1. Fourier Analyses
Time series
Sampling interval
Total period
Question: How perturbations with different frequencies contribute to the turbulent kinetic energy?

2. Decomposition illustration

3. Time-Amplitude domain
Frequency-Amplitude domain
3Hz
10Hz
50Hz

4. Wave basics
(Euler’s formula)

5. c. Discrete Fourier Transform
Observations: N
Sampling interval:
Period
First harmonic frequency:
All frequency:
nth harmonic frequency:
time at kth observation:
Time-Amplitude domain
Frequency-Amplitude domain
If time series F(k) is known, then, the coefficient c(n) can be found as:

6. Forward Transform:
Example:
Index (k):
Time (UTC):
Q(g/kg): n
c(n) i i i i
For frequencies greater than 4, the Fourier transform is just the complex
Conjugate of the frequencies less than 4.

7. c(0) =7.0
c(1)= i
c(2)=0.5
c(3)= i
c(4)=1.0
c(5)= i
c(6)=0.5
c(7)= i

8. Aliasing
We have ten observations (10 samples) in a second and two
different sinusoids that could have produced the samples.
Red sinusoid has 9 cycles spanning 10 samples, so the frequency
Blue sinusoid has 1 cycle spanning 10 samples, so the frequency
Which one is right?
Two-point rule
Two data points are required per period to determine a wave.
2 observations: 1 wave
4 observations: 2 waves
6 observations: 3 waves
N observations: N/2 waves

9. If there were a true signal of f=9 Hz that was sampled at
fs=1Hz, then, one would find that the signal has been
interpreted as the signal of f=1 Hz. In other words, the
real signal f=9 Hz was folded into the signal f=1Hz.
This is because the maximum frequency that can be
resolved by sampling rate fs=1.0Hz is 0.5 Hz!
If sampling rate is , the highest wave frequency can be
resolved is , which is called
Nyquist frequency

10. Folding
Folding occurs at Nyquist frequency.
What problem does folding cause?

11. What will cause aliasing or folding?
The sensor can respond to frequencies higher than the rate
that the sensor is sampled.
The true signal has frequencies higher than the sampling rate.
How can we remove aliasing?
We cannot resolve frequencies higher than Nyquist frequency,
i.e., if we have N data points, we can only resolve wave cycles
of N/2, but why Fourier transform gives amplitude c(n) up to
wave cycle up to n=N-1?
C(n) for n>Nf is just the complex conjugate of c(n) for n<Nf.
So, half of c(n) for n>Nf gives no new information.

12. Leakage

14. Spectral characteristics
(1) Wave with
different
frequencies

15. (2) Wave
frequencies
between
resolvable

16. (3) Unresolvable
low and high
frequencies

17. (4) Red noise

18. (5) Red, white,
and blue noise

19. Leakage

20. Multiple
waves

21. Fast Fourier Transform (FFT)
FFT is nothing more than a discrete Fourier transform that has been
restructured to take advantage of the binary computation processes
of digital computer. As a result, everything is the same but faster!
Relationship between decimal and binary numbers
The decimal numbers n and k can be represented by
If N=8, then, j=0, 1, 2, 3
7: binary 1 1 1; 5: binary 101; 3: binary 11

22. Energy Spectrum
Note that n starts from 1, because the mean (n=0) does not contribute any
information about the variation of the signal.
For frequencies higher than Nyquist frequency, values are identially
equal to those at the lower frequencies. They are folded back and added to
the lower frequencies.
Discrete spectral intensity (or energy)
Spectral energy density

23. Example
Index (k):
Time (UTC):
Q(g/kg):

24. N observations

25. Graphical presentation of atmospheric turbulence spectra
Linear-linear presentation
Semi-log presentation
Log-log presentation
Log(fS(f)) vs. log(f)

26. Blackhar Window:
N=9000

27. Turbulent spectral similarity
Energy associated with large-scale
motion eventually is transferred to
the large turbulent eddies.
The large eddies then transport this
energy to small-scale eddies.
These smaller scale eddies then
transfer the energy to even
small-scale eddies..., and so on
Eventually, the energy is dissipated
into heat via molecular viscosity.
Inertial sub-range: An intermediate range of turbulent scales that is
smaller than the energy-containing eddies but larger than viscous eddies.
In the inertial subrange, the net energy coming from the energy-containing
eddies is in equilibrium with the net energy cascading to smaller eddies
where it is dissipated.

28. wavelength wave-number Kolmogorov's Energy Spectrum
Inertial sub-range is in an equilibrium state, Kolmogorov assumes that the energy
density per unit wave number depends only on the wave number and the rate of
energy dissipation.
wavelength
wave-number 3 3 5 5

29. are the angular wavenumber in x and y direction.
2D FFT .
Let χ(m,n) be a generic scalar on a 2D grid, where m=0,1,…,M-1; n=0,1,…N-1 are the grid number index; and M and N are the number of grids in x and y direction, the 2D-FFT of χ(m,n), then, may be written as,
where k and l are the 2D-FFT wavenumber index in x and y direction, and
are the angular wavenumber in x and y direction.
χ(m,n), then, can be reconstructed as
Spectral energy can be written as
Again, energy is conserved

30. Project 2 .
Using 1D FFT to decompose the provided hurricane data in terms of wavenumbers.