Purdue digital signal processing labs ece 438 4 PDF

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Questions or comments concerning this laboratory should be directed to Prof. Charles A. Bouman, School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907; (765) 494-0340; [email protected]...

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Purdue Digital Signal Processing Labs (ECE 438)

By: Charles A. Bouman

Purdue Digital Signal Processing Labs (ECE 438)

By: Charles A. Bouman

Online: < http://cnx.org/content/col10593/1.4/ >

CONNEXIONS Rice University, Houston, Texas

∼ ∼ ∼ ∼

a=

h

1

2

3

i

a

a

1×3

b = a. a c =a∗b

b

a

b

b

3×1 c

14 = 1 ∗1 + 2 ∗ 2 + 3 ∗ 3

a c = a. ∗ a a

a. a

a

a

Z

2π

sin2 (5t) dt

0

Z

1

et dt

0

sin t/6 sin t/6 sin n/6

sin t/6

∆t

sin2 5t N

I

I (N ) N exp (t)

1 ≤ N ≤ 100

[0, 1] I (N )

J (N )

I 5 = I 10 = 0

N

0, 2π

=

Z

∞

−∞

• •

(t) (t)

t t

[−10π, 10π ] [−2, 2]

y = rect t <

|

2

(t)| dt .

a = 0.8 a = 1.0

• an (u (n) − u (n − 10)) • cos (ωn) an u (n)

n

a = 1.5

[−20, 20]

ω = π/4

n

[−1, 10]

y = u (n) >

f (t) = sin 2πt

xn

Ts x (n) = f (Ts n) = sin (2πTs n) . f (Ts n)

Ts Ts Ts Ts

Ts

= 1/10 0 ≤ n ≤ 100 = 1/3 0 ≤ n ≤ 30 = 1/2 0 ≤ n ≤ 20 = 10/9 0 ≤ n ≤ 9

Ts = 1/10 Ts = 1/2

Ts = 10/9

Ts = 1/3

n

n

f (m, n) = 255| −50 ≤ m ≤ 50

−50 ≤ n ≤ 50

(0.2m) sin (0.2n) |

y (n) = y (n − 1) + x (n) + x (n − 1) + x (n − 2) y (n)

x (n)

y = S [x]

S

y (n) x (n)

d x (t) y (t) = dt Rt y (t) = −∞ x (τ ) dτ

x (n)

1 ( 3

=

)

= 0.8 ∗

+ 0.2 ∗ +

=

1 3

• •

−10 ≤ n ≤ 20 • δ (n) − δ (n − 5) • u (n) − u (n − (N + 1))

N = 10 u (n)

>

−10 ≤ n ≤ 20

y = S1 [x]

y (n) = x (n) − x (n − 1) y = S2 [x] y (n) =

1 y (n − 1) + x (n) 2

xn

y n

S1

S2

S2 (S1 )

S2

S1 S2 S1 (S2 ) S1

S1 + S2

S1

S2

y = S2 [x] δ (n)

y = S3 [x] S2 x = S3 [S2 [x]]

δ = S3 [S2 [δ]] x

S2 y = S3 [x] y (n) = ax (n) + bx (n − 1) a

b S3

S3

S3 S2 [δ] S3 S3

S3 S2

δ

S3 S3

x M

x

M

M

x x

M

T0 T0 = 2

t ∈ [0, 2] s (t) =

  1 t− 2

T0 = 1

  t ∈ − 21 , 12

s (t) = rect (2t) −

s (t) = a0 +

∞ X

1 2

Ak sin (2πkf0 t + θk )

k=1

f0 = 1/T0

sin (2πkf0 t + θk )

[0, T0 ]

y (n) = 0.9y (n − 1) + 0.3x (n) + 0.24x (n − 1)

H (z) ≡

−π < ω < π

Y (z ) X (z)   |H ejω |

  ∠H ejω

x (t) = 0 +

13 X 4 sin (2πkt) . kπ k=1 k

  X ejω

x (n) ∞ X   x (n) e−jωn . X ejω = n=−∞

ω

f s > 2f c

S (f ) =

1 X (f ) . Ts

1 2Ts

.

|f | ≤

1 2Ts

.

1 Ts

Ts 

p (t) =

1 t − Ts 2

p (t)



.

P (f ) |P (f ) | = Ts |

(f /fs ) | . N

|Hb (f ) | =

|H (f ) | = = | Ts = 1 • • •



1+

1  N . f fc

|Hb (f ) P (f ) T1 Hb (f ) | s 2 1 | (f /fs ) | . 1+(ff )N c

(f /fs ) | fc = 0.45

N = 20

0.1

0.3 0.1 0.1 0.3

0.8

0.8 0.5

0.45

L x (n)

  z (n) 2π Z ejω [−π/L, π/L]

L −1

    Z ejω = X ejωL .   X ejω 2π/L   Z ejω

z (n)

x (n)

  X ejω [−π, π]

x (n)

L−1

x (n) x (n)

Dth y (n) x (n)

y (n)

D−1   1 X X Y ejω = D k=0

  X ejω

2π

[−π, π]



ω − 2πk D

x (Dn)



.

  Y ejω

1/D  [−π, π] X ejω

  1  ω Y ejω = X ej D D π/2

D y (n)

[−π/D, π/D] ω ∈ [−π, π] .

π/2

M

x (n)

y (n) =

N−1 X i=0

bi

ak

bi x (n − i) −

M X k=1

y (n)

ak y (n − k) N

M

y (n) h (n)

δ (n)

h (n) =

N−1 X i=0

bi δ (n − i) −

ak = 0

M X k=1

ak h (n − k) .

k

h (n) =

N−1 X bi δ (n − i) . i=0

N

ak 6= 0 ak 6= 0

n→∞

bi x (n) X (z) =

∞ X

x (n) z −n .

n=−∞

z

  X ejω = =

z

X (z)|z=ej ω P∞

n=−∞

x (n) e−jωn

x (n − K )

Z

↔

=

=

K

m = n−K

z −K P∞

n=−∞

P∞

x (n − K ) z −n

−(m+K ) m=−∞ x (m) z P∞ −K −m z m=−∞ x (m) z

z −K X (z )

=

PM PN−1 Y (z) = i=0 bi z −i X (z) − k=1 ak z −k Y (z )   PN−1 PM Y (z) 1 + k=1 ak z −k = X (z) i=0 bi z −i △ Y (z) X (z)

H (z) =

=

PN−1

1+

i=0 P M

k=1

bi z −i ak z −k

H (z)   H ejω =

PN−1 i=0

1+

PM

bi e−jωi

k=1

ak e−jωk

.

z −1

z1 = ejθ z2 = e−jθ θ

z1

θ ∈ [0, π]

   1 − z1 z −1 1 − z2 z −1    = 1 − ejθ z −1 1 − e−jθ z −1

Hf (z ) =

1 − 2cosθz−1 + z −2 .

=

h (n) h (n) ±∞ Hf (z)

ω

  Hf e±jθ = 0

e±jθ ω=θ

θ

−π < ω < π θ = π/6

  |Hf ejω |

θ = π/3

θ = π/2

Hf (z) θ

θ Hf (z)

n = 0

M

|ω| < π θ

Hf (z)

θ Hf (z)

• •

θ

Hi (z) = =

1−r (1−rej θ z −1 )(1−re−j θ z −1 ) 1−r 1−2rcos(θ)z −1 +r 2 z −2

p1 = rejθ r

θ

p2 = re−jθ p1 |r| < 1

±∞ r

  |Hi ejw |

θ

|ω| < π

θ = π/3

•

r = 0.99

•

r = 0.9

•

r = 0.7

Hi (z) r

r Hi (z)

|ω| < π

θ

Hi (z)

2π Hi (z) θ

r = 0.995

n

ω=θ

ω θ

[θ − 0.02, θ + 0.02]

• • •

ω

[θ − 0.02, θ + 0.02] r

r = 0.9999999

r ω=θ

ωp ≤ ω ≤ ωs

  H ejω

δp

δs

|ω| < ωp ωp < ωs

ωs < |ω| ≤ π

  δp |H ejω − 1| ≤  jω  |H e | ≤ δs

|ω| < ωp

ωs < |ω| ≤ π ωp ωs δ p

δs

|ω| < 1.8

|ω| > 2.2

ωc   1 |ω| ≤ ωc Hideal ejw = { 0 ωc < |ω| ≤ π hideal (n) =

ω n c

ωc π

π

−∞

>

π N

N

N

N = 20

x (n) = { N = 200 N = 50

sin (0.1πn) 0 ≤ n ≤ 49 0 x (n) x (n)

N = 50

N = 200 x (n) = 0 n ≥ 50

N = 200 X (k)

x (0) , · · · , x (N − 1)

N = 50

N = 200

N2

N

N

N/2

N−1

X

X (k) =

x (n) e−j2πkn/N .

n=0

N

X (k) = N

X (k) N

[x (n)]

x (n) n n X (k) =

N−1 X

x (n) e−j2πkn/N +

n=0 n

m

n n 2m N −2 N/2 − 1 n = 2m + 1

X

n m

X

x (2m + 1) e−j2πk(2m+1)/N .

m=0

N/2−1

X

N/2 − 1 n = 2m n 2m + 1 N −1

N/2−1

x (2m) e−j2πk2m/N +

m=0

X (k) =

x (n) e−j2πkn/N .

n=0 n

N/2−1

X (k) =

N−1 X

N/2−1

x (2m) e−j2πkm/(N/2) + e−j2πk/N

m=0

X

x (2m + 1) e−j2πkm/(N/2) .

m=0

N

N/2

N/2

N/2 N

e−j2πk/N

N/2 N/2 x0 (n)

x1 (n)

N x0 (n) =

x (2n)

x1 (n) = x (2n + 1) ,

N

n = 0, ..., N/2 − 1 x (n) X (k ) = X0 (k ) + e−j2πk/N X1 (k ) X0 (k)

X1 (k)

k = 0, ..., N − 1.

N/2

X0 (k) = DFTN/2 [x0 (n)] X1 (k) = DFTN/2 [x1 (n)]

X0 (k)

X1 (k)

N N/2

N/2 N/2

k

N

e−j2πk/N k

−e−j2π N = e−j2π

k+N/2 N

. N

X (k ) = X0 (k ) + WNk X1 (k ) X (k + N/2) = X0 (k) − WNk X1 (k)

}

k = 0, ..., N/2 − 1

WNk = e−j2πk/N

N N/2

(N/2)

X0

(k)

(N/2)

X1

(k)

N/2

N

N/2 k WN = e−j2πk/N X

x (n) = δ (n) N = 10 x (n) = 1 N = 10 x (n) = ej2πn/N N = 10

N N/2

N/2 N/4

N/2 N/2

N N = 2p

p

X (0) = x (0) + x (1) X (1) = x (0) − x (1)

x (n) = δ (n) N =8 x (n) = 1 N =8 x (n) = ej2πn/8 N =8

x (n) = 1

N =8

N = 2p N

p

>

FX (x) = P (X ≤ x) X (−∞, x]

x ∈ (−∞, ∞) .

FX (x) x

X FX (x)

fX (x) R t1 t0

fX (x) dx

FX (t1 ) − FX (t0 )

= P (t0 < X ≤ t1 ) .

X

X

=

(−∞, ∞)

g (X) E [g (X)] E [g (X)]

= =

g (X)

R∞

−∞

g (x) fX (x) dx

P∞

x=−∞

g (x) P (X = x)

µX = E [X] =

µX Z ∞

2 σX

xfX (x) dx

−∞

i Z h 2 2 = E (X − µX ) = σX

∞

−∞

2

(x − µX ) fX (x) dx .

  1 2 exp − 2 (x − µX ) . fX (x) = √ 2σX 2πσ X 1

X

  X ∼ N µ, σ 2

σ2

µ

X v ǫ

X

X =v+ǫ X

X {X1 , X2 , · · · , XN } N

Xi FX (x)

Xi {X1 , X2 , · · · , XN }

X

{X1 , X2 , · · · , XN } µX =

N 1 X Xi N i=1

2

σX=

N  X

1 Xi − µ X N − 1 i=1

2

.

µX # "  N N 1 X 1X µ Xi = E X =E E [Xi ] = µX N N i=1 i=1   i i h P hP N 1 = X X V ar µX = V ar N1 N V ar i i i=1 i=1 N2 2 PN σX 1 = i=1 V ar [Xi ] = N N2 

a

a   V ar a → 0

  E a = a N →∞

X X

X1 X2

X

X Y = aX + b a

b

a 6= 0

Y

X1000

"

2

#

2 E σ X = σX

X

X

Y σY2

µY

2 σX

Y

a b µX

X a

b Y a

X b

Y Y Y Y Y Y

Y Y

X

FX (x)

FX (x) FX (x)

{X1 , X2 , ..., XN } F X (x) F X (x) = I{Xi ≤x} F X (x)

1 N

= {

PN

i=1 I{X i ≤x}

1,

0,

Xi ≤ x

Xi

x

F X (x) Nx

Xi

x Nx =

N X i=1

I{Xi ≤x} = N F X (x)

P (Xi ≤ x) = FX (x)   P I{Xi ≤x} = 1 = FX (x)   P I{Xi ≤x} = 0 = 1 − FX (x)

F X (x)   = E F X (x)

1 E N

[Nx]   1 = N i=1 E I{X i ≤x}   1 N E I{Xi ≤x} = N     = 0 · P I{Xi ≤x} = 0 + 1 · P I{Xi ≤x} = 1 PN

=

F X (x)

FX (x) .

FX (x) 

 1 V ar F X (x) = FX (x) (1 − FX (x)) . N F X (x)

F X (t)

X

t <

X

s N = 20

FX (x) FX (x)

X Y = FX (X) X

Y

X

N = 200

FX (·)

Y [0, 1]

X

Y

FX (·)

P (Y ≤ y) = { FX (x) y = FX (x)

0,

y1 x

{Y ≤ y }

0≤y≤1 FY (y) = =

{X ≤ x}

P (Y ≤ y)

P (FX (X) ≤ FX (x))

= P (X ≤ x)

=

FX (x)

=

y .

Y FX (·)

FX−1 (U )

U

FX (·)

FX (·)

Z ∼ FZ (z)

FZ (·)

X ∼ FX (x)

FX (·)

FZ (·)

  FX (x) = 1 − e−3x u (x) . N = 20 N = 200

• •

Xi

fX (x)

x

L fX (x) L

x0 [x0 , x1 ] (x1 , x2 ]

(xL−1 , xL ]

xL

bin (k)

∆

(xk−1 , xk ] k = 1, 2, · · · , L bin (k) = (xk−1 , xk ]

k = 1, 2, · · · , L

xL −x0 L

∆ = f˜ (k)

X

bin (k)

f˜ (k ) = P (X ∈ bin (k )) R xk f (x) dx = xk−1 X

≈ fX (x) ∆

x ∈ bin (k) ∆ {X1 , X2 , · · · , XN } Xn

bin (k) In (k) = { Xn

bin (k)

Xn ∈ bin (k)

1, 0,

Xn ∈ / bin (k)

H (k)

bin (k) H (k) =

N X

In (k) .

n=1

H (k) /N

X

bin (k) E

h

H(k) N

i

1 N

= = =

1 N

PN

n=1 {1

PN

n=1

E [In (k )]

· P (Xn ∈ bin (k)) + 0 · P (Xn ∈ / bin (k))} f˜ (k) f˜ (k)

Xn H (k)

V ar N

h

H(k) N

i

=

1 N

  f˜ (k) 1 − f˜ (k) .

f˜ (k)

H (k) /N H (k) f˜(k) ≈ N fX (x) fX (x) ≈

H (k) N∆ x

bin (k)

x ∈ bin (k) . bin (k)

U FU (u) 0,

u1

1

X = U3 FX (x) =

P(X ≤ x)  1 P U3 ≤ x   P U ≤ x3

= = =

FU (u)|u=x3

0,

x1

fX (x)

f˜ (k)

xL = 1

x ∈ [0, 1] bin (k)

X

f˜ (k) = FX (xk ) − FX (xk−1 ) k = 1, · · · , L FX (x) fX (x) fX (x)

f˜ (k)

f˜(k)

f˜ (k) U X

1

X = U3

X [0, 1]

H (k) /N

H (k) /N

f˜(k)

f˜ (k)

X

Y

FX (x) X

FX,Y (x, y ) = P (X ≤ x, Y ≤ y) .

fX,Y (x, y) =

FX,Y (x, y) =

Z

∂2 FX,Y (x, y ) . ∂x∂y

y

−∞

Z

x

fX,Y (s, t) ds dt. −∞

Y

FY (y)

X

Y

fX,Y (x, y) = fX (x) fY (y) X

Y

E [XY ] = E [X] E [Y ] X

Y X

•

E [XY ] =

•

E [(X − µX ) (Y − µY )] =

Z Z

∞ −∞ ∞ −∞

Z Z

∞

xyfX,Y (x, y) dx dy

−∞ ∞ −∞

(x − µX ) (y − µY ) fX,Y (x, y) dxdy

• ρX Y =

E [(X − µX ) (Y − µY )] E [XY ] − µX µY = σX σY σX σY X

Y

(Xi , Zi ) X

Y X

Z

Z Z=Y

Z = (X + Y ) /2

Z = (4 ∗ X + Y ) /5 Z = (99 ∗ X + Y ) /100 Z X Y

Y1 Y2

Y1000

X1 X2

Z X1000 Z

Y

ρX Z X X

Y Z

(X2 , Z2 )

X Z E [XY ] = E X E Y

Y X

Y (Xi , Zi )

(X1 , Z1 )

(X1000 , Z1000 ) Z Xi

ρX Z = s

Zi

   µ µ X − Z − i i Z X i=1

PN

2 2   PN µ µ i=1 Zi − Z i=1 Xi − X

PN

ρX Z ρX Z ρX Z

ρX Z (Xi , Zi )

ρX Z ρX Z

Xn

n Xn

X rX X (m) = E [Xn Xn+m ]

m = · · · , −1, 0, 1, · · · . n

E [Xn+m Xn ]

m m

X 2 σX

rX X (m) = = { =

E [Xn Xn+m ] E [Xn ] E [Xn+m ]   E Xn2

2 σX δ (m) .

m 6= 0

m=0

E [Xn Xn+m ] =

Xn

Xn+m

m 6= 0

Xn Yn

rY Y (m) rX X (m)

h (m) rY Y (m) = h (m) ∗ h (−m) ∗ rX X (m)

Xn y (n) = x (n) − x (n − 1) + x (n − 2) Yn X Yi i = 1, 2, · · · , 1000 (n = 1, 2, 3, 4) (Yi , Yi+1 ) (i = 1, 2, · · · , 900)

(Yi , Yi+2 ) (Yi , Yi+3 ) (Yi , Yi+4 ) (i = 1, 2, · · · , 900) rY Y (m) rY Y (m)

r Y Y (m) = N

1 N − |m|

N−|m|−1 n=0

Y rY Y (m)

X

Y (n) Y (n + |m|) Yn rY Y (m)

− (N − 1) ≤ m ≤ N − 1

m

−20 ≤ m ≤ 20

rY Y (m) rY Y (m) rY Y (m) m

X

rY Y (m)

m m

rY Y (m)

r Y Y (m)

Y c X Y (m) = E [Xn Yn+m ]

Xn

m = · · · , −1, 0, 1, · · · .

Yn c X Y (m) =

cX Y (m) =

1 N −m

N−m−1

X

n=0

X (n) Y (n + m) 0 ≤ m ≤ N − 1

N−1 X 1 X (n) Y (n + m) 1 − N ≤ m < 0 N − |m| n=|m|

N m X (n) D/2 Y (n)

α

Y (n) = αX (n − D) + W (n) W (n) D cX Y c X Y (m) = =

E [X (n) Y (n + m)] E [X (n) (αX (n − D + m) + W (n + m))]

= αE [X (n) X (n − D + m)] + E [X (n)] E [W (n + m)] =

αE [X (n) X (n − D + m)]

X (n)

W (n + m)

c X Y (m) = αrX X (m − D)

rX X (m − D)

m=D

D

c X Y (m)

X (n)

rX X (m)

m=0

X Xn

Zn X

Y

m

Y...