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Measurements for
water
Rolf Hut
Meten aan water: Fourier
Challenge the future
DelftUniversity ofTechnology
Meten aan WaterFourier
Rolf Hut
Friday, November 18, 2011
Signalen
Fourier
2
Friday, November 18, 2011
Signalen
Fourier
2
Friday, November 18, 2011
Signalen
Fourier
2
Friday, November 18, 2011
f (t) =∞�
k=−∞ake
iω0kt
Signalen
Fourier
•Fourier series
•for periodic signals!!!
3
f (t+ nT ) =∞�
k=−∞ake
i 2πT kt
Friday, November 18, 2011
f (t) =∞�
k=−∞ake
iω0kt
Signalen
Fourier
•Fourier series
•for periodic signals!!!
3
f (t+ nT ) =∞�
k=−∞ake
i 2πT kt
Friday, November 18, 2011
f (t) =∞�
k=−∞ake
iω0kt
Signalen
Fourier
•Fourier series
•for periodic signals!!!
3
f (t+ nT ) =∞�
k=−∞ake
i 2πT kt
Friday, November 18, 2011
f (t) =∞�
k=−∞ake
iω0kt
ak =1
T
�
Tf (t) e−iω0ktdt
Signalen
Fourier
•Fourier coefficients
4
Friday, November 18, 2011
f (t) =1
2π
� ∞
−∞F (iω) eiωtdω
F (iω) =
� ∞
−∞f (t) e−iωtdt
Signalen
Fourier
•Fourier transformation
5
Friday, November 18, 2011
•Linear
•Time Shift
•Convolution
g (t) = f1 (t) + f2 (t) ↔ G (iω) = F1 (iω) + F2 (iω)
Signalen
Fourier Transformatie
6
f (t− t0) ↔ e−iωtoF (iω)
g (t) =
� ∞
−∞f (τ)h (t− τ) dτ ↔ G (iω) = F (iω)H (iω)
Friday, November 18, 2011
Signalen
Fourier Transformatie
•duality!
• terug naar sampling:
7
g (t) = f (t)h (t) ↔ G (ω) =1
2π
� ∞
−∞F (θ)H (ω − θ) dθ
fp (t) =∞�
−∞f (nT ) δ (t− nT )
Friday, November 18, 2011
Signalen
Nyquist criteria
•Sampling Theorem
•sample frequentie moet 2 keer zo hoog zijn als de hoogste frequentie in het signaal
•anders: aliasing
8
ωs > 2ωm
Friday, November 18, 2011
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