Bab 7 Sampling

∗

T 

x(t)

 p(  p(t) ∗

T   p(  p(t) ωs = 2π/T  2 π/T 

 p(t)

x(t)

P ( jω)

2π

x p (t)

×

T 

x(t) −3ωs −2ωs −ωs

T 

2ωs

3ωs

ω

X  p ( jω)

t

0

ωs

0

 p(t)

1

T 

t

0 x(0)

ωs

−ωs −ωM 0 ωM 

−2ωs

ω

2ωs

(ωs − ωM )

x p (t)

X  p ( jω) t

−T  0 T 

1

T 

x p (t) = x(t) p(t),

−ωs

−2ωs

ωs

0

ω

2ωs

(ωs − ωM ) X  p ( jω)

+∞



 p(t) =

δ (t − nT ).

ω X ( jω )

1/T 

n=−∞

ωM  < (ωs −ωM ) X ( jω ) ωs < 2ωM  x(t)

x(t) x(t)δ (t − t0 ) = x(t0 )δ (t − t0 ) x p (t) x(t)

x p (t) x(t)

T 

x p (t)

+∞

x p (t) =



x(t)

x(nT )δ (t − nT ),

ωs > 2ωM 

x p (t)

x p (t)

n=−∞

xr (t) X r ( jω) = X ( jω)

1 X  p ( jω) = 2π

+∞

 

x(t)

X ( jθ)P ( j(ω − θ))dθ.

xr (t)

−∞

2π P ( jω) = T 

+∞



δ (ω − kω s ).

k=−∞

X ( jω ) ∗ δ (ω − ω0 ) = X ( j(ω − ω0 )) X  p ( jω) =

1 T 

+∞



X ( j(ω − kω s )).

x(t)

k=−∞

X ( jω)

x(t)

1

x(t) x p (t)

−ωM  ωM 

ω

x p (t) ho (t)

x0 (t)

x(t) +∞

xr (t) =

t

0 x(0)



x(nT )

n=−∞

ωc T  sin(ωc (t − nT )) . π ωc (t − nT )

x p (t)

t

−T  0 T 

x(t)

h0 (t) 1 T 

t

0

t

x(0)

x p (t)

x0 (t)

t

0 x(t) hr (t)

−T 

x0 (t)

0

t

T 

xr (t)

x0 (t)

H r ( jω) r(t) = x(t). r (t) = x(t) h0 (t) hr (t)

t

0

x(t) = cos 200πt + 3 cos 400πt 800

x p (t) xr (t) T  =

x(nT ) = cos

t

0

200πn 400πn + 3 cos 800 800

x(nT ) = cos

x0 (t)

+∞

x p (t) =



cos

n=−∞

1 800

πn πn + 3 cos 4 2

πn πn n + 3 cos δ  t − 4 2 800



xr (t) = x(nT )|n=

x(t)

xr (t) = cos

h(t) =

ωc T sin(ωc t) , πωc t

t T 

800πt 800πt + 3 cos 4 2

xr (t) = cos 200πt + 3cos400πt



x(t) X ( jω ) = 0

|ω| > ωM 

x(t) x(nT )

x(nT ) = cos

2πn 4πn + 3 cos 3 3

x(nT ) = cos

2πn 1 + 3cos 1 πn 3 3

n = 0, ±1, ±2,... x(nT ) = cos ωs > 2ωM 

ωs =

 

2πn 2 x(nT ) = cos + 3 cos (2 − )πn 3 3

 

2πn 2 + 3 cos 2πn − πn 3 3

n

2π . T 

 

2πn 2 x(nT ) = cos + 3 cos − πn 3 3

x(nT ) x(t) T 

x(nT ) = 4 cos ωM 

ωs − ωM  x(t) 2ωM 

+∞

x p (t) =

= cos

2πn 3

   4cos

n=−∞

2ωM 

2πn 3

δ  t −

xr (t) = x(nT )|n= xr (t) = 4cos

2πn 2πn + 3 cos 3 3

n 300



t T 

2πn.300 3

xr (t) = 4 cos 200πt 100

ωs < 2ωM  x(t)

X ( jω )

X  p ( jω)

200

x p (t)

100 300

xr (t)

400

x(t) ω0 (ωs −ω0 )

ωs /2 < ω0 < ω s ωs ωs = ω0

ω0 (ωs − ω0 ) x(t) = sin 200πt

(ω0 = 0)

200

x p (t) xr (t) T  =

x(t) = cos 200πt + 3 cos 400πt 300

x(nT ) = sin x p (t) xr (t)

T  =

1 300

200πn 400πn + 3 cos x(nT ) = cos 300 300

1 200 200πn 200

x(nT ) = sin πn n 0 x(nT ) = 0

x p (t) = 0 xr (t) = 0 100 200

Y c ( jω) = X c ( jω)H d (ejωt ).

200 H c ( jω) H d (ej Ω) H c ( jω) =

xc (t)

H d ej Ω

 

xc (t) xc (nT ) xc (nT )



H d (ejωT  ), 0,

|ω| < ωs /2 |ω| > ωs /2

yc (t)

xd [n] H c ( jω) = jω . xd [n] = xc (nT ) xc (t)

xd [n]

ωc H c ( jω) =

yc (t)



 jω, |ω| < ωc . 0, |ω| > ωc

ωs = 2ωc

yd [n] yd [n] = yc (nT )

jΩ

H d (e ) = j



Ω , T 

|Ω| < π

|H c ( jω)| ωc

ωc

−ωc

ω

 H c ( jω)

−ωc

π 2

π

−2

ωc

ω

|H c ( jω)|

jΩ

|H d (e )| ωc

1

π

−π

−2π

2π

ω

ω

ωc

−ωc  H c ( jω)

∆ jΩ d (e )

 H 

ωc

−ωc

ω

π 2

−π

−2π

π

π

−2

2π

ω

|H d (ej Ω )| 1 π

−π

ω

 H d (ej Ω )

π∆ T 

yc = xc (t − ∆) −π

xc (t) ∆ Y c ( jω) = e−jω ∆ X c ( jω).

H c ( jω) =



e−jω ∆ , 0,

|ω| < ωc

.

ωc H c ( jω) ωc

ωs

ωs = 2ωc H d (ej Ω ) = e−j Ω∆/T , ∆

xd [n]

|Ω| < π , yd [n]

T 

 

yd [n] = xd n −

∆ . T 

∆

T 

xc (t) yc (t) yd [n] yd [n] ∆

T 

xd [n] H d (ej Ω ) xd [n]

= 1/2

− πT ∆

π

ω