
2Ï€
M
k

X

e
j
2Ï€
M
k

= X
zp
[k] , k = {0, . . . , M − 1}
• N N
X

e
jω

=
N−1
X
n=0

x [n] e
(−j)ωn

X

e
jω

=
N−1
X
n=0

1
N

!
N−1
X
k=0

X [k] e
j
2Ï€
N
kn
e
(−j)ωn

X

e
jω

=
N−1
X
k=0
(X [k])
!
1
N
N−1
X
k=0

e
(−j)
(
ω−
2Ï€
N
k
)
n

X

e
jω

=
N−1
X
k=0
(X [k])
!
1
N
sin

ωN−2πk
2

sin

ωN−2πk
2N

e
(−j)
(
ω−
2Ï€
N
k
)
N−1
2
!
1
0 2pi/N 4pi/N 2pi
D
.
1
N
sin
(
ωN
2
)
sin
(
ω
2
)
• W
N
= e
(−j)
2Ï€
N







X [0]
X [1]
X [N − 1]







=







W
0
N
W
0
N
W
0
N
W
0
N
. . .
W
0
N
W
1
N
W
2
N
W
3
N
. . .
W
0
N
W
2
N
W
4
N
W
6
N
. . .














x [0]
x [1]
x [N − 1]







X = W (x) W
· W n W W
n
N
· W W = W
T
·
1
√
N
W

1
√
N
W

1
√
N
W

H
=

1
√
N
W

H

1
√
N
W

= I
·
1
N
W
∗
= W
−1
• N
N
2
log
2
N N
2
N
N
2
log
2
N N
2
•
[0, N − 1]
•

−

1
2

,
1
2

[0, 1]
f =
k
K
k ∈ {0, . . . , K − 1}
S (k) =
N−1
X
n=0

s (n) e
−
(
j2Ï€nk
K
)

, k ∈ {0, . . . , K − 1}
S (k) S

e
j2Ï€
k
K

S (k)
k = {0, . . . , K − 1} s (n) n = {0, . . . , N − 1}
s (n) =
P
K−1
k=0

S (k) e
j2Ï€nk
K

s (n) =
K−1
X
k=0
N−1
X
m=0

s (m) e
−
(
j
2Ï€mk
K
)
e
j
2Ï€nk
K

!
K−1
X
k=0

e
−
(
j
2Ï€km
K
)
e
j
2Ï€kn
K

=



K m = {n, (n ± K) , (n ± 2K) , . . . }
0
K
K
P
l
(δ (m − n − lK))
s (n) =
N−1
X
m=0
s (m) K
∞
X
l=−∞
(δ (m − n − lK))
!
n m {0, . . . , N − 1}
m n s (n)
l = 0 K
m = n + K m n = {0, . . . , N − 1}
s (n) + s (n + K) n
K ≥ N
s (0) + s (1) + ··· + s (N − 1) = S (0)
s (0) + s (1) e
(−j)
2Ï€
K
+ ··· + s (N − 1) e
(−j)
2π(N −1)
K
= S (1)
s (0) + s (1) e
(−j)
2π(K−1)
K
+ ··· + s (N − 1) e
(−j)
2π(N −1)(K−1)
K
= S (K − 1)
K N
K < N K ≥ N
K N
S (k) =
P
N−1
n=0

s (n) e
−
(
j
2Ï€nk
N
)

s (n) =
1
N
P
N−1
k=0

S (k) e
j
2Ï€nk
N

e
−(at)
u (t)
1
a+jω
a > 0
e
at
u (−t)
1
a−jω
a > 0
e
−(a|t|)
2a
a
2
+ω
2
a > 0
te
−(at)
u (t)
1
(a+jω)
2
a > 0
t
n
e
−(at)
u (t)
n!
(a+jω)
n+1
a > 0
δ (t) 1
1 2πδ (ω)
e
jω
0
t
2πδ (ω − ω
0
)
cos (ω
0
t) π (δ (ω − ω
0
) + δ (ω + ω
0
))
sin (ω
0
t) jπ (δ (ω + ω
0
) − δ (ω − ω
0
))
u (t) πδ (ω) +
1
jω
sgn (t)
2
jω
cos (ω
0
t) u (t)
Ï€
2
(δ (ω − ω
0
) + δ (ω + ω
0
)) +
jω
ω
0
2
−ω
2
sin (ω
0
t) u (t)
Ï€
2j
(δ (ω − ω
0
) − δ (ω + ω
0
)) +
ω
0
ω
0
2
−ω
2
e
−(at)
sin (ω
0
t) u (t)
ω
0
(a+jω)
2
+ω
0
2
a > 0
e
−(at)
cos (ω
0
t) u (t)
a+jω
(a+jω)
2
+ω
0
2
a > 0
u (t + τ ) − u (t − τ) 2τ
sin(ωτ)
ωτ
= 2τsinc (ωt)
ω
0
Ï€
sin(ω
0
t)
ω
0
t
=
ω
0
Ï€
sinc (ω
0
) u (ω + ω
0
) − u (ω − ω
0
)

t
Ï„
+ 1

u

t
Ï„
+ 1

− u

t
Ï„

+

−

t
Ï„

+ 1

u

t
Ï„

− u

t
Ï„
− 1

=
triag

t
2Ï„

Ï„sinc
2

ωτ
2

ω
0
2Ï€
sinc
2

ω
0
t
2


ω
ω
0
+ 1

u

ω
ω
0
+ 1

− u

ω
ω
0

+

−

ω
ω
0

+ 1

u

ω
ω
0

− u

ω
ω
0
− 1

=
triag

ω
2ω
0

P
∞
n=−∞
(δ (t − nT )) ω
0
P
∞
n=−∞
(δ (ω − nω
0
)) ω
0
=
2Ï€
T
e
−
“
t
2
2σ
2
”
σ
√
2Ï€e
−
“
σ
2
ω
2
2
”
X (ω) =
∞
X
n=−∞

x (n) e
−(jωn)

x (n) =
1
2Ï€
Z
2Ï€
0
X (ω) e
jωn
dω
l
2
2Ï€ L
2
l
2
(Z) L
2
([0, 2Ï€))
a
1
s
1
(n) + a
2
s
2
(n) a
1
S
1

e
j2Ï€f

+ a
2
S
2

e
j2Ï€f

s (n) S

e
j2Ï€f

= S

e
−(j2πf )

∗
s (n) = s (−n) S

e
j2Ï€f

= S

e
−(j2πf )

s (n) = −(s (−n)) S

e
j2Ï€f

= −

S

e
−(j2πf )

s (n − n
0
) e
−(j2πf n
0
)
S

e
j2Ï€f

e
j2Ï€f
0
n
s (n) S

e
j2π(f −f
0
)

s (n) cos (2Ï€f
0
n)
S
(
e
j2π(f −f
0
)
)
+S
(
e
j2Ï€(f +f
0
)
)
2
s (n) sin (2Ï€f
0
n)
S
(
e
j2π(f −f
0
)
)
−S
(
e
j2Ï€(f +f
0
)
)
2j
ns (n)
1
−(2jπ)
d
df

S

e
j2Ï€f

P
∞
n=−∞
(s (n)) S

e
j2Ï€0

s (0)
R
1
2
−
(
1
2
)
S

e
j2Ï€f

df
P
∞
n=−∞

(|s (n) |)
2

R
1
2
−
(
1
2
)

|S

e
j2Ï€f

|

2
df
1
2
1
2T
s
cos

2Ï€
1
2T
s
nT
s

= cos (Ï€n)
= (−1)
n
1
2
e
−(j2πn)
2
= e
−(jπn)
= (−1)
n
f
D
= f
A
T
s
f
D
f
A
p
T
s
(t)
p
T
s
(t)
1
T
s
1
2T
s
1
2
Z
1
2
−
(
1
2
)
e
−(j2πf m)
e
+jπf n
df =



1 m = n
0 m 6= n
R
1
2
−
(
1
2
)
S

e
j2Ï€f

e
+j2Ï€f n
df =
R
1
2
−
(
1
2
)
P
m

s (m) e
−(j2πf m)
e
+j2Ï€f n

df
=
P
m

s (m)
R
1
2
−
(
1
2
)
e
(−(j2πf ))(m−n)
df

= s (n)
S

e
j2Ï€f

=
X
n

s (n) e
−(j2πf n)

s (n) =
Z
1
2
−
(
1
2
)
S

e
j2Ï€f

e
+j2Ï€f n
df
s (n) =
a
n
u (n) u (n)
S

e
j2Ï€f

=
P
∞
n=−∞

a
n
u (n) e
−(j2πf n)

=
P
∞
n=0


ae
−(j2πf )

n

∆ c
0
|c
0
| =
A∆
T
A
1
∆
∆ 0.1T
s
∞
X
n=0
(α
n
) =
1
1 − α
, |α| < 1
|a| < 1
S

e
j2Ï€f

=
1
1 − ae
−(j2πf )
|S

e
j2Ï€f

| =
1
q
(1 − acos (2πf))
2
+ a
2
sin
2
(2Ï€f)
∠

S

e
j2Ï€f

= −

arctan

asin (2Ï€f)
1 − acos (2πf)

a
−

1
2

1
2
a > 0
0
1
2
a a < 0
-2 -1 0 1 2
1
2
f
|S(e
j2Ï€f
)|
-2 -1 1 2
-45
45
f
∠S(e
j2Ï€f
)
a = 0.5
[−2, 2]
f
a = 0.9
a = 0.5
a = –0.5
Spectral Magnitude (dB)
-10
0
10
20
0.5
a = 0.9
a = 0.5
a = –0.5
Angle (degrees)
f
-90
-45
0
45
90
0.5
a = 0.5 a = −0.5
N
s (n) =



1 0 ≤ n ≤ N − 1
0
S

e
j2Ï€f

=
N−1
X
n=0

e
−(j2πf n)

N+n
0
−1
X
n=n
0
(α
n
) = α
n
0
1 − α
N
1 − α
α
α
S

e
j2Ï€f

=
1−e
−(j2πf N )
1−e
−(j2πf )
= e
(−(jπf ))(N−1)
sin(Ï€fN )
sin(Ï€f)
sin(Nx)
sin(x)
dsinc (x) S

e
j2Ï€f

= e
(−(jπf ))(N−1)
dsinc (Ï€f)
N
f
0
5
10
Spectral Magnitude
0.5
-180
-90
0
90
180
f
0.5
Angle (degrees)
α
N+n
0
−1
X
n=n
0
(α
n
) −
N+n
0
−1
X
n=n
0
(α
n
) = α
N+n
0
− α
n
0
F (Ω) =
Z
∞
−∞</