brief notes on signals and systems 7.2
TRANSCRIPT
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Brief Notes on Signals andSystems
By:
C. Sidney Burrus
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Brief Notes on Signals and
Systems
By:C. Sidney Burrus
Online:
C O N N E X I O N S
Rice University, Houston, Texas
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x (t) t {0 ≤ t ≤ T } x (t)
x (t) =
a (0)
2 +
∞k=1a (k) cos
2πT kt
+ b (k) sin
2πT kt
.
xk (t) = cos (2πkt/T ) yk (t) = sin (2πkt/T )
L2 [0, T ]
L2 [0, T ]
cos
2πT
kt
, cos2πT
t
= T 0
cos
2πT
kt
cos2πT
t
dt = δ (k − )
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cos
2π
T kt
, sin
2π
T t
=
T 0
cos
2π
T kt
sin
2π
T t
dt = 0
δ (t) δ (0) = 1 δ (k = 0) =0 k x (t) k
a (k) = 2T T 0
x (t) cos
2πT kt
dt
b (k) = 2
T
T 0
x (t) sin
2π
T kt
dt
T
x (t)
x (t) = a (0)
2 +
N k=1
a (k) cos
2π
T kt
+ b (k) sin
2π
T kt
,
= 1
T T
0
|x (t) − x (t) |2
dt
a (k) b (k)
x (t) ∈ L2 [0, T ] x (t) → 0 N → ∞
f (x)
f (x)
f (x)
x (t) q a (k) b (k)
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1kq+1 k → ∞
x (t) = d (0)
2 +
∞
k=1d (k) cos
2π
T kt + θ (k)
j =√ −1
ejx = cos (x) + jsin (x) ,
x (t) =∞
k=−∞c (k) ej
2πT kt
c (k) = a (k) + j b (k) .
c (k) = 1
T
T 0
x (t) e−j2πT ktdt
|d|2 = |c|2 = a2 + b2
θ = arg{c} = tan−1
b
a
c (k) d (k) θ (k) a (k) b (k)
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x (t)
x (t)
x (t)
{0 ≤ t ≤ T }
F{x (t)} = c (k)
x̃ (t)
x (t)
F{x + y} = F{x} + F{y} F{ax} = aF{x}
x (t) x̃ (t) = x̃ (t + T )x̃ (t)
x (t) = u (t) + j v (t) C (k) = A (k) + jB (k) = |C (k) | ejθ(k)
u v A B |C | θ
π/2
π/2
y (t) = h (t) ◦ x (t) = T 0
h̃ (t − τ ) x̃ (τ ) dτ
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F{h (t) ◦ x (t)} = F{h (t)}F{x (t)}
e (n) = d (n) ∗ c (n) =∞
m=−∞
d (m) c (n − m)
F{h (t) x (t)} = F{h (t)} ∗ F {x (t)}
1T T 0 |x (t) |
2
dt = ∞
k=−∞ |C (k) |2
F{x̃ (t − t0)} = C (k) e−j2πt0k/T
F{x (t) ej2πKt/T } = C (k − K )
T 0
e−j2πmt/T ej2πnt/T dt = T δ (n − m) = { T n = m0 n = m.
•
2π
x (t) = 4
π sin (t) + 1
3sin (3t) +
1
5sin (5t) · · · .
x (t) a (k) = 0 k
1k
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• 2π
x (t) = 4
π
sin (t) − 1
32sin (3t) +
1
52sin (5t) + · · ·
.
1k2
•
• •
jω •
δ (t)
L2
• f (x) (
−π, π)
f (x) f (x)
12 [f (x + 0) + f (x − 0)]
π, −π
12 [f (−π + 0) + f (π − 0)]
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• f (x) (−π, π)
f (x)
(a, b)
f (x)
a b •
f (x) (−π, π)
12 [f (x + 0) + f (x − 0)]
(−π, π) • f (x)
f (x) f (x) 12 [f (x + 0) + f (x − 0)]
• f (x) |f (x) | f (x)
12 [f (x + 0) + f (x − 0)] (−π, π)
π−π f (x) dx •
x [f (x + 0) + f (x − 0)] /2 I = (a, b) I
• a (k) b (k) f (x)
• a (k) b (k)
f (x)
• f (x) X [0, X ] f [0, X ] f [0, X ] [0, X ] f [0, X ] f [0, X ]
f (x)
x
f (x)
f
12 [f (x
−) + f (x+)] x
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• 1 ≤ p
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x (t)
t
F{x} = X (ω)
F{x + y} = F{x} + F{y} x (t) = u (t) + jv (t) X (ω) = A (ω) +
jB (ω)
u v A B |X | θ
π/2
π/2
y (t) = h (t) ∗ x (t) = ∞−∞
h (t − τ ) x (τ ) dτ = ∞
−∞h (λ) x (t − λ) dλ
F{h (t) ∗ x (t)} = F{h (t)}F{x (t)}
F{h (t) x (t)} = 12πF{h (t)} ∗ F {x (t)}
∞−∞
|x (t) |2dt = 12π ∞−∞
|X (ω) |2dω
F{x (t − T )} = X (ω) e−jωT
F{x (t) ej2πKt
} = X (ω
−2πK )
F{dxdt } = jωX (ω) F{x (at)} = 1|a|X (ω/a)
∞−∞
e−jω1tejω2t = 2πδ (ω1 − ω2)
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• x (t) = δ (t) X (ω) = 1• x (t) = 1 X (ω) = 2πδ (ω)• x (t) T
x (t) =
∞n=−∞ δ (t − nT )
2π/T 2π/T X (ω) =
2π
∞k=−∞ δ (ω − 2πk/T ) •
s
t
F (s) = ∞
−∞
f (t) e−st dt
f (t) = 1
2πj
c+j∞c−j∞
F (s) est ds
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s = σ + jω
s
s = jω jω
L{x + y
} =
L{x}
+L{
y}
y (t) = h (t) ∗ x (t) = h (t − τ ) x (τ ) dτ L{h (t) ∗ x (t)} = L{h (t)}L{x (t)}
L{dxdt } = sL{x (t)} L{dxdt } = sL{x (t)} − x (0) L{ x (t) dt} = 1sL{x (t)} L{x (t − T )} = C (k) e−Ts L{x (t) ejω0t} = X (s − jω0)
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x (n)
x (n) n
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x (n) N n x (n) x (n) N C (k)
C (k) =N −1
n=0
x (n) e−j2πN nk
j =√ −1
x (n) C (k)
x (n) = 1
N
N −1k=0
C (k) ej2πN nk
N −1k=0
e−j2πN mkej
2πN nk = { N n = m
0 n = m.
e−j2πN k
k ∈ {0, N − 1} N N (s − 1)N
W N = e−j 2π
N
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C (k) =N −1n=0
x (n) W nkN
N
N
C 0C 1
C 2
C N −1
=
W 0
W 0
W 0
· · · W 0
W 0 W 1 W 2
W 0 W 2 W 4
W 0 · · · W (N −1)(N −1)
x0x1
x2
xN −1
C = Fx.
F FTF = kI k FT = kF−1
N N
k
k F x C N F x
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ejx
= cos (x) + jsin (x)
C (k)
C (k)
n x (n) k C (k)
x (n)
{0 ≤ n ≤ (N − 1)} C (k)
x (n) −∞ +∞ n k
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N x (n)
N − 1
N
x (n)
2N x (−n) x (n) 2N
x (n)
x̃ (n)
y (n) =∞
m=−∞
h (m) x (n − m) = h (n) ∗ x (n)
x (n) h (n) y (n)
ỹ (n) =
N −1m=0
h̃ (m) x̃ (n − m) = h (n) ◦ x (n)
N N
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h (n)
H =
h0 hL−1 hL−2 · · · h1h1 h0 hL−1
h2 h1 h0
hL−1 · · · h0
,
Y = HX
X Y
N M N + M − 1
N x (n) F{x (n)} = C (k)
F{x (n) + y (n)} = F{x (n)} + F{y (n)} F−1 = 1N F
T
C (k) = C (k + N ) x (n) x (n) = x (n + N ) x (n) = u (n) + j v (n)
C (k) = A (k) + jB (k)
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u v A B |C | θ
π/2
π/2
F{h (n) ◦ x (n)} = F{h (n)}F{x (n)} F{h (n) x (n)} = F{h (n)} ◦ F {x (n)}
N −1n=0 |x (n) |2 = 1N
N −1k=0 |C (k) |2
F{x (n − M )} = C (k) e−j2πMk/N F{x (n) ej2πKn/N } = C (k − K ) F{x (Kn)} = 1K
K −1m=0 C (k + Lm)
N = LK xs (2n) = x (n) n
F{xs (n)
} = C (k) k = 0, 1, 2, ..., 2N
−1
W kN
N = 1 k = 0, 1, 2,...,N − 1
N −1k=0
e−j2πmk/N ej2πnk/N = { N n = m0 n = m.
y = Hx H H FTHF = Hd Hd N h (n) F N H h (n)
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δ (n)
δ (n) = { 1 n = 00
M (n)
M (n) = { 1 n = 0, 1, · · · , M − 10
• DF T {δ (n)} = 1
• DF T
{1}
= N δ (k) • DF T {ej2πKn/N } = N δ (k − K )• DF T {cos (2πMn/N ) = N 2 [δ (k − M ) + δ (k + M )]• DF T {M (n)} = sin(
πN Mk)
sin( πN k)
N
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f (n)
F (ω) =∞−∞
f (n) e−jωn
f (n) = 1
2π π
−π
F (ω) ejωn dω.
ω
x (n) F{x (n)} = X (ω)
F{x + y} = F{x} + F{y} X (ω) = X (ω + 2π) x (n) = u (n) + j v (n)
X (ω) = A (ω) + jB (ω)
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u v A B |X | θ
π/2
π/2
y (n) = h (n) ∗ x (n) = ∞m=−∞ h (n − m) x (m) =∞k=−∞ h (k) x (n − k)
F{h (n) ∗ x (n)} = F{h (n)}F{x (n)}
Y (ω) = H (ω) ◦ X (ω) = T 0
H̃ (ω − Ω) X̃ (Ω) dΩF{h (n) x (n)} = 12πF{h (n)} ◦ F {x (n)}
∞n=−∞ |x (n) |2 = 12π
π
−π|X (ω) |2dω
F{x (n − M )} = X (ω) e−jωM
F{x (n) ejω0n} = X (ω − ω0) F{x (Kn)} = 1K
K −1m=0 X (ω + Lm) N = LK
F{xs (n)} = X (ω) −Kπ ≤ ω ≤ Kπ xs (Kn) = x (n) n
∞n=−∞ e
−jω1ne−jω2n = 2πδ (ω1 − ω2)
ω N N
X (ω) = DTFT {x (n)} =∞
n=−∞
x (n) e−jωn
X (ω) =N −1n=0
x (n) e−jωn .
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ω N
X
2π
N k
=
N −1n=0
x (n) e−j2πN kn
x (n) x (n)
N x (n) = 0 0 ≥ n ≥ N − 1 L ≥ N
N −1n=0
|x (n) |2 = 1L
L−1k=0
|X (2πk/L) |2 = 1π
π0
|X (ω) |2 dω.
{0 ≤ n ≤ (N − 1)}
2π
• DTFT {δ (n)} = 1 •
DTFT {1} = 2πδ (ω)
•DTFT {ejω0n} = 2πδ (ω − ω0)
•DTFT {cos (ω0n)} = π [δ (ω − ω0) + δ (ω + ω0)]
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•DTFT {M (n)} = sin (ωMk/2)sin (ωk/2)
z
z
F (z) =∞
n=−∞
f (n) z−n
f (n) = 1
2πj
ROC
F (z) zn−1dz.
f (0)
z−1
F (z)
f (1)
F (z) f (2) zF (z) f (n) zn−1F (z)
z
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x (n) Z{x (n)} = X (z)
Z{x + y} = Z{x} + Z{y} Z{x}|z=ejω = DT FT {x} X
ejω
= X
ejω+2π
x (n) = u (n) + j v (n)
X
ejω
= A
ejω
+ jB
ejω
u v A B
even 0 even 0
odd 0 0 odd
0 even 0 even
0 odd odd 0
y (n) = h (n) ∗ x (n) = ∞m=−∞ h (n − m) x (m) =∞k=−∞ h (k) x (n − k)
Z{h (n) ∗ x (n)} = Z{h (n)}Z{x (n)} Z{x (n + M )} = zM X (z) Z{x (n + m)} = zmX (z) − zmx (0) −
zm−1x (1) − · · · − zx (m − 1) Z{x (n − m)} = z−mX (z) − z−m+1x (−1) −
· · · − x (−m) Z{x (n) an} = X (z/a)
Z{nmx (n)} = (−z)mdm
X(z)dzm
x (n)
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u (n) = { 1 n ≥ 00 n 1 • Z{u (n) an} = zz−a |z| > |a|
• z
• 1zF (z)
• F (z) = P (z)Q(z) f (n)
z
z − a = 1 + a z−1 + a2z−2 + · · ·
u (n) a
n
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x (n) x (n − 1) x (n + 3)
x (t)
x (n)
a x (n) + b x (n − 1) + c x (n − 2) = f (n)
n x (n)
x (n) = Kλn
xh (n) = K 1λn1 + K 2λ
n2
f (n)
K i x (n)
a X (z ) + b [z −1X (z ) + x (−1)] +c [z −2X (z ) + z −1x (−1) + x (−2)] = Y (z )
X (z)
X (z) = z2 [Y (z) − b x (−1) − x (−2)] − z c x (−1)
a z2 + b z + c
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x (n)
n n
y (n)
nλn
t est
a x (n + 2) + b x (n + 1) + c x (n) = f (n + 2)
f (n) = u (n) an
F (z) = 1 + a z−1 + a2 z−2 + a3 z−3 + · · · + aM z−M . a z−1
a z −1F (z ) = a z −1 + a2z −2 + a3z −3 + a4z −4 +
· · ·+
aM +1z −M −1
1 − a z−1F (z) = 1 − aM +1z−M −1
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F (z)
F (z) = 1 − aM +1z−M −1
1 − a z−1 = z − aaz M
z − a M → ∞
F (z) = z
z − a
|z| > |a| f (n) z
n
f (n) = u (−n − 1) an = { an n
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F (z) =∞
n=−∞
x (n) z−n = ZT {x (n)}
F
ejω
=∞
n=−∞
x (n) e−jωn = DT FT {x (n)}
x (n) N
F
ej2πN k
=N −1n=0
x (n) e−j2πN kn = DFT {x (n)}
x (n) X (ω)
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φj,k (t) = φ
2jt − k j, k
f (t) =j,k
cj,k φj,k (t) .
cj,k
f (t)
k
j j
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φ (t)
φ (t) =n
h (n) φ (2t − n) .
h (n) φ (t)
Nlog (N ) N 2
M = 2
φ (t) =
nh (n) φ (M t − n)
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M = 2
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Nlog (N )
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x (n) y (n) x (n) → y (n) n
• x (n) → y (n) a x (n) → a y (n) a
• x1 (n) → y1 (n) x2 (n) → y2 (n) (x1 (n) + x2 (n)) → (y1 (n) + y2 (n)) x1 x2
x (n + k) → y (n + k) k
n
n
→ −∞
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δ (n) = { 1 n = 00
δ (n) →
h (n) y (n) x (n)
y (n) = h (n) ∗ x (n) =∞
m=−∞
h (n − m) x (m)
δ (n − m) m = 0, 1, 2, · · · , ∞ x (n)
(x (n) , δ (n − m)) = x (m) δ (n − m) δ (n − m) → h (n − m)
x (m) δ (n − M ) → x (m) h (n − m) x (n)
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x (n) = {
x (n) δ (n) = x (0) δ (n)
→ x (0) h (n)
x (n) δ (n − 1) = x (1) δ (n − 1) → x (1) h (n − 1)x (n) δ (n − 2) = x (2) δ (n − 2) → x (2) h (n − 2)
x (n) δ (n − m) = x (m) δ (n − m) → x (m) h (n − m)
} =
y (n)
y (n) =∞
m=−∞
x (m) h (n − m)
y (n) =∞
m=−∞
h (n − m) x (m)
n = m δ (n − m) → h (n, m)
y (n) = h (n, m) ∗ x (n) =∞
m=−∞
h (n, m) x (m) .
y (n) = h (n) ∗ x (n) =∞
m=−∞h (m) x (n − m) .
h (n) = 0 n < 0 m =n
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y (n) = h (n) ∗ x (n) =n
m=0
h (n − m) x (m)
y (n) = h (n) ∗ x (n) =n
m=0
h (m) x (n − m) .
L
y0
y1
y2
yL−1
=
h0 0 0
· · · 0
h1 h0 0
h2 h1 h0
hL−1 · · · h0
x0
x1
x2
xL−1
x N h M L = N + M − 1
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N = 4 M = 3
y0
y1
y2
y3
y4
y5
=
h0 0 0 0
h1 h0 0 0
h2 h1 h0 0
0 h2 h1 h0
0 0 h2 h1
0 0 0 h2
x0
x1
x2
x3
h (n) = 0 n
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N h (n)
H h (n)
X = Fx Y = Fy
X H
x x (n) y
y = Hx
x
Fy = Y = FHx = FHF−1X
Hd L L h (n)
Y = HdX
Hd = FHF−1
H = F−1HdF
h (n) F
H N + M − 1 N − 1 h (n) M − 1
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x (n)
N + M −1
y0
y1
y2
y3
y4
y5
=
h0 0 0 0 h2 h1
h1 h0 0 0 0 h2
h2 h1 h0 0 0 0
0 h2 h1 h0 0 0
0 0 h2
h1
h0
0
0 0 0 h2 h1 h0
x0
x1
x2
x3
0
0
H
x y
h x N + M − 1
h x H h (n)
y (n)
y (n) =∞
m=−∞
h (n − m) x (m)
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h (n)
x (n) = zn
y (n) =∞
m=−∞
h (n − m) zm
k = n − m
y (n) =∞
k=−∞
h (k) zn−k = ∞
k=−∞
h (k) z−k zn
y (n) = H (z) zn
x (n) = zn
z
zn
H (z)
n z x (n) y (n)
Y (z) = H (z) X (z)
x (n)
zn H (z)
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x (n) = cos (ωn)
y (n) = M (ω) cos (ωn + φ (ω)) + T (n)
T (n) ω M (ω) φ (ω) T (n) n
→ ∞
zn z
z = ejω
ejx = cos (x) + jsin (x)
x (n) = ejωn = cos (ωn) + jsin (ωn)
zn z
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