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Communication Systems

Collection Editor:

Janko Calic





Communication Systems

Collection Editor:

Janko Calic

Authors:

Thanos Antoulas

Richard Baraniuk

Dan Calderon

Catherine Elder

Anders Gjendemsjø

Michael Haag

Matthew Hutchinson

Don Johnson

Stephen Kruzick

Robert Nowak

Ricardo Radaelli-Sanchez

Justin Romberg

Phil Schniter

Melissa Selik

JP Slavinsky

Online:< http://legacy.cnx.org/content/col10631/1.3/ >

OpenStax-CNX











T

t

T

f (t) = f (t + T )

f (t) T

T 0



f (t)

t1 < t < t2

t1 < t2 f (t)

−∞ < t < ∞



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)



t

f (t) =

sin (2πt) /t t ≥ 1

0 t < 1



f (t− T )

f

T

f (at) f a

f (t) f (at− b)



f (t)

t

at

f (at)

t

t− b

a

f `a`t− b

a

´´ = f (at− b)





Acos(ωt + φ)

A ω φ

T = 2π

ω

A = 2 w = 2 φ = 0

Aest

s = σ + jω σ ω

δ (t)



u (t) = 0

t < 01 t ≥ 0

t

1



a − 2

a + 2

1

δ (t) ∞−∞

δ (t) dt = 1



• δ (αt) = 1|α|δ (t)

• δ (t) = δ (−t)• δ (t) = d

dtu (t) u (t)

• f (t) δ (t) = f (0) δ (t)

∞−∞

f (t) δ (t) dt =

∞−∞

f (0) δ (t) dt = f (0)

∞−∞

δ (t) dt = f (0)



Aest

s = σ + iω σ ω

ejx = cos (x) + jsin (x)

ez

z = 0

z z = jx

ejx =∞

k=0(jx)k

k!

=

∞k=0 (−1)

k x2k

(2k)! + j

∞k=0 (−1)

k x2k+1

(2k+1)!

= cos (x) + jsin (x)

cos (x) sin (x) t = 0

x

x = ωt

ejωt = cos (ωt) + jsin (ωt)

cos (ωt) = 1

2ejωt +

1

2e−jωt

sin (ωt) =

1

2 j ejωt

− 1

2 j e−jωt



s = σ + ωj σ

ω

θ

e(σ+jω)t+jθ = eσt (cos (ωt + θ) + jsin (ωt + θ)) .

est

Ree(σ+jω)t+jθ = eσtcos (ωt + θ)

Ime(σ+jω)t+jθ = eσtsin (ωt + θ)

t = 0 σ

σ

σ





s (n) n = . . . ,−1, 0, 1, . . .

δ (n− m) n = m

n

sn

1

…

…



s (n) = ei2πfn

s (n) = Acos(2πf n + φ)

f

−12

, 12

ei2π(f +m)n = ei2πfnei2πmn

= ei2πfn

2π

δ (n) =

1 n = 0

0

1

n

δn

m

s (m)

m

δ (n−m)

s (n) =∞

m=−∞s (m) δ (n −m)



s (n) a1, . . . , aK

A



H

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

H

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

H (k1f 1 (t) + k2f 2 (t)) = k1H (f 1 (t)) + k2H (f 2 (t))



S T (f (t)) =f (t − T ) T

HS T = S T H

T



t0 y (t0)

t0

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



H 1

H 1 (f (t)) = tf (t)

f f, g a, b

H 1 (af (t) + bg (t)) = t (af (t) + bg (t)) = atf (t) + btg (t) = aH 1 (f (t)) + bH 1 (g (t))

t H 1

H 2

H 2 (f (t)) = (f (t))2

f

H 2 (2t) = 4t2

= 2t2 = 2H 2 (t)

t H 2

t

t0

t0

x (t) x (t − t0)

x (t) x (t − t0)

x (t − t0) t0



H 1

H 1 (f (t)) = tf (t)

f

S T (H 1 (f (t))) = S T (tf (t)) = (t − T ) f (t − T ) = tf (t − T ) = H 1 (f (t− T )) = H 1 (S T (f (t)))

T H 1

H 2

H 2 (f (t)) = (f (t))2

f

T

f

S T (H 2 (f (t))) = S T f (t)2 = (f (t

−T ))

2= H 2 (f (t

−T )) = H 2 (S T (f (t)))

t H 2







H 3

H 3 (f (t)) = 2f (t)

f f, g a, b

H 3 (af (t) + bg (t)) = 2 (af (t) + bg (t)) = a2f (t) + b2g (t) = aH 3 (f (t)) + bH 3 (g (t))

t H 3 T f

S T (H 3 (f (t))) = S T (2f (t)) = 2f (t − T ) = H 3 (f (t − T )) = H 3 (S T (f (t)))

t

H 3

H 3

H 1 (f (t)) = tf (t)

H 2 (f (t)) = (f (t))2



(f ∗ g) (t) =

∞−∞

f (τ ) g (t − τ ) dτ

f, g R

f ∗ g = g ∗ f

f, g R

(f ∗ g) (t) =

∞−∞

f (t − τ ) g (τ ) dτ

f, g R

H

h

x

H (x)

x (t) =

∞−∞

x (τ ) δ (t − τ ) dτ

x (t) = lim∆→0

n

x (n∆) δ ∆ (t − n∆) ∆



δ ∆ (t) =

1/∆ 0 ≤ t < ∆

0 otherwise

δ (t)

Hx (t) = lim∆→0

n

x (n∆) Hδ ∆ (t − n∆) ∆

Hx (t) =

∞−∞

x (τ ) Hδ (t − τ ) dτ.

Hδ (t − τ ) h (t − τ )

Hx (t) = ∞

−∞x (τ ) h (t

−τ ) dτ = (x

∗h) (t) .

f, g

(f ∗ g) (t) =

∞−∞

f (τ ) g (t − τ ) dτ =

∞−∞

f (t − τ ) g (τ ) dτ.

τ = 0

t

t

h (t) = 1

RC e−t/RC u (t) ,

x (t) = u (t) .

y (t) = x (t) ∗ h (t) .

x (t) = u (t)

y (t) =

∞−∞

1

RC e−τ/RC u (τ ) u (t − τ ) dτ.



y (t) = max0,t

0

1

RC e−τ/RC

dτ.

y (t) = 0 t ≤ 0

1− e−t/RC t > 0

y (t) =

1− e−t/RC

u (t) .

(f ∗ g) (t) =

T 0

f (τ )

g (t − τ ) dτ

f, g R [0, T ]

f ,

g f g

f ∗ g = g ∗ f

f, g R [0, T ]

(f ∗ g) (t) =

T 0

f (t− τ )

g (τ ) dτ

f, g R [0, T ]

f ,

g f g

(f ∗ g) (t) =

t0

f (τ ) g (t − τ ) dτ +

T t

f (τ ) g (t − τ + T ) dτ

f, g R [0, T ]



H h

x

H (x)

x (t) =

T 0

x (τ )

δ (t − τ ) dτ

x (t) = lim∆→0

n

x (n∆)

δ ∆ (t− n∆) ∆

δ ∆ (t) = 1/∆ 0

≤t < ∆

0 otherwise

δ (t)

Hx (t) = lim∆→0

n

x (n∆) H

δ ∆ (t− n∆) ∆

Hx (t) =

T 0

x (τ ) H

δ (t − τ ) dτ.

Hδ (t − τ ) h (t − τ )

Hx (t) = T

0

x (τ )

h (t − τ ) dτ = (x ∗ h) (t) .

f, g

(f ∗ g) (t) =

T 0

f (τ )

g (t − τ ) dτ =

T 0

f (t − τ )

g (τ ) dτ.

τ = 0

t ∈ R [0, T ]

t

R [0, T ]





f 1, f 2, f 3

f 1 ∗ (f 2 ∗ f 3) = (f 1 ∗ f 2) ∗ f 3

(f 1 ∗ (f 2 ∗ f 3)) (t) = ∞−∞ ∞−∞ f 1 (τ 1) f 2 (τ 2) f 3 ((t − τ 1)− τ 2) dτ 2dτ 1

= ∞−∞ ∞−∞ f 1 (τ 1) f 2 ((τ 1 + τ 2)− τ 1) f 3 (t − (τ 1 + τ 2)) dτ 2dτ 1

= ∞−∞ ∞−∞ f 1 (τ 1) f 2 (τ 3 − τ 1) f 3 (t − τ 3) dτ 1dτ 3

= ((f 1 ∗ f 2) ∗ f 3) (t)

τ 3 = τ 1 + τ 2

f 1, f 2

f 1 ∗ f 2 = f 2 ∗ f 1

(f 1 ∗ f 2) (t) = ∞−∞ f 1 (τ 1) f 2 (t − τ 1) dτ 1

= ∞−∞ f 1 (t − τ 2) f 2 (τ 2) dτ 2

= (f 2∗

f 1) (t)

τ 2 = t − τ 1

f 1, f 2, f 3

f 1 ∗ (f 2 + f 3) = f 1 ∗ f 2 + f 1 ∗ f 3

(f 1 ∗ (f 2 + f 3)) (t) =

∞−∞ f 1 (τ ) (f 2 (t − τ ) + f 3 (t − τ )) dτ

= ∞−∞ f 1 (τ ) f 2 (t − τ ) dτ +

∞−∞ f 1 (τ ) f 3 (t − τ ) dτ = (f 1 ∗ f 2 + f 1 ∗ f 3) (t)



f 1, f 2

a

a (f 1 ∗ f 2) = (af 1) ∗ f 2 = f 1 ∗ (af 2)

(a (f 1 ∗ f 2)) (t) = a ∞−∞ f 1 (τ ) f 2 (t − τ ) dτ

= ∞−∞ (af 1 (τ )) f 2 (t − τ ) dτ

= ((af 1) ∗ f 2) (t)

= ∞−∞ f 1 (τ ) (af 2 (t − τ )) dτ

= (f 1 ∗ (af 2)) (t)

f 1, f 2

f 1 ∗ f 2 = f 1 ∗ f 2

f 1 ∗ f 2

(t) =

∞−∞ f 1 (τ ) f 2 (t − τ ) dτ

= ∞−∞ f 1 (τ ) f 2 (t − τ )dτ

= ∞−∞ f 1 (τ ) f 2 (t

−τ ) dτ

=

f 1 ∗ f 2

(t)

f 1, f 2 S T

S T (f 1 ∗ f 2) = (S T f 1) ∗ f 2 = f 1 ∗ (S T f 2)

S T (f 1 ∗ f 2) (t) = ∞−∞ f 2 (τ ) f 1 ((t − T )− τ ) dτ

= ∞−∞ f 2 (τ ) S T f 1 (t − τ ) dτ

= ((S T f 1) ∗ f 2) (t)

= ∞−∞ f 1 (τ ) f 2 ((t − T )− τ ) dτ

= ∞−∞ f 1 (τ ) S T f 2 (t − τ ) dτ

= f 1 ∗ (S T f 2) (t)



f 1, f 2

ddt (f 1 ∗ f 2) (t) =

df 1dt ∗ f 2

(t) =

f 1 ∗ df 2dt

(t)

ddt (f 1 ∗ f 2) (t) =

∞−∞ f 2 (τ ) d

dtf 1 (t − τ ) dτ

=df 1dt ∗ f 2

(t)

= ∞−∞ f 1 (τ ) d

dtf 2 (t − τ ) dτ

=

f 1 ∗ df 2dt

(t)

f δ

f ∗ δ = f

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

= f (t) ∞−∞ δ (t − τ ) dτ

= f (t)

f 1, f 2

Duration (f ) f

Duration (f 1 ∗ f 2) = Duration (f 1) + Duration (f 2)

(f 1 ∗ f 2) (t)

t

τ

f 1 (τ ) f 2 (t− τ )

τ t Duration (f 1) + Duration (f 2)



s (t) T

T

1T

kT

k = . . . ,−1, 0, 1, . . .

s (t) =∞

k=−∞ckei

2πktT

ck =

1

2 (ak − ibk)

ck

c0 = a0

ei

2πktT

∼

∼

∼



T

k

T 0

ei2πktT e(−i)

2πltT dt =

T k = l

0

k = l

e−(i2πlt) [0, T ]

ck = 1T

T 0

s (t) e−(i 2πktT )dt

c0 = 1T

T 0

s (t) dt

sqT (t)

sqT (t) =

1 0 < t < T 2

−1

T 2

< t < T

ck = 1

T

T 2

0

e−(i 2πktT )dt − 1

T

T T 2

e−(i 2πktT )dt

i

ck = −2i2πk

(−1)

k − 1

=

2

iπk

k

0 k

sq (t) =

k∈...,−3,−1,1,3,...

2

iπke(i)

2πktT

1T

k k



ck

s (t)

ck = c−k

ck

(ck) = (c−k)

(ck) = − (c−k)

ck = c−k

s (−t) = s (t)

c−k = ck

s (−t) = −s (t)

c−k = −ck

τ

s (t − τ )

cke−i2πkτ T

ck

s (t)

τ

−2πkτ T

1T

T 0

s (t − τ ) e(−i)2πktT dt = 1

T

T −τ −τ s (t) e(−i)

2πk(t+τ )T dt

= 1T e

(−i) 2πkτ T

T −τ

−τ s (t) e(−i)

2πktT dt

T −τ −τ (·) dt =

T 0

(·) dt

1

T T

0

s2 (t) dt =

∞

k=−∞

(|ck|)2



∆

T

∆

t

p(t)A

……

∆ T A

ck = 1

T ∆0

Ae−i2πkt

T dt = − A

i2πk

e−i2πk∆

T − 1

1− e−(iθ) = e−iθ2

eiθ2 − e−

iθ2

= e−

iθ2 2isin

θ

2

ck = Ae−iπk∆T

sinπk∆T

πk

ck = c−k

|ck| = A|sinπk∆T

πk

|

∠ (ck) = −πk∆

T + πneg

sinπk∆T

πk

sign(k)

neg(·)

π



∆

T = 0.2

A = 1

∠ (ck)

kT

∆2

c0

π

2π

−π

−π

−π

− (2π) 2π

[−π, π)

2π



T

s (t) = a0 +∞k=1

akcos

2πkt

T

+

∞k=1

bksin

2πkt

T

ak

bk

ck

ck = 1

2 (ak − ibk)

∀k , l , k ∈ Zl ∈ Z :

T 0

sin

2πkt

T

cos

2πlt

T

dt = 0

T 0

sin

2πkt

T

sin

2πlt

T

dt =

T 2 (k = l) ∧ (k = 0) ∧ (l = 0)

0 (k = l) ∨ (k = 0 = l)

T 0

cos2πkt

T

cos2πlt

T

dt =

T 2 (k = l) ∧ (k = 0) ∧ (l = 0)

T k = 0 = l

0 k = l

sin(α)sin(β ) = 12 (cos (α − β )− cos(α + β ))

cos(α)cos(β ) = 12 (cos (α + β ) + cos (α − β ))

sin(α)cos(β ) = 12 (sin (α + β ) + sin (α− β ))

•

• T T 2

l

cos2πltT

al T

0 s (t)cos

2πlt

T

dt =

T

0 a0cos

2πlt

T

dt +

∞k=1

ak

T

0 cos2πkt

T

cos2πlt

T

dt +∞

k=1 bk

T

0 sin2πkt

T

cos2πlt

T

dt



k

l

alT 2

k = 0 = l

a0T

∀l, l = 0 :

al = 2

T T 0

s (t)cos2πlt

T

dt

a0 = 1T

T 0

s (t) dt

∀k, k = 0 :

ak = 2T

T 0

s (t)cos2πktT

dt

bk = 2T

T 0

s (t)sin2πktT

dt

a0

s (t)

s (t) =

sin

2πtT

0 ≤ t < T

2

0

T 2 ≤ t < T

bk

bk = 2

T

T 2

0

sin

2πt

T

sin

2πkt

T

dt

T 2

0 sin2πtT

sin2πktT

dt = 1

2

T 2

0 cos2π(k−1)t

T

− cos

2π(k+1)t

T

dt

=

12 k = 1

0

b1 = 1

2

b2 = b3 = · · · = 0

a0

1

π

ak =

−

2π

1k2−1

k ∈ 2, 4, . . .0



k

kT k = 1

k = 2

k

ak

-0.5

0

0.5

bk

0 2 4 6 8 10

0

0.5

k



rms(s) = 1

T T

0

s2 (t) dt

power (s) = rms2 (s)

= 1T

T 0

s2 (t) dt

power (s) = 1

T T

0a0 +

∞

k=1

akcos2πkt

T +∞

k=1

bksin2πkt

T 2

dt

power (s) = a02 +

1

2

∞k=1

ak2 + bk

2

0 2 4 6 8 10

P s(k)

0

0.1

0.2

k

k

ak2+bk

2

2 P s (k)



sK (t)

K + 1

sK (t) = a0 +K k=1

akcos

2πkt

T

+

K k=1

bksin

2πkt

T



k

ak

-0.5

0

0.5

bk

0 2 4 6 8 10

0

0.5

k

0

0.5

1

K=0

t

0

1

K=1

t

0.5

0

0.5

1

K=2

t

0 0.5 1 1.5 2

0

0.5

1

K=4

t

K + 1

K + 1



K + 1

K (t) =∞

k=K +1

akcos2πkt

T +∞

k=K +1

bksin2πkt

T

rms(K ) =

1

2

∞k=K +1

ak2 + bk2

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

R e l a t i v e r m s e r r o r

K



0 2 4 6 8 100

0.5

1P s(k)

k

R e l a t i v e r m

s e r r o r

0 2 4 6 8 100

0.5

1

K

K

1k2

1k



0

1K=49

-1

t

-1

0

1K=1

t

-1

0

1K=5

t

-1

0

1 K=11

t

sq (t)

t

∼



t

limitK →∞

rms(K ) = 0

s1 (t) s2 (t)

s1 (t) = s2 (t) t

rms(s1 − s2) = 0

x (t)

T

T

x (t) =K

k=−K

ckei2πktT

ck

ck

x (t)

N a1, . . . , aN

an cn = 1 ck = 0 k = n nth

1T

N T

a13



13 13 = 11012

N

T

T

0 T 2T 3T

-2

-1

0

1

2

t

x(t)

T



T

L (a1s1 (t) + a2s2 (t)) = a1L (s1 (t)) + a2L (s2 (t))

x (t) = ei2πktT

y (t) = H

kT

ei2πktT

f = kT

x (t)

y (t) =∞

k=−∞ckH

k

T

ei

2πktT

ckH kT



∆

T

∆

t

p(t)

A

……

0

0

0.2

0 1 20

1

0

0

0.2

0 1 20

1

0 1 20

1

0 10 20

0

0.2

p e c t r a

a g n t u

efc: 100 Hz fc: 1 kHz fc: 10 kHz

Frequency (kHz) Frequency (kHz) Frequency (kHz)

Time (ms) Time (ms) Time (ms)

A m p l i t u d e

10 2010 20

∆

T = 0.2

RC

RC

H (f ) =

1

1 + i2πfRC



RC

sT (t) T

ck (T )

ck (T ) = 1

T

T 2

−T 2

sT (t) e−i2πktT dt

f

kT k

S T (f )≡

T ck (T ) = T 2

−T 2

sT (t) e−(i2πft)dt

sT (t) =

∞k=−∞

S T (f ) ei2πft 1

T

limitT →∞

sT (t) ≡ s (t) =

∞−∞

S (f ) ei2πftdf

S (f ) = ∞

−∞

s (t) e−(i2πft)dt

S (f ) s (t)

p (t)

P (f ) =

∞−∞

p (t) e−(i2πft)dt =

∆0

e−(i2πft)dt = 1

− (i2πf )

e−(i2πf ∆) − 1



P (f ) = e−(iπf ∆) sin(πf ∆)

πf

-20 -10 0 10 200

0.2T=5

Frequency (Hz)

S p e c t r a l M a g n i t u d e

0

0.2T=1

S p

e c t r a l M a g n i t u d e

T = 1

p (t)

T = 5

sin(t)t

sinc (t)

|∆sinc (πf ∆) |

F (s) F −1 (S )

F (s) = S (f )

= ∞−∞ s (t) e−(i2πft)dt



F −1 (S ) = s (t)

= ∞−∞ S (f ) ei2πftdf

s (t) = F −1 (F (s (t))) S (f ) = F F −1 (S (f ))

F (S (f ))

∞−∞

s2 (t) dt =

∞−∞

(|S (f ) |)2df

s (t)

F (· · · (F (s))) = s (t)

s (t) ↔ S (f )



s (t) S (f )

e−(at)u (t) 1i2πf +a

e−(a|t|) 2a4π2f 2+a2

p (t) =

1 |t| < ∆

2

0 |t| > ∆2

sin(πf ∆)πf

sin(2πWt)πt

S (f ) =

1 |f | < W

0

|f |

> W

a1s1 (t) + a2s2 (t) a1S 1 (f ) + a2S 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−(i2πfτ )S (f )

ei2πf 0ts (t) S (f − f 0)

s (t)cos(2πf 0t) S (f −f 0)+S (f +f 0)2

s (t)sin(2πf 0t) S (f −f 0)−S (f +f 0)2i

ddts (t) i2πf S (f )

t−∞ s (α) dα 1

i2πf S (f )

S (0) = 0

t ts (t) 1−(i2π)

dS (f )df

∞−∞ s (t) dt S (0)

s (0) ∞−∞ S (f ) df

∞−∞ (|s (t) |)2dt

∞−∞ (|S (f ) |)2df

s (t)

(1 + s (t))cos(2πf ct)



(1 + s (t))cos(2πf ct) = cos(2πf ct) + s (t)cos(2πf ct)

cos (2πf ct)

1f c

c±1 = 12 s (t)

s (t)cos(2πf ct) =

∞−∞

S (f ) ei2πftdf cos(2πf ct)

(s (t)cos(2πf ct)) = 1

2

∞−∞

S (f ) ei2π(f +f c)tdf + 1

2

∞−∞

S (f ) ei2π(f −f c)tdf

(s (t)cos(2πf ct)) = 1

2

∞−∞

S (f − f c) ei2πftdf + 1

2

∞−∞

S (f + f c) ei2πftdf

(s (t)cos(2πf ct)) = ∞−∞

S (f − f c) + S (f + f c)2

ei2πftdf

F (s (t)cos(2πf ct)) = S (f − f c) + S (f + f c)

2

S(f)

fW –W

X(f)

f –fc+W –fc –W fc+Wfc –W –fc fc

S(f–fc)S(f+fc)

W

± (f c)

± (f c)

1

2

s (t)

s (t)



x (t)

s (t)

W x (t)

f c W

2W Hz

x (t)

H (f )

X (f ) H (f )

RC

P (f ) = e−(iπf ∆) sin(πf ∆)

πf

H (f ) = 1

1 + i2πfRC

Y (f ) = e−(iπf ∆) sin(πf ∆)

πf

1

1 + i2πfRC

Y (f )

e−(iπf ∆) sin(πf ∆)πf = e−(iπf ∆) eiπf ∆−e−(iπf ∆)

i2πf

= 1

i2πf 1− e−(i2πf ∆)

Y (f ) = 1

i2πf

1− e−(iπf ∆)

1

1 + i2πfRC

•

1i2πf



• e−(i2πf ∆) ∆

• 1− e−(i2πf ∆)

•

1RC e

− tRC u (t)

1i2πf

1

i2πf

1

1 + i2πfRC ↔

1− e− tRC

u (t)

Y (f )

1 e−(i2πf ∆)

Y (f ) ↔ 1− e− tRC

u (t)− 1− e−t−∆RC

u (t − ∆)

t = ∆

1e

1

i2πf e−(i2πf ∆)

1i2πf

e−(i2πf ∆)

Y (f ) = X (f ) H (f )

X H



X (f ) H 1 (f ) X (f ) H 1 (f ) H 2 (f )

X (f ) H 2 (f ) H 1 (f )

H 1 (f ) H 2 (f )

s (t) = sin(t) s (t) = sin2 (t)

s (t) = cos(t) + 2cos (2t) s (t) = cos(2t)cos(t)

s (t) = cos

10πt + π6

(1 + cos (2πt))

s (t)

t

s(t)

1

18

14 13

8



t

s t

1 2 3

1

t

s(t)

1 2 3

1

t

s t

1 2 3

1

4



+ –

1

1

+

–

1

1

vin(t)vout(t)

vin (t) T

T = 0.01 T = 2 T

τ

T 4

s (t)

2 = 1

T

T 0

(s (t)− s (t))2dt

s (t) s (t)

s (t) =

K k=−K

ckei2πkT t

2K + 1 c0 ck

K



1 2

1s(t)

t

0 50 100 150 200 250 300 35050

55

60

65

70

75

80

85

90

95

Day

A v e r a g e H i g h T e m p e r a t u r e

10

11

12

13

14

D a y l i gh t H o ur s

Temperature

Daylight





t

1

1 2

1

2

t

1

1 2

1

2

t

1

1 2

1

2

3 4 5 6 7 8 9

T



tT

tT

T

T 2

τ



τ

Soundwave

s(t)s(t-τ)

∆l ∆r

s (t) y (t)

τ ∆l ∆r y (t)

τ

x (t)

y (t)



x(t) y(t)H1(f)

H2(f)

H1(f)

H2(f)

x(t)

x(t)

x(t)

y(t)

H1(f)x(t) e(t) y(t)

H2(f)

–

s (t) = sin(πt)πt

y (t)

f

H(f)

14

14

1



H (f ) =

i2πf

4 + i2πf

τ

f 0 f 0 τ l τ h

C du (t)

C p 1− e−(at)u (t)

C dtu (t)

C p

1− e−(bt)

u (t)

b

a



x (t) = Acos(2πf ct)

Bcos(2π ((f c + ∆f) t + ϕ)) ∆f

10v v

∆f

m (t)

x (t) = A (1 + m (t))sin(2πf ct)

m (t)

f c

x (t)

φ

φ

X (f )

X (f )

x (t)

x (t) cosπt2

cos(πt) cos

3πt2



1/4

S(f)

f

1

/4

t

1

1 3

r (t) = A (1 + m (t))cos(2πf ct + φ) φ

m (t) W |m (t) | < 1 φ

r(t)

cos 2πfct

sin 2πfct

LPFW Hz

LPFW Hz

?

?

xc(t)

xs(t)

?

xc (t) xs (t)



jam (t)

… …

t

jam(t)

T 2T –T

A

T

l (t)

r (t)

s+ (t) = l (t) + r (t)

s− (t) = l (t) − r (t)

2W

W

f c

–W W

L(f)

f

–W W

R(f)

f



W

0 1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

1.5

Time

A m p l i t u d e

Example Transmitter Waveform

m1 (t)

m2 (t)

W

f c W x (t)

x (t) =

A (1 + am1 (t))sin(2πf ct) sin(2πf ct) ≥ 0

A (1 + am2 (t))sin(2πf ct) sin(2πf ct) < 0

0 < a < 1 m1 (t) =sin(2πf mt) m2 (t) = sin(2π2f mt) 2f m < W

x (t)

m1 (t) m2 (t) x (t)





H est

H (s) ∈ C

y (t) = H (s) est

H

y (t)

c1es1t + c2es2t → c1H (s1) es1t + c2H (s2) es2t

n

cnesnt →n

cnH (sn) esnt

H H

esnt

H (sn) ∈ C

f (t)

s (t)

s (t) =

∞n=−∞

cnejω0nt

ω0 = 2πT s (t) cn

s (t)

s (t) ∈ L2

[0, T ]

s (t) s (t)

cn

ejω0nt

s (t)

s (t)

∀n, n ∈ Z :

ejω0nt

∼



sT (t) T

cn (T )

cn = 1

T

T 0

s (t) exp (−ßω0t) dt

ω0 = 2πT

n

S T (f ) ≡ T cn = 1

T

T 0

(S T (f ) exp (ßω0t) dt

sT (t) =

∞−∞

f (t) exp (ßω0t) 1

T

limT →∞

sT (t) ≡ s (t) =

∞ −∞

S (f ) exp (ßω0t) df

S (f ) =

∞

−∞s (t) exp (−ßω0t) dt

F (Ω) =

∞−∞

f (t) e−(iΩt)dt

f (t) = 1

2π

∞−∞

F (Ω) eiΩtdΩ

Ω Ω = 2πf

i2πf t

cos (ωt) + sin (ωt) = 1−j2 ejωt + 1+j

2 e−jωt .



f (t) =

e−(αt) t ≥ 0

0

X (Ω) = 1 |Ω| ≤ M

0



f (t) =

∞

n=−∞

cnejω0nt

cn = 1

T

T 0

f (t) e−(jω0nt)dt

ω0 = 2πT

z (t) = af 1 (t) + bf 2 (t)

Z (ω) = aF 1 (ω) + bF 2 (ω)

2π



z (t) = f (t − τ )

z (t)

Z (ω) =

∞−∞

f (t − τ ) e−(iωt)dt

σ = t − τ

Z (ω) = ∞−∞ f (σ) e−(iω(σ+τ )t)dτ

= e−(iωτ ) ∞−∞ f (σ) e−(iωσ)dσ

= e−(iωτ )F (ω)

y (t) = (f 1 (t) , f 2 (t))

= ∞−∞ f 1 (τ ) f 2 (t − τ ) dτ



∞−∞

(|f (t) |)2dt =

∞−∞

(|F (ω) |)2df

z (t) = 1

2π

∞−∞

F (ω − φ) eiωtdω

z (t) = f (t) eiφt



f (t) F (ω)

a (f 1, t) + b (f 2, t) a (F 1, ω) + b (F 2, ω)

αf (t) αF (ω)

F (t) 2πf (−ω)

f (αt) 1|α|F ωα

f (t − τ ) F (ω) e−(iωτ )

(f 1 (t) , f 2 (t)) F 1 (t) F 2 (t)

f 1 (t) f 2 (t) 12π (F 1 (t) , F 2 (t))

dn

dtn f (t) (iω)nF (ω)

∞−∞ (|f (t) |)2dt

∞−∞ (|F (ω) |)2df

f (t) eiφt F (ω − φ)

•

•



•

•

•

•

•

•

•

•

•

•

•

∼



•

x (t)• F s• T s F s = 1

T s

• xs (n) xs (n) = x (nT s)

• Ω• ω ω = ΩT s

x (t) x (t)

x (t)

x (t)

•

•

•

• 2× πF s



x (t) T s xs (n)

xs (n) = 12π

π−π

X s

eiω

eiωndω

Ω ω = ΩT s

xs (n) = T s2π

πT s

−πT s

X s

eiΩT s

eiΩT sndΩ

x (t) = 1

2π

∞−∞

X (iΩ) eiΩtdΩ

x (nT s)

x (nT s) = 12π

∞−∞

X (iΩ) eiΩnT sdΩ

2πT s

x (nT s) = 1

2π

∞k=−∞

(2k+1)πT s

(2k−1)πT s

X (iΩ) eiΩnT sdΩ

Ω = η + 2×πkT s

x (nT s) = 1

2π

∞

k=−∞

πT s

−πT s

X iη + 2 × πk

T s ei(η+

2×πkT s

)nT sdη

ei2×πkn = 1 η = Ω

T sT s

x (nT s) = T s2π

πT s

−πT s

∞k=−∞

1

T sX

i

Ω +

2 × πk

T s

eiΩnT sdΩ

xs (n) = x (nT s)

n

X s

eiΩT s

= 1

T s

∞k=−∞

X

i

Ω +

2πk

T s

ω = ΩT s

X s

eiω

= 1

T s

∞k=−∞

X

i

ω + 2 × πk

T s

x (t) xs (n)



x (t) = 12π

πT s

−πT s

X (iΩ) eiΩtdΩ

X s

eiΩT s

= X(iΩ)T s

x (t) = T s2π

πT s

−πT s

X s

eiΩT s

eiΩtdΩ

X s

eiΩT s

x (t) = T s2π

πT s

−πT s

∞n=−∞

xs (n) e−(iΩnT s)eiΩtdΩ

x (t) = T s2π

∞n=−∞

xs (n)

πT s

−πT s

eiΩ(t−nT s)dΩ

x (t) =∞

n=−∞xs (n)

sin

πT s

(t − nT s)

πT s

(t − nT s)



X s

eiΩT s

= 1

T s

∞k=−∞ X

i

Ω + 2πkT s

x (t) =∞

n=−∞ xs (n) sin( πT s (t−nT s))

πT s (t−nT s)

T s

t = nT s



X (iΩ)



X (iΩ) [−Ωg, Ωg]

Ωg F g = Ωg2π

X s

eiΩT s

= 1

T s

∞k=−∞

X

i

Ω +

2πk

T s

x (t)

F s ≥ 2F g

Ω ∈ [−Ωg, Ωg]

1T s

2πF s = 2πT s

X s

x (t) F s < 2F g

xs (n)



X s

k = 0

∼



H (iΩ)

xs (n)

h (t)

h (t) = sinc tT s

xs (n) h (t)

xs (t) =∞

n=−∞ xs (n) δ (t − nT )

∼



x (t) = x (t)

x (t) = x (t)

F s

x (t)

x (t) = s (t)

F s

s (t)

x (t)

x (t)



xc (t) x [n]

xs (t) =

∞n=−∞

xc (nT ) δ (t − nT )

X s (Ω) = ∞−∞∞

n=−∞ xc (nT ) δ (t− nT ) e(−i)Ωtdt

=

∞n=−∞ xc (nT )

∞−∞ δ (t− nT ) e(−i)Ωtdt

= ∞

n=−∞ x [n] e

(

−i)ΩnT

= ∞

n=−∞ x [n] e(−i)ωn

= X (ω)

ω ≡ ΩT X (ω) x [n]

X s (Ω) = 1

T

∞k=−∞

X c (Ω− kΩs)

X (ω) = 1T

∞k=−∞ X c (Ω− kΩs)

= 1T

∞k=−∞ X c ω−2πkT

2π



±



T Ωs

xs (t) =nn

x (nT ) δ (t− nT )



x (αt) ↔ 1

α X Ω

α

α

1α

X s (Ω) ≡ X (ΩT )

s (n)

n0

s (n− n0)

n0 > 0

n0

s (n− n0) ↔ e−(i2πfn0)S

ei2πf

S (a1x1 (n) + a2x2 (n)) = a1S (x1 (n)) + a2S (x2 (n))



S (x (n)) = y (n)

S (x (n

−n0)) = y (n

−n0)

y (n) = a1y (n− 1) + · · ·+ a py (n− p) + b0x (n) + b1x (n− 1) + · · ·+ bqx (n− q )

y (n)

y (n − l)

l = 1, . . . , p

x (n)

p

q a1, . . . , a p b0, b1, . . . , bq

a0

y (n)

a0

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

∀n,n > 0 : (y (n) = ay (n− 1))

y (n) = ay (n− 1) + bδ (n)

n x (n) y (n)

−1 0 0

0 1 b

1 0 ba

2 0 ba2

0

n 0 ban

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

n0

2

4

y(n)a = 1.1, b = 1

x(n)

n

n

a

n

n

a

b = 1

|a| < 1

y (n) = a1y (n− 1) + · · ·+ a py (n− p) + b0x (n) + b1x (n− 1) + · · · + bqx (n− q )

y (n + 1) x (n + 1)

y(n)

n

15



a

y (n) = 1

q (x (n) +

· · ·+ x (n

−q + 1))

1q

n = 0, . . . , q − 1

q

1q

q

q = 7

(f ∗ g) [n] =∞

k=−∞f [k] g [n− k]

f, g Z

f ∗ g = g ∗ f



f, g Z

(f ∗ g) [n] =

∞k=−∞ f [n − k] g [k]

f, g Z

H h

x H (x)

x [n] =∞

k=−∞

x [k] δ [n−

k]

H (x [n]) =

∞k=−∞

x [k] H (δ [n− k]) .

H (δ [n− k]) h [n− k]

H (x [n]) =

∞k=−∞

x [k] h [n− k] = (x ∗ h) [n] .

f, g

(f ∗ g) [n] =

∞k=−∞

f [k] g [n− k] =

∞k=−∞

f [n − k] g [k] .

k = 0 n

n

a

h [n] = anu [n] ,

x [n] = bnu [n] .



y [n] = x [n]

∗h [n] .

y [n] =∞

k=−∞aku [k] bn−ku [n − k] .

y [n] = 0

n < 0

y [n] =n

k=0

[ab]k

n ≥ 0

ab = 1

y [n] = 0 n < 01−(ab)n+11−(ab) n ≥ 0

.

(f g) [n] =N −1k=0

f [k]

g [n− k]

f, g Z [0, N − 1]

f ,

g f g

f g = g f

f, g Z [0, N − 1]

(f g) [n] =N −1k=0

f [n− k]

g [k]

f, g Z [0, N − 1]

f ,

g f g



(f g) [n] =

nk=0

f [k] g [n− k] +

N −1k=n+1

f [k] g [n − k + N ]

f, g Z [0, N − 1]

H h

x

H (x)

x [n] =N −1k=0

x [k]

δ [n− k]

H (x [n]) =N −1k=0

x [k] H

δ [n− k]

.

H (δ [n

−k])

h [n

−k]

H (x [n]) =N −1k=0

x [k]

h [n− k] = (x h) [n] .

f, g

(f

g) [n] =

N −1

k=0

f [k]

g [n − k] =

N −1

k=0

f [n− k]

g [k] .

k = 0 n ∈ Z [0, N − 1] n





x [n] h [n]

y [n] = x [n] ∗ h [n]

= ∞

k=−∞ x [k] h [n− k]

N

n ≥ N

n ≥ N

n = 0, 1, . . . , N − 1

f [n] ck g [n] dk

v [n]

v [n] = f [n] g [n]

v [n]

ak

ak = ckdk f [n] g [n]

f [n] g [n] =N

n=0

N η=0 f [η] g [n− η]

ak = 1N

N n=0 v [n] e−(jω0kn)

= 1N 2

N n=0

N η=0 f [η] g [n− η] e−(ωj0kn)

= 1N N

η=0 f [η] 1N N

n=0 g [n

−η] e−(jω0kn)

= ∀ν, ν = n − η :

1N

N η=0 f [η]

1N

N −ην =−η g [ν ] e−(jω0(ν +η))

= 1

N

N η=0 f [η]

1N

N −ην =−η g [ν ] e−(jω0kν )

e−(jω0kη)

= 1N

N η=0 f [η] dke−(jω0kη)

= dk

1N

N η=0 f [η] e−(jω0kη)

= ckdk



Y [k] y [n]

Y [k] = F [k] H [k]

0 ≤ k ≤ N − 1 y [n]

y [n] = 1

N

N −1k=0

F [k] H [k] ej2πN kn

F [k] =N −1

m=0 f [m] e(−j)2πN kn

y [n] = 1

N N

−1

k=0N

−1

m=0 f [m] e(

−j) 2π

N kn

H [k] ej 2πN kn

= N −1

m=0 f [m]

1N

N −1k=0 H [k] ej

2πN k(n−m)

h [((n− m))N ] =1N

N −1k=0 H [k] ej

2πN k(n−m) y [n] =

N −1m=0 f [m] h [((n− m))N ]

0 ≤ n ≤ N − 1

y [n] ≡ f [n] h [n]

• f [n] F [k] h [n] H [k]

• Y [k] = F [k] H [k]• Y [k] y [n]

2 N y [n] =N −1

m=0 f [m] h [((n− m))N ] n N

N − 1

N N 2 N (N

−1) O N 2



m

f (n + m)

m

m = −2

m = −2



f (n)

N = 8



m

f [n] → f [((n + m))N ]

m = −3

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

˜



• 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

ck =

1N

k = 0

12

sin(π2 k)π2 k

ak = ck2 = 1

4sin2

(π2 k)

f [n] ↔ F [k] f [((n− m))N ]

↔ e−(i 2πN km)F [k]



f [n] = 1

N

N −1

k=0

F [k] ei2πN kn

f [n] = 1N

N −1k=0 F [k] e−(i 2πN kn)ei

2πN kn

= 1N

N −1k=0 F [k] ei

2πN k(n−m)

= f [((n −m))N ]



H eiωn

ω0 = 2πkN H [k] ∈ C

y [n] = H [k] eiωn

H y [n]

c1eiω1n + c2eiω2n → c1H [k1] eiω1n + c2H [k2] eiω1n

l

cleiωln →

l

clH [kl] eiωln

H H

eiωln H [kl] ∈ C

y [n]



f [n]

f [n] =N −1k=0

ckeiω0kn

ω0 = 2πN

f [n] cn

f [n] f [n] ∈ L2 [0, N ]

cn

ejω0kn

f [n]

f [n]

∀k, k ∈ Z :

ejω0kn

sT (t)

T

cn (T )

cn = 1

T

T 0


ω0 = 2πT

n

S T (f ) ≡ T cn = 1

T

T 0

(S T (f ) exp (ßω0t) dt

sT (t) =∞−∞

f (t) exp (ßω0t) 1

T

limT →∞

sT (t) ≡ s (t) =

∞ −∞

S (f ) exp (ßω0t) df

S (f ) =

∞ −∞


F (ω) =∞

n=−∞f [n] e−(iωn)



f [n] = 1

2π π

−π

F (ω) eiωndω

ω

ω = 2πf

i2πf t

F

eiω

F (Ω)

F (ω) =∞

n=−∞f [n] e−(iωn)



f [n] =

1

2π π−π F (ω) e

iωn

dω

z [n] = af 1 [n] + bf 2 [n]

Z (ω) = aF 1 (ω) + bF 2 (ω)

2π



z [n] = f [n− η]

z [n]

Z (ω) =

∞−∞

f [n− η] e−(iωn)dn

σ = n − η

Z (ω) = ∞−∞ f [σ] e−(iω(σ+η)n)dη

= e−(iωη) ∞−∞ f [σ] e−(iωσ)dσ

= e−(iωη)F (ω)

y [n] = (f 1 [n] , f 2 [n])

= ∞

η=−∞ f 1 [η] f 2 [n− η]



∞n=−∞

(|f [n] |)2 =

π−π

(|F (ω) |)2dω

z (t) = 1

2π

∞−∞

F (ω − φ) eiωtdω

z (t) = f (t) eiφt



a1s1 (n) + a2s2 (n) a1S 1

ei2πf

+ a2S 2

ei2πf

s (n) S ei2πf

= S e−(i2πf ) s (n) = s (−n) S

ei2πf

= S

e−(i2πf )

s (n) = −s (−n) S

ei2πf

= −S

e−(i2πf )

s (n − n0) e−(i2πfn0)S

ei2πf

ns (n) 1−(2iπ)

dS (ei2πf )df

∞n=−∞ s (n) S

ei2π0

s (0) 1

2

− 12

S

ei2πf

df

∞n=−∞ (|s (n) |)2 1

2

− 12

|S

ei2πf |2df

ei2πf 0ns (n) S

ei2π(f −f 0)

s (n)cos(2πf 0n) S (ei2π(f −f 0))+S (ei2π(f +f 0))

2

s (n)sin(2πf 0n) S (ei2π(f −f 0))−S (ei2π(f +f 0))

2i

12

12T s

cos

2π × 1

2T s nT s

= cos (πn)= (−1)

n

12

e−(i2πn)

2 = e−(iπn) = (−1)n

f D = f AT s



f D

f A

pT s (t)

pT s (t)

1T s

12T s

12

12

− 12

e−(i2πfm)eiπfndf =

1 m = n

0

m = n

12

− 12

S

ei2πf

ei2πfndf = 1

2

− 12

mm s (m) e−(i2πfm)ei2πfndf

= mm s (m) 12

− 12

e(−(i2πf ))(m−n)df

= s (n)

S

ei2πf

=nn

s (n) e−(i2πfn)

s (n) =

12

− 12

S

ei2πf

ei2πfndf

s (n) =anu (n) u (n)

S

ei2πf

= ∞

n=−∞ anu (n) e−(i2πfn)

=

∞n=0

ae−(i2πf )

n

∀α, |α| < 1 :

∞n=0

αn = 1

1− α

∆ c0

|c0| = A∆

T A

1

∆

∆

0.1T s



|a| < 1

S ei2πf

= 1

1

−ae−(i2πf )

|S

ei2πf | =

1 (1− acos(2πf ))

2+ a2sin2 (2πf )

∠

S

ei2πf

= −arctan

asin(2πf )

1− acos(2πf )

a

−12

12

a > 0

0

12

a a < 0

-2 -1 0 1 2

1

2

f

|S(e j2πf)|

-2 -1 1 2

-45

45

f

∠S(e j2πf)

a = 0.5

[−2, 2]



f

a = 0.9

a = 0.5

a = –0.5

p e c t r a

a g n t u

e

-10

0

10

20

0.5

a = 0.9

a = 0.5

a = –0.5

n g e

e g r e e s

f

-90

-45

0

45

90

0.5

a = 0.5

a = −0.5

N

s (n) = 1

0 ≤ n ≤ N − 10

S

ei2πf

=N −1n=0

e−(i2πfn)

N +n0−1n=n0

αn = αn0 1− αN

1− α

α

α

S

ei2πf

= 1−e−(i2πfN )1−e−(i2πf )

= e(−(iπf ))(N −1) sin(πfN )sin(πf )



sin(Nx)sin(x)

dsinc (x) S

ei2πf

= e(−(iπf ))(N −1)dsinc (πf ) N

f0

5

10

S p e c t r a l M a g n i t u d e

0.5

-180

-90

0

90

180

f0.5

A n g l e ( d e g r e

e s )

X [k] =N −1n=0

x [n] e(−i)2πn kn∀k, k = 0, . . . , N − 1 : (k = 0, . . . , N − 1)

x [n] = 1

N

N −1k=0

X [k] ei2πn kn∀n, n = 0, . . . , N − 1 : (n = 0, . . . , N − 1)

• X [k]

ω = 2πN k∀k, k = 0, . . . , N − 1 : (k = 0, . . . , N − 1)

•

x [n]

M

M

X

ei2πM k

=N −1n=0

x [n] e(−i)2πM k



X

ei2πM k

=N −1n=0

xzp [n] e(−i)2πM k

X

ei2πM k

= X zp [k]∀k, k = 0, . . . , M − 1 : (k = 0, . . . , M − 1)

• N N

X

eiω

=N −1n=0

x [n] e(−i)ωn

X

eiω

=

N −1n=0

1

N

N −1k=0

X [k] ei2πN kne(−i)ωn

X eiω =N −1

k=0

X [k] 1

N

N −1

k=0

e(−i)(ω−2πN k)n

X

eiω

=N −1k=0

X [k] 1

N

sinωN −2πk

2

sinωN −2πk

2N

e(−i)(ω−2πN k)N −12

1

0 2pi/N 4pi/N 2pi

D.

1

N

sin(ωN 2 )sin(ω2 )

•

W N = e(−i)2π

N

X [0]

X [1]

X [N − 1]

=

W 0N W 0N W 0N W 0N . . .

W 0N W 1N W 2N W 3N . . .

W 0N W 2N W 4N W 6N . . .

x [0]

x [1]

x [N − 1]

X = W (x) W



· W n W W nN

· W

W = W T

· 1√ N

W

1√ N

W 1√ N

W H

= 1√ N

W H

1√ N

W = I

· 1N W = W −1

•

N

N 2 log2N

N 2

N N 2 log2N N 2

•

[0, N − 1]

•

−12

, 12

[0, 1]

f = kK

k ∈ 0, . . . , K − 1

∀k, k ∈ 0, . . . , K − 1 :

S (k) =N

−1

n=0

s (n) e− i2πnkK

S (k) S

ei2π kK





S (k) =

N −1n=0 s (n) e−(i 2πnkN )

s (n) =

1

N N

−1

k=0 S (k) ei 2πnk

N

e−(at)u (t) 1a+iω a > 0

eatu (−t) 1a−iω a > 0

e−(a|t|) 2aa2+ω2 a > 0

te−(at)u (t) 1(a+iω)2

a > 0

tne−(at)u (t) n!

(a+iω)n+1 a > 0

δ (t) 1

1 2πδ (ω)

eiω0t 2πδ (ω − ω0)

cos(ω0t) π (δ (ω − ω0) + δ (ω + ω0))

sin(ω0t) iπ (δ (ω + ω0)− δ (ω − ω0))

u (t) πδ (ω) + 1iω

sgn(t) 2iω

cos(ω0t) u (t) π2 (δ (ω − ω0) + δ (ω + ω0)) +

iωω02

−ω2



sin(ω0t) u (t) π2i (δ (ω − ω0)− δ (ω + ω0)) +

ω0ω02−ω2

e−(at)sin(ω0t) u (t) ω0(a+iω)2+ω02

a > 0

e−(at)cos(ω0t) u (t) a+iω(a+iω)2+ω02

a > 0

u (t + τ )− u (t− τ ) 2τ sin(ωτ )ωτ = 2τ sinc(ωt)

ω0π

sin(ω0t)ω0t

= ω0π sinc (ω0) u (ω + ω0)− u (ω − ω0)

tτ + 1

utτ + 1

− utτ

+− t

τ + 1

utτ

− utτ − 1

=

triag t2τ

τ sinc2

ωτ 2

ω02π sinc2

ω0t2

ωω0

+ 1

u

ωω0

+ 1− u

ωω0

+

− ωω0

+ 1

u

ωω0

− u

ωω0− 1

=

triag ω2ω0∞n=−∞ δ (t − nT ) ω0

∞n=−∞ δ (ω − nω0) ω0 = 2πT

e− t2

2σ2 σ√

2πe−σ2ω2

2

n

= 1 + n

− 1 ≤ n ≤ 0

1− n 0 < n ≤ 1

0

r

r

y (n) = (1 + r) y (n− 1)

Cy (n) = f (n)



C

C = cN DN + cN −1DN −1 + ... + c1D + c0

D

D (y (n)) = y (n)− y (n − 1) .

c0,...,cn

N

k=0

aky (n

−k) =

M

k=0

bkx (n

−k)

x

y

y (n)

y (n) = 1

a0

−

N k=1

aky (n − k) +M k=0

bkx (n − k)

y (n) n

y (n) = y (n− 1) + y (n− 2)

y (0) = 0

y (1) = 1



N

N

Ay (n) = f (n)

A

A = aN DN + aN −1DN −1 + ... + a1D + a0

D

D (y (n)) = y (n)− y (n − 1) .

yh (n) y p (n) Ayh (n) = 0 Ay p (n) = f (n)

A

L (yh (n) + y p (n)) = 0 + f (n) = f (n)

yg (n)

yh (n)

Ay (n) = 0 y p (n) f (n)

N k=0 aky (n− k) = f (n)

N k=0 aky (n− k) = 0

cλn c, λ

N k=0 akcλn−k = 0

cλn−N N

k=0

akλN −k = 0

a0λN + ... + aN = 0.



yh (n) = c1λn1 + ... + c2λn

2 .

n

λ1 λ2

yh (n) = c1λn1 + c2nλn

1 + c3λn2 + c4nλn

2 + c5n2λn2 .

y (n) n

y (n)− y (n − 1)− y (n − 2) = 0

y (0) = 0 y (1) = 1

λ2 − λ − 1 = 0

λ1 = 1+√ 5

2 λ2 = 1−√ 52

y (n) = c1

1 +

√ 5

2

n

+ c2

1−√ 5

2

n

.

c1 =

√ 5

5

c2 = −√

5

5 .

y (n) =

√ 5

5

1 +

√ 5

2

n

−√

5

5

1−√ 5

2

n

.



y (n)

−ay (n

−1) = x (n) .

y (n)− ay (n − 1) = 0.

λ − a = 0 λ = a

yh (n) = c1an.

x (n) = δ (n)

y (n)− ay (n − 1) = δ (n) .

a

n

u (n)

x (n)

y p (n) = x (n) ∗ (anu (n)) .

yg (n) = yh (n) + y p (n) = c1an + x (n) ∗ (anu (n)) .





\



sin(2πf t) = 12i

ei2πft − 12i

e−(i2πft)

c1 = 12i

c−1 = − 12i

c0 = A∆T

ck = Ak + iBk

∞k=−∞

ckei2πktT =

∞k=−∞

(Ak + iBk) ei2πktT

(Ak + iBk) ei2πktT

(Ak + iBk)

cos2πktT

+ isin

2πktT

Akcos

2πktT

− Bksin2πktT

+ i

Aksin2πktT

+ Bkcos

2πktT

c−k = ck

A−k = Ak

B−k = −Bk

2Akcos2πktT

− 2Bksin2πktT

2Ak = ak

2Bk = −bk

ck = 2iπk

ak = 0

bk = − (2 (ck))

bk =

4πk k

0 k

sq (t) =

k∈1,3,...

4

πksin

2πkt

T

√ 22

A2

P∞

k=2 ak2+bk

2

a12+b12

1− 12

4π

2= 20%



N

N T

log2N T

N = 128

7128 = 0.05

N

F (S (f )) =

∞−∞

S (f ) e−(i2πft)df =

∞−∞

S (f ) ei2πf (−t)df = s (−t)

F (F (F (F (s (t))))) = s (t) F (S (f )) = ∞−∞ S (f ) e−(i2πft)df =

∞−∞ S (f ) ei2πf (−t)df =

s (−t)

s (t)

s (−t)

s (t) =

W sin(πWt)

πWt

2

x (t)

s (t)

1RC e

− tRC u (t)

1 − e−(i2πf ∆)

∆

1RC e

−(t−∆)RC u (t −∆)

1RC e

−tRC u (t)

− 1RC e

−(t−∆)RC u (t

−∆)

F (Ω) = ∞−∞ f (t) e−(iΩt)dt

= ∞0

e−(αt)e−(iΩt)dt

= ∞0

e(−t)(α+iΩ)dt

= 0− −1α+iΩ

F (Ω) = 1

α + iΩ

t = 0

x (t) = 12π

M

−M ei(Ω,t)dΩ

= 12πei(Ω,t)|Ω,Ω=eiw

= 1πt sin(Mt)



x (t) = M

π

sinc

M t

π

ak = k = 0

18

sin3[π2 k][π2 k]

3

α

N +n0−1n=n0

αn −N +n0−1n=n0

αn = αN +n0 − αn0





















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