Brief Notes on Signals and Systems 7.2

8/21/2019 Brief Notes on Signals and Systems 7.2

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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 − )


9/77

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


47/77

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


48/77

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)


49/77

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)


50/77

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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Brief Notes on Signals and Systems 7.2

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