signaux et systemes numerique freddy mudry

7/28/2019 Signaux Et Systemes Numerique Freddy Mudry

1/28

x[n] < n < +

x(t)

x[n] = x(n Te)

x[n] n n x[n]

10 5 0 5 10 15

0.4

0.2

0

0.2

0.4

0.6

instants n

signalx[n]

[n] =

1 si n = 0

0 si n = 0


2/28

[nk]

x[k]

x[n]

x[n] =+

k=

x[k] [n k]

[n] =

1 si n 0

0 si n < 0

[n] =+k=0

[n k]

[n] = [n] [n 1]

x[n] = Rn [n]

0 < R < 1

|R| > 1

n

x[n] = cos (n 0 + )

0 = 2 f0Te

x[n] = (a + jb)n [n]

a + jb

a + jb =

a2 + b2 arctanb

a R exp(j0)

x[n] = Rn exp(jn 0) [n]

n

R > 1

R < 1


3/28

10 5 0 5 10 15

1

0

1Impulsion unit

10 5 0 5 10 15

1

0

1Saut unit

10 5 0 5 10 15

1

0

1Exponentielle

10 5 0 5 10 15

1

0

1 Sinusode

0

x[n] = exp (jn 0)

0

0

x[n] = x[n + N]

N

x[n] = A cos(n0 + ) = A cos(n0 + N0 + )

2

N0 = k 2


4/28

0/

0 = 1 N = 2 k

N

k

0 = 3/11

N0 = N3/11 = k 2 N = 223

k

N k

0

0 5 10 15 20 25

1

0.5

0

0.5

1

Sinus numrique de pulsation 3 / 11

0 5 10 15 20 25

1

0.5

0

0.5

1

Sinus numrique de pulsation 1

TanlTnum

Tanl Tnum

0

x[n] = A cos(n0 + )

0 0 2

= 2

= 0


5/28

x[n]

y[n]

T

y[n] = T{x[n]}

y[n] = x[n]

y[n] = x[n 1]

y[n] = x[n + 1]

y[n] =

{x[n 1], x[n], x[n + 1]}

y[n] =n x[k]

y[n] = x[n] x[n 1]

n = 0

x[n] = {0, 1, 2, 3, 4, 5, 0, 0, 0, 0, }

y[n] = { 0, 0, 0, 1, 2, 3, 4, 5, 0, 0, 0, 0, }


6/28

y[n] = { 0, 0, 0, 0, 1, 2, 3, 4, 5, 0, 0, 0, }

y[n] = { 0, 0, 1, 2, 3, 4, 5, 0, 0, 0, 0, 0, }

n

n 1

n + 1

y[n] = { 0, 0, 1, 2, 3, 4, 5, 5, 5, 0, 0, 0 }

y[n] = { 0, 0, 0, 1, 3, 6, 10, 15, 15, 15, 15, 15 }

y[n] = { 0, 0, 0, 1, 1, 1, 1, 1,5, 0, 0 }

2 0 2 4 6 81

0

1

2

3

4

5

6

(a)

2 0 2 4 6 81

0

1

2

3

4

5

6

(b)

2 0 2 4 6 81

0

1

2

3

4

5

6

(c)

2 0 2 4 6 81

0

1

2

3

4

5

6

(d)

2 0 2 4 6 8

0

5

10

15

20

(e)

2 0 2 4 6 8

6

4

2

0

2

4

6(f)


7/28

y[n] =1

5 (x[n 2] + x[n 1] + x[n] + x[n + 1] + x[n + 2])

n

x[n]

n

5 0 5 10 15 20 25 30 35 40

0

10

20

30

x[

n]

5 0 5 10 15 20 25 30 35 400.1

0

0.1

0.2

h[n

10]

5 0 5 10 15 20 25 30 35 40

0

10

20

30

y[n]

instants n

n

y[n] =1

5(x[n] + x[n 1] + x[n 2] + x[n 3] + x[n 4])


8/28

z

z1

y[n] = 0.5 (x1[n] + x1[n 1]) x2[n] + 0.9 y[n 1]

y[n] = b0x[n] + b1x[n 1] + b2x[n 2] a1y[n 1] a2y[n 2]

z-1

z-1

x1[n]

x2[n]

x1[n-1]

y[n]

y[n-1]

0.9

0.5

z-1 z-1

y[n]

x[n] x[n-1]

b0

x[n-2]

b1 b2

z-1 z-1

y[n-1]y[n-2]

- a1- a2

Moyenneur dordre 2

Filtre passe-bas dordre 1


9/28

y[n]

n

y[n] = a x[n] + n x[n]2

y[n] =1

3(x[n 1] + x[n] + x[n + 1])

y[n] = T{a x1[n] + b x2[n]}= a T{x1[n]} + b T{x2[n]}= y1[n] + y2[n]

si T{x[n]} = y[n]alors T{x[n + d]} = y[n + d]

yD,T[n] = yT,D [n]


10/28

y[n] =n

k=

x[k]

[n]

[n]

n

y[n] = x[n + 1] x[n]

y[n] = x[n] x[n 1]

y[n] = x2[n]

x[n] x2[n] x2[n d] = yT,D [n]

x[n]

x[n

d]

x2[n

d] = yD,T[n]


11/28

y[n] = x[2n]

x[n] x[2n] x[2n d] = yT,D [n]

x[n] x[n d] x[2(n d)] = yD,T[n]

x[n + 1] x[n]

x[n] x[n 1] n x[k]

a x[n]

(x[n + 1] + x[n] + x[n 1]) /3

x[n2]

x[2n]

x[n]

n x[n]

x2[n]

a x[n] + b, b = 0

y1[n] = H1 {x[n]}

y[n] = H2 {y1[n]} y[n] = H2 {H1 {x[n]}}


12/28

y[n] = y1[n] + y2[n] y[n] = H1 {x[n]} + H2 {x[n]}

y[n] = (H1 H2) {x[n]} = (H2 H1) {x[n]}y[n] = (H1 + H2) {x[n]} = H1 {x[n]} + H2 {x[n]}

x[n] y1[n]H1 H2

H1

H2

y2[n]

y1[n]

y2[n]

y[n]x[n]

[n]

h[n]

h[n] T{[n]}

x[n]

x[n] =

+k=

x[k] [n k]


13/28

x[n]

x[k] [n k]

yk[n] = T{x[k] [n k]} = x[k] h[n k]

y[n] = T{x[n]} =+

k=

yk[n] =+

k=

x[k] h[n k]

[n] h[n][n k] h[n k]

x[k] [n k] x[k] h[n k]x[k] [n k]

x[k] h[n k]

x[n] y[n]

y[n] =

+

k=x[k] h[n k] =

+

k=h[k] x[n k]

y[n] = x[n] h[n] = h[n] x[n]

h[n]

h[n]

x[n]

n < 0

x[n]

n = 0

y[n] =

+nk=0 x[k] h[n k] =

+nk=0 h[k] x[n k] 0 n < +


14/28

5 0 5 10 15 20

0

0.5

1

[n]

5 0 5 10 15 20

0

0.5

1

h[n]

5 0 5 10 15 20

0

0.5

1

[n1]

5 0 5 10 15 20

0

0.5

1

h[n1]

5 0 5 10 15 20

0

0.5

1

[n2]

5 0 5 10 15 20

0

0.5

1

h[n2]

5 0 5 10 15 200

2

4

6

n

x[n] = k

x[k] [nk] avec x[k] = [k]

5 0 5 10 15 200

2

4

6

n

y[n] = k

x[k] h[nk]


15/28

n < 0

n N x[n < 0] = 0

N

x[n]

y[n] =N1k=0

h[k] x[n k] 0 n < +

z1

z-1 z-1 z-1 z-1

y[n]

x[n] x[n-1]

h[0]

x[n-2] x[n-3] x[n-N+1]

h[1] h[2] h[3] h[N-1]

h[k] x[n-k]k = 0

N-1

y[n] =N1k=0

h[k] x[n k]

x[k]

x[k]

n

x[k] n x[n k]

h[k]

h[k] x[n k]

n = 10

10

k=0

h[k] x[n k] = 1 + 0.9 + 0.92 + + 0.910 = 6.86 = y[10]


16/28

5 0 5 10 15 20 25 300

0.5

1

Convolution entre x[k] et h[k] pour n=10

x[nk], n=10

5 0 5 10 15 20 25 30

0

0.5

1h[k]

5 0 5 10 15 20 25 30

0

0.5

1h[k] x[nk]

instants k

y[10] = h[k] x[nk] = 6.86

5 0 5 10 15 20 25 300

5

10

y[n] = h[k] x[nk]

h[k]

x[n k]

N

h[k]

0 k N 1

N

x[n]

h(t)

y(t)

h(t) =1

et/ pour t 0

y(t) =

t0

h() x(t ) d

y[n] =n

k=0

Te h[k] x[n k]

Te

h[k]


17/28

FARA

N

FL

k

x[n-k]

n

0

N-1 N-1

0

EPROMRAM

h[k]

k

1

11

1

1

11

1

000

00000

h[0]

h[1]

h[2]

h[3]

h[N-1]

xn[k] hn[k]

k=0

N-1

xn[k] hn[k]

y[n] y(t)

x(t) x[n]

Te

Tp

t

x(t)

t

y(t)

A

N

k=0

N-1

x[n-k] h[k]y[n] =


18/28

Te

h[n] Te h(t = nTe)

h[n] = Te1

enTe/ =

Te

eTe/

n

R = eTe/

h[n] =Te

Rn pour n 0

N

y[n] =N1k=0

h(k) x[n k]

y[n] =N1k=0

Te

Rk x[n k]


19/28

N

x[n]

N

h[n]

x[n]

2 N

h0[n] = 1 |n| N/20

N/2 < |n| N

h1[n] =N |n|

N

|n| N

h2[n] =

2

N2(n + N)2

N < n < N/2

1 2N2

n2

N/2 n +N/22

N2(n N)2 +N/2 < n < +N

N = 15

N 1

y[n] = h[n] x0[n]

x[n]

x0[n] = {0, 0, 0, 0, , 1, 0, 0, 0, , 4, 0, 0, 0, , 2}


20/28

0

0.5

1

N 0 +N

0

0.5

1

N 0 +N

0

0.5

1

N 0 +N

N

x0[n]

h[n] y[n]

x[n]

h[n]

n

y[n]

y[n] =N1k=0

h(k) x[n k]


21/28

0 5 10 15 20 25 30 35 40 45

0

2

4

0 5 10 15 20 25 30 35 40 45

0

2

4

0 5 10 15 20 25 30 35 40 45

0

2

4

N = 15

0 20 40 60 80 100 120

0

5

10

0 20 40 60 80 100 120

0

5

10

0 20 40 60 80 100 120

0

5

10


22/28


23/28

y[n] =n

k=0

x[k]

y[n] =n

k=0

x[k]

=n1

k=0

x[k] + x[n]

y[n] = y[n 1] + x[n]

y[n] =N

1k=0

Te

Rk x[n k]

y[n] =N1k=0

Te

Rk x[n k] = Te

N1k=0

Rk x[n k]

=Te R

0x[n] + R1 x[n 1] + R2 x[n 2] + =

Te

x[n] + RTe

R0 x[n 1] + R1 x[n 2] + R2 x[n 3)] +

y[n] =Te

x[n] + R y[n 1]

y[n]

n

N


24/28

z-1

x[n]

n

1/(n+1)

y[n]

Moyenneur cumulatif

z-1

y[n]x[n]

Accumulateur

z-1

y[n]x[n]

R

Filtre passe-bas (R < 1)

Te

y[n] =1

n + 1

nk=0

x[k]

n + 1

(n + 1) y[n] =n

k=0

x[k] = x[n] +n1k=0

x[k]

y[n] =1

n + 1x[n] + n 1

n

n1

k=0

x[k]

y[n] =1

n + 1(x[n] + n y[n 1])


25/28

x1[n] = [n 2] x4[n] = 0.9n [n]x2[n] = +[n + 1] + [n] x5[n] = sin(n/6)x3[n] = 2[n + 2] [3 n] x6[n] = sin(n/8) [n]

x[n]

y1[n] = x[n 2] y4[n] = x[n] [n]y2[n] = x[3 n] y5[n] = x[n + 1] [n]y3[n] = x[n + 1] [n] y6[n] = x[3 n] [n 2]

5 0 5 10

0

0.5

1

1.5

2

.....

x1[n]

5 0 5 10

2

1

0

1

2

3

x2[n]

5 0 5 100.2

0

0.2

0.4

0.6

0.8

1

.....

1, 0.9, 0.81, 0.73, 0.656, .....

x3[n]

5 0 5 100.2

0

0.2

0.4

0.6

0.8

1

.....

1, 0.9, 0.81, 0.73, 0.656, .....

x4[n]

0

N


26/28

x1[n] = cos(n /20) x4[n] = (j n /4 /2)x2[n] = cos(n 3/8) x5[n] = 3 sin(5 n + /6)x3[n] = cos(n 13/8 /3) x6[n] = cos(n 3/10) sin(n /10) + 3 cos(n /5)

x1[n], x2[n] y1[n], y2[n]

5 0 5 10

0

0.5

1

1.5

2

x1[n]

5 0 5 10

0

1

2

3

4

y1[n]

5 0 5 10

0

0.5

1

1.5

2

x2[n]

5 0 5 10

0

1

2

3

4

y2[n]

h[n] = {4, 3, 2, 1, 0, 0, 0, } n 0

x1[n] = [n 1] x3[n] = [n] [n 5]x2[n] = +2[n] [n 1] x4[n] = [n + 5]

xk[n]

yk[n]


27/28

h[n 0] = {0, 1, 1, 1, 1,2,2, 0, 0, }

x[n] = [n]

h[n] = 0.8n [n]

y[n] = 2 x[n 1] + 34

y[n 1] 18

y[n 2]

h[n]

y[n]

x[n]

5 0 5 10

0

0.5

1

1.5

2

2.5

3

x[n]

5 0 5 10

3

2

1

0

1

2

y[n]

x[n] = {1, 6, 3}

h[n]

h[0] =1

h[N] = 0 h[n]

x0[n] y[n]

h[n] = 12

1 + cos

nN

N n +N


28/28

h[n] x0[n]

h1[n] h6[n]

h3[n]

y[n]

signaux et systemes numerique freddy mudry

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