Signaux Et Systemes Numerique Freddy Mudry
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Transcript of Signaux Et Systemes Numerique Freddy Mudry
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7/28/2019 Signaux Et Systemes Numerique Freddy Mudry
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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
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[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
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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
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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
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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, }
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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)
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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])
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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
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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]
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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]
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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]}}
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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]
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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 < +
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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]
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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]
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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]
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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] =
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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]
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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}
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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]
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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
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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
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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])
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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
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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]
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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
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h[n] x0[n]
h1[n] h6[n]
h3[n]
y[n]