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. a. What is a Fourier Series?
Fourier Transform
a. What is a Fourier Series?
Decompose a single using a series of sine and cos waves
Time-Amplitude
domain
Frequency-Amplitude
domain
3Hz
10Hz
50Hz

4. How to find frequencies of a signal?

5. (Euler’s formula)

6. Discrete Fourier Transform
Observations: N
Sampling interval:
Period
First harmonic frequency:
nth harmonic frequency:
All frequency:
time at kth observation:

7. Time-Amplitude domain
Frequency-Amplitude domain
If time series F(k) is known, then, the coefficient c(n) can be found as:
Forward Transform:

8. 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.

9. 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

11. 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
If sampling rate is , the highest wave frequency can be
resolved is , which is called
Nyquist frequency

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

13. Leakage

15. 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

16. 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 identically equal to those at the lower frequencies. They are folded back and added to the lower frequencies.
Discrete spectral intensity (or energy)

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