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Article publié par EDP Sciences et disponible sur le site http://www.j3ea.org ou http://dx.doi.org/10.1051/j3ea/2017001

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 0n/8) W⁰ⁿN N = 8 Xx = 8 Xy = 0 Abs(X) = 8 F_e = 0.4 f₀ = 0

cos(2 0n/8) n

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 1n/8) W¹ⁿN N = 8 Xx = 0 Xy = 0 Abs(X) = 0 F_e = 0.4 f₁ = 0.05

cos(2 1n/8) n

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 2n/8) W²ⁿ_N N = 8 Xx = 0 Xy = 0 Abs(X) = 0 Fe = 0.4 f2 = 0.1

cos(2 2n/8) n

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 3n/8) W³ⁿ_N N = 8 Xx = 0 Xy = 0 Abs(X) = 0 Fe = 0.4 f3 = 0.15

cos(2 3n/8) n

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 4n/8) W⁴ⁿN N = 8 X_x = 0 X_y = 0 Abs(X) = 0 Fe = 0.4 f4 = 0.2

cos(2 4n/8) n

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 5n/8) W⁵ⁿN N = 8 X_x = 0 X_y = 0 Abs(X) = 0 Fe = 0.4 f5 = 0.25

cos(2 5n/8) n

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 6n/8) W⁶ⁿN N = 8 Xx = 0 Xy = 0 Abs(X) = 0 F_e = 0.4 f₆ = 0.3

cos(2 6n/8) n

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 7n/8) W⁷ⁿN N = 8 Xx = 0 Xy = 0 Abs(X) = 0 F_e = 0.4 f₇ = 0.35

cos(2 7n/8) n

−1

−0.5 0

0.5 1−1

−0.5 0

0.5 0 1

10 20 30

− sin(2 14n/32) PGCD(m, N) = PGCD(14, 32) = 2

cos(2 14n/32) n

E¹⁴N N = 32 f14 = 0.7

−1

−0.5 0

0.5 1−1

−0.5 0

0.5 0 1

100 200

− sin(2 20n/256) PGCD(m, N) = PGCD(20, 256) = 4

cos(2 20n/256) n

E²⁰N N = 256 f20 = 1

−1

−0.5 0

0.5 1−1

−0.5 0

0.5 0 1

200 400

− sin(2 102n/512) PGCD(m, N) = PGCD(102, 512) = 2

cos(2 102n/512) n E102

N N = 512 f₁₀₂ = 5.1

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 0n/8) W⁰ⁿ_N N = 8 Xx = 8 Xy = 0 Abs(X) = 8 Fe = 0.4 f0 = 0

cos(2 0n/8) n

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 4n/8) W⁴ⁿ_N N = 8 Xx = 0 Xy = 0 Abs(X) = 0 Fe = 0.4 f4 = 0.2

cos(2 4n/8) n

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 1n/8) W¹ⁿN N = 8 X_x = 0 X_y = 0 Abs(X) = 0 Fe = 0.4 f1 = 0.05

cos(2 1n/8) n

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 5n/8) W⁵ⁿN N = 8 X_x = 0 X_y = 0 Abs(X) = 0 Fe = 0.4 f5 = 0.25

cos(2 5n/8) n

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

5 10 15

− sin(2 2n/16) Fe = 0.8 f2 = 0.1 W²ⁿN N = 16 X_x = 0 X_y = 0 Abs(X) = 0

cos(2 2n/16) n

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

10 20 30

− sin(2 2n/32) Fe = 1.6 f2 = 0.1 W²ⁿN N = 32 X_x = 0 X_y = 0 Abs(X) = 0

cos(2 2n/32) n

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

50 100

− sin(2 2n/128) F_e = 6.4 f₂ = 0.1 W²ⁿN N = 128 Xx = 0 Xy = 0 Abs(X) = 0

cos(2 2n/128) n

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

200 400

− sin(2 2n/512) F_e = 25.6 f₂ = 0.1 W²ⁿN N = 512 Xx = 0 Xy = 0 Abs(X) = 0

cos(2 2n/512) n

−1

−0.5 0

0.5 1−1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 1n/8) W¹ⁿN N = 8 Xx = 0 Xy = 0 Abs(X) = 0 Fe = 0.4 f1 = 0.05

cos(2 1n/8) n

−1

−0.5 0

0.5 1−1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 2n/8) W²ⁿN N = 8 Xx = 0 Xy = 0 Abs(X) = 0 Fe = 0.4 f2 = 0.1

cos(2 2n/8) n

−1

−0.5 0

0.5 1−1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 3n/8) E¹8 * E²8 = E³8

cos(2 3n/8) n

Fe = 0.4 f3 = (f1 + f2) mod(Fe) = 0.15

−1

−0.5 0

0.5 1−1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 2n/8) W2n

N N = 8 Xx = 0 Xy = 0 Abs(X) = 0 Fe = 0.4 f2 = 0.1

cos(2 2n/8) n

−1

−0.5 0

0.5 1−1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 7n/8) W7n

N N = 8 Xx = 0 Xy = 0 Abs(X) = 0 Fe = 0.4 f7 = 0.35

cos(2 7n/8) n

−1

−0.5 0

0.5 1−1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 1n/8) E²8 * E⁷8 = E¹8

cos(2 1n/8) n

Fe = 0.4 f₁ = (f₂ + f₇) mod(Fe) = 0.05

0 5 10 15 20

−1.5

−1

−0.5 0 0.5 1 1.5

Nf_s/F_e = 2

sin(2 f_st) f_s = 0.1 N = 8 ^{0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4}

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

TFD et FFT Nf_s/F_e = 2

Fréquence f

TFD FFT fs Fe − fs

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 1 2 3 4 5 6 7 8 9

La fréquence fs est multiple de Fe/N

Fréquence f TFD FFT f_s F_e − f_s

"SinCard" centrés en f_s et en F_e − f_s

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 1 2 3 4 5 6 7 8 9

La fréquence fs n’est pas multiple de Fe/N

Fréquence f TFD FFT f_s F_e − f_s

"SinCard" centrés en f_s et en F_e − f_s

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 0n/8) cos(2 0n/8)

Fe = 0.4 f0 = 0

n

s(n)W⁰ⁿN N = 8 Xx = 0 Xy = 0 Abs(X) = 0

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 1n/8) cos(2 1n/8)

Fe = 0.4 f1 = 0.05

n

s(n)W¹ⁿN N = 8 Xx = 0 Xy = 0 Abs(X) = 0

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 2n/8) cos(2 2n/8)

Fe = 0.4 f2 = 0.1

n

s(n)W²ⁿ_N N = 8 Xx = 0 Xy = −4 Abs(X) = 4

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 3n/8) cos(2 3n/8)

Fe = 0.4 f3 = 0.15

n

s(n)W³ⁿ_N N = 8 Xx = 0 Xy = 0 Abs(X) = 0

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 4n/8) cos(2 4n/8)

F_e = 0.4 f₄ = 0.2

n

s(n)W⁴ⁿN N = 8 X_x = 0 X_y = 0 Abs(X) = 0

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 5n/8) cos(2 5n/8)

F_e = 0.4 f₅ = 0.25

n

s(n)W⁵ⁿN N = 8 X_x = 0 X_y = 0 Abs(X) = 0

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 6n/8) cos(2 6n/8)

F_e = 0.4 f₆ = 0.3

n

s(n)W⁶ⁿN N = 8 Xx = 0 Xy = 4 Abs(X) = 4

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 7n/8) cos(2 7n/8)

F_e = 0.4 f₇ = 0.35

n

s(n)W⁷ⁿN N = 8 Xx = 0 Xy = 0 Abs(X) = 0

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 1n/8) W¹ⁿ_N N = 8 X_x = 0 X_y = 0 Abs(X) = 0 Fe = 0.4 f1 = 0.05

cos(2 1n/8) n

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 1n/8) cos(2 1n/8)

F_e = 0.4 f₁ = 0.05

n

s(n)W¹ⁿN N = 8 Xx = 0 Xy = 0 Abs(X) = 0

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 2n/8) W²ⁿN N = 8 Xx = 0 Xy = 0 Abs(X) = 0 F_e = 0.4 f₂ = 0.1

cos(2 2n/8) n

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 2n/8) cos(2 2n/8)

Fe = 0.4 f2 = 0.1

n

s(n)W²ⁿ_N N = 8 Xx = 0 Xy = −4 Abs(X) = 4

0 5 10 15 20

−1

−0.5 0 0.5 1

Nf_s/F_e = 2.1

sin(2 f_st) f_s = 0.105 N = 8 ^{0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4}

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

TFD et FFT Nf_s/F_e = 2.1

Fréquence f

TFD FFT fs Fe − fs

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 0n/8) cos(2 0n/8)

F_e = 0.4 f₀ = 0

n

s(n)W⁰ⁿN N = 8 Xx = −0.206 Xy = 0 Abs(X) = 0.206

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 1n/8) cos(2 1n/8)

F_e = 0.4 f₁ = 0.05

n

s(n)W¹ⁿ_N N = 8 X_x = −0.173 X_y = −0.265 Abs(X) = 0.316

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 2n/8) cos(2 2n/8)

F_e = 0.4 f₂ = 0.1

n

s(n)W²ⁿN N = 8 Xx = 0.919 Xy = −3.746 Abs(X) = 3.857

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 3n/8) cos(2 3n/8)

Fe = 0.4 f3 = 0.15

n

s(n)W³ⁿN N = 8 Xx = −0.445 Xy = 0.331 Abs(X) = 0.555

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

5 10 15

− sin(2 1n/16) cos(2 1n/16)

n

s(n)W¹ⁿN N = 16 Xx = 0 Xy = 0 Abs(X) = 0

Fe = 0.8 f1 = 0.05

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

200 400

− sin(2 1n/512) cos(2 1n/512)

n

s(n)W¹ⁿN N = 512 Xx = 0 Xy = 0 Abs(X) = 0

Fe = 25.6 f1 = 0.05

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

5 10 15

− sin(2 2n/16) cos(2 2n/16)

n

s(n)W²ⁿ_N N = 16 Xx = 0 Xy = −8 Abs(X) = 8

F_e = 0.8 f₂ = 0.1

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

200 400

− sin(2 2n/512) cos(2 2n/512)

n

s(n)W²ⁿ_N N = 512 Xx = 0 Xy = −256 Abs(X) = 256

F_e = 25.6 f₂ = 0.1

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

5 10 15

− sin(2 3n/16) cos(2 3n/16)

n

s(n)W³ⁿN N = 16 X_x = 0 X_y = 0 Abs(X) = 0

Fe = 0.8 f3 = 0.15

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

200 400

− sin(2 3n/512) cos(2 3n/512)

n

s(n)W³ⁿN N = 512 X_x = 0 X_y = 0 Abs(X) = 0

Fe = 25.6 f3 = 0.15

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

−4

−3

−2

−1 0 1 2 3 4

Spectre de Phase f_s = 0.1 Nf_s/F_e = 2

Phase en Radians

Fréquence f ^{0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4}

−4

−3

−2

−1 0 1 2 3 4

Spectre de Phase f_s = 0.105 Nf_s/F_e = 2.1

Phase en Radians

Fréquence f

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

−4

−3

−2

−1 0 1 2 3 4

tr = 1Te

Phase en Radians

Fréquence f ^{0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4}

−4

−3

−2

−1 0 1 2 3 4

Spectres de Phase fs = 0.1 Nfs/Fe = 2

tr = 2Te

Phase en Radians

Fréquence f ^{0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4}

−4

−3

−2

−1 0 1 2 3 4

tr = 3Te

Phase en Radians

Fréquence f

−1

−0.5 0

0.5 1−1

−0.8

−0.6

−0.4

−0.2 00.2

0.4 0.6

0.8 1 0

2 4 6 8

− sin(2 2n/8) cos(2 2n/8)

Fe = 0.4 f2 = 0.1

n

s(n)W²ⁿN N = 8 Xx = −4 Xy = 0 Abs(X) = 4

tr = 1Te r = − 2 0.25

−1

−0.5 0

0.5 1−1

−0.8

−0.6

−0.4

−0.2 00.2

0.4 0.6

0.8 1 0

2 4 6 8

− sin(2 2n/8) cos(2 2n/8)

Fe = 0.4 f2 = 0.1

n

s(n)W²ⁿN N = 8 Xx = 0 Xy = 4 Abs(X) = 4

tr = 2Te r = − 2 0.5

−1

−0.5 0

0.5 1−1

−0.8

−0.6

−0.4

−0.2 00.2

0.4 0.6

0.8 1 0

2 4 6 8

− sin(2 2n/8) cos(2 2n/8)

Fe = 0.4 f2 = 0.1

n

s(n)W²ⁿN N = 8 Xx = 4 Xy = 0 Abs(X) = 4

tr = 3Te r = − 2 0.75

−1

−0.5 0

0.5 1−1

−0.8−0.6

−0.4

−0.20 0.2

0.40.6 0.81 0

200 400

− sin(2 2n/512) cos(2 2n/512)

Fe = 25.6 f2 = 0.1

n

s(n)W²ⁿN N = 512 Xx = −256 Xy = 0 Abs(X) = 256

t_r = 64T_{e r} = − 2 0.25

−1

−0.5 0

0.5 1−1

−0.8−0.6

−0.4

−0.20 0.2

0.40.6 0.81 0

200 400

− sin(2 2n/512) cos(2 2n/512)

Fe = 25.6 f2 = 0.1

n

s(n)W²ⁿN N = 512 Xx = 0 Xy = 256 Abs(X) = 256

t_r = 128T_{e r} = − 2 0.5

−1

−0.5 0

0.5 1−1

−0.8−0.6

−0.4

−0.20 0.2

0.40.6 0.81 0

200 400

− sin(2 2n/512) cos(2 2n/512)

Fe = 25.6 f2 = 0.1

n

s(n)W²ⁿN N = 512 Xx = 256 Xy = 0 Abs(X) = 256

t_r = 192T_{e r} = − 2 0.75

0 5 10 15 20

−1.5

−1

−0.5 0 0.5 1 1.5

N = 8 sin(2 fs1t) + (2/3)sin(2 fs2t)

f_s1 = 0.05 f_s2 = 0.15 Nf_s1/F_e = 1 Nf_s2/F_e = 3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

TFD et FFT Nf_s1/F_e = 1 Nf_s2/F_e = 3

Fréquence f TFD FFT f_sk F_e − f_sk

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

−4

−3

−2

−1 0 1 2 3 4

Spectre de Phase Nf_s1/F_e = 1 Nf_s2/F_e = 3

Phase en Radians

Fréquence f

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 0n/8) cos(2 0n/8)

Fe = 0.4 f0 = 0

n

s(n)W⁰ⁿ_N N = 8 Xx = 0 Xy = 0 Abs(X) = 0

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 1n/8) Fe = 0.4 f1 = 0.05

n

s(n)W¹ⁿ_N N = 8 Xx = 0 Xy = −4 Abs(X) = 4

cos(2 1n/8)

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 2n/8) cos(2 2n/8)

F_e = 0.4 f₂ = 0.1

n

s(n)W²ⁿN N = 8 X_x = 0 X_y = 0 Abs(X) = 0

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 3n/8) F_e = 0.4 f₃ = 0.15

n

cos(2 3n/8)

s(n)W³ⁿN N = 8 Xx = 0 Xy = −2.667 Abs(X) = 2.667

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 4n/8) cos(2 4n/8)

Fe = 0.4 f4 = 0.2

n

s(n)W⁴ⁿN N = 8 Xx = 0 Xy = 0 Abs(X) = 0

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 5n/8) cos(2 5n/8)

Fe = 0.4 f5 = 0.25

n

s(n)W⁵ⁿN N = 8 Xx = 0 Xy = 2.667 Abs(X) = 2.667

−1

−0.5 0

0.5 1−1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 1n/8) E¹8(s1)

n

Fe = 0.4 f1 = 0.05 s(n)W1n

N N = 8 Xx = 0 Xy = −4 Abs(X) = 4

cos(2 1n/8)

−1

−0.5 0

0.5 1−1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 1n/8) (2/3)E1

8(s₂)

cos(2 1n/8) n

Fe = 0.4 f1 = 0.05 s(n)W¹ⁿN N = 8 Xx = 0 Xy = 0 Abs(X) = 0

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 1n/8) E1

8(s₁) + (2/3)E1 8(s₂)

n

Fe = 0.4 f1 = 0.05

cos(2 1n/8)

−1

−0.5 0

0.5 1−1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 3n/8) E³8(s1)

cos(2 3n/8) n

Fe = 0.4 f3 = 0.15 s(n)W³ⁿN N = 8 Xx = 0 Xy = 0 Abs(X) = 0

−1

−0.5 0

0.5 1−1

−0.5 0

0.5 1 0

2 4 6 8

− sin(2 3n/8) (2/3)E3

8(s₂)

n

Fe = 0.4 f3 = 0.15 s(n)W³ⁿN N = 8 Xx = 0 Xy = −2.6667 Abs(X) = 2.6667

cos(2 3n/8)

−1

−0.5 0

0.5

1 −1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 3n/8) E3

8(s₁) + (2/3)E3 8(s₂)

n

Fe = 0.4 f3 = 0.15

cos(2 3n/8)

−1

−0.5 0

0.5 1−1

−0.5 0

0.5 0 1

10 20 30

− sin(2 8n/32) cos(2 8n/32)

n E8

N N = 32 f₈ = 0.4

−1

−0.5 0

0.5 1−1

−0.5 0

0.5 0 1

5 10 15

− sin(2 4n/16) cos(2 4n/16)

n E4

N N = 16 f₄ = 0.2

−1

−0.5 0

0.5 1−1

−0.5 0

0.5 0 1

2 4 6 8

− sin(2 2n/8) cos(2 2n/8)

n E2

N N = 8 f₂ = 0.1