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