señales-y-sistemas-Baraniuk.pdf

8/17/2019 señales-y-sistemas-Baraniuk.pdf

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Señales y Sistemas

Collection Editor:

Richard Baraniuk


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Señales y Sistemas

Collection Editor:

Richard Baraniuk

Authors:

Thanos Antoulas

Richard Baraniuk

Steven J. Cox

Benjamin Fite

Roy Ha

Michael Haag

Don Johnson

Ricardo Radaelli-Sanchez

Justin Romberg

Phil Schniter

Melissa Selik

JP Slavinsky

Ricardo von Borries

Translated By:

Fara Meza

Erika Jackson

Online:

C O N N E X I O N S

Rice University, Houston, Texas


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T

t T

f (t) = f (T + t)

f (t) T


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T 0


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f (t) f (t) = f (−t)

f

f (t) = −f (−t)


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f (t) = 1

2 (f (t) + f (

−t)) +

1

2 (f (t)

−f (

−t))

f (t) + f (−t) f (t) − f (−t)


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e (t) = 12

(f (t) + f (−t))

o (t) = 12

(f (t) − f (−t))

e (t) + o (t) = f (t)


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f (t) = 0 t < t1 < ∞ f (t) = 0 t > t1 > −∞ t1


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f (t)

t1 < f (t) < t2

t1 > −∞ t2 < ∞ f (t)

∞ ≤ f (t) ≤ −∞


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f (t − T )

f

T

f (at) f a


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f (t) f (− (at))

f (t)

t

at

f (at)

t

t − ba

f `

a`

t − ba

´´ = f (at − b)


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x (t) = Acos(ωt + φ)

A ω φ ωt 2πf t

T = 2π

ω


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A = 2

w = 2

φ = 0

f (t) = Best

s σ ω

s = σ + jω

f (t) = Beαt

B α α

• α 0


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t = 0

δ (t)

∞−∞

δ (t) dt = 1

u (t) =

0 t < 01 t ≥ 0


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t

1

t

1

r (t) =

0

t < 0tt0

0 ≤ t ≤ t01 t > t0


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t

1

t0

a − 2

a + 2

1

limit→0

δ (t) ∞−∞

δ (t) dt = 1


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f (t) δ (t) = f (0) δ (t)


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∞−∞ f (t) δ (t) dt =

∞−∞ f (0) δ (t) dt

= f (0) ∞−∞ 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

dtu (t) u (t)

δ [n] =

1 n = 00


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k

s [k]

k

δ [n

−k]

s [n] =∞

k=−∞s [k] δ [n − k]

δ (t) δ [n]

x

ex = 1 + x1

1! +

x2

2! +

x3

3! + . . .

ex =∞k=0

1

k!xk


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ex1+x2 = ex1ex2

s

f (t) = Aest

= Aejωt

A t s s = jω

Aejωt = Acos(ωt) + j (Asin(ωt))

s s = σ + jω

f (t) = Aest

= Ae(σ+jω)t

= Aeσtejωt

S

S = Aeσt

f (t) = Aeσt (cos(ωt) + jsin(ωt))

Re (f (t)) = Aeσt

cos(ωt)

Im (f (t)) = Aeσtsin(ωt)

∼


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f [n] = BesnT

= BejωnT

nT

cos(ωt) = ejwt + e−(jwt)

2

sin(ωt) = ejwt − e−(jwt)

2 j

ejwt = cos(ωt) + jsin(ωt)


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t = 0

σ

σ

σ

σ ω ω

s


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s

s


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s (n) n = {. . . , −1, 0, 1, . . . }

δ (n − m) n = m

n

sn

1

…

…

s (n) = ej2πfn

s (n) = Acos(2πf n + φ)

f − 1

2, 12

ej2π(f +m)n = ej2πfnej2πmn

= ej2πfn

2π


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δ (n) = 1 n = 00

1

n

δn

m s (m) m δ (n − m)

s (n) =∞

m=−∞

s (m) δ (n−

m)

s (n) {a1, . . . , aK } A


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t = 0


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H

H (kf (t)) = kH (f (t))


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H

H (f 1 (t) + f 2 (t)) = H (f 1 (t)) + H (f 2 (t))

H (k1f 1 (t) + k2f 2 (t)) = k2H (f 1 (t)) + k2H (f 2 (t))

H H (f (t)) = y (t) H T

H (f (t − T )) = y (t − T )


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t0 y (t0)

t0

x (t) y (t)

|y (t) | ≤ M y < ∞

|x (t) | ≤ M x < ∞ M x M y

t


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x

L

y

x

α α


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α

β


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t

t0

t0

x (t) x (t − t0)

x (t) x (t − t0)

x (t − t0) t0


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t0

h (t) t < 0


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f (t) y (t)

dny (t)

dtn + an−1

dn−1y (t)dtn−1

+ · · · + a1 dy (t)dt

+ a0y (t) = bmdmf (t)

dtm + · · · + b1 df (t)

dt + b0f (t)

ni=0

aidiy (t)

dti =

mi=0

bidif (t)

dti

an = 1

y (t) f (t)

y (t) = yi (t) + ys (t)

yi (t) ys (t) f (t)


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yi (t)

y0 (t)

n

i=0ai

diy0 (t)

dti = 0 , an = 1

D Dn + an−1Dn−1 + · · · + a0

y0 (t) = 0

y0 (t) 0 t y0 (t) ,

dy0(t)dt

, d2y0(t)dt2

, . . .

est s ∈ C

y0 (t) = cest , c = 0

c

s

ddty0 (t) = csest d2

dt2 y0 (t) = cs2est . . .

Dn + an−1Dn−1 + · · · + a0

y0 (t) = 0

c

sn + an−1sn−1 + · · · + a1s + a0

est = 0

t

sn + an−1sn−1 + · · · + a1s + a0 = 0


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s {s1, s2, . . . , sn}

(s

−s1) (s

−s2) (s

−s3) . . . (s

−sn) = 0

c1es1t

c2es2t

. . .

cnesnt

y0 (t) = c1es1t + c2e

s2t + · · · + cnesnt {c1, . . . , cn}

yi (t) = 0 yi (t)

ni=0

aidiy (t)

dti =

mi=0

bidif (t)

dti

f (t) f (t)

f (t)

h (t)

m < n

dny (t)

dtn + an−1

dn−1y (t)dtn−1

+ · · · + a1 dy (t)dt

+ a0y (t) = bmdmf (t)

dtm + · · · + b1 df (t)

dt + b0f (t)

QD [y (t)] = P D [f (t)]

QD [·] y (t)

QD [y (t)] = dny (t)

dtn + an−1

dn−1y (t)dtn−1

+ · · · + a1 dy (t)dt

+ a0y (t)

P D [·] f (t)

h (t) = bnδ (t) + P D [yn (t)] µ (t)

m < n bn = 0 yn

yn−1 (0) = 1, yn−2 (0) = 1, . . . , y (0) = 0


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x (t) h (t)

y (t) =

∞

−∞

x (τ ) h (t − τ ) dτ

∗

y (t) = x (t) ∗ h (t)

τ = t − τ

x (t) ∗ h (t) = h (t) ∗ x (t)

∼


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f (t)

h (t)


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Æ

f (τ )

t

τ

f (t)

δ ∆ (t) =

1∆ − ∆2 < t < ∆20


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limit∆→0 δ ∆ (t) → h → limit∆→0 h (t)limit∆→0

δ ∆ (t − n∆) → h → limit∆→0

h (t − n∆)limit∆→0

f (n∆) δ ∆ (t − n∆) ∆ → h → limit∆→0

f (n∆) h (t − n∆) ∆limit∆→0

n f (n∆) δ ∆ (t − n∆) ∆ → h → limit∆→0

n f (n∆) h (t − n∆) ∆ ∞−∞ f (τ ) δ (t − τ ) dτ → h →

∞−∞ f (τ ) h (t − τ ) dτ

f (t) → h → y (t) = ∞−∞ f (τ ) h (t − τ ) dτ

t t t


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x (t) h (t)

τ h (t − τ )


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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) = t0

1dτ

= t

1 ≤ t < 2 h (t − τ ) t − 1


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y (t) =

1

t−1 1dτ

= 1−

(t−

1)

= 2 − t

y (t) =

0 t < 0

t 0 ≤ t < 1

2 − t 1 ≤ t < 20 t ≥ 2

x (t) ∗ h (t)

x (t)


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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) =

∞−∞

f (τ ) h (t − τ ) dτ

τ = t − τ

y (t) = ∞−∞ f (t − τ ) h (τ ) dτ

= ∞−∞ h (τ ) f (t − τ ) dτ

f (t) ∗ h (t) = h (t) ∗ f (t)


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f 1 (t)

∗(f 2 (t) + f 3 (t)) = f 1 (t)

∗f 2 (t) + f 1 (t)

∗f 3 (t)


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c (t) = f (t) ∗ h (t)

c (t − T ) = f (t − T ) ∗ h (t)

c (t − T ) = f (t) ∗ h (t − T )


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f (t) ∗ δ (t) = f (t)

δ (t)

f (t) ∗ δ (t) = ∞−∞

δ (τ ) f (t − τ ) dτ

δ (τ ) = 0 τ

= 0

f (t) ∗ δ (t) = ∞−∞ δ (τ ) f (t) dτ = f (t)

∞−∞ δ (τ ) dτ

δ (τ ) τ = 0

f (t) ∗ δ (t) = f (t)


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Duracin (f 1) = T 1 (f 2) = T 2

Duracin (f 1

∗f 2) = T 1 + T 2


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(f 1) = N 1 (f 2) = N 2

Duracin (f 1 ∗ f 2) = N 1 + N 2 − 1


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f h f ∗ h

A

|f (t) | < A

h (t)

∞

−∞ |h (t)

|dt <

∞

h (n)

∞n=−∞

|h (n) | < ∞


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jω

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